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 Shore-breaking wave energetics
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FGS








State of Florida
Department of Environmental Protection
David B. Struhs, Secretary




Division of Resource Assessment and Management
Ed Conklin, Director



Florida Geological Survey
Walter Schmidt, State Geologist and Chief





Special Publication No. 45


On the Breaking of Nearshore Waves

by

James H. Balsillie

and

Shore-Breaking Wave Energetics

by


James H. Balsillie





Florida Geological Survey
Tallahassee, Florida
1999
























































Printed for the
Florida Geological Survey

Tallahassee, Florida
1999

ISSN 0085-0640






ii







LETTER OF TRANSMITTAL


Florida Geological Survey
Tallahassee

Govemor Jeb Bush
Florida Department of Environmental Protection
Tallahassee, Florida 32304-7700

Dear Governor Bush:

The Florida Geological Survey, Division of Administrative and Technical Services,
Department of Environmental Protection is publishing two papers in Special Publication No. 45:
"On the breaking of shore-breaking waves" and "Shore-breaking wave energetics.

The first paper identifies where waves shore-break. When or where waves shore-break
has been a controversial topic for over a century. Where they break, significantly affects wave
energy constraints. This work has finally settled the issue. It is found that nearshore waves are
water depth limited and that the water depth in which they break is 1.28 times the mean shore-
breaker height. The second paper provides a numerical methodology for determining how energy
is distributed across the wavelength for waves at the shore-breaking position. It was found that
energy density of the breaker crest is 14 times that of the breaking wave trough.

Practical uses for the results presented in these works include the determination of
dynamic impact pressures necessary in the design of coastal structures, increased precision in
identifying breaking wave parameters leading to increased precision in predicting sediment
transport, sediment budgets, beach and coast erosion during storm/hurricane impacts, etc.

Respectfully yours,


Walter Schmidt, Ph.D., P.G.
State Geologist and Chief
Florida Geological Survey









FOREWORD

Whether rocks of our planet (or others for that matter) are igneous, metamorphic, or
sedimentary, the final form of the rock deposit is dependent upon forces which led to its
formation. The former we refer to as a response element (i.e., the final form of the deposit),
the latter as force elements (i.e., forces which led to deposition and induration). Force
elements include wind, hydraulic forces, gravity, pressure, temperature, chemical reactions,
etc.

This work is concerned with sedimentary deposits. The majority of geologists have
been involved in studying and describing insitu unconsolidated or lithified sedimentary
deposits. Unless fossils are present, they will have little idea of the conditions leading to
deposition. Moreover, fossils may not indicate specific conditions of transport and deposition.
While relatively small in number, there are, however, geologists who have adopted an
expanded Earth Science perspective, and have sought to study first currently occurring forces
(e.g., water waves, water currents, wind, gravity, etc.) and then describe the resulting
sedimentary deposits (e.g., based on their granulometry and/or bedding characteristics). This
does, after all, constitute an underlying and basic Geological or Earth Science concept
proposed in 1785 by James Hutton, termed the Principle of Uniformitarianism. It states: "The
present is the key to the past."

It is from the latter school of geology that this work surfaces. It has always been an
errand of keen professional interest to the author for two reasons: 1) it allows for the study
of natural environmental processes using the robust scientific method that, in combination,
2) provides results that can not only be used to interpret ancient sedimentary deposits and
rocks, but provides information that can have significant value for current and future
environmental concerns. After all, if Hutton's principle is true, then the corollary must also
be true that ... the present is the key to the future.

The work presented in this study is concerned with numerically quantifying where
waves break and wave energy. Alaska, California, Florida, and Texas have the longest ocean-
fronting shorelines in the United States. Florida has approximately 1,253 miles of shoreline
that front directly upon the Atlantic Ocean, Gulf of Mexico, and Straits of Florida. Moreover,
annual average wave energy levels range from near zero for Florida's Big Bend Gulf Coast, low
to moderate for the remainder of the Gulf Coast, to high along Florida's Atlantic shores. Only
Alaska experiences such wave energy variability, not only because of its large waves but
because of zero-energy occurring along ice-wedged shores during a significant portion of the
year. Alaska does not, however, have a coastal population of any significant proportions.
California has a significant coastal population and large Pacific Ocean waves, but not the wave
height variability. Coastal Florida and Texas also experience tropical storm and hurricane wave
impacts that far exceed annual average wave energy levels. Texas does not, however, have
Florida's wave energy variability. Florida, then, has perhaps the longest shoreline in terms of
wave energy variability and socio-economic characteristics. Such wave energy variability
requires those of us in Florida to be sensitively precise in wave characteristics and wave
energy assessment and application. We must remind ourselves that it was and is marine
forces (i.e., primarily waves) that, for the most part, formed and form Florida's surficial
sediment configurations.








It is with this explanation that I submit the content of this work to the record

James H. Balsillie
May 1999








TABLE OF CONTENTS


ON THE BREAKING OF NEARSHORE WAVES


ABSTRACT .......................
INTRODUCTION ....................
PREVIOUS WORK ..................
THE DATA .......................
Laboratory Data . . . . . . .
Field Data ...................
Gaillard Data ............
Scripps Leica Data ........
Scripps Special Measurements
Balsillie and Carter Data ....
W eishar Data ...........
ANALYTICAL PROCEDURE ............
Occam's Razor ... ......
nnuvl'e Prinrinlo


. . . . . . . . . . . .
Procedure .............................
STEPW ISE REGRESSION .............................
Assessment of Equation (11) .....................
Data Redundancy ........................
Net Contributions ........................
Some Observations ......................
Data Set Listings and Stepwise Regression Results
Stepwise Assessment of Individual Data Sets ....
Evaluation of a Special Data Set .............
Assessment of Equations (12) and (13) .............
A Note on Matrix Algebra .......................
COMPARATIVE ANALYSIS OF PREDICTIVE METHODS .......
A Logarithmic Relationship ......................
Variability Analysis ........ ... .................
Functional Regression Variability Analysis ........
Natural Variability Analysis ............ .....
DISCUSSION ............. ..... ............. .....
The Nelson-Gourlay Horizontal Bed Slope Breaking Anomal
CONCLUSIONS ...................................
ACKNOW LEDGEMENTS .............................
REFERENCES .....................................


y


LIST OF FIGURES
Figure 1. Water depth in which shore-breaking occurs, db, given by db = c Hb (Hb
= shore-breaking wave height), versus the relative breaker energy level.. ..

Figure 2. Three basic types of shore-breaking waves. .......................


Page








Figure 3. Relationship between the percentage of the wave crest that is involved in
shore-breaking, Hb (measured from the crest apex downward) where Hb is
the mean shore-breaking wave height, as a function of the modified surf
similarity parameter, fb. Numbers next to symbols represent number of
values from which averages were determined. (After Balsillie, 1985). ......... 4

Figure 4. Comparison of results of predicted db/Hb using the relationships of Galvin
(1969), Collins and Weir (1969), and Mallard (1978) which consider bed
slope, tan ab, only, described in text.................................. 8

Figure 5. The McCowan equation as it represents the data of Table 3 for db and
Hb. Data include 810 total points, 156 field data points, and 16 laboratory
prototype data pairs, and 624 small wave laboratory data pairs; 3 sb
indicate the limits of natural shore-breaker wave height variability within a
shore-breaking wave train. ...................................... 25

Figure 6. Relationship between the mean shore-breaking wave height, Hb, for a
single wave train and its associated standard deviation, sb, (after Balsillie and
Carter, 1984a, 1984b). ................... ..................... 27

LIST OF TABLES

Table 1. Some early general observations reported by Gailliard (1904) and Scripps
Institution of Oceanography (1945). ................................ 6

Table 2. Breaker depth studies resulting in quantifying relationships. .............. 7

Table 3. List of general characteristics of field and laboratory data used in
this w ork. ................................... ............... 10

Table 4. Stepwise regression results for equation (11). ....................... 15

Table 5. Correlation matrix for results forthcoming from equation (11). Numerical
values above and to the right of the zig-zag diagonal line are Pearson
product-moment correlation coefficients (r), those below and to the left are
100 r2. See text for explanation. ................................. 16

Table 6. Net contributions of independent variables. .......................... 17

Table 7. Net contributions of independent variables in predicting db from stepwise
regression analysis for equation (11). Results are listed for each
investigation and/or data groupings and averages associated with data
groupings .. . . . . . . . . . . . . . . . . . . .. . . . . . 19

Table 8. Characteristics of data and assessments for a special data set ............ 20

Table 9. Net contributions of independent variables for special data set. ........... 21

Table 10. Determination of final regression coefficient relating db and Hb. ........... 22








Table 11. Comparative analysis results for three prediction methods of db. .......... 23

Table 12. Determination of corrective regression coefficients for Mallard's and
Weggel's equations. ........................................... 24

Table 13. Examples of the degree on which shore-breaking wave energy levels are
dependent on the depth in which the waves break. Assessment is made in
terms of sb given by equation (22), where the wave energy is directly
proportional to the wave height squared. ........................... 28

APPENDICES


APPENDIX I. Example calculations of the effect of shore-breaking water depth
on wave energy constraints. ...................................

APPENDIX II. Stepwise regression results for data sets and subsets ............
All data. ...... ......... .................................
All data minus Gaillard's (1904) field data. ..........................
All data minus Gaillard's (1904), Scripps Leica Type I and II (1944a, 1944b,
1945), and Weishar's (1976) field data. ...........................
All Laboratory data. ...................................... ...
Laboratory data minus prototype laboratory data .....................
Prototype laboratory data. ....................................
A ll Field data. . . . . . . . . . . . . . . . . . . . . . . .

APPENDIX Ill. Data listings and stepwise regression results for each investigation. .


..37

. 41
. 43
..45

..47
. 49
..51
. 53
. 55

. 57


Gaillard (1904) field data. ........................................


Scripps (1944a, 1944b, 1945) Leica Type I field data..
Scripps (1944a, 1944b, 1945) Leica Type II field data.
Balsillie and Carter (1980)field data. .............
Weishar (1976) original field data. ...............
Munk (1949) Beach Erosion Board laboratory data . .
Munk (1949) Berkeley laboratory data. .............
Putnam and others (1949) laboratory data. ..........
Iversen (1952) laboratory data. ..................
Morison and Crooke (1953) laboratory data. .........
Galvin and Eagelson (1965) laboratory data. .........
Eagleson (1965) laboratory data. .................
Horikawa and Kuo (1967) laboratory data. .........
Bowen and others (1968) laboratory data ..........
Komar and Simmons (1968) laboratory data.........
Galvin (1969) laboratory data. ...................
Weggel and Maxwell (1970) laboratory data ........
Iwagaki and others (1974) laboratory data..........
Walker (1974) laboratory data .................
Van Dorn 1978) laboratory data. .................
Hansen and Svendsen (1979) laboratory data .......
Singamsetti and Wind (1980) laboratory data........
Nadaoka (1986) laboratory data. .................
Smith and Kraus (1990) laboratory data ..........


S. 63
... 67
... 69
... 73
... 77
. 81
... 83
... 87
. 91
. 93
... 97
... 99
. 103
. 105
. 109
.. 111
. 113
. 117
. 119
. 121
. 123
. 127
. 129


.............

.............








Small laboratory wave sets combined. .........................

APPENDIX IV. Stepwise regression results for a special data set ..........

APPENDIX V. Stepwise regression results where db\Hb is the independent
param eter. . . . . . . . . . . . . . . . . . . . . . ..

APPENDIX VI. Stepwise regression results where tan ab is the independent
variable . . . . . . . . . . . . . . . . . . . . . . ..


APPENDIX VII. Additional data needed to improve the statistical fit of Figure 5
of the m ain text. .......................................

APPENDIX VIII. Field data for determination of additional values of Hb"/Hb and


fb for Figure 3 ...................................


..... 149


. . . . . 153


SHORE-BREAKING WAVE ENERGETIC


A BSTRA CT ..................................
INTRODUCTION.................................
AIRY WAVE THEORY ...... .....................
Potential Energy, EP .......................
Classical Derivation .. ...............
A Different Approach .................
Kinetic Energy, EK ........................
Classical Derivation ..................
Total and Specific Wave Energy Over a Wavelength
Total and Specific Wave Trough Energy .........
Total and Specific Wave Crest Energy ..........
Discussion of Airy Results ...................
DISTORTED SHORE-BREAKING WAVES ..............
Potential Energy, EP .......................
Kinetic Energy, EK ........................
Total and Specific Wave Energy Over a Wavelength
Total and Specific Wave Trough Energy .........
Total and Specific Wave Crest Energy ..........
DISTORTED SHORE-BREAKING WAVES ON A HORIZONTAL
DISTORTED SHORE-BREAKING WAVES ON SLOPING BEDS
DISCUSSION AND CONCLUSIONS .................
ACKNOWLEDGEMENTS .........................
REFERENCES ..................................


...... 157
...... 157
...... 158
...... 159
...... 159
...... 160
...... 160
...... 16 1
. .... 165
. .... 165
. .... 165
..... 165
. . 165
...... 169
...... 170
.. .... 17 1
...... 173
...... 173
...... 173
...... 174
...... 176
...... 180
...... 180


BED


LIST OF FIGURES

Figure 1. Definition sketch of wave conditions for an Airy sine wave at the shore-
breaking position; Hb = 1.0 m, T = 10 s. .......................... 159


.... 133


137


141


145








Figure 2. Figure 2. Eulerian horizontal (top) and vertical (bottom) water particle
velocity (m/s) fields for an Airy shore-breaker where Hb = 1.0 m, T = 8.0 s,
and Lb = 39.68 m (Note: the terminology "bed" is used since this work does
not extend into the bottom eddy layer (BEL) nor is it concerned at this time
w ith BEL kinem atics) ......................................... 163

Figure 3. The average of the sum of horizontal and vertical water particle velocities
squared (from equations (17) and (18), respectively) versus the wave celerity
squared for Airy waves at the shore-breaking position ................. 164

Figure 4. Tabulated (Table 1) values of wave crest and wave trough kinetic
energies versus total kinetic energy for one wavelength. .............. 164

Figure 5. Definition sketch of distorted waves at the shore-breaker position. ...... 166

Figure 6. Laboratory data of Hansen and Svendsen (1979) illustrating the
attenuation of wave crest length, Ic, at the shore-breaking position ........ 168

Figure 7. Wave crest length at shore-breaking, Ic (Lb = wave length at shore-
breaking) versus the surf similarity parameter, Fb (values following symbols
are bed slopes; numbers associated with plotted points are number of waves
representing plotted averages). ................................... 168

Figure 8. Determination of the front face angle, tan ab of the shore-breaking wave
crest; data from Iverson (1952). ................................. 169

Figure 9. Data from Adeymo (1970) for determination of the length of the breaking
wave trough front, Itf, where It = length of the wave trough, db = water
depth at shore-breaking, and Lb = wave length at shore-breaking. ........ 170

Figure 10. Eulerian horizontal water particle velocity (m/s) field for a distorted
shore-breaker where Hb = 1.0 m, T = 8.0 s, tan ab = 0.02, cb = 3.92
m/s, Lb = 31.39 m, Ic = 9.73 m, It = 21.66 m (Note: the terminology "bed"
is used since this work does not extend into the bottom eddy layer (BEL) nor
is it concerned at this time with BEL kinematics). ................... .. 172

Figure 11. Eulerian wave energy density ratios for distorted and Airy waves (the
latter on a horizontal bed) at shore-breaking, illustrating the effect of bed
slope, tan ab and the surf similarity parameter, fb ; bed slope wave groups
represent breaking heights ranging from 0.25 to 2.0 m with periods ranging
from 4to 12 s. ... .......................................... 177

Figure 12. Ratios of Eulerian distorted shore-breaker crest to trough, and crest to
wavelength total energies, illustrating the effect of bed slope, tan ab and
surf similarity parameter, 4b ; bed slope wave groups represent breaker
heights ranging from 0.25 to 2.0 m with periods ranging from 4 to 12 s. ...... 177

Figure 13. Energy density relative dispersion for energy associated with water
particle columns for distorted shore-breakers. ......................... 178








LIST OF TABLES

Table 1. Some examples of total kinetic wave energies for Airy shallow water waves
assumed to be shore-breaking. .................................. 162

Table 2. Comparison of distorted shore-breaking wave and Airy wave energies on a
horizontal bed. ............................................. 175

Table 3. Energy density relative dispersion analysis results for distorted
shore-breakers. ............................................. 178

Table 4. Summary of wavelength, wave crest, and wave trough energy constraints
for Airy and distorted breaking waves. ................. ........... 179









ON THE BREAKING OF NEARSHORE WAVES

by

James H. Balsillie, P. G. No. 167

Coastal Engineering Geologist, Geologic Investigations Section
The Florida Geological Survey

ABSTRACT

This investigation considers data for 624 small laboratory shore-breaking waves,
16 prototype (large) laboratory wave tank shore-breaking waves, and from 131 to 172
(depending on the number of variables available for analysis) field shore-breaking
waves, for the determination as to where shore-breaking occurs. The original, formal
definition of McCowan (1894) suggested that nearshore waves are depth limited (i.e.,
related to water depth only). Subsequent investigators, feeling the answer must be
more complicated have, in addition, included bed slope, tan ab, equivalent wave
steepness, Hb/(g T2), the surf similarity parameter, fb, etc., in endeavors to "refine"
predictive power. Results of this study, to a highly significant level, however, confirm
that nearshore waves are depth limited. It was found that db = 1.277 Hb, where db
is the water depth at the shore-breaking position and Hb is the average shore-breaking
wave height. This result is so closely related to McCowan's original result of db = 1.28
Hb, that McCowan's relationship remains as the standard instrument for prediction.


INTRODUCTION

Where or when waves shore-break
affects our ability to predict longshore
currents, water levelss, sediment transport,
wave impact pressures, longshore bar
dynamics, wave reformation, etc. It is, in
fact, an issue of fundamental importance in
coastal applications, even when rated in the
context of the most basic of Earth-bound
Newtonian physics. It is, perhaps, no where
more importantly sensitive than when we
consider wave energy. Following are
some examples concerning its importance.
Golob and Brus (1993) state that at any
moment the "... total amount of power in
waves breaking along the world's shorelines
has been estimated at about 2 million to 3
million megawatts, which is about equal to
the generating capacity of 3,000 large power
plants". They also note that at any moment
"... the total wave-power potential along


Europe's Atlantic Coast is about 111,000
megawatts, or about 85 percent of the
European Community's current demand for
electricity".

In Florida (1990-1992 data; Florida
Department of Community Affairs (1995), U.
S. Department of Energy, 1997)), the
average residential structure uses an average
of 39 kilowatt-hours in one day or 1188
kilowatt-hours in one average month (30.42
days). Suppose that for your beach-front
community a specialized power plant placed
in the surf could precisely harness all the
breaking wave energy along but a 10 m
length of beach. For a 0.25 m breaker
height with a 4 s wave period, enough
energy would be harnessed in one day
(133,013,388 joules or 36.95
kilowatt-hours) to power your home for
almost one day (22 .7 hours). For a 0.5 m
breaker height with a period of 6 s, enough









energy would be
harnessed in one day
(752,439,744 joules or
209.1 kilowatt-hours) to
provide power for more
than 5 homes (5.36
homes to be precise). For
a 1.0 m breaker height
with a period of 8 s,
enough energy would be
harnessed in one day
(4,256,442,000 joules or
1182 kilowatt-hours) to
provide power for more
than 30 homes (30.28
homes to be precise).
Several breaker


RELATIVE
BREAKER 2
ENERGY
LEVEL
0.5 ---

0.2
S 0.5 1.0 1.5 2.0 2.5 3.0

Figure 1. Water depth in which shore-breaking occurs, db,
given by db = c Hb (Hb = mean shore-breaking wave
height), versus the relative breaker energy level.


height/period examples are
given since the ocean-generated wave field
is almost always comprised of multiple wave
trains. Hence, if all the above waves were
present, then enough energy would be
available to power over 36 homes (36.3
homes to be precise).

In addition, where waves shore-break
has remarkable influence on wave energy
constraints. Let us assume, for the moment,
that nearshore waves are simply water depth
limited. That is, water depth only
determines where they shore-break. The
criterion of McCowan (1894) is used to
illustrate this simple example which is
plotted in Figure 1, where wave energy is
generally considered to be proportional to
the breaker height squared. Hence, a wave
with a height of 2 m will contain 4 times the
energy of a 1 m wave, a wave height of 3 m
has 9 times the energy of a 1 m wave, a
wave with a height of 4 m has 16 times the
energy of a 1 m wave, and so on. The value
of c (Figure 1) as given by McCowan (1894)
is 1.28 (although its value has been
questioned by many investigators). Let us
assume it is valid for the following examples.
If a value of c = 0.5 is used, the breaker
energy level becomes 6.55 times that for c
= 1.28. If the value of c = 3, the wave
energy level becomes 0.182 times the


energy level where c = 1.28 (i.e., 5.5 times
less). One can easily see from Figure 1 that
wave energy constraints are sensitively
dependent upon the water depth in which
breaking occurs (see Appendix I for example
calculations).

In Florida, out of necessity, it
becomes an important issue because wave
conditions range from zero energy to high
energy (Tanner, 1960), and so we must be
sensitively precise in our assessment of
wave breaking. Almost every coastal
application, in one way or another, is
dependent on where or when waves shore-
break. It is an issue that is of vital
importance to every professional coastal
practitioner.

The breaking of ocean waves (i.e.,
gravity waves) is defined to occur when the
internal horizontal water particle velocities,
ub, in any part of the wave crest exceed the
wave phase speed, cb (e.g., McCowan,
1894; Munk, 1949; Kinsman, 1965), or:

b > 1.0 (1)
Cb
termed the kinematic stability parameter
(Dean, 1968). There is also involved the
dynamic stability parameter (Dean, 1968)









which involves vertical water particle
accelerations necessary to conserve the
integrity of the wave form as regards its
height. Its treatment is beyond the scope of
this work, nor is it necessary (i.e., we shall
assume wave height integrity is maintained,
which in reality it is, at least until shore-
breaking is final).

The author is careful to use the
terminology shore-breaking waves (including
bar-breaking waves) which are produced due
to nearshore shoaling conditions (i.e., depth
limitations), so named to distinguish them
from fully forced waves breaking in deeper
water (Balsillie and others, 1976; Mooers,
1976) which may break due to critically high
wind stresses (e.g., "white caps" or
"horses"), seismic impulses, explosive
events of anthropic origin, or even, perhaps,
in the laboratory by improperly sited wave
generators.

For the bulk of the history (- 1859 to
present) in endeavors to determine the
causes) of shore-breaking, technical means
were not available to measure the criterion
of equation (1). Nor are they available to
many of us today. We have had to or do,
therefore, rely upon an approximating
surrogate set of visual definitions, which
more nearly identify when waves shore-
break.

By 1946, Dean M. P. O'Brien had
identified and suggested conditions required
to produce spilling and plunging type shore-
breakers (Beach Erosion Board, 1949).
Subsequent evaluations (e.g., Galvin, 1968)
led to a set of "standardized" visual
definitions of shore-breakers in profile view.
The author (JHB), however, for the principal
types of shore-breakers (Balsillie, 1985;
1999), prefers the following.

Spilling occurs where the top of the
unstable wave crest results in aerated,
turbulent water slipping down the front face
of the wave (Figure 2). Up to 25% of the


-SWL----------


Figure 2. Three basic types of shore-
breaking waves.

upper portion of the breaking wave crest is
considered to be involved in spilling.
Aerated water at the crest apex signifies that
the breaking position has been reached.

Plunging occurs (Figure 2) where the
upper portion of the wave crest (> 25%)
curls over, forms an air pocket and the
curling crest eventually falls into the trough
fronting the breaking wave crest. Shore-
breaking for a plunger is defined to occur
when the front face of the crest becomes
vertical.

Surging occurs at the point when the
basal portion of the wave crest rushes
forward from beneath the crest, sliding up
the foreshore slope with a minimum of
bubble production (Figure 2).

Collapsing occurs when wave crest


Direction of Wave Travel
Plunging



Defined shore-breaker
position for plungers.
------------------------ ------- --SWL--


Spilling


Surging




















100
Hb

(%)


-- SPILLING


Figure 3. Relationship between the percentage of the wave crest that is
involved in shore-breaking, Hb" (measured from the crest apex downward)
where Hb is the mean shore-breaking wave height, as a function of the modified
surf similarity parameter, Fb. Numbers next to symbols identify the number of
values from which averages were determined. (After Balsillie, 1985).


advance suddenly terminates, and the wave
crest simply collapses. In all his years of
observing shore-breaking waves on sandy
shores in Florida, the author has seen this
occur but once.

A shore-breaking continuum (Figure
3) has been proposed (Balsillie, 1985)
according to:

tanh (2)
H, 8
where Hb" is the amount of the shore-
breaking wave crest, measured from the
crest apex downward, involved in translator


shore-breaking, Hb is the mean breaker
height, and fb is the modified surf similarity
parameter of Iribarren and Nogales (1949).
Surrogate definitions fit into the continuum
of equation (2) according to:
< 0.64, Spilling
tan ab 0.64 5.0, Plunging (3)
S i 05.0 ?, Surging
Mb/l(f ) > ?, Collapsing
in which tan ab is the bed slope, g is the
acceleration of gravity, and T is the wave
period. It was determined that the breaker
with the most destructive potential (i.e., in
terms of the horizontal impact pressure) at
the highest elevation occurs where fb = 1.0.
This correlates with a plunging shore-breaker









where 38% of the breaker crest top is
involved in translator shore-breaking.

Scripps Institution of Oceanography
(1945) in its World War II field experiments
on shore-breaking was careful to note that it
is not the bed slope that appears to be so
important in influencing breaking but, rather,
a sudden change in the bed slope. The
author posits that not even that is important
as it concerns the physical processess.
More nearly, a wave-encountered sudden
change in the bed slope induces a change in
wave profile shape which aids the observers'
cognitive abilities to more readily identify a
change in the wave shape that he perceives
to have reached the shore-breaking position.
In short, the surrogate definitions provide but
discontinuous, albeit useful, clues as to
where shore-breaking occurs. In fact, for
such conditions, the author can provide a
precise definition for the equilibrium
nearshore profile, when the shore-breaker is
of the spilling type. This occurs for a profile
upon which the spilling shore-breaker
continuously breaks until, at the shoreline
it's energy is completely dissipated and the
wave crest form is no longer evident.
Moreover, because uc is only very slightly
greater than cb, the observer may be unable
to detect according to definition that the
wave has reached shore-breaking conditions
anywhere along its entire shore-breaking
journey.

PREVIOUS WORK

Interest in where waves shore-break,
at least in terms of published work on the
subject, has been a subject of serious
interest for well over a century. The first
published formal account known to the
author was the theoretical work of
McCowan (1881) which he subsequently
modified (McCowan, 1894), resulting in:

d, = 1.28 H, (4]
where db is the water depth at shore-
breaking (measured as the vertical distance


from the still water level (SWL) to the bed),
and Hb is the mean shore-breaking wave
height.

The first comprehensive field data
collection effort was compiled by Gaillard
(1904). He also identified an earlier wave
tank study conducted by Henri Bazin in
1859, predating McCowan's work by 25
years. Other earlier studies, both in the field
and (perhaps) laboratory, were also identified
by Gaillard. These and some other early
studies identified by the Scripps Institution of
Oceanography (1945), leading to some
general observations, are listed in Table 1.

Interest in the issue accelerated
during World War II when landing craft
operations became of serious concern. The
U. S. Government initiated a program of
research through the Joint Army-Navy
Intelligence Service (JANIS) with the U. S.
Army, Beach Erosion Board (BEB) assigned
as the lead agency (Quinn, 1977). While the
BEB conducted much of its own research, it
also contracted with such institutions as
Wood's Hole Oceanographic Institution and
Scripps Institution of Oceanography (Scripps
Institution of Oceanography, 1945). A
considerable number of confidential works
were completed. They remained virtually
unknown to the public until the summer of
1976 when the author, then on the staff of
the U. S. Army, Coastal Engineering
Research Center (CERC, successor to the
Beach Erosion Board), found them in a
secured section of the agency and had them
released (ironically, these documents had
been declassified in the summer of 1950,
but were apparently forgotten and not made
available to the public). The Scripps (1945)
Leica Type 1 and II data, and special
measurements listed in these secured
documents were published by Munk
(1949). In both the Scripps (1945) and
Munk's (1949) works the special
measurements data were plotted in support
of equation (4).

Through ensuing years investigators









Table 1. Some early general observations reported by Gaillard (1904) and Scripps
Institution of Oceanography (1945)
INVESTIGATION RESULTS
J. Scott Russell, wave tank db/Hb = 1.0, tan ab = 0.02
J. Scott Russell, field data db/Hb = 1.0, tan ab = 0.02
Thomas Stevenson, Firth of Forth field data* (db/Hblavg = 0.715
Thomas Stevenson, Scarborough field data, 1870* db/Hb = 2.3
Henri Bazin, wave tank, 1859 db/Hb = 3/2, tan ab = 0.015
William Shield, Peterhead field data, 1888* db/Hb = 1.27
British Admiralty, Swell Forecast Section, Comdr. (db/Hb)avg = 1.5
Suthons, R. N., 1944
British Superintendent of Mine Design" db/Hb = 4/3
Canadian Report on "Beach intelligence in the 1.2 S db/Hb b 1.45
Mediterranean area"
University of California wave tank study, 1945* Decrease of db/Hb with decreasing wave steepness
and increasing bed slope.
Estero Bay, California landing craft field study, Lt. (db/Hb)avg = 1.30, although data showed a
Munch decrease in db/Hb with decreasing wave steepness.
Scripps Institution field study on July 1, 1944* (db/Hb)avg = 1.30, but show an increase in db/Hb
with decreasing wave steepness.
Monterey Harbor, California photo study, 1945** db/Hb = 1.14
Oceanside, California field study, 1945** db/Hb = 1.55


variously conducted experiments to
numerically quantify where shore-breaking
occurs. Largely, these studies were
conducted by engineers apparently
convinced that waves were simply not depth
limited . that the correct answer had to be
more complicated. Often only small data
sets were considered in their quests. A list
of studies and wave parameters considered
is given in Table 2.

Galvin (1969) suggested that where
at the shore-breaking position, the bed slope,
tan ab, is tan ab > 0.07, then:


=d,
0.92
H,


(5a)


and where tan ab < 0.07:


1.4 6.85 tan a,


(5b)


which are both referenced to the mean water
level (MWL) rather than SWL. Galvin (1969)
suggested that for tan ab on the order of
from 0.05 to 0.1, SWL is higher than MWL
by a factor of 0.04 Hb, and where tan ab is
about 0.2 by a factor of 0.08 Hb.

Collins and Weir (1979) found:


-= (0.72 + 5.6 tan a,)-1

and Mallard (1978) concluded:









Table 2. Table of independent variables related to the water depth
at shore-breaking.
Wave Parameter types Wave parameters used
and Investigators in investigations

Wave parameters at Hb T tan Ob H/(g T2) b
shore-breaking.
McCowan (1894) X
Scripps (1945), Munk (1949) X
Galvin (1969) X X
Collins and Weir (1969) X X

Weggel (1972a, 1972b) X X X
Mallard (1978) X X
Balsillie (1983) X X X X
Deep water wave parameters. Ho Ho/Lo Fo
Smith and Kraus (1990, 1991) X X X X

Nelson (1994), and Gourlay (1994) X X X
Notes: Ho = deep water wave height, Lo = deep water wave length,
Fo = deep water surf similarity parameter.


c, = 4 g (1.0 e-'9 tan ab)


= [0.73 + 2.87 (tan a)'997]-1

Equations (5a) through (7) are plo
in Figure 4. For these equations, value
db/Hb are close for tan ab less than al
0.01. However, for tan ab > 0.01 their
disparity.

An additional parameter which
been earlier considered as a pos!
influential factor was the equivalent v
steepness, Hbl(g T2), whose derivatio
given by Battjes (1974, p. 469). We
(1972a, 1972b) introduced this param
into his investigations, and suggested:


- c3 + C,


Hr max


where:


S1-1

9 T 2


tted
s of
bout
re is


(8a)


in which c2 = 4.462 m2/s = 1.36 ft2/sec
are constants for use in assuring unit
consistency, and:


S= 1.56 1.0 le-195 tan a)-1
Ca= 1.56 (tO+ e


(8b)


Balsillie (1983) undertook an
investigation to compare the relative validity
had of the preceding predictive methods. He
sibly excluded Galvin's (1969) equation because:
nave 1) it was referenced to MSL, 2) had a
n is discontinuous behavior for tan ab > 0.07,
ggel and 3) was represented by other predictive
Qeter
methods that were more comprehensive in
their data coverage. The Collins and Weir
(1979) method was not considered because
(8) Mallard's (1978) result was so similar, and
included more comprehensive data coverage.
The result was that McCowan's equation
was slightly more successful in quantifying
the relationship for shore-breaker water





















0.001 0.01 0
0.001 0.01 0.1


1.0


tan atb
Figure 4. Comparison of results of predicted
dblHb using the relationships of Galvin (1969),
Collins and Weir (1969), and Mallard (1978)
which consider bed slope, tan ab, only,
described in text.


depth. Since that work, considerably more
data has been collected, primarily in the
laboratory range of possibilities, although (
some prototypical laboratory results have a'
surfaced (Maruyama and others, 1983; "
Stive, 1985; Nadaoka, 1986; Takikama and P
others ,1997). For this and other reasons, a
the subject of where shore-breaking occurs o
has been revisited. tt

Smith and Kraus (1990, 1991)
conducted laboratory studies of shore- M
breaking over longshore bars from which o
they (Smith and Kraus, 1991) suggest: o
s
Si
b = (0.41 + 0.98 to)- (9a) d
Hb is
where 0.3 < o < 0.85, and: si

d a
Hb (1.45- 0.22 ,)-' (9b) e
*b in
where 1.6 < o 3.5, in which f is a w
hybrid form of the surf similarity parameter b
(comprised partly of deep water 1
characteristics and partly of breaker zone a
characteristics) evaluated as: b


tan a,

FH 1L;


(9c)


in which Ho and Lo are the deep water
wave height and wave length,
respectively. They note a correlation
coefficient of 0.85 between measured
and predicted values of db/Hb for the
above regions of validity for equation (9a)
and (9b). For the region 0.85 < f <
1.6, which they term the transition
region, they seem to suggest the use of
equation (9b) although they do note the
highly scattered nature of their data.

Based on laboratory data,
Kaminsky and Kraus (1993) confirmed
the use of equation (4) as an average
representation for typical field beach
slopes.

Nelson (1982, 1994) and Gourlay
1994) have published rather convoluted
accounts using mostly laboratory data (one
Field" study by Nelson, 1994) in which they
osit that . for waves propagating over
horizontal bottom . the maximum value
f Hb/db never exceeds 0.55." Recasting
is yields:


d, 1.82 H,


(10)


moreover, Gourlay (1994) notes that the use
f equation (4) rather than (10) ".
verestimates the wave energy reaching a
structure by a factor of 2 . .", which it
oes. He further suggests that equation (4)
, therefore, .unnecessarily
conservative Both assertions are
gnificantly surprising, The depth of water
t shore-breaking significantly affects the
energy content of the wave form, particularly
Slight of new findings concerning how
rave energy is partitioned between the
breaking wave crest and trough (Balsillie,
997, 1999). Hence, this has prompted the
uthor to revisit the issue of where shore-
reaking occurs. We shall term this last
result the Nelson-Gourlay horizontal bed
fope breaking anomaly which will be









readdressed later in this work.

THE DATA

General characteristics of the data
considered in this work are listed in Table 3.
They constitute the largest data compilation
amassed to date for assessment of where
shore-breaking occurs. All the major shore-
breaking wave types (i.e., spilling, plunging,
and surging breakers) are represented by
these data.

Laboratory Data

This work includes data from 27
laboratory investigations, to include 640 data
sets, 1 6 of which have prototype dimensions
and 624 of which are much smaller waves
of typical laboratory size. Generally,
laboratory investigators are quite specific
about reporting experimental physical
attributes and measurement techniques.
Occasionally, a researcher may fail to report
a crucial factor because it seemed obvious at
the time of final document preparation. By-
and-large, however, final documentation is
usefully complete. Rather than reiterating
such information here, which would require
significant space, the reader is referred to
each referenced study.

Field Data

It is the author's impression, because
of the lack of inclusion of field data in
existing treatments on this subject, that field
data are not accepted with the veracity
accorded data collected in the laboratory
wave tank. Field data significantly increase
the domain of data coverage. As it turns
out, the laboratory data occupy but only
3.36% of the dimensional domain of the
data available. Scientific pursuits require
consideration of all viable data. Given some
of the characteristics of all the data which
requires anthropic observation (to be
discussed later), the author does not quite
understand why the exclusion of field data in


most works published on this subject should
be so. And so, it appears necessary to
provide some detailed discussion about field
data considered in this account.

Data from six field investigators are
used in this work, totalling 131 data sets for
db, Hb, T, tan ab, and 172 data sets for db
and Hb.

GallFard Data

The first comprehensive compilation
of the data on the subject was authored by
Gaillard (1904). There is no reported record
of how he measured his data. Even so, his
results are sincerely and sensitively complete
with regard to documentation and global
extent. They are, at the very least, viable
observations which, in the ensuing 94 years
have not been discredited. His data in the
scientific sense, therefore, cannot be
dismissed.

Scnpps Leia Data

These data, collected as part of the
war effort during World War II were
photographed at Scripps Institution of
Oceanography (1944a, 1944b, 1945) at La
Jolla, California. The Leica Type I and II data
were collected on a daily basis from January
9 to April 15, 1944, and depended upon
profile conditions where the breakers were
measured. Type II measurements were
made for breakers occurring over the stoss
slope of longshore bars. Type I
measurements represent waves shore-
breaking on the linear nearshore slope just
offshore from the shoreline. Photographs
were taken at a point about 7.62 m above
the pier deck at a distance of about 75 m
south of the shore end of the pier. A large
clock suspended from a boom was in the
field of view and clearly visible in the photos.
Wide planks clearly marked at 0.61 m
vertical intervals were affixed to the Scripps
Pier piles. Daily profiles were surveyed along
the pier and the SWL was determined using










Table 3. List of general characteristics of field and laboratory data used in this work.

db Hb T
Investigator n b Hb T tan ab
(m) (m) (s)

FIELD DATA

Gaillard (1904) 25 0.792- 5.349 0.61 -3.353 3.85- 10.98 0.0154- 0.029
12 1.189-5.456 0.61 -3.962 ----- ---

Scripps (1944a,
1944b,1945) Leica 56 1.554- 4.450 1.219 3.475 6.5- 13.7 0.0159
Type [

Scripps (1944a,
1944b, 1945) 18 1.68-3.72 1.28-2.74 7.0- 13.0 0.049
Leica Type II

Scripps (1945) 29 0.49- 1.26 0.27- 1.13 -- -----
Spec. Meas.

Balsillie and Carter 30 0.11 0.758 0.057 0.541 1.33 8.57 0.017 0.462
(1980)

Weishar (1978) 2 0.94, 1.01 0.60, 0.77 7.7 0.05

Total 172

LABORATORY DATA PROTOTYPE DIMENSIONS

Maruyamaand 1 2.0 1.29 3.1 0.0340
others (1983)

Stive (1985) 2 0.2, 1.90 0.18, 1.50 1.8, 5.0 0.0250

Nadaoka (1986) 12 0.14-0.605 0.103- 0.509 0.92- 2.99 0.05

Takikama and
Takikamaand 1 0.232 0.215 2.08 0.05
others (1997)

Total 16

LABORATORY DATA

Munk (1949)BEB 37 0.043 0.187 0.031 -0.13 0.73- 1.09 003, 0.049,
data 0.159

Munk (1949) 0.0541,
Munkl(1949) 16 0.0631- 0.1451 0.068-0.1 0.86- 1.98 0.01,
Berkeley data 0.0719, 0.09

Putnam and others 0.066, 0.098,
(1949) 37 0.058- 0.229 0.037- 0.143 0.72- 2.32 0.10, 0.139,
0.143, 0.241

Iverson (1952) 63 0.043- 0.165 0.043- 0.128 0.74- 2.67 0.0, 03,
0.05, 0.10

Morison and
Morison and 6 0.07-0.129 0.056- 0.113 0.78-2.62 0.02,0.10
Crooke (1953)










Table 3. List of general characteristics of field and laboratory data used in this work
(cont).
dr Hb T
Investigator n b Hb tan ab
(m) (m) (s)

LABORATORY DATA (CONT.)
Galvin and
Galvin 24 0.021- 0.081 0.03-0.091 1.0- 1.50 0.10
Eagleson (1965)

Eagleson (1965) 7 0.058- 0.123 0.044- 0.095 0.79- 1.57 0.10

Horikawa and Kuo 0.0125,
(1o ad Ko 97 0.06- 0.263 0.06- 0.182 1.2-2.3 0.033 05
(1967)* 0.0333, 0.050

Bowen and others 11 0.042- 0.097 0.04- 0.13 0.82- 2.37 0.082
(1968)

Komar and 0.036, 0.07,
mKomar and (44 0.034- 0.213 0.03- 0.17 0.81 -2.37 0.036,0.07,
Simmons (1968)** 0.086, 0.105

Galvin (1969) 17 0.039- 0.114 0.038- 0.115 1.0-6.0 0.05,0.10
0.20

Weggel and
Mwll (0 9 0.087- 0.169 0.089- 0.162 1.27-2.05 0.051
Maxwell (1970)

Saeki and Sasaki
Sak d asaki 2 0.097- 0.164 0.099- 0.106 1.3-2.5 0.020
(1973)

wagaki and others 23 0.06- 0.158 0.044- 0.128 1.0- 2.0 0.03, 005,
(19741 0.10

Walker (1974) 15 0.031 -0.125 0.024- 0.116 1.17-2.33 0.033

Van Dorn (19781 0.022, 0.04,
Van Dorn (1978) 12 0.093- 0.217 0.108- 0.166 1.65-4.8 00 04'
0.083
Hansen and
Svendsen979 16 0.047- 0.149 0.043- 0.14 0.83-3.33 0.0282
Svendsen 01979)

Singamsetti and 0.025, 0.05,
Wind (1980) 95 0.078- 0.22 0.073- 0.193 1.03- 1.73 0.10, 0.20
Wind (19801 0.10,0.20

Mizugauchi (1981) 1 0.083 0.10 1.2 0.10

Visser (1982) 7 0.088 0.122 0.58- 0.108 0.7- 2.01 0.05, 0.10

Watanabe and
Dibajnia ( ) 3 0.1-0.11 0.075- 0.082 0.94- 1.19 0.05
Dibajnia (1988)

Smith and Kraus
h and Kraus 77 0.091 -0,271 0.088- 0.216 1.02-2.49 0.08-0.412
(1990) bars

Smith and Kraus
Saneeach 5 0.131 0.216 0.082- 0.165 1.02- 2.49 0.033
(1990) plane beach

Total 624

*Data listed in Smith and Kraus (1990); **data listed in Gaughan and others (1973).








a tide gauge. General bed slope conditions
for the field experiments and other details of
the field effort are documented by the
Scripps Institution of Oceanography (1944a,
1944b).

Scripps Special Measurements

A number of precise simultaneous
measurements of db and Hb were made near
the Scripps Pier and included in the Scripps
Institution of Oceanography (1945, p. 22-
23) report. Water depth at breaking was
measured using a hollow tube (internal to
which was a float) in the surf to measure the
still water level. Breaker height was
simultaneously measured using a scale
affixed to the outside of the tube. Three
observers would make simultaneous readings
of water depth, elevation of trough, and of
breaker crest. Unless the wave broke right
at the tube, measurements were eliminated.

BalsIe and Carter Data

Beach and nearshore shore-normal
profile surveying was the first task in the
quantitative measurement portion of these
field experiments (Balsillie and Carter, 1980;
Balsillie and Carter, 1984a, 1984b). Next, a
site was selected and surveyed at a location
considerably seaward of the surf zone,
where waves were as undistorted as
possible. Thirty elevations of wave crest
and trough measured from the bed were
recorded. These data, when averaged,
became as closely as was possible an
estimate of the SWL for the experiment.
The investigators were careful to note the
presence of separate wave trains, and
breaker zone widths for each wave train
were identified on the surveyed shore-normal
profile. For each wave train, 30 shore-
breaker crest height and trough elevations
were measured using a special rod fitted
with a foot that would not sink into the
sandy bed. Wave period for each wave train
was determined by measuring the time it
took for 11 wave crests to pass a stationary


point in the surf zone. When divided by 10,
an estimate of the wave period resulted.
Several such sets of these measurements
were taken by each participant to ensure a
more representative quantification. Bed
slopes were determined from the surveyed
profile data.

Weishar Data

Weishar (1976), Weishar and Byrne
(1979) analyzed the contents of a one-hour
16 mm film of wave activity from the
Virginia Beach, Virginia, pier, filmed by
Robert Byrne eight years earlier. The filming
was focused upon a rectangular grid
comprised of 0.61 m by 0.61 m segments
mounted on steel pipes jetted into the
nearshore (from the foreshore to 30.5 m
offshore) along a shore-normal azimuth at 3
m centers. Every 20 minutes the nearshore
profile was surveyed. During the period of
filming a relatively well defined swell was
present, with the wave crest traces being
almost parallel with the shoreline; very little
local wind wave activity was present.
Weisher reported the characteristics of 120
"consecutive" individual waves for the one-
hour film. Certainly, his procedure was
highly selective, since it can be estimated
that some 400 to 500 waves were probably
present during the filming. He did not,
however, identify his data by wave train.
The author, therefore, conducted a modal
analysis by decomposing the cumulative
probability distribution using the Method of
Differences (Tanner, 1959; see also Balsillie,
1995, p. 59-61) which, in the author's
opinion, is far superior to spectral analysis.
From this analysis, it was determined that
Weishar primarily measured waves for two
wave trains for which quantitatively
representative statistics were determined.

ANALYTICAL PROCEDURE

Prior to engaging in analytical pursuits
there are two scientific analytical tools that
the author should like to introduce to the









study.


Occam's Razor


Most of us dealing with science are
familiar with the terminology "Occam's
Razor". How many have more detailed
knowledge about the terminology is
unknown. The concept, however, is of
such importance to the present treatment
that some discussion is here presented.

William (of) Ockham or Occam
(Ockham is near Guildford, southwest of
London, England) was a medieval scholastic
monk (1285-1349)who is generally credited
with the law of parsimony which states
pluaralitas non est ponenda sine necessitas
(plurality should not be posited without
necessity). Others before Occam (e.g.,
French Dominican theologian Durand de
Saint-Porcain and Occam's teacher Duns
Scotus) and after Occam (e.g., Galileo
Galilei) also posited the principle but Occam
is credited as its originator because he used
it so frequently and employed it so sharply.
(e.g., Thorburn, 1915, 1918; Burns, 1915;
Encyclopaedia Britannica, 1981.) So it is to
be sharply employed in this work. Simply
stated . if multiple mechanisms explain
the facts equally well then the simplest
mechanism is that, in the scientific sense,
which requires acceptance.

Doyles' Principle

Sir Arthur Conan Doyle (1859-1930)
coined the phrase through his super sleuth
Sherlock Holmes, which states: When you
have eliminated the impossible, whatever
remains, however improbable, must be the
truth. This, the reader shall come to
understand, has merit in this work.

Procedure

Two analytical procedures are utilized
in this work. The first is stepwise
regression. The second is a comparative


analysis of several selected methods used to
predict where shore-breaking occurs.

STEPWISE REGRESSION

Stepwise regression is a powerful
statistical tool for ranking variables by their
relative importance (Harrison and Krumbein,
1964; Krumbein and Graybill, 1965) when
considered synergistically. In addition, a
basic contribution of this statistical tool is
that it readily provides for the evaluation of
data redundancy. Redundant variables are
those that, in large part, restate what some
other variable has already measured.
Redundancy is common in the early stages
of scientific quantification, when the physical
meaning of multiple interrelated variables
may not be clearly discernible (Harrison and
Krumbein, 1964; Krumbein and Graybill,
1965). Matrix operations performed to
derive relating equation coefficients are less
affected by roundoff errors; moreover,
derived coefficients are less sensitive to
small errors in the data (Miesch and Conner,
1968).

Three sets of data were selected for
stepwise regression analysis. In the first,
the water depth at shore-breaking, db, is
selected as the dependent variable and
based on the preceding accounts of
attempted identification of input variables,
Hb, T, tan ab, Hb/(g T2), and fb are selected
as independent variables. Note that
predictive methods which use deep water
wave parameters are not considered in this
analysis. It is the author's opinion that they
introduce additional complications that are
simply not necessary. Also, please
understand that, at this stage in the
investigation, one is interested in using the
stepwise regression tool as the means for a
substantive search for identifying the
contributory importance of variables, rather
than for finding a formal predicting equation
(e.g., Krumbein and Graybill, 1965, p. 395
and 398). Hence, in the first data set the
dependent variable (db) is in units of length









as is one independent variable (Hb), another
independent variable is in units of time (T),
and the remainder (tan ab, Hb/(g T2), and fb)
are dimensionless. Determination of a formal
equation (i.e., unit consistency across the
equation) is a later step.

The stepwise regression application
used in this work (all possible regressions)
follows the development of Krumbein and
Graybill (1965, p. 391-399). The computer
application (written by the author) is
described by Balsillie and Tanner (1999).
The first stepwise regression equation for
consideration has the form:


d= Po+(P1 H) + (P2 T)

+ (P3 tan a,) + (P4 -

+ (Ps Ub)


(11)


In the second data selection, the ratio
db/Hb was selected as the dependent
variable (actually a parameter), and the
stepwise regression equation assumes the
form:


P + (Po T) + (P tan lC )

+ (3, P -H) + (3b)
^ ^}e 5


+ (tanh 0.4 t,)
In the third data selection, the bed
slope, tan ab, was selected as the dependent
variable, and the general equation for
stepwise regression assessment is:
tan, = o + (P da) + (p2 H,)

g r
+(,, AT) (13+

Stepwise regression, through
algebraic matrix inversion, allows for the
determination of values of f so that the best
fit predictive outcomes as closely as possible
approach the measured independent variable
or parameter. All possible nonrecurring
combinations of independent variables are


processed, and relative assessment statistics
are then determined. The order in which
input independent variables are introduced to
the statistical procedure makes no
difference, nor do the units involved unless,
perhaps, they are transformed in some
unusual manner. Assessment statistics can
be the Pearson product-moment correlation
coefficient r, or r2, the sum of squares of db
accounted for. The later expressed as a
percentage (i.e., 100 r2) is preferred for this
method (e.g., Krumbein and Graybill, 1965;
Draper and Smith, 1980). Both, however,
indicate the degree of success of prediction,
i.e., the degree of agreement between
predicted and measured values of db (simply
put: 0 or 0% for no agreement, 1.0 or
100% for perfect agreement).

Assessment of Equation (11)

There are 771 sets of data (the
largest data base amassed to date for this
subject) available for analytical treatment
(Table 3). Only the Scripps (1945) Special
Measurements and part of Gaillard's (1904)
data set are excluded from stepwise
regression, because only db and Hb were
measured. Stepwise regression results for
equation (11) are listed in Table 4 in which
each line in Table 4 represents results for a
predictive equation and where assessment
statistics 100 r2 and r are given in columns
(7) and (8). In addition, the relative percent
contributory importance of independent input
variables is given in columns (2) through (6).
Note that in this later relative assessment,
some variables can have a contributory value
exceeding 100%. This occurs because
others have negative values (since f
coefficients can have negative assignments).
Each line, however, totals to 100%. The
later allows the reader the advantage to
readily determine the magnitude of
contribution, or to see how secondary
contributions are "cancelled out".

Now, we need only to decide upon an
upper limit value relative to which we select










Table 4. Stepwise regression results for equation (11).

Mean percent contributions of variables
in predicting db 7)
Percent of (8)
(2) (3) (4) (5) (6) Sum of Correlation
(1) Squares of db Coefficient,
Eq. Hb T tan ob Hb/(g T2) [b Accounted for r
(m) (s) (100 r2)


Independent Variables Taken One at a Time

1 100 94.26* 0.9709*
2 100 74.25 0.8617
3 100 6.79 0.2606
4 100 3.66 0.1914
5 100 4.87 0.2208
Independent Variables Taken Two at a Time

6 104.72 -4.72 94.27* 0.9709*
7 101.15 -1.15 94.26* 0.9709*
8 97.99 2.01 94.26* 0.9709*
9 100.83 -0.83 94.26* 0.9709*
10 103.25 -3.25 74.31 0.8620
11 71.72 28.28 78.58 0.8865
12 119.72 -19.72 76.01 0.8719
13 43.06 46.94 8.43 0.2904
14 92.21 7.79 6.81 0.2029
15 58.43 41.57 9.90 0.3146

Independent Variables Taken Three at a Time

16 106.50 -5.07 -1.43 94.27* 0.9710*
17 104.65 -4.69 0.04 94.27* 0.9709*
18 104.93 -4.57 -0.36 94.27* 0.9709*
19 98.98 -1.50 2.52 94.27* 0.9709*
20 101.13 -1.29 -0.16 94.26* 0.9709*
21 98.73 1.85 -0.58 94.26* 0.9709*
22 74.20 -5.17 30.97 78.92 0.8884
23 108.19 37.22 -45.41 73.38 0.8853
24 79.34 28.96 -8.30 79.43 0.8912
25 -4.53 59.93 44.60 9.91 0.3149

Independent Variables Taken Four at a Time

26 105.49 -4.59 -1.48 0.58 94.27* 0.9710"
27 108.06 -7.00 -4.15 3,08 94.28* 0.9710*
28 104.96 -4.59 -0.02 -0.36 94.27* 0.9709*
29 96.89 -4.44 4.38 3.17 94.27* 0.9709*
30 83.07 11.79 23.66 -18.53 79.71 0.8928

Independent Variables Taken Five at a Time

31 103.99 -5.5 -5.53 2.74 4.36 94.28* 0.9710"

NOTES: Each line represents results for a predicting equation. The correlation coefficient
measures the degree of agreement between predicted db outcomes from step-wise regression
equations and the measured db values.









Table 5. Correlation matrix for results forthcoming from equation (11). Numerical
values above and to the right of the zig-zag diagonal line are Pearson product-moment
correlation coefficients (r), those below and to the left are 100 r2. See text for
application.
db Hb T tan b Hb(g T2) b
(m) (m) (S)
db (m) 1%\00 0.9709 0.8617 -0.2606 -0.1914 -0.2208
Hb (m) 94.26% 100 1.00 0.8926 -0.2631 -0.2037 -0.2233
T (s) 74.25% 79.68% 100% \.00 -0.2759 -0.4391 -0.1031
tan ab 6.79% 6.92% 7.61% 100%\.00 0.2593 0.8204
Hbl(g T2) 3.66% 4.15% 19.28% 6.72% 10 00 -0.1387

5b 4.87% 4.99% 1.06% 67.30% 1.92% 100%.00


appropriate contributing combinations. For
this investigation, the selection of r2 > 94%
( or r > 0.97) is straightforward (values are
tagged with asterisks). All others can be
dismissed as being predicting combinations
with subservient import. Normally, the next
task is to determine the net contributions of
ranked variables. However, for the data
assessed in this work, not even that is
necessary (although we shall do so later).

For equation (11) there are 31
mechanisms, or equations in Table 4, from
which to choose. The equations become, in
groups of independent variables taken n at a
time (1 n 5), progressively more complex
from equations (1) to (31). More complex
successful equations (i.e., where 100 r2 >
0.94, or where r > 0.97) deviate from a
successful solution given by the simplest a
maximum of two-hundredths of a percent.
A deviation of 0.02% is simply not grounds
for selecting a more convoluted predicting
solution. We can, then, without the
slightest hesitation, employ Occam's Razor
and select equation (1) given by:

db = P + (P1 Hb) (14)
as the best solution (see Appendix II for the
R value table). We can, in addition, because
of the small magnitude of f/ (/ = -0.017),


dispense with the term and suggest:

db = P1 Hb = 1.372H,


(15)


although we shall use another statistical
fitting procedure to determine a formal form
for the relationship between measured data
and predicted results.

Data Redundancy

We can inspect for data redundancy
among the independent variables and
parameters (IVs and IPs) using the
correlation matrix of Table 5. As an
example, the relationship between Hb and
tan ab is 6.92%. For this case, we state
that Hb accounts for 6.92% of the sum of
squares of tan ab (we can also state it the
other way around). There is, therefore, little
redundancy between the two variables.
However, a rather large 100 r2 value of
67.03% exists between tan ab and fb. This
might be expected since fb is an IP
containing tan ab The effect is evident, in
terms of the percent contribution of IVs, for
equations 15, 25, 28, and 30 of Table 4.
All, however, are not in the acceptance
region (i.e., 100 r2 > 94%), except for
equation 28 of Table 4 in which the
contributory percentages are both very small.
The only other "noticeable" indication of








data redundancy is between T and Hb/(g T2),
which may be applicable to equation 13 of
Table 4; equation 13 of Table 4 is not,
however, within the acceptance region.

Net Contrbutions

Let us now, to be complete in our
analysis, determine the net contribution of
ranked independent variables for equation
(11) following the method of Krumbein and
Graybill (1965, p. 397-398).

Determination of net contributions of
the independent variables or parameters
starts with the selection of the strongest of
the independent variables taken one at a
time which is equation 1 of Table 4 for Hb
where 100 r2 = 94.26%.

Next, from the independent variables
taken two at a time series, we select the
highest 100 r2 value of 94.27% which
represents equation 6 of Table 4 in which T
is in the presence of Hb. Hence, the net
contribution of T becomes: 94.27% minus
the 100 r2 value for equation 1 of Table 4, or
94.27% 94.26% = 0.01%.

Now, from the independent variables
taken three at a time series, we select in the
presence of Hb and T, the next highest 100
r2 value for a IV or IP not yet selected.
(Note that as this procedure progresses we
can be faced with the situation that the
choice becomes less clear, so one may have
to make a tentative selection.) We select
equation 16 of Table 4 for tan ab which has
a 100 r2 value of 94.27%, and the net
contribution of tan ab becomes: 94.27%
minus the 100 r2 value for equation 6 of
Table 4 or 94.27% 94.27% = 0.00%.

Applying the same technique for
independent variables taken four at a time,
we find that equation 27 is the strongest
selection in which fb is in the presence of
Hb, T, and tan ab where 100 r2 = 94.28%.
The net contribution of tb becomes 94.28%


Table 6. Net contributions
of independent variables.
Variable Net
Contribution

Hb 94.26%
T 0.01%
tan ab 0.00%
fb 0.01%
Hb/(g T2) 0.00%


minus the 100 r2 value for equation 16 of
Table 4, or 94.28% 94.27% = 0.01%.
Finally, the last remaining variable
needing net contributory assessment is Hb/(g
T2) in the presence of Hb, T, tan ab, and fb.
Only one equation remains in Table 4 since
it represents independent variables taken five
at a time. The net contribution of Hb/(g T2)
becomes 94.28% minus the 100 r2 value for
equation 16 of Table 4 or 94.28% 94.28%
= 0.00%.

The net contributions are listed in
Table 6. Two of the values appear with a
net contribution of 0.00% because we have
not carried the results enough places to the
right of the decimal point. The exercise
makes little difference, however, because
except for Hb, contributions of the other
variables or parameters are negligible. These
results tell us, for the example at hand, no
more than the conclusion previously reached
that the best predictor, by far, are the results
given by equations (14) or (15). On the
other hand, the exercise is highly useful
because one would not submit to "high level
agency or corporate management", Table 4
and accompanying discussion as the final
assessment. Managers are concerned with
making, with all dispatch possible, decisions
based on the best simply defendable facts
available, and the Table 4 assessment would
probably still be too involved. Certainly, it
would require some time to explain and,
probably, one they would not want to
reiterate. Hence, while the preceding needs









to be formalized in a support document, the
final results presented to management is to
be reported as for Table 6, because they are
straightforwardly the simplest.

Some Observations

For those skeptics who doubt the
accuracy of field measurements in favor of
"more accurate" laboratory wave tank data,
here are some results that may be surprising.
If Gaillard's data (n = 25) are excluded
because we do not know how they were
measured and, therefore, may be unsure of
their accuracy, please be advised that
stepwise regression results become less
improved by an "across the board" average
value of 2.82% relative to Table 4 values
(see Appendix II for data and results).
Further, if Scripps Leica Type I and II data (n
= 74) are excluded because they are
represented by general bed slope conditions,
and Weishar's data (n = 2) are excluded
because of the manner in which the author
included them in this account, stepwise
regression results become less improved
"across the board" by an average value of
1.15% relative to Table 4 results. If all field
data are excluded (n = 131), stepwise
regression results "across the board"
become less improved by an average of
0.99% than those of Table 4. Finally, if
laboratory data only are considered and data
of prototypical dimensions (Maruyama and
others (1983), Stive (1985), Nadaoka
(1986), and Takikama and others (1997)) are
excluded (n = 16), stepwise regression
results "across the board" become less
improved by an average value of 20.22%
than those of Table 41 This results in an
100 r2 value of 74.04% which certainly is
not a convincing magnitude. These results
(particularly the last one) are significant
because it is the larger waves (field and
prototypical laboratory waves) which
significantly increases the strength of the
fitted relationship, even though they
comprise but 19.07% of the data. In terms
of the ranges in the domains of the data


(3.45 m for Hb, and 5.33 m for db),
however, the field plus prototype laboratory
data represent close to 96.64% of the
domain, the small laboratory waves but the
remaining 3.36%.

Data Set Listings and Stepwise
Regression Results

Appendix II provides additional
stepwise regression results for various data
groupings used in this work (see Table of
Contents). Data listings by investigator and
stepwise regression results are given in
Appendix III. All of Weishar's (1976) data
are listed and analyzed rather than the two
data groupings used in this work. Several
sets of laboratory investigations were so
small, or had no inverse matrix algebraic
solutions (see A Note on Matrix Algebra to
follow), and were grouped together.

Stepwise Assessment of Individual
Data Sets

The preceding has dealt with
stepwise regression analysis for all available
data. In addition, it would appear to be of
value to assess each individual data set
using stepwise regression. The results
(using Appendix III data listings) are given in
Table 7 as net contributions, as are simple
and weighted averages (the latter based on
sample size, n) for various data groupings.

From the results listed in Table 7, it is
apparent, for the most part, that there is no
guarantee that any single randomly selected
data set will result in a predictive outcome in
which one can have confidence. A strong
outcome is given by the prototype laboratory
data, and it can be used as a clue. This data
set, however, has but 16 data entries.

The clue alluded to is that the
strength of the previous result using all the
data is because of the large range in the data
domain.










Table 7. Net contributions of independent variables in predicting db from stepwise
regression analysis for equation (11). Results are listed for each investigation and/or
data groupings and averages associated with data groupings.


Investigator


Net Contribution in Predicting db {%)

H) ( tanab Hbl/(g T2) b Total
(m) (sELD D

FIELD DATA


Gaillard (1904) 25 81.80 0.62 2.81 0.01 5.47 90.71
Scripps (1944a) Leica Type I 56 71.00 0.00 1.20 0.07 0.51 73.32
Scripps (1944a) Leica Type II 18 37.15 10.99 5.86 7.17 7.08 68.25
Balsillie and Carter (1980) 30 83.84 0.05 0.92 0.04 0.22 85.07

Total 129
Simple Averages 68.45 2.92 2.70 1.82 3.32 79.34
Weighted Averages 71.36 1.67 2.10 1.04 2.32 78.72

LABORATORY DATA PROTOTYPE DIMENSIONS

Various investigators* 16 98.00 0.86 0.64 0.17 0.13 99.80

Field and Prototype Laboratory Data

Total 145
Simple Averages 74.36 2.50 2.29 1.49 2.68 83.43
Weighted Averages 74.30 1,58 1.94 0.95 2.08 81.04

LABORATORY DATA

Munk (1949) BEB data 37 91.27 0.17 0.04 0.80 0.12 92.40
Munk (1949) Berkeley data 16 52.69 21.46 5.38 0.01 6.03 85.57
Putnam and others (1949) 37 90.48 3.85 0.02 3.86 0.03 98.26
Iverson (1952) 63 77.23 1.26 8.37 2.03 0.27 89.16
Morison and Crooke (1953) 6 53.96 4.92 7.70 9.41 24.01 100.00
Galvin and Eagleson (1965) 24 2.43 2.73 8.44 67.66 0.37 81.54
Eagleson (1965) 7 2.83 2.44 0.83 93.43 0.00 99.53
Horikawa and Kuo (1967) 97 70.88 0.00 13.19 0.29 0.08 84.44
Bowen and others (1968) 11 89.74 0.96 1.21 5.53 0.27 97.44
Komar and Simmons (1968) 44 91.15 0.02 0.65 2.74 2.83 97.39
Galvin (1969) 17 70.18 0.18 0.44 2.34 6.68 79.82
Weggel and Maxwell (1970) 9 25.64 0.01 0.20 0.40 72.85 99.10
Iwagaki and others (1974) 23 83.59 0.06 5.26 4.31 0.38 93.60
Walker (1974) 15 91.82 0.05 0.59 0.00 1.57 94.03
Van Dorn (1978) 12 64.62 0.38 16.79 10.87 0.50 93.16
Hansen and Svendsen (1979) 16 93.02 0.00 0.34 5.19 0.16 98.71
Singamsetti and Wind (19801 95 70.53 0.71 9.64 2.73 5.81 89.42
Smith and Kraus (1990) 82 35.39 0.79 3.77 0.42 0.05 40.42
Small laboratory data sets** 13 18.76 6.83 2.26 1.17 31.15 80.51

Total 624
Simple Averages 61.91 2.47 4.48 11.22 8.06 89.18
Weighted Averages 66.57 1.48 6.08 5.62 3.49 83.67

Field, Prototype Laboratory, and Laboratory Data

Total 769
Simple Averages 64.50 2.47 4.02 9.19 6.94 87.99
Weighted Averages 68.03 1.50 5.30 4.74 3.23 83.17

*Maruyama and others (1983), Stive (1985), Nadaoka (1986), Takikama and others (1997).
**Saeki and Sasaki (1973), Mizugauchi (1981), Visser (1982), Watanabe and Dibajnia (1988).









Evaluation of a Special Data Set

It has been identified that where or
when waves shore-break is a surrogate
qualitative determination, not a quantitative
measurement. Let us assume, for the
moment, that some investigators are more
precise in their surrogate assessment than
are others, and that this is indicated by the
stepwise regression results (it must be
understood, however, that there is no way
to prove this to be so). The result might be
that db is, for such data sets, predicted by a
more complex equation.

The author screened individual data
sets of Appendix II1, selecting those which
exhibit significantly large values and ranges
in values for 100 r2. Twelve data sets (all


laboratory data) were found. Values of 100
r2 are listed in Table 8 for the simplest
predictive equation (i.e., equation 1 of the
stepwise regression tables for db versus Hb)
and the most complex predicting equation
(i.e., equation 31 of the stepwise regression
tables for db versus all independent
variables). All acceptable values for other
combinations of independent variables will
have 100 r2 values lying between the two.
Also listed in Table 8 are the differences for
the above, and sample size for each data
set.

Three data sets (data set I.D. Nos. 1,
2, and 3 of Table 8; n = 69) would, without
any question, result in db versus Hb being
the best outcome. These data sets would
not, therefore, be appropriate for


Table 8. Characteristics of data and assessments for a special data set.
100r 100 r2 for
for Eq. 1 of Eq. 31 of Differences
I.D. Stepwise Stepwise of preceding
Imrnvestigator n o ."
No. Regression Regression two columns
Table Table (%)
(%) (%) _
Range of minimum and maximum values of 100 r2 is minimal.

1 Munk (1949) BEB 91.27 92.40 1.13 37
2 Walker (1974) 91.82 94.03 2.21 15
3 Small Lab Data Sets* 98.53 99.48 0.95 17

Average 1.43
Total 69

Range of minimum and maximum values of 100 r2 is large.
4 Putnam and others (1949) 90.46 98.26 7.80 37
5 Eagleson (1965) 55.39 99.63 44.14 7
6 Bowen and others (1968) 89.74 97.44 7.70 11
7 Komar and Simmons (1968) 91.15 97.89 6.74 44
8 Weggel and Maxwell (1970) 89.90 99.10 9.20 9
9 Iwagaki and others (1974) 83.59 93.60 10.01 23
10 Van Dorn (1978) 64.62 93.16 28.54 12
11 Hansen and Svendsen (1979) 93.02 98.71 5.69 16
12 Nadaoka (1986) 91.61 99.69 8.08 12

Average 14.21
Total 171

Includes the small data sets of Saeki and Sasaki (1973), Mizugauchi (1981), Visser (1982),
Maruyama and others (1983), Watanabe and Dibajnia (1988) and Takikama and others (1997).









consideration.

Nine data sets (data set I.D. Nos. 4-
12 of Table 8; n = 171) were found to have
differences between the simplest and most
complex prediction outcomes ranging from
5.69% to 44.14%, with an average
difference of 14.21%. Stepwise regression
results for these nine combined data sets are
detailed in Appendix IV; net contributions are
listed in Table 9. Results of Table 9 indicate
that the independent variable Hb remains
most strongly related to db with a small
contribution attributable to Hb/(g T2) and but
a minor contribution due to tan ab. It would
seem reasonable that secondary contributing
variables or parameters should have
contributions of from 10% to 40%. The
results do not, therefore, appear to be
convincing that a more complex predicting
equation for determining where breaking
occurs can be identified with certainty.
Moreover, this combined data set comprises
but 26.73% of all laboratory data or but
22.18% of all data considered elsewhere in
this work.


Table 9. Net contribution
of independent variables for
special data set.
Variable Net
Contribution

Hb 88.23%
Hb/(g T2) 4.82%
f, 0.03%
tan ab 0.16%
T 0.00%


Assessment of Equations (12) and (13)

The percent sum of squares of the
independent variable (or parameter)
accounted for are, in every instance so low,
that it is inappropriate to even consider any
viably causative relationships) forthcoming
from equations (12) or (13). Stepwise


regression analysis results for equation (12)
where the dependent parameter is given by
db/Hb are given in Appendix V. Results for
equation (13) in which tan ab is the
dependent variable are given in Appendix VI.

A Note on Matrix Algebra

There is one important occurrence of
which the reader should be aware. Not all
matrices have an inverse (see Krumbein and
Graybill, 1965, p. 269-275). The reason
may or may not be obvious. If, for instance,
for a data set. all the values for a column of
data representing a variable or parameter are
identical (e.g., tan ab), there may be no
inverse. A small introduced change to one
of the values can correct the problem. For
other data sets, the answer may not be
obvious, and another solution will be
required (e.g., combine it with some other
data set or data sets).

COMPARATIVE ANAL YSIS
OF
PREDICTIVE METHODS

In addition to the stepwise regression
analysis we shall conduct comparative
analyses for some of the predictive
methodologies identified earlier. This was
the analytical procedure used in an earlier
work (Balsillie, 1983) except that now more
data are available.

As in the earlier work (Balsillie,
1983) we shall employ a highly useful,
although not commonly applied, statistical
tool. The commonly utilized statistical tool
for applications is predictive regression in
which one parameter or variable is defined to
be independent upon which the other is
dependent. Under these circumstances
regression is determined using the sum of
squares of the distances of data points from
the regressed line assessed only in the
direction parallel to the axis of the
independent variable. In actual practice,
however, we utilize db and Hb









interchangeably, or more precisely, we
consider them both to be independent (i.e.,
a bivariate distribution). For such
circumstances, we need to use the
regression techniques as detailed by Ricker
(1973) which he calls functional regression
but which can also be termed cubic least
squares regression. It is based on the
consideration (e.g., Tessier, 1948) that the
central tendency of the data is more
precisely determined by minimizing the sum
of the products of both the vertical and
horizontal distances of each data point from
the regressed line. Further, if it is
determined that the data pass through the
origin (i.e., 0, 0) of the plot, as they should
for our data, the slope of the regressed cubic
least squares (CLS) fit precisely becomes:


(16)


E H-
Y5Ht


which for all the data of this study becomes:


db = 1.3533 H,


(17)


It is, however, to be recognized,
when dealing with regression techniques,
that larger data values introduce greater
"weight" (i.e., more influence) over the
resulting regression outcome. For many data
sets, the effect is inconsequential. For
others, it is sensitively influential. The
influence is evident for the data sample of
this work because it encompasses over 2.5
orders of dimensional magnitude. Hence,
the much larger field data magnitudes exert
more numerical influence upon the resulting
regression outcome than do the small
laboratory wave tank data. How, then, does
one provide for a fair assessment?

A sensible solution is to subdivide the
sample into logical subsets with smaller
domains and determine representative
regression coefficients for each subset. For
the data of this work it would seem
appropriately fair to consider three data
subsets: 1) small scale laboratory wave tank
data, 2) prototype scale laboratory wave


tank data, and 3) field data. (Note:
Laboratory waves were considered to have
prototypical dimensions at about Hb > 0.20
m . Data sets were not split up. Only
those data sets for which the majority of
entries met the criterion are considered to
have prototypical magnitudes.) Results are
listed in Table 10, which include simple and
weighted average regressions and the
regression result given by equation (17)
which is a viable outcome. By taking
combinations, it is seen that averages
converge, from which a grand average
regression emerges. Hence, the final
regression equation becomes:


db = 1.2767 Hb


(18)


which is 0.26% less than McCowan's
coefficient of 1.28 (r = 0.9709).

Table 10. Determination of final reg-
ression coefficient relating db and Hb.
Cubic
Least
Data Grouping ares n
Squares
Regression
Prototype Laboratory Data 1.2423 16
Small Laboratory Data 1.1548 624
Field Data 1.4045 172

Simple Average 1.2672 812
Weighted Average 1.2094 812
Equation (17) 1.3533 812
Grand Average 1.2767 812


For reasons identified earlier, we shall
be selective in choosing the methods to be
compared. They are: 1) the McCowen form
of the equation, 2) Weggel's equation, and
3) Mallard's equation.

Functional regression results are listed
in Table 11 for all data, field data, and
laboratory data. As in the earlier
comparative analysis (Balsillie, 1983), the
form of the equation of McCowan does best
in predicting db than do either of Mallard's











Table 11. Comparative analysis results for three prediction methods of db,

F .... .. 1 BFunctional Regesson Natural VariablTry
Functional Regression Results Function ess.on Na.rl V
No Variability Analysis Resuits Analysis Results
Po n s Poubst Pub Points Ponb Pot
Dies
n m r a WMhin Above Below Witrin Above Below
Envelope Envelop En Envelope velope Envelope Envelope
: Cowan (1894) form of elating Equaton: db = 1.277 Hb
All Data
1 239 200 383 452 115 245
812 1.353 0.9697 0.0764 + 2 430 124 258 713 34 65
S3 594 66 152 791 14 7

Field Data

S1 93 63 16 102 56 14
172 1.405 0.9186 0.1451 2 132 38 2 142 30 0
3 151 21 0 158 14 0
Laboratory Data
1 277 84 279 350 59 231
640 1.162 0.9665 0.1234 2 487 16 137 571 4 65
3 599 2 39 633 0 7

lard's (1978) Formula: db = 1.094 (fn JH4 b *

All Data

1 230 242 299 500 146 125
771 1.103 0.9651 0.0665 2 434 163 174 683 78 10
3 590 121 60 728 43 0
Field Data
1 60 44 27 71 39 21
131 1.107 0.8922 0.1306 2 106 23 2 111 20 0
3 119 12 0 122 9 0

Laboratory Data
1 359 122 159 429 107 104
640 1.089 0.9644 0.1248 2 531 73 34 572 58 10
3 592 46 2 606 34 0



All Data

1 289 223 249 527 148 96
771 1.251 0.9600 0.0802 2 531 147 93 689 75 7
3 639 106 26 729 41 1
Field Data
1 57 56 18 63 54 14
131 1.251 0.8812 0.1546 2 96 35 0 97 34 0
3 111 20 0 112 19 0

Laboratory Data
1 319 144 177 464 92 82
640 1.136 0.9298 0.1008 2 525 74 41 592 41 7
3 587 45 8 617 22 1









which considers Hb and
tan ab, or Weggel's
equation which
considers Hb, tan ab,
and Hb/(g T2). Mallard's
and Weggel's equations
do not differ greatly in
terms of the correlation
coefficient from the best
solution. The stepwise
regression analysis
shows, however, that
this occurs because
variables other than Hb
do not introduce
important contributions.
(Note, however, that
Mallard's equation


Table 12. Determination of corrective regression
coefficients for Mallard's and Weggel's equations.
Mallard's Equation Weggel's Equation
Cubic Least Cubic Least
Data Grouping Squares n Squares n
Regression Regression
Prototype Laboratory Data 1.0475 16 0.9403 16
Small Laboratory Data 1.0937 624 1.1602 624
Field Data 1.1068 131 1.2513 131

A. Simple Average 1.0827 771 1.1173 771
B. Weighted Average 1.0950 771 1.1711 771
C. CLS Regression Coeff. 1.1030 771 1.2247 771
Gr average of A B, ,& C 1.0936 771'1 1.1979 771


underestimates measured db by 8.56%, and
Weggel's equation underestimates measured
db by 16.52%. See Table 12.) Hence,
again, we are responsibly bound to apply
Occam's Razor, and accept that the simplest
approach be selected as the best predicting
equation. Equation (18) is only 0.26% less
than McCowan's solution given by equation
(4). It would, in the author's opinion, be
improper to suggest equation (20) in lieu of
equation (4), the later of which it is posited,
stands as the standard instrument for
prediction.

The plotted results (Figure 5)
encompass 812 data points, 172 of which
represent field data, 16 of which are
laboratory data with prototypical dimensions,
and 624 that are of the small laboratory
variety. The domain of values for each of db
and Hb, covers 2.5 orders of magnitude.

A Logarithmic Relationship

Visual inspection of Figure 5 suggests
that the trend of data might be better
represented (relative to equation (4)) by a
logarithmic relationship. The relationship for
the data of Figure 5 (dash-dot-dash line) is
given by:


0.275 + 1.058 In H,
e


(19)


whose goodness of fit is represented by a
significantly large product-moment
correlation coefficient, r, of 0.9689, which is
comparable to the r attained for equations
(17) and (18). It is to be noted, however,
that the departure of equation (19) from
equation (4) is minimal for the domain of
plotted data. Given the data variability,
small departure, and comparable
correlations, the use of equation (19) instead
of equation (4) does not appear (again, in
terms of Occam's Razor) to be justified.

Variabbty Analysis

Functional Regression Variabilty
Analysis

Following the functional regression
analysis, the plus and minus standard
deviation, s, about the regression line is
given (Ricker, 1973) by:


(20)


s = [2 (n 2)] 1 -rM2
n 2


where ta/2 is the Student's t value for a two-
tailed test, n is the sample size, r is the


























db
(m)


0.1






Ol-


0.01
0.01


0.1 1
Hb (m)


Figure 5. The McCowen equation as it represents the data of Table 3 for db and Hb.
Data include 812 total points, 172 field data points, 16 laboratory prototype data
pairs, and 624 small wave laboratory data pairs; 3 sb indicate the maximum limits
of natural shore-breaker wave height variability within a shore-breaking wave train
(applies to McCowan equation).


Pearson product-moment correlation
coefficient, and m is the regression slope.
Equation (20) cannot be effectively evaluated
if the sample size becomes too large, at least
where large values of r are concerned (i.e., r
approaching unity). Following Ricker (1973),


therefore, s was determined by calculating
the grand average of 60 random samples
taken from a data grouping with n number of
data points, where the number of random
data selections was 2 n. For example, for
the fifth entry of Table 11 where n = 131,









60 random samples of were taken 2n or 262
times from which a grand average for s was
determined.

If we define envelopes 1, 2, and
+ 3 three standard deviations about the
regressed line then, assuming the data are
Gaussian, 68.27% of the data should lie
within 1 s of the mean, 95.45% should lie
within 2 s of the mean, and 99.73% of
the data should lie within 3 s of the mean.
These are statistical expectations. We apply
it in a relative manner. That is, the larger the
amount of data included within the
envelopes, the more confidence we have in
the regression (i.e., in addition to the
correlation coefficient). While variously large
values for data inclusion within the envelope
do occur (Table 11), the statistical variability
analysis is limited because it is self-
containedwithin its own statistical system
and because it is not comparable across data
groupings (i.e., all data, field data, and
laboratory data). That is, even though we
have kept the functional regression
coefficient constant for each predictive
method (see Table 11), the standard
deviations and correlation coefficients
associated with the regression coefficient
differs for each data grouping; hence, s will
be different. Fortunately, there is an
alternative, more realistic approach.

Natural Variability Analysis

This method is more applicable
related to the issue at hand because it
assesses the natural variability associated
with ocean wave heights. Let us, first,
inspect the case where a wave gauge is
sited in a water depth of, say, 10 m. The
gauge is not capable of selectively measuring
each of the several to many wave trains
(generally non-breaking) which encounter it.
It, therefore, measures the spectral record.
The analysis of many records indicate that
the standard deviation, ss, associated with
the mean wave height, H, becomes (e.g., U.
S. Army, 1984):


s, = 0.4 H


(21)


In the surf zone, however, conditions
are distinctly different. We have learned that
at shore-breaking waves are depth limited.
Hence, a wave train characterized by a
shore-breaker height of 1 m will break in a
water depth of close to 1.28 m. A wave
train with 0.5 m shore-breakers will break in
a water depth of 0.64 m. Wave
superposition interference between shore
propagating or reflected waves can occur
periodically in which the wave heights
become precisely additive. If the
interference is not precisely in phase then
partial precise additive superposition occurs.
The point is that there can be variability in
results due to wave interference, but if
observations are carefully conducted it can
be kept to a minimum. Balsillie and Carter
(1984a, 1984b) investigated the natural
variability of shore-breaker heights (single
wave trains) in the field by measuring at the
shore-breaker position 30 elevations each of
the distance from the bed to breaker trough
and breaker crest and found:


s, = 0.21 H,


(22)


in which Hb is the mean shore-breaker height
for a wave train, and sb is its associated
standard deviation. Field data for 47 wave
trains (2,820 individual measurements)
leading to the quantification of equation (22)
are plotted in Figure 6.

Quantities such as ss/H and sb/Hb are
termed the relative dispersion (or coefficient
of dispersion). For natural distributions
relative dispersions of 0.5 or less are
considered as having excellent homogeneity,
i.e., a "tight" distribution (see Balsillie, 1995,
p. 17 and 38-39), which lends much greater
credence to our assessment.

If we now define envelopes of 1,
2, and 3 standard deviations about the
regressed line, again assuming the data are
Gaussian, then 68.27% of the data for 1
sb, 95.45% of the data for 2 s, and









0.25


Sb

(m)


0.15

0.1


0.05 1 n = 47 wave trains

01 0.2 03 0.4 0.5 0.6 0.7 0.8 0.9
Hb (m)

Figure 6. Relationship between the mean shore-breaking wave height, Hb, for
single wave trains and their associated standard deviations, sb, (after Balsillie
and Carter, 1984a, 1984b).


99.73% of the data for 3 sb should lie
within each of the envelopes. The data,
listed in Table 8 (Natural Variability Analysis)
show that this is a much better variability
analytical treatment.

In fact, for the most part the amount
of data lying within the envelopes (Table 8)
are all sufficiently large that, in addition to
the significantly large correlation
coefficients, allow us to have increased
confidence in the regressed results (Ricker,
1973).

DISCUSSION

The regression line conforming to
McCowan's equation is plotted through the
data of Figure 4, as are lines (dashed) for
plus and minus 3 sb (the 1 sb and 2 sb
envelopes are not plotted to keep the plot
simplified). It is evident that the bulk of data
lies between these indicators of expected
natural breaker height variability, which we
have noted lends greater credibility to our
results. It is also recognized that the field
data (and prototype laboratory data) show
less variability than do the small wave
laboratory data. This consideration is not
trivial in terms of the location of the
regression line and its effect on wave


energy, E. Examples of how the wave
energy (note: E oc Hb2) is affected are listed
in Table 13. If one takes the position that a
more conservative approach (i.e., responsibly
higher versus arbitrarily lower energy) is
more desirable, we can posit some
comments about the resulting outcome. We
are not greatly concerned with data lying
above (and to the left) of the regression line
because 1) they yield lower breaking wave
energies not conducive to a responsibly
conservative design solution, and 2) waves
can break in this region due to critically high
wind stresses (and other esoteric causes).
We become, in fact, more concerned with
the data lying below (and to the right) of the
regression line. Note that the field and
laboratory prototype data are less "ragged"
in their distribution below than above the
regression line, and plot closer to the line
than do the small wave laboratory data. The
latter are about evenly distributed about the
regression line. Based on his years of
experience in the field, the auther has found
that larger waves (up to about 1 m in height)
are more precisely measurable than very
small waves. In fact, only 14 of the 172
field and prototype waves (8.13%) lie below
the -1 sb envelope line, while 231 of the 640
small laboratory waves (36.09%) lie below
the line. None of the field or prototype









Table 13. Examples of the degree on which shore-breaking wave energy levels are dependent
on the depth in which the waves break. Assessment is made in terms of sb given by equation
(20), where the wave energy is directly proportional to the wave height squared.
Wave Energy Additional Ramifications for
Position Relative to Regression Line of Relative to Final Shore-Breaking Wave
Regression Line of Figure 4 n b McCowan Swash Energy Dissipation
Figure 4Equation Potential

db = 1.28 Hb + 3 Sb = 1.91 Hb 0.45 Waves break further offshore,
regresn swash potentially wider
Above regression db = 1.28 Hb + 2 sb = 1.70 Hb 0.57 allowing greater energy
ne' dissipation; damage potential is
db = 1.28 Hb + 1 sb = 1.49 Hb 0.74 lessened.

On regression line. db = 1.28 Hb 1.00 Status quo.

db = 1.28 Hb 1 sb = 1.07 Hb 1.43 Waves break closer to shore,
Below regression swash narrower and energy
line. db = 1.28 Hb 2 sb = 0.86 Hb 2.22 dissipation less; damage
db =.28 Hb 3 sb = 0.65 Hb 3.87 potential increases.
db = 1.281Hb 3sb = 0.65 Hb 3.87


waves lie below the -2 sb line, while 65
small laboratory waves (10.16) lie below 2
sb. The author posits, given the large
sample size, that this result strengthens the
position of the plotted regression line,
particularly since the field and prototype
waves comprise 96.64% of the measured
domain of the data.

With the proper measurement
equipment and techniques, whether in the
field or the laboratory, we can, within
acceptable limits, measure each shore-
breaking characteristic such as Hb, db, tan
ab, cb, and uc with precision. It is the
cinematic stability criterion (equation (1))
that provides us with the better answer as to
when breaking occurs. With rare exception,
however, we do not use this approach.
Rather, we have relied upon a set of
surrogate definitions concerning the
appearance of various shore-breaking wave
types which aid us more nearly to determine
where shore-breaking occurs. It is an
observed determination, not a measured
quantity; no gauge can measure where
waves break in terms of the surrogate
definitions. Employing Sir A. C. Doyle's
principle, then, this must be the principle


cause of the bulk of variability or error of the
data.

Low slope conditions greatly increase
the cognitive difficulty in determining
precisely when shore-breaking happens.
This is particularly true for small laboratory
waves, and is why laboratory nearshore
slopes are usually exaggerated relative to
natural nearshore bed slopes. That is, the
steeper bed slopes induce a response in
wave behavior that is easier to perceive and,
therefore, to "measure". This was, in fact,
recognized in early field studies (e.g., Scripps
Institution of Oceanography, 1945).

It has been demonstrated that where
or when nearshore waves shore-break,
remarkably affects wave energy constraints.
Let us, also, illustrate an analog to energy
constraints by looking at wave induced
impact pressures. Pressures produced by
waves are described to consist of a first
extremely high pressure of very short
duration termed shock or impact pressure or
gifle by Larras (1937), followed by a second
pressure that is significantly less in
magnitude and longer in duration termed
dynamic pressure or bourrage by Larras









(1937). As an example, short of
constructing a Galveston, Texas, seawall,
manmade structures simply cannot be
designed to withstand impact pressures
within reasonable funding constraints.
Results from field studies (Miller and others,
1974a, 1974b; Miller, 1976) indicate that
impact forces from shore-breaking and
broken waves significantly exceed those
from non-breaking waves. Highest impact
pressures occur in post-breaking waves (i.e.,
bores), with greater pressures occurring from
plunger-generated than spilling-generated
bores. Shore-breaking waves produced next
highest impact pressures, with greater
pressures occurring for plunging than spilling
shore-breakers. The difference between
breaking and post-breaking impact pressures
is the elevation at which the maximum
horizontal impact occurs. For post-breaking
bores, the maximum pressure is nearer the
still water level. For waves at the shore-
breaking position, it occurs in the upper
portion of the wave crest. (Balsillie, 1985).
The author (Balsillie, 1985) found that at the
shore-breaking position the shore-breaker
with the greatest impact pressure had a
value of fb = 1.0 which occurred at an
elevation of 0.45 Hb above the still water
level or 1.7 Hb above the bed.

It should, therefore, be evident that
while nearshore waves are depth limited, the
breaker type is dependent upon both bed
slope and equivalent wave steepness.

The Nelson-Gourlay Horizontal
Bed Slope Breaking Anomaly

It is acknowledged that the subject of
shore-breaking on a horizontal bed is one
deserving of study. It is very difficult to
measure such an occurrence, even in the
field, let alone in the laboratory. In fact, as
previously noted, laboratory bed slopes over
which breaking is induced are exaggerated
so that the researcher is better able to
recognize a sudden change in crest
deformation aiding in identifying when shore-


breaking occurs.

Bearing this in mind, the results of
Nelson (1982, 1994) and Gourlay (1994)
pose some difficulties. While the list is
longer, following are some of the major
concerns.

1) Their laboratory wave tank
designss, while termed a reef, are what are
more commonly described as a step profile.
As nearly as can be determined, the first
encountered "seaward" or stoss slope of the
step ranged from 0.0467 < tan ab <
0.2222. The step top, or "test slope" over
which breaking was assessed, ranged from
tan ab = 0.0358 to horizontal. It is the
slope immediately preceding breaking which
clearly causes shore-breaking. It is also clear
that it was the stoss slope which caused
their waves to break. These are not, then,
the proper conditions for assessing shore-
breaking on a horizontal bed.

2) Their wave tank was some 30 m
in length. However, the distance from the
wave generator to the front of the step stoss
slope was but 5 m. This does not seem to
be an adequate distance for waves to
undergo wave dispersion mechanics. They
may, then, represent forced waves, which
explains breaking in greater water depths.

3) They did not define their criteria
for identifying when shore-breaking occurs.

4) Their relating coefficient (i.e.,
db/Hb = 1.82) lies to the left and above the
regression line of Figure 4. It, therefore,
results in a non-conservative assessment of
wave energy. Results of this study represent
a much larger and dimensionally broader data
set than they used. Even so, their
coefficient lies within + 3 sb (precisely at
2.48 sb), which may not be an anomalous
value when assessed in terms of natural
shore-breaker height variability except for the
previous concerns.









5) Gourlay (1994) modeled a field
site in a small-scale laboratory wave tank,
then scaled his waves to prototype
dimensions. Such scaling is subject to
question. It would have been much more
useful had he reported the original wave tank
data. It is not, for example, clear if the
error/variability associated with his
determination of gauge determined wave
heights is on the order of variability
associated with d/H.

6) Nelson's (1994) reported field
study results are not clearly reported in
support of his conclusionss.

CONCLUSIONS

This work has been conducted to find
a least equivocal, best representative
predicting numerical relationship identifying
where shore-breaking occurs. It may well be
that variables or parameters other than Hb
will provide a refining and strengthening role
in determining where shore-breaking occurs
(although it is the authors' suspicion they
will be of secondary, tertiary, etc.,
importance). The results of this study,
however, indicate that as long as we persist
to use the surrogate method (i.e., visually
determined shore-breaking wave types to
determine where shore-breaking occurs),
please be advised that the issue of
identifying a more complex predicting
relationship for determining when or where
waves shore-break will not be forthcoming (a
suggestion as to why not is given in
Appendix VII). To ultimately resolve the
issue will require measurement of the
kinematic stability parameter (equation (1)),
perhaps even the dynamic stability
parameter. The former has, to some extent,
been measured under controlled conditions
by various researchers (e.g., Adyemo, 1970;
Divoky and others, 1970; Iwagaki and
others, 1974; Van Dorn, 1978; Sakai and
Iwagaki, 1978; Easson and others, 1988;
etc.). Hence, while we know that it is
possible, such a program of research will be


resource intensive. Given, however, the
importance of where shore-breaking occurs
on wave energy constraints, such a research
effort would seem justifiable.

There is the possibility, of course,
that measurement of the kinematic stability
parameter will do no better than the
surrogate methodology. The reality is, of
course, that such work has not been
accomplished, and we must posit our
conclusions based on available data. Many
investigators have dealt with the subject of
the cause of shore-breaking, quite often
using only their own data sets. To ignore
existing data or exclude existing data
without sound reasoning and justification is
simply not part of the scientific process. It
is fortunate that a significantly large amount
of data are available (i.e., all data on the
subject known to the author) upon which it
is possible to conclude, in terms of reliability,
statistically significant results. Two lines of
statistical assessment were employed in this
work: 1) stepwise regression, and 2)
comparative analysis of selected existing
prediction methodologies using functional
regression. It was determined that the best
predicting equation is given by db = 1.277
Hb. This result is but 0.26% larger than
McCowan's (1894) result given by equation
(4). Hence, it is concluded that McCowan's
relationship remains as the standard
instrument for prediction.

The establishment of a standard,
particularly one of import, is more often than
not, a tricky proposition at best. This one,
however, significantly affects wave energy
and impact pressures, both of which are
fundamental shore-breaking outcomes of
significant and pragmatically important
proportions. It is anologous to filing our
annual U. S. income tax forms in which we
must first know of the number of deductions
to be claimed before we can proceed.
Hence, before we can do anything else
concerning a littoral design solution or
coastal processes result, we must know









where the waves shore-break; in one way or
another it is a requirement of almost all
coastal geology and engineering applications.
We must ask, therefore, the following
question. Do we become tolerant of
resulting multiple prediction methods based
on small data sets which will pose future
accounting problems, or do we establish a
standard based on all available data that in
the future, should it be refined, can easily be
transformed?

It must be concluded, therefore,
based on the existing evidence, that
nearshore waves are water-depth limited.
The consideration of other variables or
parameters are highly useful in determining
the type of shore-breaking wave (i.e.,
spilling, plunging, surging, etc.) that occurs
(see equation (3)). It is, in fact, here
suggested that McCowan's equation (i.e.,
equation (4)) be formally termed McCowan's
Lemma.

ACKNOWLEDGEMENTS

This work may not have been pursued
had it not been for the encouragement of
William F. Tanner, Regents Professor,
Department of Geology, Florida State
University and Visiting Scientist, Florida
Geological Survey. His guidance, both
personal and through published work on the
application and merits of stepwise
regression, was invaluable. His concerted
interest and countless hours of discussion
are more than acknowledged with thanks.
Special thanks are also to be extended to
Paulette Bond for her attention to the subject
and advice, and to Paulette Bond, Don L.
Hargrove, Jacqueline M. Lloyd, and William
C. Parker for their necessary presence in
collecting field data for extending the data
coverage of Figure 3 of this work.

Florida Geological Survey staff
reviews were conducted by Paulette Bond,
Kenneth Campbell, L. James Ladner, Ed
Lane, Jacqueline Lloyd, Deborah Mekeel,


Walter Schmidt, and Thomas Scott. The
outside review and encouragement of
Nicholas C. Kraus with the Coastal
Engineering Laboratory, is also
acknowledged.

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APPENDIX 1

Example calculations of the effect of shore-breaking water depth
on wave energy constraints.









Example Calculations of the
Water Depth on Wave



First, let us evaluate all shore-
breaking cases at a water depth, db, of 1.28
m. Second, we wish to evaluate final results
in terms of the energy density, Eb, which for
Airy waves is given by:


Effect of Shore-Breaking
Energy Constraints


E2
EbI


6.5536
1.00


= 6.55


That is, the wave energy occurring at C2 =
0.50 is 6.55 times that where c, = 1.28.


(I-1) EXAMPLE 2.


p,g H:
8


in which pf is the fluid mass density, and g
is the acceleration gravity assigned the value
of 9.8 m/s2 for Earth-bound applications.
Hence, for our purposes Eb oc Hb2. Third,
we wish to assess the examples relative to
some standard which has been arbitrarily


selected to be given by:

d, = C, H = 1.28 H,,


Let us next evaluate the case where:


dw, = c H, = 3.00 Hb


(1-4)


Height and energy conditions for Hbl apply
precisely here as they appear for Example 1.
For equation (1-4):


dc
(1-2) H -, 3.00


EXAMPLE 1.


1.28
3.00


= 0.427m


and


Let us evaluate the case where:


d6 = C2 H, = 0.50 H,2


H -2 = E,


(1-3)


Now, for equation (1-2):

c, 1.28
Hb 1.28 1.0m
1.28 1.28
and for equation (1-3)


db
H, 0= 0
0.50


1.28
0.05


0.4272 = 0.1823


Comparatively, then:


H,2
"3
H 2
-M


0.1823
- 0.182
1.00


and the wave energy at c3 is 0.182 times
that where cl = 1.28.


2.56m


Moreover, the following can be stated:

H, = Eb = 1.002 = 1.00
and

H2 = E,,, = 2.562 = 6.5536


Comparatively, then:












APPENDIX II

Stepwise regression results for data sets and subsets



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (11) of the text.

100r*2 = 100r2.

The dependent variable, db, is the water depth at the shore-breaking position.

Independent variables:

(1) Shore-breaking wave height, Hb
(2) Wave period, T
(3) Bed slope leading to shore-breaking, tan ab
(4) Equivalent wave steepness, Hb/g T2
(5) Surf similarity parameter, tan ab/[Hb/(g T2)105

Correation Matrix.

Correlation matrix treatment is given by Table 5 of the text.

Variables:

(1) db (2) Hb (3) T (4) tan ab

(5) Hb/(g T2) (6) tan ab/[Hb/(g T2)]0.5










ALL DATA
n = 771

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 94.26 0.9709
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 74.25 0.8617
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 6.79 0.2606
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 3.66 0.1914
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 4.87 0.2208
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 104.72 -4.72 0.00 0.00 0.00 0.00 0.00 0.00 94.27 0.9709
7 101.15 0.00 -1.15 0.00 0.00 0.00 0.00 0.00 94.26 0.9709
8 97.99 0.00 0.00 2.01 0.00 0.00 0.00 0.00 94.26 0.9709
9 100.83 0.00 0.00 0.00 -0.83 0.00 0.00 0.00 94.26 0.9709
10 0.00 103.25 -3.25 0.00 0.00 0.00 0.00 0.00 74.31 0.8620
11 0.00 71.72 0.00 28.28 0.00 0.00 0.00 0.00 78.58 0.8865
12 0.00 119.72 0.00 0.00 -19.72 0.00 0.00 0.00 76.01 0.8719
13 0.00 0.00 53.06 46.94 0.00 0.00 0.00 0.00 8.43 0.2904
14 0.00 0.00 92.21 0.00 7.79 0.00 0.00 0.00 6.81 0.2609
15 0.00 0.00 0.00 58.43 41.57 0.00 0.00 0.00 9.90 0.3146
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 106.50 -5.07 -1.43 0.00 0.00 0.00 0.00 0.00 94.27 0.9710
17 104.65 -4.69 0.00 0.04 0.00 0.00 0.00 0.00 94.27 0.9709
18 104.93 -4.57 0.00 0.00 -0.36 0.00 0.00 0.00 94.27 0.9709
19 98.98 0.00 -1.50 2.52 0.00 0.00 0.00 0.00 94.27 0.9709
20 101.13 0.00 -1.29 0.00 0.16 0.00 0.00 0.00 94.26 0.9709
21 98.73 0.00 0.00 1.85 -0.58 0.00 0.00 0.00 94.26 0.9709
22 0.00 74.20 -5.17 30.97 0.00 0.00 0.00 0.00 78.92 0.8884
23 0.00 108.19 37.22 0.00 -45.41 0.00 0.00 0.00 78.38 0.8853
24 0.00 79.34 0.00 28.96 -8.30 0.00 0.00 0.00 79.43 0.8912
25 0.00 0.00 -4.53 59.93 44.60 0.00 0.00 0.00 9.91 0.3149
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 105.49 -4.59 -1.48 0.58 0.00 0.00 0.00 0.00 94.27 0.9710
27 108.06 -7.00 -4.15 0.00 3.08 0.00 0.00 0.00 94.28 0.9710
28 104.96 -4.59 0.00 -0.02 -0.36 0.00 0.00 0.00 94.27 0.9709
29 96.89 0.00 -4.44 4.38 3.17 0.00 0.00 0.00 94.27 0.9709
30 0.00 83.07 11.79 23.66 -18.53 0.00 0.00 0.00 79.71 0.8928
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 103.99 -5.56 -5.53 2.74 4.36 0.00 0.00 0.00 94.28 0.9710


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9709 0.8617 -0.2606 -0.1914 -0.2208

(2) 94.2587 1.0000 0.8926 -0.2631 -0.2037 -0.2233

(3) 74.2542 79.6819 1.0000 -0.2759 -0.4391 -0.1031

(4) 6.7908 6.9235 7.6131 1.0000 0.2593 0.8204

(5) 3.6632 4.1475 19.2788 6.7219 1.0000 -0.1387

(6) 4.8738 4.9881 1.0629 67.3045 1.9247 1.0000

Numbers in Parentheses are the Variable or Parameter Number










ALL DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + .,, + (BB X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 -0.017 1.372 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 -0.340 0.000 0.322 0.000 0.000 0.000 0.000 0.000 0.000
3 0.765 0.000 0.000 -3.543 0.000 0.000 0.000 0.000 0.000
4 0.794 0.000 0.000 0.000 -57.112 0.000 0.000 0.000 0.000
5 0.715 0.000 0.000 0.000 0.000 -0.188 0.000 0.000 0.000
6 -0.005 1.402 -0.009 0.000 0.000 0.000 0.000 0.000 0.000
7 -0.010 1.370 0.000 -0.075 0.000 0.000 0.000 0.000 0.000
8 -0.028 1.374 0.000 0.000 1.970 0.000 0.000 0.000 0.000
9 -0.012 1.371 0.000 0.000 0.000 -0.004 0.000 0.000 0.000
10 -0.308 0.000 0.319 -0.336 0.000 0.000 0.000 0.000 0.000
11 -0.805 0.000 0.360 0.000 69.113 0.000 0.000 0.000 0.000
12 -0.192 0.000 0.317 0.000 0.000 -0.114 0.000 0.000 0.000
13 0.939 0.000 0.000 -3.075 -39.614 0.000 0.000 0.000 0.000
14 0.768 0.000 0.000 -3.305 0.000 -0.018 0.000 0.000 0.000
15 1.105 0.000 0.000 0.000 -67.551 -0.215 0.000 0.000 0.000
16 0.004 1.402 -0.010 -0.090 0.000 0.000 0.000 0.000 0.000
17 -0.005 1.402 -0.009 0.000 0.037 0.000 0.000 0.000 0.000
18 -0.003 1.401 -0.009 0.000 0.000 -0.002 0.000 0.000 0.000
19 -0.022 1.371 0.000 -0.100 2.439 0.000 0.000 0.000 0.000
20 -0.011 1.370 0.000 -0.084 0.000 0.001 0.000 0.000 0.000
21 -0.024 1.373 0.000 0.000 1.800 -0.003 0.000 0.000 0.000
22 -0.746 0.000 0.355 -0.828 72.222 0.000 0.000 0.000 0.000
23 -0.314 0.000 0.338 3.881 0.000 -0.308 0.000 0.000 0.000
24 -0.657 0.000 0.353 0.000 62.701 -0.080 0.000 0.000 0.000
25 1.117 0.000 0.000 0.366 -70.608 -0.235 0.000 0.000 0.000
26 0.000 1.399 -0.009 -0.094 0.541 0.000 0.000 0.000 0.000
27 0.007 1.414 -0.013 -0.260 0.000 0.013 0.000 0.000 0.000
28 -0.003 1.401 -0.009 0.000 -0.015 -0.002 0.000 0.000 0.000
29 -0.034 1.373 0.000 -0.302 4.339 0.014 0.000 0.000 0.000
30 -0.609 0.000 0.354 1.680 49.090 -0.172 0.000 0.000 0.000
31 -0.010 1.408 -0.011 -0.359 2.588 0.018 0.000 0.000 0.000











ALL DATA MINUS GAILLARDS (1904) FIELD DATA
n = 746
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 97.00 0.9849
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 81.40 0.9022
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 5.75 0.2397
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 5.13 0.2264
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 3.73 0.1930
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 86.59 13.41 0.00 0.00 0.00 0.00 0.00 0.00 97.10 0.9854
7 101.11 0.00 -1.11 0.00 0.00 0.00 0.00 0.00 97.00 0.9849
8 101.32 0.00 0.00 -1.32 0.00 0.00 0.00 0.00 97.00 0.9849
9 100.14 0.00 0.00 0.00 -0.14 0.00 0.00 0.00 97.00 0.9849
10 0.00 101.40 -1.40 0.00 0.00 0.00 0.00 0.00 81.41 0.9023
11 0.00 73.00 0.00 27.00 0.00 0.00 0.00 0.00 85.32 0.9237
12 0.00 118.23 0.00 0.00 -18.23 0.00 0.00 0.00 82.98 0.9109
13 0.00 0.00 42.78 57.22 0.00 0.00 0.00 0.00 8.64 0.2939
14 0.00 0.00 103.51 0.00 -3.51 0.00 0.00 0.00 5.75 0.2398
15 0.00 0.00 0.00 63.34 36.66 0.00 0.00 0.00 10.32 0.3212
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 86.99 13.40 -0.39 0.00 0.00 0.00 0.00 0.00 97.10 0.9854
17 76.77 17.03 0.00 6.21 0.00 0.00 0.00 0.00 97.13 0.9856
18 87.29 14.51 0.00 0.00 -1.80 0.00 0.00 0.00 97.11 0.9854
19 101.99 0.00 -0.97 -1.02 0.00 0.00 0.00 0.00 97.00 0.9849
20 100.80 0.00 -2.85 0.00 2.06 0.00 0.00 0.00 97.00 0.9849
21 101.75 0.00 0.00 -1.42 -0.33 0.00 0.00 0.00 97.00 0.9849
22 0.00 74.97 -4.01 29.04 0.00 0.00 0.00 0.00 85.52 0.9248
23 0.00 106.51 38.96 0.00 -45.47 0.00 0.00 0.00 85.76 0.9261
24 0.00 80.52 0.00 27.54 -8.05 0.00 0.00 0.00 86.10 0.9279
25 0.00 0.00 -13.21 67.43 45.78 0.00 0.00 0.00 10.45 0.3232
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 77.19 17.26 -1.10 6.65 0.00 0.00 0.00 0.00 97.14 0.9856
27 84.17 16.87 4.46 0.00 -5.51 0.00 0.00 0.00 97.12 0.9855
28 77.46 17.82 0.00 6.11 -1.39 0.00 0.00 0.00 97.14 0.9856
29 100.41 0.00 -3.14 0.40 2.33 0.00 0.00 0.00 97.00 0.9849
30 0.00 85.93 17.11 19.82 -22.87 0.00 0.00 0.00 86.67 0.9310
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 77.51 18.08 0.95 5.67 -2.21 0.00 0.00 0.00 97.14 0.9856


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9849 0.9022 -0.2397 -0.2264 -0.1930

(2) 96.9971 1.0000 0.9019 -0.2386 -0.2260 -0.1953

(3) 81.3960 81.3346 1.0000 -0.2543 -0.4474 -0.0749

(4) 5.7465 5.6927 6.4689 1.0000 0.2598 0.8173

(5) 5.1263 5.1073 20.0174 6.7481 1.0000 -0.1432

(6) 3.7260 3.8155 0.5604 66.7958 2.0497 1.0000

Numbers in Parentheses are the Variable or Parameter Number










ALL DATA MINUS GAILLARDS (1904) FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (83 X3) + ... + (B8 X8)

EQ BO B1 82 83 84 BS B6 B7 88
NO
1 -0.008 1.274 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 -0.312 0.000 0.291 0.000 0.000 0.000 0.000 0.000 0.000
3 0.620 0.000 0.000 -2.729 0.000 0.000 0.000 0.000 0.000
4 0.705 0.000 0.000 0.000 -56.452 0.000 0.000 0.000 0.000
5 0.571 0.000 0.000 0.000 0.000 -0.138 0.000 0.000 0.000
6 -0.039 1.187 0.024 0.000 0.000 0.000 0.000 0.000 0.000
7 -0.003 1.273 0.000 -0.057 0.000 0.000 0.000 0.000 0.000
8 -0.002 1.273 0.000 0.000 -1.008 0.000 0.000 0.000 0.000
9 -0.007 1.274 0.000 0.000 0.000 0.000 0.000 0.000 0.000
10 -0.300 0.000 0.290 -0.125 0.000 0.000 0.000 0.000 0.000
11 -0.685 0.000 0.323 0.000 55.250 0.000 0.000 0.000 0.000
12 -0.195 0.000 0.288 0.000 0.000 -0.090 0.000 0.000 0.000
13 0.812 0.000 0.000 -2.209 -43.887 0.000 0.000 0.000 0.000
14 0.619 0.000 0.000 -2.811 0.000 0.006 0.000 0.000 0.000
15 0.948 0.000 0.000 0.000 -64.667 -0.164 0.000 0.000 0.000
16 -0.037 1.187 0.024 -0.022 0.000 0.000 0.000 0.000 0.000
17 -0.084 1.159 0.034 0.000 5.695 0.000 0.000 0.000 0.000
18 -0.033 1.179 0.026 0.000 0.000 -0.006 0.000 0.000 0.000
19 0.001 1.272 0.000 -0.049 -0.773 0.000 0.000 0.000 0.000
20 -0.004 1.273 0.000 -0.147 0.000 0.007 0.000 0.000 0.000
21 0.000 1.273 0.000 0.000 -1.083 -0.001 0.000 0.000 0.000
22 -0.647 0.000 0.321 -0.534 57.384 0:000 0.000 0.000 0.000
23 -0.305 0.000 0.307 3.498 0.000 -0.266 0.000 0.000 0.000
24 -0.567 0.000 0.318 0.000 50.273 -0.064 0.000 0.000 0.000
25 0.978 0.000 0.000 0.959 -72.747 -0.217 0.000 0.000 0.000
26 -0.080 1.157 0.034 -0.068 6.058 0.000 0.000 0.000 0.000
27 -0.043 1.163 0.031 0.252 0.000 -0.020 0.000 0.000 0.000
28 -0.077 1.153 0.035 0.000 5.526 -0.006 0.000 0.000 0.000
29 -0.006 1.273 0.000 -0.163 0.307 0.008 0.000 0.000 0.000
30 -0.510 0.000 0.319 1.979 34.045 -0.172 0.000 0.000 0.000
31 -0.076 1.151 0.035 0.057 5.119 -0.009 0.000 0.000 0.000











ALL DATA MINUS GAILLARDS (1904), SCRIPPS LEICA TYPE I AND II (1944a, 1944b,
1945), AND WEISHARS (1976) FIELD DATA
n = 670

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2
No


(1)

100.00
0.00
0.00
0.00
0.00

105.38
106.13
96.78
105.64
0.00
0.00
0.00
0.00
0.00
0.00

113.76
104.09
107.21
101.23
106.56
103.63
0.00
0.00
0.00
0.00

106.75
116.29
104.46
100.04
0.00

108.05


(2)

0.00
100.00
0.00
0.00
0.00

-5.38
0.00
0.00
0.00
101.82
58.70
131.38
0.00
0.00
0.00

-6.95
-4.75
-1.85
0.00
0.00
0.00
61.07
108.92
65.95
0.00

-3.51
-9.68
-0.56
0.00
70.14

-6.23


INDEPENDENT VARIABLES TAKEN FIVE AT A
-10.92 4.71 4.39 0.00 0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
-6.13 0.00 0.00
0.00 3.22 0.00
0.00 0.00 -5.64
-1.82 0.00 0.00
0.00 41.30 0.00
0.00 0.00 -31.38
-45.59 145.59 0.00
58.25 0.00 41.75
0.00 104.35 -4.35
INDEPENDENT VARIABLES
-6.81 0.00 0.00
0.00 0.65 0.00
0.00 0.00 -5.35
-6.64 5.41 0.00
-3.65 0.00 -2.91
0.00 1.68 -5.31
-5.64 44.57 0.00
63.13 0.00 -72.05
0.00 42.74 -8.69
-95.26 118.19 77.08
INDEPENDENT VARIABLES
-6.81 3.56 0.00
-9.74 0.00 3.13
0.00 1.39 -5.28
-8.22 6.44 1.75
13.51 36.86 -20.52


(6) (7)
TAKEN ONE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


2.3242 1.7788 33.2640 4.3432


1.0000 -0.2069


5) 0.1670 0.0408 10.6303 65.3993 4.2787 1.0000

Numbers in Parentheses are the Variable or Parameter Number


93.18
16.22
0.18
2.32
0.17

93.28
93.56
93.24
93.55
16.23
38.40
19.54
2.89
0.19
2.33

93.69
93.28
93.56
93.70
93.59
93.56
39.59
29.75
41.32
4.50

93.72
93.71
93.56
93.71
42.51


0.9653
0.4027
0.0421
0.1525
0.0409

0.9658
0.9673
0.9656
0.9672
0.4029
0.6197
0.4420
0.1701
0.0436
0.1528

0.9679
0.9658
0.9672
0.9680
0.9674
0.9673
0.6292
0.5454
0.6428
0.2120

0.9681
0.9680
0.9673
0.9680
0.6520


93.75 0.9683


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9653 0.4027 -0.0421 0.1525 -0.0409


93.1835


1.0000 0.4453 0.0201 0.1334 0.0202


16.2205 19.8285 1.0000 -0.0755 -0.5767 0.3260

0.1768 0.0403 0.5707 1.0000 0.2084 0.8087











ALL DATA MINUS GAILLARDS (1904), SCRIPPS LEICA TYPE I AND II (1944a, 1944b,
1945), AND WEISHARS (1976) FIELD DATA (CONT)

TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (81 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 82 B3 B4 B5 B6 B7 B8


NO
1 -0.016
2 0.042
3 0.142
4 0.104
5 0.141
6 -0.011
7 -0.008
8 -0.021
9 -0.008
10 0.044
11 -0.156
12 0.053
13 0.112
14 0.142
15 0.106
16 -0.001
17 -0.012
18 -0.007
19 -0.014
20 -0.007
21 -0.011
22 -0.148
23 -0.020
24 -0.144
25 0.086
26 -0.008
27 0.001
28 -0.010
29 -0.015
30 -0.147
31 -0.007


0.000 0.000
0.057 0.000
0.000 -0.071
0.000 0.000
0.000 0.000
-0.005 0.000
0.000 -0.103
0.000 0.000
0.000 0.000
0.057 -0.020
0.104 0.000
0.066 0.000
0.000 -0.130
0.000 -0.044
0.000 0.000
-0.006 -0.109
-0.004 0.000
-0.002 0.000
0.000 -0.117
0.000 -0.061
0.000 0.000
0.105 -0.188
0.102 1.151
0.112 0.000
0.000 -0.554
-0.003 -0.115
-0.008 -0.153
0.000 0.000
0.000 -0.146
0.120 0.448
-0.005 -0.183


0.000 0.000
0.000 0.000
0.000 0.000
5.664 0.000
0.000 -0.004
0.000 0.000
0.000 0.000
0.897 0.000
0.000 -0.006
0.000 0.000
21.417 0.000
0.000 -0.020
6.261 0.000
0.000 -0.002
5.589 -0.001
0.000 0.000
0.171 0.000
0.000 -0.006
1.437 0.000
0.000 -0.003
0.438 -0.006
22.444 0.000
0.000 -0.087
21.228 -0.019
10.400 0.030
0.909 0.000
0.000 0.003
0.359 -0.006
1.729 0.002
18.515 -0.045
1.195 0.005


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


EQ


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


1.322
0.000
0.000
0.000
0.000
1.343
1.324
1.318
1.324
0.000
0.000
0.000
0.000
0.000
0.000
1.349
1.339
1.330
1.317
1.324
1.321
0.000
0.000
0.000
0.000
1.333
1.358
1.324
1.315
0.000
1.341










ALL LABORATORY DATA
n = 640
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 93.42 0.9665
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 5.15 0.2270
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.23 0.0480
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 7.29 0.2700
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 1.70 0.1305
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 115.82 -15.82 0.00 0.00 0.00 0.00 0.00 0.00 93.87 0.9689
7 106.15 0.00 -6.15 0.00 0.00 0.00 0.00 0.00 93.81 0.9686
8 94.71 0.00 0.00 5.29 0.00 0.00 0.00 0.00 93.57 0.9673
9 106.93 0.00 0.00 0.00 -6.93 0.00 0.00 0.00 93.89 0.9690
10 0.00 104.05 -4.05 0.00 0.00 0.00 0.00 0.00 5.19 0.2278
11 0.00 57.67 0.00 42.33 0.00 0.00 0.00 0.00 33.14 0.5757
12 0.00 151.59 0.00 0.00 -51.59 0.00 0.00 0.00 8.85 0.2974
13 0.00 0.00 -34.24 134.24 0.00 0.00 0.00 0.00 8.51 0.2918
14 0.00 0.00 -203.04 0.00 303.04 0.00 0.00 0.00 2.96 0.1720
15 0.00 0.00 0.00 128.22 -28.22 0.00 0.00 0.00 7.96 0.2822
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 128.80 -20.24 -8.56 0.00 0.00 0.00 0.00 0.00 94.41 0.9717
17 127.94 -22.82 0.00 -5.12 0.00 0.00 0.00 0.00 93.91 0.9691
18 118.63 -12.68 0.00 0.00 -5.95 0.00 0.00 0.00 94.15 0.9703
19 98.99 0.00 -6.98 7.99 0.00 0.00 0.00 0.00 94.11 0.9701
20 107.18 0.00 -1.69 0.00 -5.50 0.00 0.00 0.00 93.90 0.9690
21 102.15 0.00 0.00 4.05 -6.20 0.00 0.00 0.00 93.97 0.9694
22 0.00 59.64 -5.10 45.45 0.00 0.00 0.00 0.00 34.50 0.5873
23 0.00 113.23 78.16 0.00 -91.39 0.00 0.00 0.00 22.14 0.4705
24 0.00 63.94 0.00 43.79 -7.74 0.00 0.00 0.00 36.06 0.6005
25 0.00 0.00 -54.90 124.69 30.21 0.00 0.00 0.00 8.75 0.2958
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 132.87 -22.55 -8.66 -1.65 0.00 0.00 0.00 0.00 94.42 0.9717
27 142.23 -34.64 -21.86 0.00 14.26 0.00 0.00 0.00 94.58 0.9725
28 129.27 -18.55 0.00 -4.43 -6.29 0.00 0.00 0.00 94.17 0.9704
29 97.34 0.00 -9.12 9.31 2.47 0.00 0.00 0.00 94.12 0.9702
30 0.00 69.02 13.45 37.96 -20.43 0.00 0.00 0.00 37.34 0.6110
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 137.38 -32.02 -21,99 2.08 14.55 0.00 0.00 0.00 94.59 0.9726


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9665 0.2270 -0.0480 0.2700 -0.1305

(2) 93.4155 1.0000 0.3017 0.0156 0.2404 -0.0636

(3) 5.1543 9.1028 1.0000 -0.1307 -0.6267 0.2456

(4) 0.2308 0.0243 1.7080 1.0000 0.2222 0.8374

(5) 7.2874 5.7807 39.2754 4.9361 1.0000 -0.1846

(6) 1.7027 0.4045 6.0342 70.1186 3.4068 1.0000

Numbers in Parentheses are the Variable or Parameter Number











ALL LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (BB X8)

BO B1 B2 B3 B4 B5 B6 B7 BB


0.000 0.000
0.042 0.000
0.000 -0.074
0.000 0.000
0.000 0.000
-0.013 0.000
0.000 -0.098
0.000 0.000
0.000 0.000
0.041 -0.029
0.119 0.000
0.050 0.000
0.000 -0.176
0.000 0.317
0.000 0.000
-0.015 -0.115
-0.017 0.000
-0.010 0.000
0.000 -0.117
0.000 -0.026
0.000 0.000
0.120 -0.185
0.107 1.337
0.126 0.000
0.000 -0.354
-0.016 -0.113
-0.024 -0.272
-0.014 0.000
0.000 -0.156
0.137 0.483
-0.023 -0.283


0.000 0.000
0.000 0.000
0.000 0.000
9.269 0.000
0.000 -0.014
0.000 0.000
0.000 0.000
1.369 0.000
0.000 -0.007
0.000 0.000
23.308 0.000
0.000 -0.021
10.135 0.000
0.000 -0.032
8.739 -0.009
0.000 0.000
-1.028 0.000
0.000 -0.006
1.971 0.000
0.000 -0.006
0.971 -0.007
24.256 0.000
0.000 -0.107
23.003 -0.019
11.796 0.013
-0.318 0.000
0.000 0.012
-0.871 -0.006
2.335 0.003
20.024 -0.050
0.393 0.013


-0.017
0.062
0.133
0.073
0.144
0.000
-0.009
-0.024
-0.007
0.066
-0.193
0.075
0.083
0.140
0.087
0.012
0.010
0.004
-0.017
-0.007
-0.012
-0.183
-0.019
-0.178
0.072
0.015
0.020
0.012
-0.019
-0.179
0.017


1.319
0.000
0.000
0.000
0.000
1.349
1.321
1.306
1.313
0.000
0.000
0.000
0.000
0.000
0.000
1.355
1.368
1.337
1.302
1.315
1.305
0.000
0.000
0.000
0.000
1.361
1.387
1.354
1.302
0.000
1.381


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











LABORATORY DATA MINUS PROTOTYPE LABORATORY DATA
n = 624

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 74.04 0.8605
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.40 0.0631
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0023
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 15.17 0.3895
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 2.93 0.1712
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 115.32 -15.32 0.00 0.00 0.00 0.00 0.00 0.00 75.57 0.8693
7 106.78 0.00 -6.78 0.00 0.00 0.00 0.00 0.00 75.70 0.8700
8 89.42 0.00 0.00 10.58 0.00 0.00 0.00 0.00 75.95 0.8715
9 109.37 0.00 0.00 0.00 -9.37 0.00 0.00 0.00 76.89 0.8768
10 0.00 93.20 6.80 0.00 0.00 0.00 0.00 0.00 0.41 0.0640
11 0.00 51.63 0.00 48.37 0.00 0.00 0.00 0.00 35.93 0.5994
12 0.00 515.34 0.00 0.00 -415.34 0.00 0.00 0.00 4.23 0.2056
13 0.00 0.00 -18.78 118.78 0.00 0.00 0.00 0.00 16.15 0.4018
14 0.00 0.00 -474.15 0.00 574.15 0.00 0.00 0.00 9.97 0.3158
15 0.00 0.00 0.00 123.26 -23.26 0.00 0.00 0.00 16.28 0.4035
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 129.26 -19.92 -9.34 0.00 0.00 0.00 0.00 0.00 77.83 0.8822
17 95.02 -3.80 0.00 8.77 0.00 0.00 0.00 0.00 75.99 0.8717
18 119.44 -10.78 0.00 0.00 -8.65 0.00 0.00 0.00 77.52 0.8805
19 94.36 0.00 -8.23 13.87 0.00 0.00 0.00 0.00 78.60 0.8866
20 109.02 0.00 2.76 0.00 -11.78 0.00 0.00 0.00 76.96 0.8772
21 98.46 0.00 0.00 9.29 -7.75 0.00 0.00 0.00 78.08 0.8836
22 0.00 53.53 -5.26 51.72 0.00 0.00 0.00 0.00 37.67 0.6137
23 0.00 115.66 110.17 0.00 -125.83 0.00 0.00 0.00 24.81 0.4981
24 0.00 58.40 0.00 50.13 -8.53 0.00 0.00 0.00 40.20 0.6340
25 0.00 0.00 -2.27 123.14 -20.87 0.00 0.00 0.00 16.28 0.4035
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 98.98 -3.03 -8.48 12.53 0.00 0.00 0.00 0.00 78.62 0.8867
27 129.97 -20.68 -10.06 0.00 0.77 0.00 0.00 0.00 77.83 0.8822
28 94.74 2.49 0.00 10.48 -7.71 0.00 0.00 0.00 78.10 0.8837
29 93.45 0.00 -9.44 14.59 1.40 0.00 0.00 0.00 78.61 0.8866
30 0.00 65.74 18.60 41.92 -26.26 0.00 0.00 0.00 42.98 0.6556
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 100.66 -5.66 -11.79 13.20 3.58 0.00 0.00 0.00 78.67 0.8870


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8605 0.0631 0.0023 0.3895 -0.1712

(2) 74.0435 1.0000 0.2136 0.1505 0.2996 -0.0030

(3) 0.3988 4.5618 1.0000 -0.1223 -0.6874 0.2708

(4) 0.0005 2.2639 1.4968 1.0000 0.2514 0.8363

(5) 15.1705 8.9761 47.2546 6.3188 1.0000 -0.1732

(6) 2.9311 0.0009 7.3352 69.9318 3.0005 1.0000

Numbers in Parentheses are the Variable or Parameter Number











LABORATORY DATA MINUS PROTOTYPE LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = 80 + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

0B B1 B2 B3 B4 85 B6 B7 B8


0.000 0.000
0.005 0.000
0.000 0.001
0.000 0.000
0.000 0.000
-0.010 0.000
0.000 -0.084
0.000 0.000
0.000 0.000
0.005 0.007
0.049 0.000
0.009 0.000
0.000 -0.066
0.000 0.311
0.000 0.000
-0.012 -0.099
-0.003 0.000
-0.007 0.000
0.000 -0.108
0.000 0.032
0.000 0.000
0.050 -0.088
0.042 0.709
0.054 0.000
0.000 -0.007
-0.002 -0.108
-0.012 -0.106
0.002 0.000
0.000 -0.126
0.061 0.304
-0.004 -0.150


0.000 0.000
0.000 0.000
0.000 0.000
5.874 0.000
0.000 -0.008
0.000 0.000
0.000 0.000
2.182 0.000
0.000 -0.008
0.000 0.000
12.377 0.000
0.000 -0.009
6.261 0.000
0.000 -0.026
5.594 -0.005
0.000 0.000
1.737 0.000
0.000 -0.006
2.758 0.000
0.000 -0.009
1.758 -0.007
13.030 0.000
0.000 -0.055
12.457 -0.010
5.661 -0.004
2.412 0.000
0.000 0.001
2.035 -0.007
2.931 0.001
10.356 -0.029
2.536 0.003


0.007
0.110
0.118
0.084
0.127
0.019
0.012
0.000
0.017
0.109
-0.028
0.115
0.088
0.123
0.092
0.027
0.005
0.023
0.004
0.017
0.010
-0.026
0.062
-0.024
0.092
0.008
0.027
0.007
0.003
-0.023
0.008


1.087
0.000
0.000
0.000
0.000
1.121
1.111
1.032
1.086
0.000
0.000
0.000
0.000
0.000
0.000
1.157
1.053
1.109
1.049
1.076
1.042
0.000
0.000
0.000
0.000
1.065
1.160
1.029
1.050
0.000
1.081


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











PROTOTYPE LABORATORY DATA
n = 16
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2
No


(1)

100.00
0.00
0.00
0.00
0.00


(2)

0.00
100.00
0.00
0.00
0.00


147.96 -47.96
97.30 0.00
94.50 0.00
126.16 0.00
0.00 -210.52
0.00 72.22
0.00-5719.66
0.00 0.00
0.00 0.00
0.00 0.00


171.04
467.90
136.80
99.62
103.05
125.71
0.00
0.00
0.00
0.00


-57.03
-289.62
-61.52
0.00
0.00
0.00
148.93
116.36
191.25
0.00


370.85 -230.04


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
2.70 0.00 0.00
0.00 5.50 0.00
0.00 0.00 -26.16
310.52 0.00 0.00
0.00 27.78 0.00
0.00 0.00 5819.66
111.61 -11.61 0.00
80.36 0.00 19.64
0.00 16.21 83.79
INDEPENDENT VARIABLES


-14.01
0.00
0.00
-5.81
21.83
0.00
-114.03
110.17
0.00
133.06


0.00
-78.28
0.00
6.19
0.00
0.14
65.10
0.00
36.60
-17.48


0.00
0.00
24.72
0.00
-24.88
-25.85
0.00
-126.54
-127.85
-15.59


INDEPENDENT VARIABLES
23.00 -63.81 0.00


27********24321.6121640.99 0.00********
28 339.27 -242.06 0.00 -52.97 55.75
29 104.11 0.00 52.77 -10.46 -46.42
30 0.00 115.95 101.04 2.99 -119.98
INDEPENDENT VARIABLES
31-4544.24 4134.88 2495.82 406.15-2392.60


(6) (7)
TAKEN ONE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A


0.00


0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9900 0.7561 -0.6728 0.1771 -0.5403

(2) 98.0014 1.0000 0.8177 -0.6823 0.1433 -0.4991

(3) 57.1710 66.8677 1.0000 -0.6269 -0.3571 -0.0077

(4) 45.2711 46.5568 39.2992 1.0000 0.1137 0.5752

(5) 3.1360 2.0523 12.7513 1.2939 1.0000 -0.6382

(6) 29.1949 24.9090 0.0060 33.0902 40.7358 1.0000

Numbers in Parentheses are the Variable or Parameter Number


98.00
57.17
45.27
3.14
29.19

98.86
98.00
98.13
98.29
63.68
80.08
85.74
51.79
48.78
33.94

98.88
99.43
98.99
98.13
98.35
98.29
83.50
89.73
88.10
52.24

99.44
99.63
99.56
98.44
89.76

99.80


0.9900
0.7561
0.6728
0.1771
0.5403

0.9943
0.9900
0.9906
0.9914
0.7980
0.8949
0.9260
0.7196
0.6984
0.5826

0.9944
0.9971
0.9949
0.9906
0.9917
0.9914
0.9138
0.9473
0.9386
0.7228

0.9972
0.9982
0.9978
0.9922
0.9474

0.9990










PROTOTYPE LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + ( X + (B2 X2) (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 84 B5 B6 B7 B8
NO
1 -0.067 1.423 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 -0.449 0.000 0.447 0.000 0.000 0.000 0.000 0.000 0.000
3 2.471 0.000 0.000 -43.809 0.000 0.000 0.000 0.000 0.000
4 0.311 0.000 0.000 0.000 16.164 0.000 0.000 0.000 0.000
5 1.393 0.000 0.000 0.000 0.000 -1.744 0.000 0.000 0.000
6 0.057 1.613 -0.095 0.000 0.000 0.000 0.000 0.000 0.000
7 -0.084 1.428 0.000 0.321 0.000 0.000 0.000 0.000 0.000
8 -0.095 1.416 0.000 0.000 3.287 0.000 0.000 0.000 0.000
9 0.055 1.379 0.000 0.000 0.000 -0.199 0.000 0.000 0.000
10 0.776 0.000 0.326 -21.328 0.000 0.000 0.000 0.000 0.000
11 -1.105 0.000 0.555 0.000 46.774 0.000 0.000 0.000 0.000
12 0.477 0.000 0.445 0.000 0.000 -1.725 0.000 0.000 0.000
13 2.340 0.000 0.000 -45.712 23.453 0.000 0.000 0.000 0.000
14 2.472 0.000 0.000 -35.229 0.000 -0.739 0.000 0.000 0.000
15 1.945 0.000 0.000 0.000 -25.840 -2.327 0.000 0.000 0.000
16 0.114 1.601 -0.097 -1.061 0.000 0.000 0.000 0.000 0.000
17 0.313 1.871 -0.211 0.000 -12.495 0.000 0.000 0.000 0.000
18 -0.016 1.759 -0.144 0.000 0.000 0.221 0.000 0.000 0.000
19 -0.062 1.405 0.000 -0.664 3.488 0.000 0.000 0.000 0.000
20 -0.046 1.407 0.000 2.413 0.000 -0.236 0.000 0.000 0.000
21 0.054 1.379 0.000 0.000 0.060 -0.197 0.000 0.000 0.000
22 -0.168 0.000 0.460 -15.634 44.008 0.000 0.000 0.000 0.000
23 -0.560 0.000 0.583 24.518 0.000 -2.419 0.000 0.000 0.000
24 -0.064 0.000 0.493 0.000 20.657 -1.257 0.000 0.000 0.000
25 2.280 0.000 0.000 -52.737 34.151 0.531 0.000 0.000 0.000
26 0.272 1.892 -0.214 0.950 -12.995 0.000 0.000 0.000 0.000
27 0.464 2.146 -0.353 -13.930 0.000 1.043 0.000 0.000 0.000
28 0.239 2.027 -0.264 0.000 -12.634 0.232 0.000 0.000 0.000
29 -0.047 1.426 0.000 5.851 -5.717 -0.442 0.000 0.000 0.000
30 -0.556 0.000 0.579 22.401 3.271 -2.285 0.000 0.000 0.000
31 0.479 2.197 -0.364 -9.768 -7.838 0.804 0.000 0.000 0.000











ALL FIELD DATA
n = 131
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 82.76 0.9097
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 34.48 0.5872
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 15.51 0.3938
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 16.71 0.4088
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 15.42 0.3926
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 111.40 -11.40 0.00 0.00 0.00 0.00 0.00 0.00 82.98 0.9109
7 99.24 0.00 0.76 0.00 0.00 0.00 0.00 0.00 82.77 0.9098
8 90.80 0.00 0.00 9.20 0.00 0.00 0.00 0.00 83.45 0.9135
9 100.10 0.00 0.00 0.00 -0.10 0.00 0.00 0.00 82.76 0.9097
10 0.00 114.32 -14.32 0.00 0.00 0.00 0.00 0.00 38.54 0.6208
11 0.00 64.70 0.00 35.30 0.00 0.00 0.00 0.00 74.99 0.8659
12 0.00 113.69 0.00 0.00 -13.69 0.00 0.00 0.00 41.91 0.6474
13 0.00 0.00 -76.34 176.34 0.00 0.00 0.00 0.00 27.78 0.5270
14 0.00 0.00 58.89 0.00 41.11 0.00 0.00 0.00 15.98 0.3997
15 0.00 0.00 0.00 147.65 -47.65 0.00 0.00 0.00 24.85 0.4985
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 110.45 -11.07 0.62 0.00 0.00 0.00 0.00 0.00 82.99 0.9110
17 69.36 15.77 0.00 14.87 0.00 0.00 0.00 0.00 83.65 0.9146
18 111.32 -11.46 0.00 0.00 0.14 0.00 0.00 0.00 82.98 0.9109
19 90.23 0.00 0.70 9.07 0.00 0.00 0.00 0.00 83.47 0.9136
20 98.36 0.00 5.71 0.00 -4.07 0.00 0.00 0.00 82.92 0.9106
21 90.25 0.00 0.00 9.30 0.44 0.00 0.00 0.00 83.46 0.9136
22 0.00 64.76 -0.12 35.35 0.00 0.00 0.00 0.00 74.99 0.8660
23 0.00 102.84 25.43 0.00 -28.28 0.00 0.00 0.00 44.48 0.6669
24 0.00 65.12 0.00 35.37 -0.49 0.00 0.00 0.00 75.04 0.8663
25 0.00 0.00 -117.31 177.26 40.05 0.00 0.00 0.00 28.45 0.5334
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 68.07 16.21 0.86 14.86 0.00 0.00 0.00 0.00 83.68 0.9148
27 107.26 -8.45 3.65 0.00 -2.46 0.00 0.00 0.00 83.03 0.9112
28 68.86 15.74 0.00 14.95 0.45 0.00 0.00 0.00 83.66 0.9147
29 90.33 0.00 1.23 8.87 -0.44 0.00 0.00 0.00 83.47 0.9136
30 0.00 65.34 3.82 34.02 -3.18 0.00 0.00 0.00 75.30 0.8677
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 67.08 17.16 2.54 14.58 -1.37 0.00 0.00 0.00 83.70 0.9149


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9097 0.5872 -0.3938 0.4088 -0.3926

(2) 82.7559 1.0000 0.6831 -0.4464 0.3642 -0.4292

(3) 34.4802 46.6613 1.0000 -0.3490 -0.3278 -0.2153

(4) 15.5070 19.9298 12.1827 1.0000 -0.1601 0.9353

(5) 16.7102 13.2635 10.7464 2.5616 1.0000 -0.2931

(6) 15.4164 18.4171 4.6364 87.4857 8.5882 1.0000

Numbers in Parentheses are the Variable or Parameter Number











ALL FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 82 B3 B4 B5 B6 B7 88
NO
1 0.089 1.325 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.128 0.000 0.280 0.000 0.000 0.000 0.000 0.000 0.000
3 2.705 0.000 0.000 -11.030 0.000 0.000 0.000 0.000 0.000
4 1.297 0.000 0.000 0.000 336.641 0.000 0.000 0.000 0.000
5 2.602 0.000 0.000 0.000 0.000 -0.364 0.000 0.000 0.000
6 0.218 1.389 -0.031 0.000 0.000 0.000 0.000 0.000 0.000
7 0.055 1.335 0.000 0.431 0.000 0.000 0.000 0.000 0.000
8 -0.047 1.278 0.000 0.000 73.560 0.000 0.000 0.000 0.000
9 0.094 1.323 0.000 0.000 0.000 -0.003 0.000 0.000 0.000
10 0.637 0.000 0.244 -6.023 0.000 0.000 0.000 0.000 0.000
11 -2.291 0.000 0.385 0.000 554.783 0.000 0.000 0.000 0.000
12 0.580 0.000 0.251 0.000 0.000 -0.258 0.000 0.000 0.000
13 1.794 0.000 0.000 -9.439 292.223 0.000 0.000 0.000 0.000
14 2.668 0.000 0.000 -5.940 0.000 -0.180 0.000 0.000 0.000
15 1.755 0.000 0.000 0.000 264.607 -0.276 0.000 0.000 0.000
16 0.190 1.396 -0.030 0.334 0.000 0.000 0.000 0.000 0.000
17 -0.395 1.122 0.055 0.000 136.634 0.000 0.000 0.000 0.000
18 0.212 1.391 -0.031 0.000 0.000 0.003 0.000 0.000 0.000
19 -0.080 1.287 0.000 0.423 73.524 0.000 0.000 0.000 0.000
20 0.039 1.333 0.000 3.281 0.000 -0.101 0.000 0.000 0.000
21 -0.073 1.284 0.000 0.000 75.192 0.012 0.000 0.000 0.000
22 -2.274 0.000 0.384 -0.139 553.529 0.000 0.000 0.000 0.000
23 0.126 0.000 0.288 14.042 0.000 -0.677 0.000 0.000 0.000
24 -2.210 0.000 0.381 0.000 545.942 -0.024 0.000 0.000 0.000
25 1.746 0.000 0.000 -15.957 323.146 0.236 0.000 0.000 0.000
26 -0.460 1.128 0.058 0.603 139.913 0.000 0.000 0.000 0.000
27 0.151 1.381 -0.023 1.991 0.000 -0.058 0.000 0.000 0.000
28 -0.430 1.128 0.056 0.000 139.206 0.013 0.000 0.000 0.000
29 -0.078 1.288 0.000 0.745 71.893 -0.011 0.000 0.000 0.000
30 -2.307 0.000 0.390 4.490 536.568 -0.162 0.000 0.000 0.000
31 -0.481 1.120 0.062 1.800 138.336 -0.042 0.000 0.000 0.000












APPENDIX III


Data listings and stepwise regression results for each investigation



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (11) of the text.

100r*2 = 100r2.

The dependent variable, db, is the water depth at the shore-breaking position.

Independent variables:

(1) Shore-breaking wave height, Hb
(2) Wave period, T
(3) Bed slope leading to shore-breaking, tan ab
(4) Equivalent wave steepness, Hb/g T2
(5) Surf similarity parameter, tan ab/[Hb/(g T2)]05

Correlaton Matrix.

Correlation matrix treatment is given by Table 5 of the text.

Variables:

(1) db (2) Hb (3) T (4) tan ab

(5) Hb/(g T2) (6) tan ab/[Hbl(g T2)]0.5











GAILLARD (1904) FIELD DATA
n = 25


Water
Depth
(m)

1.5390
1.8140
1.9200
1.9350
2.4380
4.5420
3.8400
4.5570
4.3890
4.2210
4.2670
5.1970
4.1450
5.1050
5.3490
4.4200
4.9990
0.7920
0.9150
0.8230
1.2500
1.4790
1.3410
3.0480
3.0480


Breaker Wave
Height Period
(m) (s)

0.9140 4.4200
1.0670 5.0700
1.1280 6.5300
1.2190 4.6600
1.3720 5.6300
1.6760 5.9800
1.8290 6.8000
1.9810 5.1700
2.0730 8.3300
2.1340 6.6100
2.3770 10.9800
2.4380 8.3900
2.5910 9.4400
2.7430 7.1900
2.8960 6.8400
3.0480 6.4700
3.3530 7.2300
0.6100 4.9600
0.7620 5.2500
0.7920 5.8000
0.8530 5.6000
0.9140 4.8400
1.0060 3.8500
2.2860 6.0000
2.4380 4.7500


Bed Equivalent Surf
SLope Wave Sim
Steepness Parm

0.0200 0.00477 0.28946
0.0200 0.00424 0.30730
0.0200 0.00270 0.38495
0.0200 0.00573 0.26426
0.0160 0.00442 0.24075
0.0250 0.00478 0.36151
0.0250 0.00404 0.39351
0.0250 0.00756 0.28748
0.0250 0.00305 0.45279
0.0250 0.00498 0.35413
0.0250 0.00201 0.55737
0.0290 0.00353 0.48782
0.0250 0.00297 0.45898
0.0290 0.00541 0.39412
0.0290 0.00632 0.36489
0.0250 0.00743 0.29003
0.0290 0.00655 0.35845
0.0250 0.00253 0.49702
0.0250 0.00282 0.47069
0.0250 0.00240 0.51006
0.0250 0.00278 0.47453
0.0250 0.00398 0.39621
0.0250 0.00693 0.30041
0.0154 0.00648 0.19131
0.0154 0.01103 0.14666











GAILLARD (1904) FIELD DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(2)

0.00
100.00
0.00
0.00
0.00

24.82
0.00
0.00
0.00
62.89
70.18
214.59
0.00
0.00
0.00

12.87
8.30
16.30
0.00
0.00
0.00
47.34
80.26
65.33
0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
36.07 0.00 0.00
0.00 -18.45 0.00
0.00 0.00 17.34
37.11 0.00 0.00
0.00 29.82 0.00
0.00 0.00 -114.59
76.72 23.28 0.00
170.72 0.00 -70.72
0.00 44.25 55.75


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(1)

100.00
0.00
0.00
0.00
0.00

75.18
63.93
118.45
82.66
0.00
0.00
0.00
0.00
0.00
0.00

57.37
104.70
76.29
73.82
63.20
110.98
0.00
0.00
0.00
0.00


29.76
0.00
0.00
32.50
39.47
0.00
28.46
104.09
0.00
106.83


0.00
-13.00
0.00
-6.32
0.00
-14.98
24.20
0.00
30.30
16.43


0.00
0.00
7.41
0.00
-2.67
4.00
0.00
-84.35
4.37
-23.26


INDEPENDENT VARIABLES
26 42.93 21.30 29.30 6.48 0.00
27 1.80 78.23 102.05 0.00 -82.07
28 100.21 7.38 0.00 -10.81 3.22
29 118.29 0.00 118.97 -49.87 -87.39
30 0.00 80.04 103.61 0.17 -83.82
INDEPENDENT VARIABLES
31 2.73 78.51 103.95 -1.05 -84.15


0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


81.80
40.76
18.38
8.31
0.05

83.31
84.61
83.47
83.15
44.47
74.04
60.21
35.67
35.05
23.09

85.23
83.49
83.40
84.95
84.63
83.48
81.40
90.69
74.13
36.26

85.34
90.70
83.50
86.67
90.69


0.9044
0.6384
0.4287
0.2883
0.0214

0.9127
0.9198
0.9136
0.9119
0.6669
0.8604
0.7759
0.5973
0.5920
0.4805

0.9232
0.9137
0.9132
0.9217
0.9199
0.9137
0.9022
0.9523
0.8610
0.6022

0.9238
0.9524
0.9138
0.9310
0.9523


90.71 0.9524


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9044 0.6384 0.4287 0.2883 -0.0214

(2) 81.7990 1.0000 0.5968 0.2970 0.4465 -0.1508


40.7607 35.6187 1.0000 0.3942 -0.3831


0.5454


4) 18.3769 8.8234 15.5382 1.0000 -0.2634 0.6630

5) 8.3114 19.9390 14.6732 6.9360 1.0000 -0.8260

5) 0.0459 2.2741 29.7449 43.9627 68.2242 1.0000

Numbers in Parentheses are the Variable or Parameter Number


INDEPENDENT VARIABLES TAKEN THREE AT











GAILLARD (1904) FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 0.000 1.739 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 -0.731 0.000 0.610 0.000 0.000 0.000 0.000 0.000 0.000
3 -0.925 0.000 0.000 169.533 0.000 0.000 0.000 0.000 0.000
4 2.063 0.000 0.000 0.000 215.952 0.000 0.000 0.000 0.000
5 3.216 0.000 0.000 0.000 0.000 -0.327 0.000 0.000 0.000
6 -0.606 1.563 0.146 0.000 0.000 0.000 0.000 0.000 0.000
7 -1.468 1.639 0.000 69.412 0.000 0.000 0.000 0.000 0.000
8 0.296 1.863 0.000 0.000 -108.119 0.000 0.000 0.000 0.000
9 -0.723 1.773 0.000 0.000 0.000 1.792 0.000 0.000 0.000
10 -2.201 0.000 0.531 82.885 0.000 0.000 0.000 0.000 0.000
11 -4.399 0.000 0.839 0.000 467.780 0.000 0.000 0.000 0.000
12 0.511 0.000 0.884 0.000 0.000 -8.017 0.000 0.000 0.000
13 -3.532 0.000 0.000 214.433 322.918 0.000 0.000 0.000 0.000
14 -1.246 0.000 0.000 312.564 0.000 -8.312 0.000 0.000 0.000
15 -3.791 0.000 0.000 0.000 637.882 10.391 0.000 0.000 0.000
16 -1.668 1.535 0.098 59.786 0.000 0.000 0.000 0.000 0.000
17 0.060 1.785 0.040 0.000 -82.577 0.000 0.000 0.000 0.000
18 -0.718 1.634 0.099 0.000 0.000 0.765 0.000 0.000 0.000
19 -1.051 1.720 0.000 56.828 -54.879 0.000 0.000 0.000 0.000
20 -1.475 1.623 0.000 76.073 0.000 -0.331 0.000 0.000 0.000
21 0.125 1.852 0.000 0.000 -93.132 0.322 0.000 0.000 0.000
22 -6.719 0.000 0.741 117.806 497.165 0.000 0.000 0.000 0.000
23 -3.559 0.000 0.852 292.082 0.000 -15.194 0.000 0.000 0.000
24 -4.805 0.000 0.823 0.000 501.005 0.935 0.000 0.000 0.000
25 -2.665 0.000 0.000 259.494 198.133 -3.627 0.000 0.000 0.000
26 -2.419 1.330 0.187 68.127 74.792 0.000 0.000 0.000 0.000
27 -3.484 0.066 0.821 283.145 0.000 -14.618 0.000 0.000 0.000
28 -0.069 1.781 0.037 0.000 -71.592 0.276 0.000 0.000 0.000
29 0.470 1.745 0.000 131.721 -274.040 -6.211 0.000 0.000 0.000
30 -3.575 0.000 0.851 291.450 2.378 -15.136 0.000 0.000 0.000
31 -3.350 0.099 0.807 282.550 -14.135 -14.682 0.000 0.000 0.000











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE I FIELD DATA
n = 56


Water
Depth
(m)

2.2555
1.6154
2.3165
2.7737
2.1946
1.5545
2.6518
2.2860
2.7737
2.0117
2.0117
1.9812
2.3165
1.7374
2.2250
2.9870
3.1394
2.1946
2.1641
2.2250
2.0117
2.4994
2.9870
2.8651
2.9870
3.8710
3.2918
3.7186
2.6822
1.8288
3.0480
2.9566
2.6822
1.8288
2.4079
3.3528
2.4079
2.4689
2.5603
3.6881
1.9507
2.5603
2.7127
2.1336
3.7186
2.8042
2.7737
2.6213
3.2004
3.6881
3.4442
3.4138
3.0175
3.0175
2.4994
4.4501


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


Breaker Wave
Height Period
(m) (s)

2.2555 13.7000
1.4630 12.0000
1.6459 13.3000
2.2555 12.7000
1.9507 12.2000
1.2192 10.2000
2.1336 11.6000
1.7678 12.0000
2.3165 11.5000
1.7069 10.0000
2.0117 10.0000
1.7069 10.0000
2.0117 11.2000
1.4021 9.2000
1.5850 9.0000
2.0117 10.2000
2.6213 10.5000
2.0117 10.0000
1.4630 9.5000
2.1336 9.6000
1.7678 9.5000
1.9507 9.4000
2.4384 9.5000
2.1946 9.6000
2.4384 10.3000
2.7432 10.5000
2.1946 10.5000
2.6822 9.6000
2.1336 9.8000
1.4021 8.1000
2.8651 10.3000
2.0726 9.0000
2.5603 9.4000
2.0726 9.0000
1.5240 7.7000
2.9870 9.0000
1.8288 8.5000
2.3774 8.8000
2.1946 8.8000
2.4384 10.0000
1.7678 8.0000
2.0117 7.2000
2.4384 9.0000
1.8898 8.0000
2.6213 9.2000
2.3774 8.8000
2.9870 8.5000
2.1946 8.0000
2.3774 7.5000
3.4747 9.0000
3.0480 8.2000
2.7432 7.5000
2.4384 7.2000
2.8651 8.0000
2.1946 6.5000
3.3528 7.8000


0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159
0.0159


0.00123
0.00104
0.00095
0.00143
0.00134
0.00120
0.00162
0.00125
0.00179
0.00174
0.00205
0.00174
0.00164
0.00169
0.00200
0.00197
0.00243
0.00205
0.00165
0.00236
0.00200
0.00225
0.00276
0.00243
0.00235
0.00254
0.00203
0.00297
0.00227
0.00218
0.00276
0.00261
0.00296
0.00261
0.00262
0.00376
0.00258
0.00313
0.00289
0.00249
0.00282
0.00396
0.00307
0.00301
0.00316
0.00313
0.00422
0.00350
0.00431
0.00438
0.00463
0.00498
0.00480
0.00457
0.00530
0.00562


0.45374
0.49381
0.51601
0.42091
0.43478
0.45980
0.39529
0.44923
0.37609
0.38099
0.35094
0.38099
0.39305
0.38673
0.35583
0.35796
0.32281
0.35094
0.39094
0.32713
0.35564
0.33500
0.30282
0.32256
0.32832
0.31555
0.35280
0.29176
0.33395
0.34049
0.30288
0.31116
0.29241
0.31116
0.31046
0.25920
0.31286
0.28408
0.29568
0.31876
0.29949
0.25268
0.28688
0.28967
0.28284
0.28408
0.24480
0.26880
0.24211
0.24032
0.23378
0.22539
0.22950
0.23525
0.21840
0.21203











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE I FIELD DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1)

100.00
0.00
0.00
0.00
0.00


(2)

0.00
100.00
0.00
0.00
0.00


103.21 -3.21
0.34 0.00
99.36 0.00
104.10 0.00
0.00 -0.43
0.00 62.18
0.00-2930.74
0.00 0.00
0.00 0.00
0.00 0.00

0.31 0.01
120.82 -14.60
103.70 19.80
0.33 0.00
0.33 0.00
120.05 0.00
0.00 0.26
0.00 0.47
0.00 306.91
0.00 0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
99.66
0.00
0.00
100.43
0.00
0.00
101.04
99.52
0.00


0.00
0.00
0.64
0.00
0.00
37.82
0.00
-1.04
0.00
-327.88


0.00
0.00
0.00
-4.10
0.00
0.00
3030.74
0.00
0.48
427.88


INDEPENDENT VARIABLES
99.67 0.00 0.00
0.00 -6.22 0.00
0.00 0.00 -23.50
99.68 -0.01 0.00
99.66 0.00 0.01
0.00 -5.79 -14.26
99.59 0.14 0.00
99.99 0.00 -0.46
0.00 39.15 -246.06
99.89 -0.22 0.33


INDEPENDENT VARIABLES
26 0.31 0.01 99.67 0.00 0.00
27 0.24 0.11 99.74 0.00 -0.10
28 117.52 6.40 0.00 -4.92 -19.01
29 0.34 0.00 99.69 -0.01 -0.01
30 0.00 0.40 99.83 0.06 -0.29
INDEPENDENT VARIABLES
31 0.21 0.15 99.74 0.01 -0.11


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


TI





TI


TAKEN THREE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A 1
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A 1
0.00 0.00


(8)
ME
0.00
0.00
0.00
0.00
0.00
ME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


71.00
3.60
0.82
34.25
33.29


0.8426
0.1897
0.0906
0.5853
0.5770


71.03 0.8428
72.05 0.8488
71.00 0.8426
71.05 0.8429
3.65 0.1911
54.58 0.7388
66.41 0.8149
34.27 0.5854
33.56 0.5793
35.03 0.5919

72.10 0.8491
71.09 0.8431
71.09 0.8432
72.09 0.8491
72.07 0.8489
71.12 0.8433
57.51 0.7583
68.23 0.8260
68.19 0.8258
35.14 0.5928

72.10 0.8491
72.28 0.8502
71.12 0.8433
72.10 0.8491
70.65 0.8406

72.32 0.8504


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.8426 -0.1897 0.0906 0.5853 -0.5770


70.9951


1.0000 -0.2038 -0.0146 0.6883 -0.6643


3.5990 4.1544


1.0000 -0.3663 -0.7932 0.8516


0.8203 0.0213 13.4164 1.0000 0.1756 -0.2438

34.2544 47.3792 62.9198 3.0821 1.0000 -0.9309


33.2898 44.1235 72.5268 5.9443 86.6506


1.0000


Numbers in Parentheses are the Variable or Parameter Number











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE I FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

B1 B2 B3 B4 B5 B6 B7 B8


EQ BO
NO
1 0.342
2 3.401
3 -603.013
4 1.806
5 4.338
6 0.425
7 -687.745
8 0.347
9 0.489
10 -159.478
11 -1.911
12 2.815
13 85.829
14 360.569
15 2.834
16 -748.012
17 0.645
18 0.659
19 -722.595
20 -718.018
21 0.757
22-1258.331
23 -974.955
24 1.293
25 232.023
26 -750.954
27 -796.122
28 0.767
29 -706.742
30-1152.452


0.000
0.000
0.000
321.226
0.000
0.000
0.000
5.527
0.000
0.000
643.562
0.000
322.437
0.000
198.128
0.000
-46.805
0.000
-15.401
0.000
-41.139
675.525
0.000
200.261
190.888
3.123
0.000
-34.757
-27.515
236.892


31 -843.777 0.804 0.13253098.787 40.998


0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-5.096 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-0.273 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-13.355 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-5.211 0.000 0.000 0.000
-2.128 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-1.445 0.000 0.000 0.000
0.000 0.000 0.000 0.000
0.157 0.000 0.000 0.000
-0.834 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-13.675 0.000 0.000 0.000
-10.360 0.000 0.000 0.000
-2.310 0.000 0.000 0.000
0.000 0.000 0.000 0.000
-2.448 0.000 0.000 0.000
-1.106 0.000 0.000 0.000
-0.225 0.000 0.000 0.000
-10.190 0.000 0.000 0.000
-2.908 0.000 0.000 0.000


0.000 0.000
-0.077 0.000
0.00038093.775
0.000 0.000
0.000 0.000
-0.008 0.000
0.00043276.234
0.000 0.000
0.000 0.000
-0.07310241.944
0.299 0.000
0.443 0.000
0.000-5284.744
0.000*********
0.000 0.000
0.01047059.845
-0.031 0.000
0.042 0.000
0.00045467.279
0.00045174.926
0.000 0.000
0.34578987.835
0.47761481.921
0.443 0.000
0.000*********
0.01147243.891
0.09550109.815
0.013 0.000
0.00044477.232
0.48372529.651


1.057
0.000
0.000
0.000
0.000
1.053
1.059
1.049
1.032
0.000
0.000
0.000
0.000
0.000
0.000
1.066
1.111
0.947
1.084
1.074
1.043
0.000
0.000
0.000
0.000
1.062
0.888
1.016
1.081
0.000











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE II FIELD DATA


Wave
Period
(s)

13.0000
12.5000
12.0000
10.5000
11.2000
10.0000
10.0000
8.8000
9.3000
9.6000
10.5000
10.0000
9.5000
8.9000
9.0000
9.0000
8.0000
7.0000


n = 18

Bed Equivalent
Slope Wave
Steepness


Water
Depth
(m)

3.1400
3.0200
3.2300
3.7200
3.4700
3.5400
3.3200
1.6800
3.6300
3.4400
3.5000
3.7200
2.8300
3.6000
3.3500
3.1700
3.4400
3.3200


Breaker
Height
(m)

1.7700
1.9500
2.3200
2.2000
2.5000
1.8300
1.8300
1.2800
2.3800
2.5600
2.3200
2.1000
2.3800
2.5000
2.3200
2.2600
2.3200
2.7400


Surf
Sim
Parm

1.49551
1.37310
1.20850
1.08589
1.08657
1.13392
1.13392
1.19313
0.92471
0.92037
1.05744
1.05852
0.94459
0.86343
0.90637
0.91833
0.80567
0.64868


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1)

100.00
0.00
0.00
0.00
0.00

72.04
-1.83
118.75
77.73
0.00
0.00
0.00
0.00
0.00
0.00

-191.45
111.95
106.52
-4.18
-21.56
147.23
0.00
0.00
0.00
0.00

-2.95
1.94
153.37
-1.44
0.00


(2)

0.00
100.00
0.00
0.00
0.00

27.96
0.00
0.00
0.00
0.28
71.25
710.20
0.00
0.00
0.00

-73.66
4.28
-295.00
0.00
0.00
0.00
1.29
-4.56
2292.32
0.00

0.39
-4.39
-312.01
0.00
-2.38


31 4.23 -8.35


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
101.83
0.00
0.00
99.72
0.00
0.00
97.31
99.20
0.00


0.00
0.00
-18.75
0.00
0.00
28.75
0.00
2.69
0.00
19.81


0.00
0.00
0.00
22.27
0.00
0.00
-610.20
0.00
0.80
80.19


INDEPENDENT VARIABLES
365.12 0.00 0.00
0.00 -16.22 0.00
0.00 0.00 288.48
103.56 0.62 0.00
127.53 0.00 -5.97
0.00 -30.06 -17.17
98.26 0.45 0.00
100.49 0.00 4.07
0.00 -72.75-2119.57
98.76 0.27 0.97
INDEPENDENT VARIABLES
102.01 0.56 0.00
97.89 0.00 4.56
0.00 -19.55 278.18
100.39 0.44 0.62
99.53 0.21 2.64
INDEPENDENT VARIABLES
96.67 -0.41 7.87


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(7)
AT A
0.00
0.00
0.00
0.00
0.00
AT A
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


TAKEN THREE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
IME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


37.15
0.16
0.60
4.41
8.07

43.12
38.42
44.32
41.18
1.42
19.98
50.63
4.41
9.02
9.62

43.12
44.34
55.85
44.76
41.19
44.53
24.02
51.47
50.90
13.43

44.90
62.77
63.31
46.23
52.91


0.6095
0.0395
0.0774
0.2099
0.2841

0.6566
0.6199
0.6657
0.6417
0.1190
0.4470
0.7116
0.2101
0.3004
0.3102

0.6567
0.6659
0.7474
0.6690
0.6418
0.6673
0.4901
0.7174
0.7134
0.3665

0.6701
0.7923
0.7957
0.6799
0.7274


68.25 0.8262


0.0489
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490


0.00107
0.00127
0.00164
0.00204
0.00203
0.00187
0.00187
0.00169
0.00281
0.00283
0.00215
0.00214
0.00269
0.00322
0.00292
0.00285
0.00370
0.00571


T





T











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE II FIELD DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.6095 0.0395 0.0774 0.2099 -0.2841

(2) 37.1489 1.0000 -0.3156 0.3034 0.6704 -0.7010

(3) 0.1558 9.9620 1.0000 -0.5011 -0.8380 0.8933

(4) 0.5991 9.2059 25.1075 1.0000 0.3296 -0.5570

(5) 4.4058 44.9444 70.2181 10.8623 1.0000 -0.9152

(6) 8.0738 49.1456 79.7898 31.0300 83.7589 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + + (BB8 X8)

EQ BO B1 82 B3 B4 85 B6 B7 88
NO
1 1.511 0.807 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 3.165 0.000 0.012 0.000 0.000 0.000 0.000 0.000 0.000
3 -64.835 0.000 0.000 1390.374 0.000 0.000 0.000 0.000 0.000
4 3.057 0.000 0.000 0.000 91.997 0.000 0.000 0.000 0.000
5 3.963 0.000 0.000 0.000 0.000 -0.652 0.000 0.000 0.000
6 0.495 0.914 0.079 0.000 0.000 0.000 0.000 0.000 0.000
7 105.636 0.854 0.000-2127.387 0.000 0.000 0.000 0.000 0.000
8 1.199 1.127 0.000 0.000 -158.193 0.000 0.000 0.000 0.000
9 0.265 1.068 0.000 0.000 0.000 0.645 0.000 0.000 0.000
10 -111.231 0.000 0.032 2330.876 0.000 0.000 0.000 0.000 0.000
11 0.210 0.000 0.220 0.000 357.579 0.000 0.000 0.000 0.000
12 2.666 0.000 0.442 0.000 0.000 -3.624 0.000 0.000 0.000
13 -5.059 0.000 0.000 165.712 90.665 0.000 0.000 0.000 0.000
14 107.326 0.000 0.000-2106.509 0.000 -0.801 0.000 0.000 0.000
15 4.973 0.000 0.000 0.000 -135.337 -1.300 0.000 0.000 0.000
16 4.335 0.915 0.078 -78.299 0.000 0.000 0.000 0.000 0.000
17 1.108 1.109 0.009 0.000 -142.866 0.000 0.000 0.000 0.000
18 6.813 -1.710 1.048 0.000 0.000 -9.768 0.000 0.000 0.000
19 63.215 1.141 0.000-1266.776 -151.109 0.000 0.000 0.000 0.000
20 14.133 1.064 0.000 -282.383 0.000 0.621 0.000 0.000 0.000
21 1.639 1.102 0.000 0.000 -200.088 -0.271 0.000 0.000 0.000
22 -208.801 0.000 0.275 4253.431 389.047 0.000 0.000 0.000 0.000
23 99.561 0.000 0.442-1974.688 0.000 -3.760 0.000 0.000 0.000
24 3.094 0.000 0.439 0.000 -55.926 -3.865 0.000 0.000 0.000
25 251.729 0.000 0.000-5008.225 -270.941 -2.305 0.000 0.000 0.000
26 93.815 1.216 -0.035-1884.851 -205.268 0.000 0.000 0.000 0.000
27 345.981 -3.029 1.513-6846.899 0.000 -14.980 0.000 0.000 0.000
28 10.482 -5.023 2.261 0.000 569.129 -19.212 0.000 0.000 0.000
29 167.566 1.075 0.000-3366.040 -289.633 -0.972 0.000 0.000 0.000
30 183.054 0.000 0.430-3651.808 -156.277 -4.551 0.000 0.000 0.000
31 300.532 -5.718 2.500-5865.058 494.481 -22.438 0.000 0.000 0.000











BALSILLIE AND CARTER (1980) FIELD DATA
n = 30

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.3750 0.2560 3.5000 0.0820 0.00213 1.77572
0.1600 0.1760 4.8300 0.1013 0.00077 3.65101
0.7580 0.4580 4.1900 0.0824 0.00266 1.59706
0.3700 0.1740 3.5300 0.0393 0.00142 1.04113
0.3500 0.2720 5.1400 0.0172 0.00105 0.53066
0.2000 0.1540 3.5300 0.1080 0.00126 3.04124
0.4000 0.3510 3.2700 0.0630 0.00335 1.08855
0.3000 0.2490 4.2900 0.0534 0.00138 1.43718
0.4500 0.3760 6.2100 0.4615 0.00099 14.63128
0.2600 0.2050 2.5700 0.1689 0.00317 3.00123
0.5400 0.4460 5.2900 0.0210 0.00163 0.52074
0.5400 0.3280 5.2900 0.0210 0.00120 0.60723
0.2000 0.1040 1.5100 0.0246 0.00465 0.36059
0.4300 0.2080 5.6300 0.0480 0.00067 1.85494
0.1600 0.1120 2.6500 0.1016 0.00163 2.51851
0.1100 0.0570 1.5700 0.0874 0.00236 1.79923
0.6300 0.5410 7.3000 0.1125 0.00104 3.49534
0.4800 0.3940 6.4300 0.0557 0.00097 1.78620
0.1800 0.1750 8.5700 0.0951 0.00024 6.09895
0.1700 0.0990 2.6700 0.0951 0.00142 2.52631
0.2100 0.1590 2.3200 0.1118 0.00301 2.03631
0.4400 0.2730 7.3200 0.0525 0.00052 2.30251
0.1800 0.1150 2.1100 0.0526 0.00264 1.02455
0.1300 0.0790 1.5800 0.0364 0.00323 0.64056
0.5600 0.4770 4.1900 0.0507 0.00277 0.96289
0.4300 0.3370 3.7500 0.0213 0.00245 0.43073
0.3300 0.3030 3.0000 0.1842 0.00344 3.14270
0.2600 0.1740 2.1700 0.0492 0.00377 0.80124
0.1600 0.0970 1.3300 0.1818 0.00560 2.43037
0.3200 0.3240 3.9100 0.0570 0.00216 1.22572











BALSILLIE AND CARTER (1980) FIELD DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(2)

0.00
100.00
0.00
0.00
0.00

0.59
0.00
0.00
0.00
108.55
71.53
118.33
0.00
0.00
0.00

1.46
-6.50
7.38
0.00
0.00
0.00
74.63
98.53
79.25
0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
-6.14
0.00
0.00
-8.55
0.00
0.00
9.06
411.01
0.00


0.00
0.00
-2.18
0.00
0.00
28.47
0.00
90.94
0.00
86.79


0.00
0.00
0.00
-4.89
0.00
0.00
-18.33
0.00
-311.01
13.21


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(1)

100.00
0.00
0.00
0.00
0.00

99.41
106.14
102.18
104.89
0.00
0.00
0.00
0.00
0.00
0.00

104.66
112.34
98.35
108.21
105.89
111.93
0.00
0.00
0.00
0.00


-6.13
0.00
0.00
-6.25
-3.85
0.00
-4.99
67.58
0.00
-71.15


0.00
-5.84
0.00
-1.97
0.00
-5.82
30.36
0.00
28.61
102.03


0.00
0.00
-5.73
0.00
-2.04
-6.12
0.00
-66.11
-7.86
69.12


INDEPENDENT VARIABLES
26 112.71 -2.81 -6.36 -3.54 0.00
27 92.83 12.01 5.46 0.00 -10.30
28 108.11 2.43 0.00 -4.44 -6.11
29 113.92 0.00 5.01 -8.53 -10.40
30 0.00 89.71 51.29 9.96 -50.95
INDEPENDENT VARIABLES
31 102.65 8.03 8.44 -5.84 -13.28


0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


83.84
28.89
0.19
6.20
0.00

83.84
84.76
83.88
84.73
29.62
36.60
33.70
6.35
1.21
6.78

84.77
83.94
84.93
84.80
84.79
84.98
38.09
52.61
41.39
8.38

84.80
84.97
84.98
85.02
53.47


0.9156
0.5375
0.0434
0.2489
0.0016

0.9156
0.9207
0.9159
0.9205
0.5443
0.6050
0.5805
0.2520
0.1101
0.2604

0.9207
0.9162
0.9216
0.9208
0.9208
0.9218
0.6172
0.7253
0.6434
0.2895

0.9209
0.9218
0.9219
0.9220
0.7312


85.07 0.9223


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.9156 0.5375 -0.0434 -0.2489 0.0016


83.8381


1.0000 0.5835 0.0574 -0.2492 0.1041


28.8871 34.0415 1.0000 0.0784 -0.7840 0.3804

0.1881 0.3295 0.6153 1.0000 0.0167 0.9133


6.1974 6.2107 61.4635 0.0281


1.0000 -0.2992


5) 0.0003 1.0838 14.4731 83.4090 8.9512 1.0000

Numbers in Parentheses are the Variable or Parameter Number


INDEPENDENT VARIABLES TAKEN THREE AT











BALSILLIE AND CARTER (1980) FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (Bl Xl) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 83 B4 B5 B6 B7 B8
NO
1 0.049 1.152 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.149 0.000 0.047 0.000 0.000 0.000 0.000 0.000 0.000
3 0.344 0.000 0.000 -0.086 0.000 0.000 0.000 0.000 0.000
4 0.405 0.000 0.000 0.000 -32.304 0.000 0.000 0.000 0.000
5 0.336 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
6 0.048 1.148 0.000 0.000 0.000 0.000 0.000 0.000 0.000
7 0.064 1.159 0.000 -0.190 0.000 0.000 0.000 0.000 0.000
8 0.057 1.145 0.000 0.000 -2.872 0.000 0.000 0.000 0.000
9 0.060 1.164 0.000 0.000 0.000 -0.006 0.000 0.000 0.000
10 0.162 0.000 0.047 -0.170 0.000 0.000 0.000 0.000 0.000
11 -0.096 0.000 0.078 0.000 58.060 0.000 0.000 0.000 0.000
12 0.151 0.000 0.055 0.000 0.000 -0.015 0.000 0.000 0.000
13 0.411 0.000 0.000 -0.077 -32.219 0.000 0.000 0.000 0.000
14 0.348 0.000 0.000 -0.533 0.000 0.016 0.000 0.000 0.000
15 0.423 0.000 0.000 0.000 -35.411 -0.005 0.000 0.000 0.000
16 0.062 1.150 0.001 -0.191 0.000 0.000 0.000 0.000 0.000
17 0.077 1.170 -0.004 0.000 -7.154 0.000 0.000 0.000 0.000
18 0.052 1.123 0.005 0.000 0.000 -0.007 0.000 0.000 0.000
19 0.071 1.153 0.000 -0.189 -2.461 0.000 0.000 0.000 0.000
20 0.063 1.161 0.000 -0.120 0.000 -0.002 0.000 0.000 0.000
21 0.080 1.149 0.000 0.000 -7.018 -0.007 0.000 0.000 0.000
22 -0.092 0.000 0.080 -0.243 61.308 0.000 0.000 0.000 0.000
23 -0.055 0.000 0.097 3.008 0.000 -0.114 0.000 0.000 0.000
24 -0.093 0.000 0.085 0.000 57.986 -0.015 0.000 0.000 0.000
25 0.450 0.000 0.000 0.919 -54.676 -0.034 0.000 0.000 0.000
26 0.079 1.163 -0.002 -0.186 -4.299 0.000 0.000 0.000 0.000
27 0.041 1.098 0.009 0.183 0.000 -0.013 0.000 0.000 0.000
28 0.073 1.140 0.002 0.000 -5.499 -0.007 0.000 0.000 0.000
29 0.085 1.146 0.000 0.143 -10.089 -0.011 0.000 0.000 0.000
30 -0.124 0.000 0.104 2.688 21.640 -0.103 0.000 0.000 0.000
31 0.066 1.112 0.005 0.259 -7.431 -0.016 0.000 0.000 0.000











WEISHAR (1976) ORIGINAL FIELD DATA
n = 111


Water
Depth
(m)

4.9000
2.3000
3.8000
2.6000
3.3000
4.3000
1.4000
2.6000
4.1000
2.3000
2.8000
4.4000
4.4000
2.7000
4.4000
4.4000
4.2000
4.6000
4.3000
3.3000
2.6000
4.1000
4.8000
3.8000
3.3000
4.2000
4.4000
4.4000
2.4000
4.3000
3.1000
2.8000
3.2000
2.3000
1.3000
2.5000
2.1000
1.6000
3.9000
2.5000
4.1000
2.8000
2.3000
2.3000
3.3000
2.3000
3.3000
4.1000
2.3000
2.7000
1.3000
4.3000
4.8000
3.2000
4.4000
2.4000
2.6000
4.4000
4.3000
2.8000
2.8000
2.5000
2.3000
4.3000
4.4000
4.2000


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


Breaker Wave
Height Period
(m) (s)

2.6000 5.0300
2.0000 11.0300
2.6000 9.1800
2.5000 6.6200
2.9000 11.1700
4.2000 7.3400
1.6000 8.1400
2.7000 5.7500
5.1000 8.3400
1.4000 8.3700
2.7000 5.9100
4.0000 3.7500
1.6000 8.5600
2.5000 5.9000
4.0000 3.9100
3.4000 7.4300
4.1000 5.7400
2.4000 6.7800
2.0000 7.6600
2.4000 11.9700
3.1000 7.8300
4.0000 3.6700
2.0000 4.7800
2.8000 9.1800
3.1000 7.8200
3.2000 5.4300
3.3000 6.0600
4.2000 9.0200
1.5000 1.2000
3.4000 5.1800
2.7000 9.0200
3.7000 7.9800
3.9000 6.3100
1.0000 7.0200
0.4000 3.2700
3.2000 2.1500
1.0000 9.3800
1.0000 4.9500
3.9000 3.4300
0.9000 12.9500
3.5000 6.3000
2.0000 8.3800
2.1000 6.1400
2.0000 5.5100
2.0000 10.3000
1.3000 8.6100
2.0000 3.6000
3.2000 4.4000
1.1000 8.0600
2.4000 6.2200
0.7000 8.2200
2.5000 2.7200
3.0000 9.6500
1.1000 8.3800
2.2000 5.9100
1.2000 7.3800
1.5000 8.8600
1.0000 3.2700
3.2000 9.5000
1.0000 13.6500
2.1000 10.9300
2.0000 4.9500
0.9000 6.5400
3.8000 5.9800
2.7000 8.3000
3.3000 9.3400


0.0000
0.0700
0.0500
0.0500
0.0600
0.0000
0.0800
0.0500
0.0100
0.0300
0.0600
0.0100
0.0400
0.0500
0.0100
0.0100
0.0000
0.0500
0.0500
0.0800
0.0500
0.0100
0.0300
0.0500
0.0800
0.0200
0.0400
0.0300
0.0400
0.0000
0.0500
0.0600
0.0500
0.0700
0.0800
0.0500
0.0700
0.0700
0.0700
0.0500
0.0100
0.0330
0.0300
0.0300
0.0800
0.0800
0.0800
0.0000
0.0200
0.0500
0.0800
0.0000
0.0500
0.0500
0.0300
0.0300
0.0500
0.0100
0.0000
0.0600
0.0600
0.0500
0.0300
0.0000
0.0200
0.0200


0.01049
0.00168
0.00315
0.00582
0.00237
0.00795
0.00246
0.00833
0.00748
0.00204
0.00789
0.02902
0.00223
0.00733
0.02670
0.00628
0.01270
0.00533
0.00348
0.00171
0.00516
0.03030
0.00893
0.00339
0.00517
0.01107
0.00917
0.00527
0.10629
0.01293
0.00339
0.00593
0.00999
0.00207
0.00382
0.07064
0.00116
0.00416
0.03383
0.00055
0.00900
0.00291
0.00568
0.00672
0.00192
0.00179
0.01575
0.01687
0.00173
0.00633
0.00106
0.03448
0.00329
0.00160
0.00643
0.00225
0.00195
0.00954
0.00362
0.00055
0.00179
0.00833
0.00215
0.01084
0.00400
0.00386


0.00000
1.70912
0.89113
0.65535
1.23202
0.00000
1.61164
0.54773
0.11561
0.66435
0.67557
0.05870
0.84740
0.58407
0.06120
0.12614
0.00000
0.68503
0.84781
1.93505
0.69609
0.05744
0.31743
0.85871
1.11232
0.19005
0.41772
0.41335
0.12269
0.00000
0.85923
0.77923
0.50013
1.53833
1.29485
0.18813
2.05548
1.08472
0.38060
2.13664
0.10542
0.61215
0.39792
0.36591
1.82400
1.89119
0.63752
0.00000
0.48115
0.62845
2.46051
0.00000
0.87207
1.25064
0.37421
0.63270
1.13233
0.10237
0.00000
2.56388
1.41669
0.54786
0.64743
0.00000
0.31626
0.32191











WEISHAR (1976) ORIGINAL FIELD DATA (CONT)


Water
Depth
(m)

4.4000
6.1000
3.3000
3.2000
3.3000
4.4000
0.8000
2.8000
4.5000
2.3000
2.4000
2.3000
1.0000
2.5000
3.9000
4.6000
3.3000
3.2000
2.4000
3.2000
2.5000
2.7000
2.8000
4.4000
4.3000
2.9000
2.5000
2.8000
2.3000
2.6000
4.0000
4.4000
3.8000
3.8000
3.8000
4.2000
2.1000
4.1000
4.7000
2.3000
2.4000
2.8000
1.4000
3.1000
2.7000


Breaker Wave
Height Period
(m) (s)

3.6000 4.3900
2.0000 6.3000
1.1000 7.1800
3.1000 7.1900
2.3000 5.1000
3.0000 7.1100
0.8000 9.8900
2.5000 6.1500
2.5000 7.1800
0.6000 7.9800
2.0000 5.3400
2.8000 7.5000
2.5000 11.0900
2.9000 7.9000
3.6000 12.5300
2.4000 7.1000
1.4000 12.2900
1.8000 11.9700
1.9000 15.0000
2.8000 11.0200
1.8000 9.7300
1.0000 7.5000
2.2000 7.6600
3.9000 5.8300
3.0000 5.5100
2.8000 11.4900
2.4000 6.7000
2.6000 11.9100
1.2000 7.6600
2.6000 3.6700
3.5000 6.6300
3.9000 7.2600
3.1000 8.8600
2.9000 7.0200
3.4000 7.0200
2.9000 7.0200
2.0000 13.0000
3.5000 4.9500
3.1000 8.1400
2.0000 4.7900
1.4000 6.5400
1.6000 2.4000
1.3000 7.3400
3.2000 8.3800
2.5000 7.7400


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm

0.0100 0.01906 0.07243
0.0200 0.00514 0.27891
0.0800 0.00218 1.71447
0.0500 0.00612 0.63919
0.0600 0.00902 0.63164
0.0100 0.00606 0.12851
0.1100 0.00083 3.80765
0.0600 0.00674 0.73058
0.0100 0.00495 0.14216
0.0700 0.00096 2.25756
0.0300 0.00716 0.35462
0.0700 0.00508 0.98219
0.0900 0.00207 1.97614
0.0400 0.00474 0.58090
0.0700 0.00234 1.44714
0.0500 0.00486 0.71736
0.0800 0.00095 2.60130
0.0500 0.00128 1.39650
0.0300 0.00086 1.02199
0.0500 0.00235 1.03083
0.0400 0.00194 0.90813
0.0300 0.00181 0.70436
0.0600 0.00383 0.97002
0.0200 0.01171 0.18483
0.0500 0.01008 0.49794
0.0600 0.00216 1.28975
0.0400 0.00546 0.54155
0.0300 0.00187 0.69368
0.0700 0.00209 1.53232
0.0500 0.01970 0.35626
0.0600 0.00812 0.66565
0.0100 0.00755 0.11508
0.0500 0.00403 0.78765
0.0500 0.00600 0.64524
0.0600 0.00704 0.71509
0.0100 0.00600 0.12905
0.0700 0.00121 2.01437
0.0100 0.01458 0.08283
0.0500 0.00477 0.72365
0.0700 0.00889 0.74222
0.0300 0.00334 0.51910
0.0600 0.02834 0.35638
0.0800 0.00246 1.61223
0.0500 0.00465 0.73325
0.0500 0.00426 0.76622











WEISHAR (1976) ORIGINAL FIELD DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(2)

0.00
100.00
0.00
0.00
0.00

-24.11
0.00
0.00
0.00
-0.64
97.42
1128.88
0.00
0.00
0.00

10.47
-54.50
72.32
0.00
0.00
0.00
4.33
-107.16
-464.02
0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
-373.98 0.00 0.00
0.00 -1.02 0.00
0.00 0.00 -83.17
100.64 0.00 0.00
0.00 2.58 0.00
0.00 0.00 1228.88
98.66 1.34 0.00
70.13 0.00 29.87
0.00 11.38 88.62


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(1)

100.00
0.00
0.00
0.00
0.00

124.11
473.98
101.02
183.17
0.00
0.00
0.00-
0.00
0.00
0.00

433.17
163.46
112.13
587.31
504.32
270.42
0.00
0.00
0.00
0.00


-343.64
0.00
0.00
-460.81
-389.98
0.00
93.84
102.08
0.00
59.54


0.00
-8.96
0.00
26.50
0.00
26.31
1.83
0.00
15.00
5.86


0.00
0.00
-84.45
0.00
-14.34
-144.10
0.00
105.08
549.03
34.60


INDEPENDENT VARIABLES
26 3117.43 -460.41-2358.53 -198.50 0.00
27 377.67 42.75 -287.18 0.00 -33.24
28 156.70 63.94 0.00 -10.96 -109.67
29 892.93 0.00 -648.92 -52.90 -91.12
30 0.00 -81.35 87.31 3.92 90.13
INDEPENDENT VARIABLES
31 1883.11 -162.99-1378.89 -121.95 -119.29


0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


35.28
3.66
38.43
1.01
35.10

36.26
50.47
35.34
44.20
38.43
3.69
40.24
38.48
40.90
37.08

50.48
37.30
45.99
50.96
50.49
45.94
38.50
42.88
40.34
41.75

51.13
50.55
46.51
51.14
42.97


0.5940
0.1912
0.6199
0.1003
0.5924

0.6021
0.7104
0.5945
0.6648
0.6199
0.1922
0.6344
0.6204
0.6395
0.6090

0.7105
0.6107
0.6781
0.7139
0.7105
0.6778
0.6205
0.6548
0.6351
0.6462

0.7150
0.7110
0.6820
0.7151
0.6555


51.17 0.7153


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.5940 -0.1912 -0.6199 0.1003 -0.5924

(2) 35.2841 1.0000 -0.1578 -0.4617 0.2076 -0.5923

(3) 3.6553 2.4905 1.0000 0.3120 -0.6057 0.6224

(4) 38.4305 21.3193 9.7328 1.0000 -0.1986 0.8055

(5) 1.0070 4.3098 36.6888 3.9424 1.0000 -0.3886

(6) 35.0969 35.0774 38.7351 64.8794 15.0995 1.0000

Numbers in Parentheses are the Variable or Parameter Number


INDEPENDENT VARIABLES TAKEN THREE AT











WEISHAR (1976) ORIGINAL FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 1.767 0.611 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 3.808 0.000 -0.073 0.000 0.000 0.000 0.000 0.000 0.000
3 4.365 0.000 0.000 -24.983 0.000 0.000 0.000 0.000 0.000
4 3.202 0.000 0.000 0.000 7.703 0.000 0.000 0.000 0.000
5 3.952 0.000 0.000 0.000 0.000 -0.859 0.000 0.000 0.000
6 2.090 0.595 -0.038 0.000 0.000 0.000 0.000 0.000 0.000
7 3.058 0.403 0.000 -17.704 0.000 0.000 0.000 0.000 0.000
8 1.769 0.616 0.000 0.000 -1.842 0.000 0.000 0.000 0.000
9 2.750 0.385 0.000 0.000 0.000 -0.537 0.000 0.000 0.000
10 4.359 0.000 0.001 -25.014 0.000 0.000 0.000 0.000 0.000
11 3.865 0.000 -0.079 0.000 -1.874 0.000 0.000 0.000 0.000
12 3.339 0.000 0.111 0.000 0.000 -1.120 0.000 0.000 0.000
13 4.388 0.000 0.000 -25.172 -1.817 0.000 0.000 0.000 0.000
14 4.293 0.000 0.000 -16.379 0.000 -0.384 0.000 0.000 0.000
15 4.117 0.000 0.000 0.000 -11.740 -0.945 0.000 0.000 0.000
16 3.038 0.403 0.003 -17.807 0.000 0.000 0.000 0.000 0.000
17 2.350 0.610 -0.067 0.000 -9.928 0.000 0.000 0.000 0.000
18 2.555 0.325 0.069 0.000 0.000 -0.751 0.000 0.000 0.000
19 3.093 0.413 0.000 -18.084 -5.539 0.000 0.000 0.000 0.000
20 3.073 0.396 0.000 -17.055 0.000 -0.035 0.000 0.000 0.000
21 2.921 0.380 0.000 0.000 -10.987 -0.622 0.000 0.000 0.000
22 4.439 0.000 -0.007 -25.028 -2.596 0.000 0.000 0.000 0.000
23 3.787 0.000 0.076 -12.099 0.000 -0.686 0.000 0.000 0.000
24 3.429 0.000 0.102 0.000 -2.962 -1.122 0.000 0.000 0.000
25 4.376 0.000 0.000 -15.030 -7.881 -0.481 0.000 0.000 0.000
26 3.233 0.417 -0.020 -17.592 -7.886 0.000 0.000 0.000 0.000
27 3.017 0.383 0.014 -16.219 0.000 -0.103 0.000 0.000 0.000
28 2.731 0.342 0.046 0.000 -7.097 -0.735 0.000 0.000 0.000
29 3.158 0.392 0.000 -15.869 -6.890 -0.123 0.000 0.000 0.000
30 3.875 0.000 0.067 -12.089 -2.889 -0.688 0.000 0.000 0.000
31 3.213 0.402 -0.012 -16.397 -7.724 -0.078 0.000 0.000 0.000











MUNK (1949) BEACH EROSION BOARD LABORATORY DATA
n = 37


Water
Depth
(m)

0.0610
0.0790
0.0460
0.0470
0.0810
0.0520
0.0510
0.0630
0.0430
0.0550
0.0430
0.0450
0.0810
0.0670
0.1050
0.0880
0.0540
0.1390
0.0980
0.0550
0.0630
0.0640
0.1870
0.1130
0.0480
0.0580
0.0440
0.1020
0.0820
0.0820
0.0440
0.0530
0.1020
0.0580
0.1110
0.1260
0.1700


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


Breaker Wave
Height Period
(m) (s)

0.0430 1.0300
0.0540 1.0300
0.0330 0.8500
0.0340 1.0300
0.0510 1.0300
0.0380 0.8500
0.0330 0.7500
0.0410 0.8500
0.0310 0.7500
0.0650 1.0800
0.0510 1.0800
0.0430 0.9600
0.0660 1.0800
0.0620 0.9700
0.0840 1.0800
0.0690 0.9500
0.0440 0.7300
0.1000 1.0800
0.0790 0.9700
0.0440 0.7500
0.0470 0.7400
0.0490 0.7500
0.1300 1.0800
0.0900 0.9700
0.0350 0.9700
0.0520 1.0800
0.0430 1.0800
0.0640 1.0800
0.0510 1.0800
0.0550 0.9700
0.0340 0.7500
0.0390 0.7400
0.0800 0.9600
0.0490 0.7400
0.0940 1.0900
0.0950 0.9700
0.1210 1.0900


0.0300
0.0300
0.0300
0.0300
0.0300
0.0300
0.0300
0.0300
0.0300
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.0490
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590
0.1590


0.00414
0.00519
0.00466
0.00327
0.00491
0.00537
0.00599
0.00579
0.00562
0.00569
0.00446
0.00476
0.00577
0.00672
0.00735
0.00780
0.00843
0.00875
0.00857
0.00798
0.00876
0.00889
0.01137
0.00976
0.00380
0.00455
0.00376
0.00560
0.00446
0.00596
0.00617
0.00727
0.00886
0.00913
0.00807
0.01030
0.01039


0.46648
0.41627
0.43944
0.52460
0.42834
0.40951
0.38774
0.39424
0.40005
0.64979
0.73358
0.71014
0.64485
0.59756
0.57160
0.55476
0.53383
0.52388
0.52938
0.54846
0.52359
0.51972
0.45947
0.49597
2.58076
2.35739
2.59239
2.12493
2.38039
2.05874
2.02457
1.86513
1.68942
1.66396
1.76959
1.56646
1.55971











MUNK (1949) BEACH EROSION BOARD LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
0.00
3.02
0.00
0.00
36.51
0.00
94.07
0.00
93.95


0.00
0.00
0.00
0.64
0.00
0.00
-4.22
0.00
-984.88
6.05


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(1)

100.00
0.00
0.00
0.00
0.00

117.29
99.21
96.98
99.36
0.00
0.00
0.00
0.00
0.00
0.00

116.53
-382.66
118.12
96.13
99.16
95.59
0.00
0.00
0.00
0.00

-339.53
138.77
-334.13
94.31
0.00

-335.01


(2)

0.00
100.00
0.00
0.00
0.00

-17.29
0.00
0.00
0.00
96.05
63.49
104.22
0.00
0.00
0.00

-17.64
322.71
-19.66
0.00
0.00
0.00
63.57
97.11
63.63
0.00

296.59
-39.63
294.85
0.00
63.93

300.86


0.00
0.00
1.54
0.00
-1.26
0.85
0.00
-69.29
-0.12
21.73


INDEPENDENT VARIABLES
-4.10 147.04 0.00
17.20 0.00 18.05
0.00 143.66 -4.38
-2.71 5.01 3.39
0.91 36.14 -0.99
INDEPENDENT VARIABLES
16.56 137.62 -20.03


0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


91.27
22.92
2.50
48.59
0.00

91.44
91.29
91.30
91.28
23.77
86.28
23.96
49.46
30.29
49.66

91.46
92.24
91.48
91.31
91.29
91.32
86.28
66.15
86.28
49.92

92.33
91.61
92.36
91.32
86.29

92.40


0.9554
0.4787
0.1580
0.6971
0.0039

0.9562
0.9554
0.9555
0.9554
0.4875
0.9289
0.4894
0.7033
0.5503
0.7047

0.9564
0.9604
0.9565
0.9556
0.9555
0.9556
0.9289
0.8133
0.9289
0.7065

0.9609
0.9571
0.9610
0.9556
0.9289

0.9613


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.9554 0.4787 0.1580 0.6971 -0.0039


91.2725


1.0000 0.5371 0.1528 0.7185 -0.0151


22.9185 28.8462

2.4970 2.3336


1.0000 0.1394 -0.1796 0.2003

1.9432 1.0000 0.0935 0.9559


48.5907 51.6187 3.2250 0.8745 1.0000 -0.1520


0.0015 0.0229 4.0124 91.3657 2.3106


1.0000


Numbers in Parentheses are the Variable or Parameter Number


0.00
0.79
0.00
0.00
3.95
0.00
0.00
5.93
1084.88
0.00


INDEPENDENT VARIABLES TAKEN THREE AT


0.00
159.94
0.00
3.06
0.00
3.55
36.53
0.00
36.49
95.45


1.12
0.00
0.00
0.80
2.10
0.00
-0.10
72.18
0.00
17.18











MUNK (1949) BEACH EROSION BOARD LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + --- + (BB X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


NO
1 -0.003
2 -0.043
3 0.069
4 0.000
5 0.077
6 0.007
7 -0.003
8 -0.004
9 -0.003
10 -0.044
11 -0.169
12 -0.043
13 -0.004
14 0.068
15 -0.007
16 0.007
17 0.111
18 0.007
19 -0.004
20 -0.003
21 -0.005
22 -0.169
23 -0.083
24 -0.169
25 -0.014
26 0.116
27 0.014
28 0.117
29 -0.006
30 -0.168
31 0.117


0.000 0.000 0.000
0.000 0.000 0.000
0.098 0.000 0.000
0.000 11.550 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.008 0.000 0.000
0.000 0.366 0.000
0.000 0.000 0.000
0.058 0.000 0.000
0.000 13.407 0.000
0.000 0.000 -0.005
0.058 11.405 0.000
1.163 0.000 -0.084
0.000 11.813 0.005
0.009 0.000 0.000
0.000 -8.008 0.000
0.000 0.000 0.001
0.008 0.373 0.000
0.020 0.000 -0.001
0.000 0.435 0.001
-0.003 13.417 0.000
1.395 0.000 -0.106
0.000 13.394 0.000
-0.190 13.052 0.019
0.019 -8.441 0.000
-0.131 0.000 0.011
0.000 -8.425 0.002
-0.027 0.621 0.003
0.027 13.220 -0.002
-0.079 -8.129 0.008


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


1.350 0.000
0.000 0.127
0.000 0.000
0.000 0.000
0.000 0.000
1.387 -0.013
1.348 0.000
1.328 0.000
1.351 0.000
0.000 0.123
0.000 0.165
0.000 0.132
0.000 0.000
0.000 0.000
0.000 0.000
1.385 -0.013
2.169 -0.115
1.392 -0.015
1.325 0.000
1.343 0.000
1.324 0.000
0.000 0.165
0.000 0.165
0.000 0.166
0.000 0.000
2.207 -0.121
1.476 -0.026
2.219 -0.123
1.323 0.000
0.000 0.166
2.241 -0.126











MUNK (1949) BERKELEY LABORATORY DATA


Wave
Period
(s)

0.8700
1.1500
1.2200
1.5000
1.5400
1.9700
0.8600
0.9650
1.3400
1.5000
1.9700
1.0500
1.0900
1.3500
1.5000
1.9800


n = 16

Bed Equivalent
Slope Wave
Steepness


Water
Depth
(m)

0.1256
0.1070
0.0951
0.0823
0.0832
0.0713
0.1384
0.1106
0.0747
0.0747
0.0631
0.1433
0.1430
0.1451
0.1445
0.1180


Breaker
Height
(m)

0.0988
0.0985
0.0994
0.0945
0.0872
0.0823
0.0917
0.0911
0.0826
0.0792
0.0683
0.0997
0.0975
0.0985
0.0939
0.0872


Surf
Sim
Parm

0.62313
0.82496
0.87114
1.09836
1.17401
1.70046
0.48086
0.54137
0.78963
0.90242
1.27687
0.93705
0.98333
1.21222
1.37932
1.88943


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
18.93
0.00
0.00
348.80
0.00
0.00
73.93
164.41
0.00


0.00
0.00
6.23
0.00
0.00
125.73
0.00
26.07
0.00
49.36


0.00
0.00
0.00
2.65
0.00
0.00
140.33
0.00
-64.41
50.64


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


INDEPENDENT VARIABLES TAKEN THREE


-111.86
0.00
0.00
112.21
448.73
0.00
95.89
19.81
0.00
76.11


0.00
11.40
0.00
40.19
0.00
20.18
24.54
0.00
-12.61
25.54


0.00
0.00
-61.28
0.00
-182.25
23.52
0.00
-130.60
-190.90
-1.65


INDEPENDENT VARIABLES
-258.72 -36.21 0.00
-11.44 0.00 -53.11
0.00 1.73 -56.85
222.69 53.77 -33.48
13.23 -7.75 -164.93
INDEPENDENT VARIABLES
-10.06 0.61 -52.52


TAKEN FOUI
0.00
0.00
0.00
0.00
0.00
TAKEN FIVI
0.00


(7) (8)
AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
EE AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
R AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
E AT A TIME
0.00 0.00


52.69
31.16
34.75
34.78
2.60

53.45
58.07
55.86
53.22
75.32
34.81
81.28
77.77
69.27
60.21

79.53
58.39
85.27
78.50
69.97
68.93
78.22
81.47
81.55
77.78

81.35
85.56
85.37
79.14
81.58


0.7259
0.5582
0.5895
0.5897
0.1613

0.7311
0.7621
0.7474
0.7295
0.8679
0.5900
0.9015
0.8819
0.8323
0.7760

0.8918
0.7641
0.9234
0.8860
0.8365
0.8302
0.8844
0.9026
0.9031
0.8819

0.9020
0.9250
0.9240
0.8896
0.9032


85.57 0.9250


0.0719
0.0719
0.0719
0.0719
0.0719
0.0791
0.0541
0.0541
0.0541
0.0541
0.0541
0.0900
0.0900
0.0900
0.0900
0.0900


0.01331
0.00760
0.00681
0.00429
0.00375
0.00216
0.01266
0.00999
0.00469
0.00359
0.00180
0.00922
0.00838
0.00551
0.00426
0.00227


(1)

100.00
0.00
0.00
0.00
0.00

107.19
81.07
93.77
97.35
0.00
0.00
0.00
0.00
0.00
0.00

125.35
70.15
52.47
-52.40
-166.48
56.30
0.00
0.00
0.00
0.00

266.89
61.62
50.81
-142.98
0.00

59.93


(2)

0.00
100.00
0.00
0.00
0.00

-7.19
0.00
0.00
0.00
-248.80
-25.73
240.33
0.00
0.00
0.00

86.50
18.45
108.80
0.00
0.00
0.00
-20.43
210.79
303.50
0.00

128.05
102.93
104.31
0.00
259.44

102.04











MUNK (1949) BERKELEY LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.7259 -0.5582 0.5895 0.5897 -0.1613

(2) 52.6872 1.0000 -0.6807 0.5439 0.6200 -0.3177

(3) 31.1605 46.3335 1.0000 0.1250 -0.9355 0.8833


(4) 34.7533 29.5804


1.5626 1.0000 -0.1060 0.5553


(5) 34.7763 38.4397 87.5134 1.1226 1.0000 -0.7937

(6) 2.6004 10.0908 78.0288 30.8311 62.9970 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 B2 B3 B4 85 B6 B7 B8


2.429 0.000
0.000 -0.045
0.000 0.000
0.000 0.000
0.000 0.000
2.157 -0.010
1.926 0.000
1.958 0.000
2.511 0.000
0.000 -0.052
0.000 -0.004
0.000 -0.152
0.000 0.000
0.000 0.000
0.000 0.000
-1.871 -0.086
2.223 0.039
-1.618 -0.223
-0.582 0.000
-0.729 0.000
1.353 0.000
0.000 -0.015
0.000 -0.172
0.000 -0.139
0.000 0.000
-1.642 -0.052
-1.844 -0.204
-1.737 -0.237
-1.018 0.000
0.000 -0.151
-1.863 -0.211


0.000 0.000 0.000
0.000 0.000 0.000
1.199 0.000 0.000
0.000 4.888 0.000
0.000 0.000 -0.012
0.000 0.000 0.000
0.563 0.000 0.000
0.000 1.881 0.000
0.000 0.000 0.006
1.362 0.000 0.000
0.000 4.482 0.000
0.000 0.000 0.116
1.341 5.467 0.000
1.997 0.000 -0.054
0.000 10.343 0.064
2.089 0.000 0.000
0.000 5.222 0.000
0.000 0.000 0.164
1.560 6.456 0.000
2.459 0.000 -0.069
0.000 7.012 0.049
1.352 3.997 0.000
-0.305 0.000 0.140
0.000 1.258 0.114
1.367 5.303 -0.002
1.991 3.221 0.000
0.428 0.000 0.138
0.000 -0.855 0.169
1.984 5.537 -0.021
-0.146 0.985 0.126
0.391 -0.276 0.142


1 -0.113
2 0.169
3 0.021
4 0.077
5 0.120
6 -0.075
7 -0.108
8 -0.082
9 -0.126
10 0.079
11 0.085
12 0.194
13 -0.024
14 0.019
15 -0.024
16 0.243
17 -0.180
18 0.387
19 0.007
20 0.068
21 -0.110
22 0.005
23 0.219
24 0.170
25 -0.023
26 0.163
27 0.379
28 0.417
29 0.043
30 0.187
31 0.389


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











PUTNAM AND OTHERS (1949) LABORATORY DATA
n = 37


Water
Depth
(m)

0.2290
0.1340
0.1710
0.1250
0.1190
0.1220
0.1130
0.0730
0.0700
0.0730
0.1460
0.1580
0.0850
0.0820
0.0980
0.0820
0.0700
0.0670
0.1310
0.1010
0.0880
0.0730
0.0760
0.1890
0.1310
0.1250
0.0790
0.0700
0.1100
0.0820
0.0610
0.0580
0.1430
0.1160
0.0820
0.0790
0.0910


Breaker Wave
Height Period
(m) (s)

0.1430 1.0000
0.0980 1.0600
0.1220 1.1400
0.0940 1.1500
0.0910 1.2500
0.0980 1.3200
0.0880 1.4000
0.0490 1.9000
0.0460 2.1300
0.0460 2.2200
0.0850 0.7200
0.1070 0.9200
0.0670 1.1400
0.0670 1.2200
0.0730 0.9900
0.0670 1.3200
0.0490 1.6300
0.0490 1.9800
0.0850 0.8300
0.0700 0.9100
0.0670 1.0000
0.0610 1.1200
0.0610 1.3500
0.1040 0.8000
0.0880 0.9000
0.0850 0.9800
0.0610 1.2300
0.0670 1.2700
0.0790 0.9500
0.0640 1.3300
0.0490 1.6700
0.0370 1.9900
0.1010 1.0800
0.0880 1.3600
0.0610 1.5800
0.0610 1.9100
0.0670 2.3200


Bed Equivalent Surf
SLope Wave Sim
Steepness Parm

0.0660 0.01459 0.54637
0.0660 0.00890 0.69960
0.0660 0.00958 0.67434
0.0660 0.00725 0.77498
0.0660 0.00594 0.85614
0.0660 0.00574 0.87120
0.0660 0.00458 0.97509
0.1440 0.00139 3.86929
0.1440 0.00103 4.47689
0.1440 0.00095 4.66605
0.2410 0.01673 1.86317
0.2410 0.01290 2.12191
0.2410 0.00526 3.32275
0.2410 0.00459 3.55592
0.1000 0.00760 1.14706
0.1000 0.00392 1.59643
0.1000 0.00188 2.30517
0.1000 0.00128 2.80014
0.1390 0.01259 1.23879
0.1390 0.00863 1.49665
0.1390 0.00684 1.68109
0.1390 0.00496 1.97324
0.1390 0.00342 2.37846
0.2600 0.01658 2.01911
0.2600 0.01109 2.46938
0.2600 0.00903 2.73592
0.2600 0.00411 4.05347
0.2600 0.00424 3.99349
0.0980 0.00893 1.03693
0.0980 0.00369 1.61288
0.0980 0.00179 2.31450
0.0980 0.00095 3.17389
0.1430 0.00884 1.52129
0.1430 0.00485 2.05233
0.1430 0.00249 2.86379
0.1430 0.00171 3.46192
0.1430 0.00127 4.01236











PUTNAM AND OTHERS (1949) LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(2)

0.00
100.00
0.00
0.00
0.00

0.02
0.00
0.00
0.00
85.61
42.53
64.90
0.00
0.00
0.00

4.48
26.42
-3.52
0.00
0.00
0.00
48.01
153.90
62.28
0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
5.37
0.00
0.00
14.39
0.00
0.00
-64.36
168.05
0.00


0.00
0.00
16.12
0.00
0.00
57.47
0.00
164.36
0.00
138.28


0.00
0.00
0.00
5.21
0.00
0.00
35.10
0.00
268.05
-38.28


(1)

100.00
0.00
0.00
0.00
0.00

99.98
94.63
83.88
94.79
0.00
0.00
0.00
0.00
0.00
0.00

89.65
52.81
97.28
84.34
93.86
79.17
0.00
0.00
0.00
0.00


INDEPENDENT VARIABLES
26 52.64 26.71 -0.85 21.50 0.00
27 82.24 13.75 12.07 0.00 -8.06
28 52.43 27.07 0.00 21.04 -0.54
29 70.92 0.00 -21.41 26.18 24.31
30 0.00 66.94 7.95 51.94 -26.82
INDEPENDENT VARIABLES
31 54.86 22.96 -4.22 22.37 4.04


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


TAKEN THREE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


T





T


(8)
*IME
0.00
0.00
0.00
0.00
0.00
*IME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


0.9512
0.6095
0.0116
0.8674
0.5936

0.9512
0.9548
0.9713
0.9539
0.6283
0.8943
0.6648
0.8970
0.7125
0.8774

0.9555
0.9910
0.9542
0.9716
0.9550
0.9748
0.9196
0.7386
0.9238
0.9061

0.9911
0.9561
0.9910
0.9881
0.9243


98.26 0.9912


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.9512 -0.6095 0.0116 0.8674 -0.5936

90.4806 1.0000 -0.6409 -0.0744 0.7835 -0.6788

37.1460 41.0738 1.0000 -0.2611 -0.8393 0.6388

0.0135 0.5531 6.8166 1.0000 0.2675 0.5394

75.2337 61.3902 70.4429 7.1574 1.0000 -0.5581


35.2366 46.0736 40.8038 29.0972 31.1425


1.0000


Numbers in Parentheses are the Variable or Parameter Nunber


INDEPENDENT VARIABLES
5.87 0.00 0.00
0.00 20.77 0.00
0.00 0.00 6.24
-2.33 18.00 0.00
4.00 0.00 2.14
0.00 14.17 6.66
-19.79 71.77 0.00
160.70 0.00 -214.60
0.00 57.63 -19.91
-82.45 134.70 47.74


90.48
37.15
0.01
75.23
35.24

90.48
91.16
94.34
90.98
39.48
79.99
44.20
80.47
50.76
76.98

91.30
98.21
91.06
94.40
91.21
95.02
84.57
54.55
85.35
82.11

98.23
91.41
98.21
97.63
85.44











PUTNAM AND OTHERS (1949) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (81 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 B2 B3 B4 B5 B6 B7 B8


1.595 0.000
0.000 -0.055
0.000 0.000
0.000 0.000
0.000 0.000
1.595 0.000
1.605 0.000
1.180 0.000
1.705 0.000
0.000 -0.059
0.000 0.036
0.000 -0.035
0.000 0.000
0.000 0.000
0.000 0,000
1.663 0.005
1.153 0.033
1.680 -0.003
1.144 0.000
1.647 0.000
1.297 0.000
0.000 0.034
0.000 0.065
0.000 0.052
0.000 0.000
1.133 0.033
1.618 0.015
1.140 0.034
1.219 0.000
0.000 0.061
1.150 0.027


0.000
0.000
0.007
0.000
0.000
0.000
0.048
0.000
0.000
-0.091
0.000
0.000
-0.136
0.269
0.000
0.057
0.000
0.000
-0.016
0.037
0.000
-0.128
0.618
0.000
-0.266
-0.010
0.124
0.000
-0.192
0.066
-0.046


0.000 0.000
0.000 0.000
0.000 0.000
7.555 0.000
0.000 -0.019
0.000 0.000
0.000 0.000
2.754 0.000
0.000 0.003
0.000 0.000
10.486 0.000
0.000 -0.011
8.108 0.000
0.000 -0.028
6.781 -0.005
0.000 0.000
5.508 0.000
0.000 0.004
2.965 0.000
0.000 0.001
2.820 0.004
10.803 0.000
0.000 -0.053
10.291 -0.010
10.109 0.010
5.619 0.000
0.000 -0.005
5.557 0.000
5.464 0.014
10.038 -0.014
5.693 0.003


1 -0.015
2 0.179
3 0.104
4 0.058
5 0.149
6 -0.015
7 -0.023
8 -0.001
9 -0.030
10 0.197
11 -0.008
12 0.178
13 0.075
14 0.129
15 0.075
16 -0.035
17 -0.059
18 -0.025
19 0.003
20 -0.027
21 -0.018
22 0.012
23 0.050
24 -0.006
25 0.059
26 -0.057
27 -0.043
28 -0.059
29 -0.024
30 -0.015
31 -0.053


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











IVERSEN (1952) LABORATORY DATA
n= 63


Water Breaker Wave


Depth
(m)

0.1250
0.1250
0.0910
0.0980
0.1370
0.1010
0.0910
0.0790
0.0610
0.0670
0.0550
0.0490
0.0550
0.0430
0.0850
0.0730
0.1560
0.1230
0.1070
0.0980
0.0930
0.1020
0.1290
0.0990
0.0690
0.1000
0.0700
0.0680
0.0650
0.1460
0.1550
0.1110
0.1070
0.1140
0.1020
0.1130
0.0820
0.0790
0.0790
0.0840
0.0850
0.0810
0.0700
0.0740
0.1620
0.1400
0.1460
0.1340
0.1190
0.1650
0.1190
0.1040
0.1040
0.1010
0.0880
0.0760
0.0880
0.0820
0.0760
0.0700
0.0640
0.0490
0.0550


Height
(m)

0.1220
0.1220
0.1130
0.1100
0.1070
0.0790
0.0940
0.0890
0.0640
0.0670
0.0670
0.0580
0.0610
0.0490
0.0700
0.0730
0.1210
0.0920
0.0910
0.0840
0.0820
0.0860
0.0980
0.0680
0.0580
0.0760
0.0610
0.0630
0.0550
0.1070
0.1260
0.0840
0.0870
0.0800
0.0770
0.0880
0.0780
0.0530
0.0550
0.0770
0.0700
0.0610
0.0580
0.0550
0.1280
0.1220
0.1220
0.1160
0.1100
0.1070
0.0940
0.1010
0.0910
0.0820
0.0830
0.0760
0.0580
0.0640
0.0550
0.0580
0.0610
0.0430
0.0460


Period
(s)

1.0000
1.0000
1.5100
1.7300
1.0000
0.9200
1.9800
1.9800
0.8000
1.1100
1.2700
1.2600
1.4500
1.2600
2.1000
2.5000
2.6500
1.0000
1.1300
1.1700
1.6200
1.7400
2.6500
0.9000
0.9500
1.3000
1.3500
2.0000
1.9000
1.0500
2.3700
1.2400
1.4600
1.8700
2.0300
2.6700
1.4900
1.6000
1.7900
2.1000
2.2900
2.5200
2.5200
2.6500
1.4000
1.5000
1.5900
1.8900
2.2400
1.0400
1.1500
1.2600
1.3300
1.4100
1.6700
1.9300
0.7400
0.9300
1.0300
1.1200
1.1700
1.3400
1.5500


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0200
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500


0.01245
0.01245
0.00506
0.00375
0.01092
0.00952
0.00245
0.00232
0.01020
0.00555
0.00424
0.00373
0.00296
0.00315
0.00162
0.00119
0.00176
0.00939
0.00727
0.00626
0.00319
0.00290
0.00142
0.00857
0.00656
0.00459
0.00342
0.00161
0.00155
0.00990
0.00229
0.00557
0.00416
0.00233
0.00191
0.00126
0.00359
0.00211
0.00175
0.00178
0.00136
0.00098
0.00093
0.00080
0.00666
0.00553
0.00492
0.00331
0.00224
0.01009
0.00725
0.00649
0.00525
0.00421
0.00304
0.00208
0.01081
0.00755
0.00529
0.00472
0.00455
0.00244
0.00195


0.89626
0.89626
1.40621
1.63291
0.95702
1.02468
2.02169
2.07770
0.98995
1.34245
1.53596
1.63783
1.83788
1.78191
2.48475
2.89662
0.47698
0.20642
0.23453
0.25275
0.35420
0.37149
0.53000
0.21609
0.24697
0.29524
0.34223
0.49889
0.50724
0.33161
0.68975
0.44199
0.51135
0.68300
0.75575
0.92982
0.55115
0.71797
0.78849
0.78181
0.89416
1.05405
1.08097
1.16732
0.61250
0.67219
0.71253
0.86859
1.05715
0.49765
0.58711
0.62057
0.69010
0.77072
0.90732
1.09581
0.48095
0.57541
0.68745
0.72793
0.74149
1.01147
1.13119











IVERSEN (1952) LABORATORY DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(2)

0.00
100.00
0.00
0.00
0.00

-0.43
0.00
0.00
0.00
-4.08
63.12
335.49
0.00
0.00
0.00

-8.67
20.15
9.72
0.00
0.00
0.00
69.25
125.04
74.63
0.00


(3) (4) (5) (6)
INDEPENDENT VARIABLES TAKEN ONE


0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
-18.95
0.00
0.00
104.08
0.00
0.00
-665.00
-81.25
0.00


0.00
0.00
3.60
0.00
0.00
36.88
0.00
765.00
0.00
-386.03


0.00
0.00
0.00
-17.11
0.00
0.00
435.49
0.00
181.25
486.03


(1)

100.00
0.00
0.00
0.00
0.00

100.43
118.95
96.40
117.11
0.00
0.00
0.00
0.00
0.00
0.00

130.48
67.60
107.91
112.68
120.48
117.86
0.00
0.00
0.00
0.00


INDEPENDENT VARIABLES
26 79.56 20.59 -16.34 16.19 0.00
27 129.41 -7.69 -20.73 0.00 -0.99
28 72.23 28.52 0.00 12.64 -13.39
29 103.62 0.00 -35.40 15.59 16.18
30 0.00 76.95 9.44 35.49 -21.88
INDEPENDENT VARIABLES
31 82.95 15.55 -24.96 18.02 8.45


0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(7) (8)
AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
AT A TIME


TAKEN THREE AT A


TAKEN FOUR AT A


0.00
0.00
0.00
0.00
0.00
TAKEN F
0.00


0.00
0.00
0.00
0.00
0.00
IVE AT A
0.00


0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


89.16 0.9442


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.8788 0.0498 -0.2092 0.3552 -0.3827


77.2275 1.0000 0.0618 0.0899 0.3261


-0.1311


0.2485 0.3822 1.0000 -0.2161 -0.8128 0.3056


4.3763 0.8078 4.6714


1.0000 0.2442 0.7810


12.6135 10.6330 66.0654 5.9657 1.0000 -0.3223


14.6431


1.7178 9.3391 60.9912 10.3846 1.0000


Numbers in Parentheses are the Variable or Parameter Number


INDEPENDENT VARIABLES
-21.81 0.00 0.00
0.00 12.25 0.00
0.00 0.00 -17.63
-21.13 8.45 0.00
-13.91 0.00 -6.57
0.00 -0.53 -17.33
-12.47 43.21 0.00
110.76 0.00 -135.80
0.00 39.00 -13.64
-586.29 652.29 34.00


77.23
0.25
4.38
12.61
14.64

77.23
85.60
77.75
84.51
4.38
46.38
17.71
21.93
16.70
20.64

86.10
79.36
85.24
87.63
86.00
84.51
54.63
36.26
57.30
21.93

88.89
86.10
87.05
88.61
57.82


0.8788
0.0498
0.2092
0.3552
0.3827

0.8788
0.9252
0.8818
0.9193
0.2093
0.6811
0.4209
0.4683
0.4087
0.4543

0.9279
0.8909
0.9232
0.9361
0.9273
0.9193
0.7391
0.6022
0.7570
0.4683

0.9428
0.9279
0.9330
0.9413
0.7604











IVERSEN (1952) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (83 X3) + ... + (B8 X8)

BO B1 B2 B3 B4 B5 B6 87 B8


1.138 0.000 0.000
0.000 0.003 0.000
0.000 0.000 -0.210
0.000 0.000 0.000
0.000 0.000 0.000
1.139 0.000 0.000
1.172 0.000 -0.291
1.106 0.000 0.000
1.092 0.000 0.000
0.000 0.000 -0.209
0.000 0.056 0.000
0.000 0.010 0.000
0.000 0.000 -0.315
0.000 0.000 0.230
0.000 0.000 0.000
1.180 -0.004 -0.307
0.978 0.015 0.000
1.080 0.005 0.000
1.111 0.000 -0.325
1.144 0.000 -0.206
1.096 0.000 0.000
0.000 0.055 -0.297
0.000 0.040 1.073
0.000 0.058 0.000
0.000 0.000 -0.335
0.997 0.013 -0.319
1.175 -0.004 -0.293
0.933 0.019 0.000
1.126 0.000 -0.600
0.000 0.061 0.223
1.035 0.010 -0.485


0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
3.410 0.000 0.000
0.000 -0.021 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.737 0.000 0.000
0.000 -0.015 0.000
0.000 0.000 0.000
11.194 0.000 0.000
0.000 -0.024 0.000
4.147 0.000 0.000
0.000 -0.030 0.000
2.484 -0.016 0.000
0.000 0.000 0.000
3.158 0.000 0.000
0.000 -0.016 0.000
1.485 0.000 0.000
0.000 -0.006 0.000
-0.088 -0.015 0.000
11.768 0.000 0.000
0.000 -0.078 0.000
10.462 -0.019 0.000
4.259 0.001 0.000
3.616 0.000 0.000
0.000 -0.001 0.000
2.910 -0.016 0.000
3.019 0.016 0.000
9.581 -0.031 0.000
4.005 0.010 0.000


1 0.003
2 0.091
3 0.107
4 0.080
5 0.114
6 0.003
7 0.015
8 0.002
9 0.020
10 0.106
11 -0.043
12 0.101
13 0.093
14 0.111
15 0.099
16 0.022
17 -0.022
18 0.014
19 0.015
20 0.018
21 0.020
22 -0.029
23 0.045
24 -0.027
25 0.093
26 -0.007
27 0.022
28 -0.010
29 0.007
30 -0.028
31 -0.006


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











MORISON AND CROOKE (1953) LABORATORY DATA
n=6


Wave
Period
(s)

2.5000
1.5100
1.0000
2.6200
1.4100
0.7800


Bed Equivalent
Slope Wave
Steepness


0.1000
0.1000
0.1000
0.0200
0.0200
0.0200


0.00119
0.00506
0.01092
0.00120
0.00431
0.00939


Surf
Sim
Parm

2.89662
1.40621
0.95702
0.57637
0.30459
0.20637


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(2)

0.00
100.00
0.00
0.00
0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


-17.56 0.00 0.00 0.00
0.00 -9.43 0.00 0.00
0.00 0.00 12.81 0.00
0.00 0.00 0.00 -10.30
420.81 -320.81 0.00 0.00
53.54 0.00 46.46 0.00
88.65 0.00 0.00 11.35
0.00 43.01 56.99 0.00
0.00 350.16 0.00 -250.16
0.00 0.00 100.07 -0.07
INDEPENDENT VARIABLES
-14.37 -9.64 0.00 0.00
25.16 0.00 22.47 0.00
-6.06 0.00 0.00 -9.30
0.00 -8.99 11.95 0.00
0.00 21.56 0.00 -24.29
0.00 0.00 10.09 -5.77
47.36 8.44 44.19 0.00
74.41 139.46 0.00 -113.88
58.48 0.00 45.96 -4.45
0.00******** 2772.66 8756.90


INDEPENDENT VARIABLES
26 55.08 29.65 -8.14 23.42 0.00
27 64.06 29.86 60.69 0.00 -54.61
28 50.90 32.07 0.00 22.53 -5.50
29 93.68 0.00 -57.47 25.41 38.38
30 0.00 68.69 78.01 18.28 -64.98
INDEPENDENT VARIABLES
31 78.00 11.41 -50.51 27.98 33.13


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


TAKEN THREE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
1IME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


53.96
6.82
10.05
17.03
3.29

57.99
56.97
63.37
61.35
17.70
27.18
7.07
24.69
52.07
17.03

60.22
68.29
61.73
66.51
63.16
65.49
30.05
63.12
28.75
56.94

75.99
64.48
73.92
82.16
64.32


0.7346
0.2611
0.3170
0.4127
0.1814

0.7615
0.7548
0.7961
0.7832
0.4207
0.5213
0.2660
0.4968
0.7216
0.4127

0.7760
0.8264
0.7857
0.8155
0.7947
0.8093
0.5482
0.7945
0.5362
0.7546

0.8717
0.8030
0.8598
0.9064
0.8020


100.00 1.0000


Water
Depth
(m)

0.0770
0.0920
0.1290
0.0910
0.1010
0.0700


Breaker
Height
(m)

0.0730
0.1130
0.1070
0.0810
0.0840
0.0560


(1)

100.00
0.00
0.00
0.00
0.00

117.56
109.43
87.19
110.30
0.00
0.00
0.00
0.00
0.00
0.00

124.01
52.36
115.36
97.04
102.73
95.69
0.00
0.00
0.00
0.00


T





T











MORISON AND CROOKE (1953) LABORATORY DATA (CONT)

CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.7346 -0.2611 0.3170

(2) 53.9640 1.0000 -0.0833 0.6173

(3) 6.8189 0.6939 1.0000 0.0478

(4) 10.0496 38.1064 0.2284 1.0000

(5) 17.0324 2.2186 85.6269 1.0253

(6) 3.2918 1.4442 28.1469 57.6086


0.4127 -0.1814

0.1489 0.1202

-0.9253 0.5305

0.1013 0.7590

1.0000 -0.4389

19.2608 1.0000


Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


0.715 0.000
0.000 -0.007
0.000 0.000
0.000 0.000
0.000 0.000
0.699 -0.005
0.848 0.000
0.670 0.000
0.747 0.000
0.000 -0.008
0.000 0.023
0.000 -0.006
0.000 0.000
0.000 0.000
0.000 0.000
0.815 -0.005
0.638 0.016
0.736 -0.002
0.805 0.000
0.562 0.000
0.701 0.000
0.000 0.018
0.000 0.014
0.000 0.026
0.000 0.000
0.850 0.024
0.317 0.008
0.679 0.022
3.591 0.000
0.000 0.021
4.435 0.034


0.000
0.000
0.150
0.000
0.000
0.000
-0.104
0.000
0.000
0.156
0.000
0.000
0.132
0.508
0.000
-0.091
0.000
0.000
-0.107
0.168
0.000
0.087
0.710
0.000
0.685
-0.179
0.429
0.000
-3.145
0.651
-4.101


0.000 0.000
0.000 0.000
0.000 0.000
2.100 0.000
0.000 -0.004
0.000 0.000
0.000 0.000
1.578 0.000
0.000 -0.006
0.000 0.000
6.057 0.000
0.000 -0.001
1.957 0.000
0.000 -0.021
2.099 0.000
0.000 0.000
4.389 0.000
0.000 -0.005
1.589 0.000
0.000 -0.011
1.184 -0.003
5.110 0.000
0.000 -0.033
6.333 -0.003
-1.866 -0.030
5.789 0.000
0.000 -0.022
4.814 -0.006
15.610 0.119
1.713 -0.031
25.495 0.153


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


NO
1 0.032
2 0.105
3 0.084
4 0.082
5 0.097
6 0.042
7 0.027
8 0.027
9 0.035
10 0.096
11 0.024
12 0.105
13 0.075
14 0.085
15 0.082
16 0.037
17 -0.011
18 0.039
19 0.022
20 0.046
21 0.031
22 0.032
23 0.063
24 0.020
25 0.094
26 -0.039
27 0.051
28 -0.021
29 -0.235
30 0.043
31 -0.394











GALVIN AND EAGLESON (1965) LABORATORY DATA
n= 24

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.0810 0.0910 1.1300 0.1000 0.00727 1.17266
0.0440 0.0440 1.2500 0.1000 0.00287 1.86551
0.0410 0.0570 1.3800 0.1000 0.00305 1.80948
0.0210 0.0300 1.2500 0.1000 0.00196 2.25924
0.0290 0.0340 1.2500 0.1000 0.00222 2.12219
0.0400 0.0550 1.2500 0.1000 0.00359 1.66856
0.0360 0.0540 1.2500 0.1000 0.00353 1.68394
0.0510 0.0510 1.0000 0.1000 0.00520 1.38621
0.0500 0.0570 1.1300 0.1000 0.00456 1.48168
0.0480 0.0570 1.2500 0.1000 0.00372 1.63903
0.0370 0.0590 1.3800 0.1000 0.00316 1.77855
0.0340 0.0480 1.5000 0.1000 0.00218 2.14330
0.0280 0.0380 1.2500 0.1000 0.00248 2.00739
0.0370 0.0450 1.2500 0.1000 0.00294 1.84466
0.0470 0.0540 1.0000 0.1000 0.00551 1.34715
0.0380 0.0620 1.1300 0.1000 0.00495 1.42068
0.0410 0.0530 1.2500 0.1000 0.00346 1.69975
0.0440 0.0510 1.3800 0.1000 0.00273 1.91297
0.0300 0.0310 1.5000 0.1000 0.00141 2.66700
0.0230 0.0480 1.2500 0.1000 0.00313 1.78609
0.0270 0.0380 1.2500 0.1000 0.00248 2.00739
0.0400 0.0450 1.2500 0.1000 0.00294 1.84466
0.0420 0.0510 1.2500 0.1000 0.00333 1.73276
0.0430 0.0720 1.2500 0.1000 0.00470 1.45833











GALVIN AND EAGLESON (1965) LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(1) (2) (3) (4) (5)
INDEPENDENT VARIABLES
100.00 0.00 0.00 0.00 0.00
0.00 100.00 0.00 0.00 0.00
0.00 0.00 100.00 0.00 0.00
0.00 0.00 0.00 100.00 0.00
0.00 0.00 0.00 0.00 100.00
INDEPENDENT VARIABLES
274.89 -174.89 0.00 0.00 0.00
89.42 0.00 10.58 0.00 0.00
55.98 0.00 0.00 44.02 0.00
88.53 0.00 0.00 0.00 11.47
0.00 121.88 -21.88 0.00 0.00
0.00 58.69 0.00 41.31 0.00
0.00 122.75 0.00 0.00 -22.75
0.00 0.00 14.90 85.10 0.00
0.00 0.00 -531.52 0.00 631.52
0.00 0.00 0.00 82.89 17.11
INDEPENDENT VARIABLES


420.65
-9.94
491.34
44.30
149.64
42.21
0.00
0.00
0.00
0.00

-5.18
12.77
-2.93
45.05
0.00


-375.00
66.60
-468.30
0.00
0.00
0.00
53.14
181.83
51.68
0.00

57.89
164.46
54.49
0.00
64.71


31 -55.96 135.99


54.35
0.00
0.00
10.05
321.36
0.00
3.74
357.65
0.00
170.69


0.00
43.34
0.00
45.65
0.00
45.84
43.12
0.00
43.69
111.02


0.00
0.00
76.97
0.00
-371.00
11.95
0.00
-439.48
4.63
-181.71


INDEPENDENT VARIABLES
3.31 43.97 0.00
345.49 0.00 -422.72
0.00 44.12 4.31
13.58 45.59 -4.22
29.61 38.69 -33.00
INDEPENDENT VARIABLES
69.35 40.28 -89.66


(6) (7)
TAKEN ONE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME


0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN FOUR AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN FIVE AT A TIME
0.00 0.00 0.00


66.03
17.45
53.74
67.66
42.99

69.04
73.42
72.34
72.32
60.98
75.38
55.44
76.10
75.76
75.88

76.30
75.69
76.16
77.90
76.96
77.88
78.83
77.12
78.69
76.52

78.87
77.12
78.70
77.90
79.20


0.8126
0.4178
0.7331
0.8226
0.6557

0.8309
0.8569
0.8506
0.8504
0.7809
0.8682
0.7446
0.8724
0.8704
0.8711

0.8735
0.8700
0.8727
0.8826
0.8772
0.8825
0.8879
0.8782
0.8871
0.8748

0.8881
0.8782
0.8871
0.8826
0.8899


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8126 -0.4178 0.7331 0.8226 0.6557

(2) 66.0347 1.0000 -0.3115 0.6471 0.8488 0.5489

(3) 17.4534 9.7016 1.0000 -0.2112 -0.7363 -0.1020

(4) 53.7397 41.8725 4.4613 1.0000 0.6118 0.9890

(5) 67.6631 72.0525 54.2184 37.4323 1.0000 0.4941

(6) 42.9914 30.1326 1.0395 97.8160 24.4171 1.0000

Numbers in Parentheses are the Variable or Parameter Number


81.54 0.9030











GALVIN AND EAGLESON (1965) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 84 B5 B6 B7 B8
NO
1 0.002 0.742 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.091 0.000 -0.041 0.000 0.000 0.000 0.000 0.000 0.000
3 0.033 0.000 0.000 0.048 0.000 0.000 0.000 0.000 0.000
4 0.014 0.000 0.000 0.000 7.472 0.000 0.000 0.000 0.000
5 0.031 0.000 0.000 0.000 0.000 0.004 0.000 0.000 0.000
6 0.027 0.690 -0.018 0.000 0.000 0.000 0.000 0.000 0.000
7 0.009 0.531 0.000 0.023 0.000 0.000 0.000 0.000 0.000
8 0.006 0.374 0.000 0.000 4.316 0.000 0.000 0.000 0.000
9 0.006 0.592 0.000 0.000 0.000 0.002 0.000 0.000 0.000
10 0.067 0.000 -0.027 0.044 0.000 0.000 0.000 0.000 0.000
11 -0.046 0.000 0.040 0.000 10.218 0.000 0.000 0.000 0.000
12 0.075 0.000 -0.035 0.000 0.000 0.004 0.000 0.000 0.000
13 0.017 0.000 0.000 0.024 5.431 0.000 0.000 0.000 0.000
14 0.046 0.000 0.000 0.253 0.000 -0.019 0.000 0.000 0.000
15 0.015 0.000 0.000 0.000 5.992 0.002 0.000 0.000 0.000
16 0.034 0.482 -0.018 0.023 0.000 0.000 0.000 0.000 0.000
17 -0.064 -0.202 0.055 0.000 12.936 0.000 0.000 0.000 0.000
18 0.034 0.523 -0.020 0.000 0.000 0.002 0.000 0.000 0.000
19 0.012 0.243 0.000 0.020 3.680 0.000 0.000 0.000 0.000
20 0.032 0.222 0.000 0.177 0.000 -0.013 0.000 0.000 0.000
21 0.009 0.254 0.000 0.000 4.058 0.002 0.000 0.000 0.000
22 -0.023 0.000 0.027 0.017 7.838 0.000 0.000 0.000 0.000
23 0.028 0.000 0.017 0.312 0.000 -0.024 0.000 0.000 0.000
24 -0.026 0.000 0.027 0.000 8.273 0.001 0.000 0.000 0.000
25 0.030 0.000 0.000 0.124 3.190 -0.008 0.000 0.000 0.000
26 -0.030 -0.071 0.032 0.017 8.842 0.000 0.000 0.000 0.000
27 0.028 0.029 0.015 0.295 0.000 -0.022 0.000 0.000 0.000
28 -0.030 -0.040 0.030 0.000 8.841 0.001 0.000 0.000 0.000
29 0.013 0.240 0.000 0.027 3.563 -0.001 0.000 0.000 0.000
30 -0.012 0.000 0.027 0.111 5.720 -0.008 0.000 0.000 0.000
31 -0.068 -1.186 0.118 0.546 12.538 -0.044 0.000 0.000 0.000


































































96











EAGLESON (1965) LABORATORY DATA
n= 7


Wave
Period
(s)

0.7900
0.8000
0.9500
1.1700
1.2000
1.3500
1.5700


Bed Equivalent
Slope Wave
Steepness


0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000


0.00899
0.01515
0.00825
0.00388
0.00425
0.00358
0.00182


Surf
Sim
Parm

1.05442
0.81253
1.10072
1.60619
1.53362
1.67054
2.34308


STEPWISE REGRESSION ANALYSIS RESULTS


Eq 100 r*2 r
No


(1)

100.00
0.00
0.00
0.00
0.00

-87.92
-0.03
-229.37
-91.43
0.00
0.00
0.00
0.00
0.00
0.00

-0.05
-176.74
557.03-
-0.04
-0.03
-134.32
0.00
0.00
0.00
0.00


(2)

0.00
100.00
0.00
0.00
0.00

187.92
0.00
0.00
0.00
-0.08
-61.87
106.94
0.00
0.00
0.00

0.07
27.55
1587.35
0.00
0.00
0.00
0.01
-0.08
-411.13
0.00


-0.04 -0.01
-0.34 0.91
520.86-1503.84
-0.04 0.00
0.00 -0.12

-0.09 0.17


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
100.03 0.00 0.00
0.00 329.37 0.00
0.00 0.00 191.43
100.08 0.00 0.00
0.00 161.87 0.00
0.00 0.00 -6.94
100.07 -0.07 0.00
99.43 0.00 0.57
0.00 84.46 15.54
INDEPENDENT VARIABLES
99.99 0.00 0.00
0.00 249.19 0.00
0.00 0.00 1130.32
100.01 0.04 0.00
100.02 0.00 0.00
0.00 206.16 28.15
100.07 -0.08 0.00
100.05 0.00 0.03
0.00 193.09 318.03
100.08 -0.06 -0.02
INDEPENDENT VARIABLES
100.01 0.03 0.00
100.09 0.00 -0.66
0.00 9.88 1073.10
100.01 0.03 0.00
100.00 0.03 0.08
INDEPENDENT VARIABLES
100.03 0.02 -0.13


(6) (7) (8)
TAKEN ONE AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN TWO AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN THREE AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN FOUR AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN FIVE AT A TIME
0.00 0.00 0.00


Water
Depth
(m)

0.0990
0.1230
0.0970
0.0720
0.0660
0.0580
0.0650


Breaker
Height
(m)

0.0550
0.0950
0.0730
0.0520
0.0600
0.0640
0.0440


55.39
76.25
8.91
93.43
70.86

82.39
77.72
96.26
73.13
78.78
93.78
76.29
93.99
70.88
93.63

82.80
96.31
98.70
97.80
78.12
96.43
94.01
82.06
97.45
94.47

97.97
98.77
98.71
97.93
98.32

99.53


0.7443
0.8732
0.2986
0.9666
0.8418

0.9077
0.8816
0.9811
0.8552
0.8876
0.9684
0.8734
0.9695
0.8419
0.9676

0.9099
0.9814
0.9935
0.9889
0.8839
0.9820
0.9696
0.9058
0.9872
0.9720

0.9898
0.9938
0.9935
0.9896
0.9916

0.9976











EAGLESON (1965) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.7443 -0.8732 -0.2986 0.9666 -0.8418

(2) 55.3914 1.0000 -0.6324 0.2184 0.8591 -0.7699

(3) 76.2463 39.9910 1.0000 0.4997 -0.8735 0.9698

(4) 8.9134 4.7692 24.9665 1.0000 -0.2337 0.3383

(5) 93.4294 73.8015 76.3001 5.4596 1.0000 -0.8916

(6) 70.8621 59.2714 94.0496 11.4425 79.5020 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 82 B3 B4 B5 B6 B7 B8
NO
1 0.016 1.061 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.163 0.000 -0.072 0.000 0.000 0.000 0.000 0.000 0.000
3 188.413 0.000 0.000-1883.333 0.000 0.000 0.000 0.000 0.000
4 0.050 0.000 0.000 0.000 5.022 0.000 0.000 0.000 0.000
5 0.140 0.000 0.000 0.000 0.000 -0.039 0.000 0.000 0.000
6 0.116 0.456 -0.055 0.000 0.000 0.000 0.000 0.000 0.000
7 305.431 1.211 0.000-3054.293 0.000 0.000 0.000 0.000 0.000
8 0.070 -0.468 0.000 0.000 6.489 0.000 0.000 0.000 0.000
9 0.106 0.337 0.000 0.000 0.000 -0.031 0.000 0.000 0.000
10 -115.635 0.000 -0.079 1158.084 0.000 0.000 0.000 0.000 0.000
11 0.065 0.000 -0.010 0.000 4.469 0.000 0.000 0.000 0.000
12 0.165 0.000 -0.079 0.000 0.000 0.004 0.000 0.000 0.000
13 48.560 0.000 0.000 -485.104 4.929 0.000 0.000 0.000 0.000
14 9.970 0.000 0.000 -98.310 0.000 -0.039 0.000 0.000 0.000
15 0.040 0.000 0.000 0.000 5.475 0.005 0.000 0.000 0.000
16 76.565 0.610 -0.045 -764.720 0.000 0.000 0.000 0.000 0.000
17 0.065 -0.503 0.004 0.000 6.842 0.000 0.000 0.000 0.000
18 0.065 1.556 -0.251 0.000 0.000 0.138 0.000 0.000 0.000
19 -149.220 -1.008 0.000 1493.134 8.466 0.000 0.000 0.000 0.000
20 248.036 0.995 0.000-2480.081 0.000 -0.008 0.000 0.000 0.000
21 0.061 -0.467 0.000 0.000 6.907 0.004 0.000 0.000 0.000
22 40.181 0.000 -0.003 -401.260 4.752 0.000 0.000 0.000 0.000
23 -242.176 0.000 -0.170 2423.821 0.000 0.048 0.000 0.000 0.000
24 0.065 0.000 -0.066 0.000 5.296 0.040 0.000 0.000 0.000
25 62.464 0.000 0.000 -624.297 5.645 0.008 0.000 0.000 0.000
26 -175.618 -1.034 -0.009 1757.254 8.110 0.000 0.000 0.000 0.000
27 30.189 1.610 -0.246 -301.322 0.000 0.137 0.000 0.000 0.000
28 0.065 1.481 -0.242 0.000 0.271 0.134 0.000 0.000 0.000
29 -179.780 -1.121 0.000 1798.874 8.425 -0.005 0.000 0.000 0.000
30 -99.586 0.000 -0.104 996.754 4.920 0.055 0.000 0.000 0.000
31 515.958 7.741 -0.787-5160.184 -19.015 0.450 0.000 0.000 0.000











HORIKAWA AND KUO (1967) LABORATORY DATA
n = 97


Water Breaker Wave
Depth Height Period
(m) (m) (s)

0.1250 0.0860 1.2000
0.1250 0.0910 1.2000
0.1380 0.1220 1.2000
0.1630 0.1420 1.2000
0.1630 0.1160 1.2000
0.1630 0.1310 1.2000
0.1750 0.1320 1.2000
0.2000 0.1330 1.2000
0.2250 0.1470 1.2000
0.2130 0.1390 1.2000
0.2130 0.1820 1.2000
0.2630 0.1630 1.2000
0.1250 0.0780 1.4000
0.1380 0.0930 1.4000
0.1500 0.0940 1.4000
0.1500 0.1160 1.4000
0.1630 0.1150 1.4000
0.1880 0.1230 1.4000
0.2000 0,1300 1.4000
0.2130 0.1350 1.4000
0.2130 0.1380 1.4000
0.2250 0.1640 1.4000
0.2500 0.1730 1.4000
0.1130 0.0790 1.6000
0.1250 0.1020 1.6000
0.1500 0.1030 1.6000
0.1630 0.1170 1.6000
0.1630 0.1370 1.6000
0.1630 0.1430 1.6000
0.1750 0.1520 1.6000
0.2130 0.1470 1.6000
0.2250 0.1500 1.6000
0.1130 0.0880 1.8000
0.1250 0.0920 1.8000
0.1500 0.0940 1.8000
0.1380 0.1000 1.8000
0.1500 0.1080 1.8000
0.1630 0.1160 1.8000
0.1630 0.1180 1.8000
0.1880 0.1150 1.8000
0.1880 0.1240 1.8000
0.2000 0.1260 1.8000
0.2130 0.1380 1.8000
0.2130 0.1380 1.8000
0.2250 0.1500 1.8000
0.1310 0.0830 2.0000
0.1380 0.0910 2.0000
0.1500 0.1040 2.0000
0.1500 0.1120 2.0000
0.1500 0.1180 2.0000
0.1680 0.1200 2.0000
0.1650 0.1190 2.0000
0.2000 0.1250 2.0000
0.2000 0.1360 2.0000
0.2130 0.1460 2.0000
0.2130 0.1540 2.0000
0.2500 0.1550 2.0000
0.1730 0.1280 2.2000
0.1450 0.1000 2.2000
0.1520 0.1150 2.2000
0.1350 0.1120 2.2000
0.1180 0.0760 2.2000
0.1350 0.1130 2.2000
0.1350 0.1080 2.2000
0.1180 0.0870 2.2000
0.1080 0.0920 2.2000


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0125
0.0333
0.0333
0.0333
0.0333
0.0333
0.0333
0.0333
0.0333
0.0333


0.00609
0.00645
0.00865
0.01006
0.00822
0.00928
0.00935
0.00942
0.01042
0.00985
0.01290
0.01155
0.00406
0.00484
0.00489
0.00604
0.00599
0.00640
0.00677
0.00703
0.00718
0.00854
0.00901
0.00315
0.00407
0.00411
0.00466
0.00546
0.00570
0.00606
0.00586
0.00598
0.00277
0.00290
0.00296
0.00315
0.00340
0.00365
0.00372
0.00362
0.00391
0.00397
0.00435
0.00435
0.00472
0.00212
0.00232
0.00265
0.00286
0.00301
0.00306
0.00304
0.00319
0.00347
0.00372
0.00393
0.00395
0.00270
0.00211
0.00242
0.00236
0.00160
0.00238
0.00228
0.00183
0.00194


0.16012
0.15566
0.13444
0.12461
0.13787
0.12974
0.12925
0.12876
0.12247
0.12595
0.11007
0.11631
0.19616
0.17964
0.17868
0.16085
0.16155
0.15621
0.15194
0.14910
0.14747
0.13528
0.13171
0.22276
0.19604
0.19509
0.18304
0.16915
0.16557
0.16059
0.16330
0.16166
0.23744
0.23222
0.22974
0.22274
0.21433
0.20681
0.20505
0.20770
0.20003
0.19843
0.18961
0.18961
0.18187
0.27165
0.25944
0.24268
0.23385
0.22783
0.22592
0.22687
0.22136
0.21222
0.20482
0.19943
0.19879
0.64103
0.72524
0.67629
0.68528
0.83190
0.68225
0.69786
0.77754
0.75611











HORIKAWA AND KUO (1967) LABORATORY DATA (CONT)


Water Breaker Wave


Depth
(m)

0.0980
0.0680
0.2020
0.1840
0.1830
0.1820
0.1820
0.0750
0.0600
0.0600
0.0800
0.0800
0.1400
0.1580
0.0750
0.1400
0.1600
0.1600
0.2050
0.2050
0.0730
0.1080
0.0930
0.0930
0.1050
0.1230
0.1280
0.1350
0.1330
0.1530
0.1650


Height
(m)

0.0840
0.0600
0.1670
0.1550
0.1510
0.1270
0.1210
0.0630
0.0750
0.0940
0.1030
0.1110
0.1170
0.1040
0.0690
0.1390
0.1310
0.1320
0.1570
0.1660
0.0650
0.0780
0.0830
0.1050
0.1230
0.1130
0.1160
0.1170
0.1230
0.1280
0.1560


Period
(s)

2.2000
2.2000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
1.4000
2.2000
2.2000
2.2000
2.2000
2.2000
2.2000
2.2000
2.2000
2.2000
2.2000
2.3000


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm

0.0333 0.00177 0.79130
0.0333 0.00126 0.93628
0.0333 0.00869 0.35713
0.0333 0.00807 0.37070
0.0333 0.00786 0.37558
0.0333 0.00661 0.40953
0.0333 0.00630 0.41956
0.0333 0.00328 0.58145
0.0500 0.00390 0.80017
0.0500 0.00489 0.71474
0.0500 0.00536 0.68280
0.0500 0.00578 0.65773
0.0500 0.00609 0.64065
0.0333 0.00541 0.45255
0.0330 0.00359 0.55059
0.0500 0.00724 0.58776
0.0500 0.00682 0.60545
0.0500 0.00687 0.60315
0.0500 0.00817 0.55305
0.0500 0.00864 0.53784
0.0500 0.00137 1.35067
0.0500 0.00164 1.23299
0.0500 0.00175 1.19527
0.0500 0.00221 1.06270
0.0500 0.00259 0.98187
0.0500 0.00238 1.02439
0.0500 0.00245 1.01106
0.0500 0.00247 1.00673
0.0500 0.00259 0.98187
0.0500 0.00270 0.96250
0.0500 0.00301 0.91148


100











HORIKAWA AND KUO (1967) LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(2)

0.00
100.00
0.00
0.00
0.00

-5.08
0.00
0.00
0.00
48.20
61.41
-207.86
0.00
0.00
0.00

9.44
-31.74
25.91
0.00
0.00
0.00
73.52
166.64
79.60
0.00


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
-20.46 0.00 0.00
0.00 0.71 0.00
0.00 0.00 -16.20
51.80 0.00 0.00
0.00 38.59 0.00
0.00 0.00 307.86
-244.14 344.14 0.00
-63.68 0.00 163.68
0.00 1664.34-1564.34
INDEPENDENT VARIABLES
-19.03 0.00 0.00
0.00 -13.78 0.00
0.00 0.00 -16.33
-20.63 -4.61 0.00
-26.13 0.00 6.33
0.00 -11.75 -18.71
-12.65 39.13 0.00
106.74 0.00 -173.38
0.00 33.22 -12.82
-277.95 659.31 -281.36


INDEPENDENT VARIABLES
26 107.37 10.70 -18.82 0.74 0.00
27 112.01 7.15 -21.26 0.00 2.10
28 61.60 42.09 0.00 10.48 -14.17
29 122.95 0.00 -24.23 -2.81 4.09
30 0.00 81.60 10.25 29.29 -21.14
INDEPENDENT VARIABLES
31 117.25 3.79 -22.59 -1.41 2.96


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(1)

100.00
0.00
0.00
0.00
0.00

105.08
120.46
99.29
116.20
0.00
0.00
0.00
0.00
0.00
0.00

109.60
145.53
90.42
125.24
119.80
130.46
0.00
0.00
0.00
0.00


T





T


(8)
IME
0.00
0.00
0.00
0.00
0.00
IME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


0.8419
0.2768
0.5006
0.5575
0.6141

0.8427
0.9169
0.8420
0.8979
0.5190
0.6825
0.6227
0.6733
0.6255
0.6740

0.9188
0.8448
0.9114
0.9185
0.9185
0.9057
0.8182
0.6721
0.8646
0.6772

0.9188
0.9189
0.9138
0.9189
0.8771


84.44 0.9189


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.8419 -0.2768 -0.5006 0.5575 -0.6141

70.8834 1.0000 -0.2879 -0.1694 0.6526 -0.3877

7.6631 8.2898 1.0000 0.2909 -0.8587 0.5866

25.0569 2.8708 8.4623 1.0000 -0.2403 0.8997

31.0751 42.5897 73.7318 5.7744 1.0000 -0.5220


(6) 37.7091 15.0326

Numbers in Parentheses


34.4096 80.9407 27.2481


1.0000


are the Variable or Parameter Number


TAKEN THREE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


70.88
7.66
25.06
31.08
37.71

71.01
84.07
70.89
80.62
26.94
46.58
38.77
45.34
39.12
45.42

84.42
71.37
83.07
84.36
84.36
82.03
66.94
45.17
74.75
45.86

84.42
84.44
83.51
84.44
76.93











HORIKAWA AND KUO (1967) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 B2 B3 B4 B5 B6 B7 B8


1.415 0.000 0.000
0.000 -0.035 0.000
0.000 0.000 -1.452
0.000 0.000 0.000
0.000 0.000 0.000
1.397 -0.005 0.000
1.310 0.000 -1.069
1.400 0.000 0.000
1.194 0.000 0.000
0.000 -0.018 -1.331
0.000 0.096 0.000
0.000 0.016 0.000
0.000 0.000 -1.129
0.000 0.000 0.790
0.000 0.000 0.000
1.336 0.008 -1.115
1.549 -0.024 0.000
1.217 0.024 0.000
1.386 0.000 -1.097
1.370 0.000 -1.436
1.345 0.000 0.000
0.000 0.115 -1.368
0.000 0.051 2.264
0.000 0.139 0.000
0.000 0.000 -0.558
1.327 0.009 -1.117
1.351 0.006 -1.231
1.024 0.049 0.000
1.396 0.000 -1.321
0.000 0.155 1.343
1.373 0.003 -1.271


0.000 0.000
0.000 0.000
0.000 0.000
9.653 0.000
0.000 -0.086
0.000 0.000
0.000 0.000
0.241 0.000
0.000 -0.048
0.000 0.000
21.077 0.000
0.000 -0.097
8.034 0.000
0.000 -0.121
5.638 -0.063
0.000 0.000
-3.560 0.000
0.000 -0.063
-1.238 0.000
0.000 0.021
-2.941 -0.055
21.372 0.000
0.000 -0.219
20.286 -0.092
6.689 -0.034
0.222 0.000
0.000 0.007
4.224 -0.067
-0.775 0.013
19.374 -0.165
-0.402 0.010


1 -0.011
2 0.216
3 0.193
4 0.110
5 0.193
6 0.000
7 0.028
8 -0.010
9 0.035
10 0.221
11 -0.110
12 0.170
13 0.146
14 0.188
15 0.156
16 0.013
17 0.031
18 -0.002
19 0.026
20 0.022
21 0.035
22 -0.110
23 0.105
24 -0.141
25 0.152
26 0.011
27 0.014
28 -0.040
29 0.023
30 -0.166
31 0.018


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











BOWEN AND OTHERS (1968) LABORATORY DATA


Wave
Period
(s)

0.8200
0.820C
1.1400
1.1400
1.140C
1.650C
1.650C
1.650C
1.650C
2.370C
2.3700


n = 11

Bed Equivalent
SLope Wave
Steepness


0.0820
0.0820
0.0820
0.0820
0.0820
0.0820
0.0820
0.0820
0.0820
0.0820
0.0820


0.00607
0.00895
0.00518
0.00675
0.00832
0.00292
0.00364
0.00431
0.00487
0.00214
0.00231


Surf
Sim
Parm

1.05234
0.86659
1.13909
0.99789
0.89883
1.51657
1.35996
1.24900
1.17473
1.77106
1.70716


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(1)

100.00
0.00
0.00
0.00
0.00

131.78
-0.14
85.19
141.00
0.00
0.00
0.00
0.00
0.00
0.00

-0.14
79.15
148.41
-0.10
-0.12
72.16
0.00
0.00
0.00
0.00

-0.09
-0.09
63.90
-0.09
0.00


(2)

0.00
100.00
0.00
0.00
0.00

-31.78
0.00
0.00
0.00
0.02
63.67
-550.17
0.00
0.00
0.00

0.03
4.42
28.08
0.00
0.00
0.00
0.88
-0.20
-542.38
0.00

-0.01
-0.04
-56.79
0.00
-0.20


31 -0.15 0.12


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
100.14
0.00
0.00
99.98
0.00
0.00
100.00
99.99
0.00


0.00
0.00
14.81
0.00
0.00
36.33
0.00
0.00
0.00
24.85


0.00
0.00
0.00
-41.00
0.00
0.00
650.17
0.00
0.01
75.15


INDEPENDENT VARIABLES


100.11
0.00
0.00
100.11
100.08
0.00
98.61
99.96
0.00
99.99


0.00
16.43
0.00
-0.02
0.00
17.70
0.50
0.00
0.51
0.00


0.00
0.00
-76.48
0.00
0.03
10.14
0.00
0.24
641.87
0.01


INDEPENDENT VARIABLES
100.12 -0.02 0.00
100.06 0.00 0.07
0.00 15.97 76.92
100.13 -0.02 -0.02
99.97 0.00 0.23
INDEPENDENT VARIABLES
100.25 -0.04 -0.17


(6) (7)
TAKEN ONE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
A TIME
0.00


89.74
37.58
25.89
5.76
8.67

93.96
90.15
95.27
94.08
45.49
71.62
91.71
28.67
29.57
10.40

94.70
95.30
94.10
96.48
94.91
95.40
71.64
92.81
91.71
29.84

96.61
95.01
96.23
96.75
92.82


0.9473
0.6130
0.5089
0.2400
0.2945

0.9694
0.9495
0.9761
0.9700
0.6745
0.8463
0.9576
0.5354
0.5438
0.3225

0.9731
0.9762
0.9701
0.9823
0.9742
0.9767
0.8464
0.9634
0.9576
0.5463

0.9829
0.9747
0.9810
0.9836
0.9634


97.44 0.9871


Water
Depth
(m)

0.0420
0.0550
0.0500
0.0680
0.0970
0.0590
0.0680
0.0950
0.0970
0.0880
0.0920


Breaker
Height
(m)

0.0400
0.0590
0.0660
0.0860
0.1060
0.0780
0.0970
0.1150
0.1300
0.1180
0.1270











BOWEN AND OTHERS (1968) LABORATORY DATA COUNT )

CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9473 0.6130

(2) 89.7368 1.0000 0.7823

(3) 37.5830 61.2062 1.0000

(4) 25.8942 35.0443 16.9769

(5) 5.7611 22.3092 75.0881

(6) 8.6736 25.1278 86.0957


0.5089 -0.2400 0.2945

0.5920 -0.4723 0.5013

0.4120 -0.8665 0.9279

1.0000 -0.1480 0.2105

2.1907 1.0000 -0.9518

4.4330 90.5868 1.0000


Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (83 X3) + ... + (B8 X8)

BO 81 82 83 B4 B5 B6 B7 B8


0.661
0.000
0.000
0.000
0.000
0.841
0.694
0.749
0.745
0.000
0.000
0.000
0.000
0.000
0.000
0.893
0.727
0.683
0.814
0.797
0.743
0.000
0.000
0.000
0.000
0.772
0.656
1.296
0.810
0.000
1.308


0.000 0.000
0.023 0.000
0.000 3489.999
0.000 0.000
0.000 0.000
-0.013 0.000
0.000 -548.188
0.000 0.000
0.000 0.000
0.019 2116.993
0.062 0.000
0.094 0.000
0.000 3319.059
0.000 3206.883
0.000 0.000
-0.013 -733.912
0.003 0.000
0.008 0.000
0.000 -951.804
0.000 -779.326
0.000 0.000
0.061 124.903
0.102 -916.873
0.094 0.000
0.000 3141.837
0.005-1000.665
0.019 -834.589
-0.072 0.000
0.000-1017.973
0.101 -927.479
-0.065 -962.026


0.000
0.000
0.000
-2.156
0.000
0.000
0.000
2.399
0.000
0.000
10.502
0.000
-1.513
0.000
3.845
0.000
2.778
0.000
2.610
0.000
3.359
10.365
0.000
-0.026
1.555
3.397
0.000
5.968
4.058
0.324
6.387


0.000
0.000
0.000
0.000
0.020
0.000
0.000
0.000
-0.016
0.000
0.000
-0.132
0.000
0.013
0.047
0.000
0.000
-0.026
0.000
-0.017
0.008
0.000
-0.143
-0.132
0.024
0.000
-0.040
0.116
0.012
-0.141
0.110


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


104


1 0.012
2 0.039
3 -286.103
4 0.085
5 0.049
6 0.014
7 44.960
8 -0.008
9 0.025
10 -173.546
11 -0.072
12 0.099
13 -272.078
14 -262.904
15 -0.004
16 60.191
17 -0.012
18 0.031
19 78.032
20 63.925
21 -0.022
22 -10.312
23 75.284
24 0.099
25 -257.592
26 82.031
27 68.471
28 -0.115
29 83.436
30 76.149
31 78.765











KOMAR AND SIMMONS (1968) LABORATORY DATA
n = 44


Water Breaker
Depth Height
(m) (m)

0.0930 0.0740
0.1090 0.0830
0.1210 0.1060
0.1250 0.0910
0.1770 0.1220
0.1330 0.1160
0.2130 0.1650
0.1580 0.1370
0.2120 0.1660
0.1580 0.1370
0.1620 0.1270
0.1640 0.1590
0.1400 0.1400
0.0780 0.0580
0.0400 0.0300
0.0740 0.0690
0.0390 0.0410
0.0890 0.0960
0.0490 0.0550
0.1090 0.1130
0.0530 0.0550
0.1100 0.0950
0.1580 0.1320
0.1510 0.1560
0.0980 0.0960
0.0510 0.0540
0.1300 0.1420
0.1620 0.1260
0.1550 0.1470
0.0410 0.0420
0.0900 0.0900
0.0580 0.0580
0.0740 0.0590
0.0340 0.0340
0.0800 0.0770
0.0840 0.0890
0.0490 0.0480
0.0800 0.0760
0.0610 0.0500
0.0360 0.0350
0.0970 0.0930
0.0590 0.0540
0.1520 0.1430
0.1700 0.1700


Wave
Period
(s)

1.1400
1.6500
2.3700
1.1400
1.1400
1.6500
1.6500
2.3700
1.6500
2.3700
1.1400
1.6500
2.3700
0.8100
0.8100
1.1400
1.1400
1.6500
1.6500
2.3700
2.3700
2.3700
1.1400
1.6500
2.3700
2.3700
2.3700
1.1400
1.6500
1.1400
1.6500
1.6500
0.8100
0.8100
1.1400
1.6500
1.6500
1.1400
0.8100
0.8100
2.3700
2.3700
1.1400
1.6500


Bed Equivalent Surf
SLope Wave Sim
Steepness Parm

0.0360 0.00581 0.47229
0.0360 0.00311 0.64545
0.0360 0.00193 0.82037
0.0360 0.00715 0.42589
0.0360 0.00958 0.36782
0.0360 0.00435 0.54597
0.0360 0.00618 0.45778
0.0360 0.00249 0.72161
0.0360 0.00622 0.45640
0.0360 0.00249 0.72161
0.0700 0.00997 0.70099
0.0700 0.00596 0.90677
0.0700 0.00254 1.38802
0.0700 0.00902 0.73702
0.0700 0.00467 1.02479
0.0700 0.00542 0.95102
0.0700 0.00322 1.23374
0.0700 0.00360 1.16697
0.0700 0.00206 1.54175
0.0700 0.00205 1.54497
0.0700 0.00100 2.21451
0.0700 0.00173 1.68499
0.0700 0.01036 0.68759
0.0700 0.00585 0.91545
0.0860 0.00174 2.05932
0.0860 0.00098 2.74576
0.0860 0.00258 1.69323
0.0860 0.00989 0.86463
0.0860 0.00551 1.15861
0.0860 0.00330 1.49759
0.0860 0.00337 1.48072
0.0860 0.00217 1.84451
0.0860 0.00918 0.89778
0.0860 0.00529 1.18265
0.0860 0.00605 1.10604
0.1050 0.00334 1.81799
0.1050 0.00180 2.47552
0.1050 0.00597 1.35925
0.1050 0.00778 1.19070
0.1050 0.00544 1.42316
0.1050 0.00169 2.55452
0.1050 0.00098 3.35238
0.1050 0.01123 0.99092
0.1050 0.00637 1.31541


105











KOMAR AND SIMMONS (1968) LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00 0.00
0.00 0.00
10.90 0.00
0.00 -20.66
0.00 0.00
39.20 0.00
0.00-1036.52
-86.97 0.00
0.00 57.80
-18.96 118.96


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


(1)

100.00
0.00
0.00
0.00
0.00

118.71
135.57
89.10
120.66
0.00
0.00
0.00
0.00
0.00
0.00

175.18
74.83
120.58
118.27
137.66
106.74
0.00
0.00
0.00
0.00

85.18
257.75
64.15
104.84
0.00

114.57


(2)

0.00
100.00
0.00
0.00
0.00

-18.71
0.00
0.00
0.00
-41.73
60.80
1136.52
0.00
0.00
0.00

-30.37
10.16
0.16
0.00
0.00
0.00
75.89
150.08
85.14
0.00

19.94
-99.74
32.56
0.00
85.00

-7.61


0.00
0.00
-20.73
0.00
-14.83
-13.48
0.00
-144.65
-25.45
-18.50


INDEPENDENT VARIABLES
-29.15 24.02 0.00
-114.36 0.00 56.36
0.00 19.14 -15.85
-54.25 27.20 22.20
7.52 37.31 -29.82
INDEPENDENT VARIABLES
-60.69 26.73 26.99


TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


91.15
3.89
21.44
15.76
29.28

92.57
93.95
93.34
93.98
23.38
69.54
56.65
36.72
31.25
29.73

95.47
93.48
93.98
96.56
94.63
94.32
79.41
65.29
84.47
36.80

96.88
96.11
95.29
97.37
84.68

97.39


0.9547
0.1971
0.4631
0.3970
0.5411

0.9621
0.9693
0.9661
0.9694
0.4836
0.8339
0.7527
0.6060
0.5590
0.5453

0.9771
0.9669
0.9694
0.9827
0.9728
0.9712
0.8911
0.8080
0.9191
0.6066

0.9843
0.9803
0.9761
0.9868
0.9202

0.9869


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


1.0000 0.9547 0.1971 -0.4631 0.3970 -0.5411

91.1509 1.0000 0.3244 -0.3190 0.2664 -0.4059

3.8861 10.5227 1.0000 -0.1274 -0.7419 0.4825

21.4437 10.1758 1.6240 1.0000 -0.0134 0.6615

15.7646 7.0973 55.0386 0.0180 1.0000 -0.6384


29.2831 16.4734 23.2829 43.7533 40.7512


1.0000


Numbers in Parentheses are the Variable or Parameter Number


106


0.00
-35.57
0.00
0.00
141.73
0.00
0.00
186.97
42.20
0.00


INDEPENDENT VARIABLES TAKEN THREE AT


0.00
15.00
0.00
17.05
0.00
6.73
50.33
0.00
40.31
105.28


-44.81
0.00
0.00
-35.32
-22.83
0.00
-26.22
94.57
0.00
223.78











KOMAR AND SIMMONS (1968) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO Bl 82 B3 B4 B5 B6 B7 B8


1 -0.002 1.137 0.000 0.000 0.000 0.000 0.000
2 0.078 0.000 0.018 0.000 0.000 0.000 0.000
3 0.177 0.000 0.000 -0.959 0.000 0.000 0.000
4 0.073 0.000 0.000 0.000 6.847 0.000 0.000
5 0.157 0.000 0.000 0.000 0.000 -0.040 0.000
6 0.011 1.186 -0.011 0.000 0.000 0.000 0.000
7 0.031 1.070 0.000 -0.366 0.000 0.000 0.000
8 -0.010 1.088 0.000 0.000 2.649 0.000 0.000
9 0.023 1.048 0.000 0.000 0.000 -0.014 0.000
10 0.154 0.000 0.013 -0.922 0.000 0.000 0.000
11 -0.149 0.000 0.097 0.000 20.839 0.000 0.000
12 0.099 0.000 0.053 0.000 0.000 -0.061 0.000
13 0.144 0.000 0.000 -0.948 6.741 0.000 0.000
14 0.174 0.000 0.000 -0.387 0.000 -0.031 0.000
15 0.144 0.000 0.000 0.000 1.502 -0.036 0.000
16 0.045 1.119 -0.012 -0.373 0.000 0.000 0.000
17 -0.025 1.026 0.008 0.000 4.091 0.000 0.000
18 0.023 1.048 0.000 0.000 0.000 -0.014 0.000
19 0.025 1.012 0.000 -0.393 2.901 0.000 0.000
20 0.034 1.040 0.000 -0.225 0.000 -0.009 0.000
21 0.012 1.047 0.000 0.000 1.315 -0.010 0.000
22 -0.080 0.000 0.089 -0.665 19.510 0.000 0.000
23 0.020 0.000 0.081 1.107 0.000 -0.099 0.000
24 -0.078 0.000 0.098 0.000 15.438 -0.037 0.000
25 0.140 0.000 0.000 -1.040 7.476 0.005 0.000
26 0.004 0.913 0.013 -0.407 5.122 0.000 0.000
27 0.059 1.268 -0.029 -0.732 0.000 0.021 0.000
28 -0.018 0.834 0.025 0.000 4.951 -0.016 0.000
29 0.013 1.021 0.000 -0.688 5.272 0.016 0.000
30 -0.083 0.000 0.101 0.194 14.711 -0.045 0.000
31 0.018 1.055 -0.004 -0.727 4.898 0.019 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


107











GALVIN (1969)


LABORATORY DATA
= 17


Water
Depth
(m)

0.0620
0.0800
0.0690
0.0630
0.0610
0.0390
0.0880
0.0770
0.0620
0.0400
0.0900
0.1000
0.0450
0.1140
0.1050
0.1020
0.1010


Bed Equivalent
Slope Wave
Steepness


Breaker
Height
(m)

0.0620
0.0910
0.0900
0.0690
0.0650
0.0380
0.1420
0.1010
0.0720
0.0430
0.1180
0.1500
0.0450
0.0940
0.1450
0.0940
0.1130


Wave
Period
(s)

1.0000
1.0000
1.0000
2.0000
1.0000
2.0000
5.0000
6.0000
1.0000
2.0000
2.0000
5.0000
2.0000
2.0000
4.0000
2.0000
4.0000


Surf
Sim
Parm

2.51447
2.07550
2.08700
4.76704
1.22788
3.21182
4.15374
5.91022
1.16667
3.01932
1.82264
4.04145
2.95146
2.04211
3.28843
1.02105
1.86253


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1)

100.00
0.00
0.00
0.00
0.00


(2)

0.00
100.00
0.00
0.00
0.00


115.99 -15.99
116.02 0.00
98.84 0.00
131.31 0.00
0.00 242.85
0.00 77.49
0.00-1916.40
0.00 0.00
0.00 0.00
0.00 0.00

168.23 -34.04
159.51 -43.86
131.29 10.85
119.11 0.00
143.72 0.00
168.38 0.00
0.00 135.70
0.00 207.77
0.00 481.81
0.00 0.00


188.03
149.01
180.73
168.75
0.00


-49.50
-6.68
-17.75
0.00
287.94


31 151.64 24.96


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
-16.02
0.00
0.00
142.85
0.00
0.00
101.82
86.73
0.00


0.00
0.00
1.16
0.00
0.00
22.51
0.00
-1.82
0.00
36.60


0.00
0.00
0.00
-31.31
0.00
0.00
2016.40
0.00
13.27
63.40


(6)
TAKEN ONE
0.00
0.00
0.00
0.00
0.00
TAKEN TWO
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00


INDEPENDENT VARIABLES TAKEN THREE AT


-34.19 0.00 0.00
0.00 -15.65 0.00
0.00 0.00 -42.14
-27.42 8.31 0.00
-10.63 0.00 -33.09
0.00 -13.01 -55.38
101.56 65.85 0.00
115.89 0.00 -223.66
0.00 55.07 -436.89
52.44 15.09 32.47
INDEPENDENT VARIABLES
-28.00 -10.52 0.00
-15.15 0.00 -27.19
0.00 -17.32 -45.65
32.41 -26.41 -74.76
277.57 -66.21 -399.31
INDEPENDENT VARIABLES
53.96 -29.25 -101.31


0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


109


0.2000
0.2000
0.2000
0.2000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.1000
0.0500
0.0500


0.00633
0.00929
0.00918
0.00176
0.00663
0.00097
0.00058
0.00029
0.00735
0.00110
0.00301
0.00061
0.00115
0.00240
0.00092
0.00240
0.00072


70.18
18.22
8.97
3.00
0.94

72.98
71.57
70.23
76.86
19.85
22.02
48.47
8.98
9.23
8.50

76.96
75.78
77.14
72.89
77.30
78.76
28.22
62.03
50.19
9.57

77.61
77.32
79.04
79.64
68.67

79.82


0.8377
0.4268
0.2996
0.1732
0.0968

0.8543
0.8460
0.8381
0.8767
0.4456
0.4693
0.6962
0.2997
0.3038
0.2916

0.8772
0.8705
0.8783
0.8538
0.8792
0.8875
0.5312
0.7876
0.7085
0.3093

0.8810
0.8793
0.8890
0.8924
0.8286

0.8934











GALVIN (1969) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8377 0.4268 -0.2996 -0.1732 -0.0968

(2) 70.1754 1.0000 0.6597 -0.2200 -0.2350 0.1876

(3) 18.2171 43.5239 1.0000 -0.4315 -0.7219 0.6972

(4) 8.9737 4.8400 18.6217 1.0000 0.6052 0.1559

(5) 2.9982 5.5231 52.1204 36.6249 1.0000 -0.5618

(6) 0.9367 3.5194 48.6043 2.4312 31.5600 1.0000

Nunbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 XI) + (B2 X2) + (83 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 85 B6 B7 88


0.563 0.000 0.000
0.000 0.006 0.000
0.000 0.000 -0.142
0.000 0.000 0.000
0.000 0.000 0.000
0.661 -0.003 0.000
0.545 0.000 -0.058
0.567 0.000 0.000
0.596 0.000 0.000
0.000 0.005 -0.067
0.000 0.009 0.000
0.000 0.014 0.000
0.000 0.000 -0.146
0.000 0.000 -0.139
0.000 0.000 0.000
0.678 -0.005 -0.106
0.740 -0.007 0.000
0.554 0.002 0.000
0.555 0.000 -0.098
0.584 0.000 -0.033
0.580 0.000 0.000
0.000 0.009 -0.149
0.000 0.023 0.274
0.000 0.016 0.000
0.000 0.000 -0.094
0.718 -0.007 -0.082
0.603 -0.001 -0.047
0.631 -0.002 0.000
0.596 0.000 0.088
0.000 0.025 0.519
0.535 0.003 0.146


0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
-1.289 0.000 0.000
0.000 -0.002 0.000
0.000 0.000 0.000
0.000 0.000 0.000
0.187 0.000 0.000
0.000 -0.005 0.000
0.000 0.000 0.000
2.099 0.000 0.000
0.000 -0.013 0.000
0.096 0.000 0.000
0.000 -0.001 0.000
-2.475 -0.005 0.000
0.000 0.000 0.000
-2.033 0.000 0.000
0.000 -0.006 0.000
1.085 0.000 0.000
0.000 -0.004 0.000
-1.256 -0.006 0.000
3.527 0.000 0.000
0.000 -0.022 0.000
1.421 -0.013 0.000
-0.988 -0.002 0.000
-1.125 0.000 0.000
0.000 -0.004 0.000
-1.694 -0.005 0.000
-2.613 -0.009 0.000
-4.531 -0.032 0.000
-2.889 -0.012 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


0.026
0.061
0.093
0.080
0.081
0.025
0.034
0.025
0.035
0.071
0.046
0.078
0.093
0.095
0.098
0.040
0.035
0.038
0.034
0.040
0.045
0.059
0.049
0.068
0.097
0.042
0.040
0.045
0.045
0.054
0.045











WEGGEL AND MAXWELL (1970) LABORATORY DATA
n=9

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.0870 0.0890 2.0500 0.0510 0.00216 1.09687
0.1200 0.1200 2.0500 0.0510 0.00291 0.94481
0.1190 0.1210 2.0500 0.0510 0.00294 0.94090
0.1300 0.1290 1.6900 0.0510 0.00461 0.75123
0.1620 0.1530 1.7000 0.0510 0.00540 0.69388
0.1690 0.1620 1.7000 0.0510 0.00572 0.67433
0.1400 0.1260 1.4400 0.0510 0.00620 0.64768
0.1470 0.1290 1.4400 0.0510 0.00635 0.64010
0.1440 0.1250 1.2600 0.0510 0.00803 0.56898


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 89.90 0.9482
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 37.69 0.6140
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 53.16 0.7291
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 55.92 0.7478
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 72.85 0.8535
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 155.11 -55.11 0.00 0.00 0.00 0.00 0.00 0.00 98.70 0.9935
7 0.31 0.00 99.69 0.00 0.00 0.00 0.00 0.00 90.53 0.9515
8 84.17 0.00 0.00 15.83 0.00 0.00 0.00 0.00 98.89 0.9944
9 168.20 0.00 0.00 0.00 -68.20 0.00 0.00 0.00 98.49 0.9924
10 0.00 -0.02 100.02 0.00 0.00 0.00 0.00 0.00 64.07 0.8005
11 0.00 62.94 0.00 37.06 0.00 0.00 0.00 0.00 78.83 0.8879
12 0.00 -479.69 0.00 0.00 579.69 0.00 0.00 0.00 94.75 0.9734
13 0.00 0.00 99.98 0.02 0.00 0.00 0.00 0.00 71.04 0.8428
14 0.00 0.00 100.07 0.00 -0.07 0.00 0.00 0.00 76.92 0.8770
15 0.00 0.00 0.00 26.02 73.98 0.00 0.00 0.00 82.34 0.9074
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 -8.02 2.85 105.17 0.00 0.00 0.00 0.00 0.00 98.70 0.9935
17 89.85 -4.45 0.00 14.61 0.00 0.00 0.00 0.00 98.90 0.9945
18 152.19 -64.95 0.00 0.00 12.75 0.00 0.00 0.00 98.71 0.9935
19 -4.50 0.00 105.35 -0.85 0.00 0.00 0.00 0.00 98.89 0.9945
20 -0.48 0.00 100.28 0.00 0.20 0.00 0.00 0.00 98.65 0.9932
21 84.43 0.00 0.00 15.79 -0.21 0.00 0.00 0.00 98.89 0.9944
22 0.00 0.23 99.63 0.14 0.00 0.00 0.00 0.00 82.79 0.9099
23 0.00 -0.18 99.97 0.00 0.21 0.00 0.00 0.00 98.30 0.9915
24 0.00 246.68 0.00 46.27 -192.95 0.00 0.00 0.00 95.86 0.9791
25 0.00 0.00 88.25 3.06 8.69 0.00 0.00 0.00 82.34 0.9074
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 -4.50 0.22 105.01 -0.73 0.00 0.00 0.00 0.00 98.90 0.9945
27 0.13 -0.23 99.89 0.00 0.21 0.00 0.00 0.00 99.10 0.9955
28 90.79 -12.00 0.00 14.03 7.18 0.00 0.00 0.00 98.90 0.9945
29 -2.74 0.00 103.14 -0.48 0.08 0.00 0.00 0.00 98.90 0.9945
30 0.00 -0.22 100.03 -0.02 0.21 0.00 0.00 0.00 98.71 0.9936
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 0.13 -0.23 99.88 0.00 0.21 0.00 0.00 0.00 99.10 0.9955











WEGGEL AND MAXWELL (1970) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9482 -0.6140 0.7291 0.7478 -0.8535

(2) 89.9037 1.0000 -0.3552 0.7104 0.5182 -0.6710

(3) 37.6940 12.6135 1.0000 -0.4338 -0.9719 0.9261

(4) 53.1579 50.4637 18.8161 1.0000 0.5361 -0.6811

(5) 55.9169 26.8557 94.4494 28.7365 1.0000 -0.9674

(6) 72.8487 45.0203 85.7730 46.3901 93.5950 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 83 84 85 B6 B7 B8
NO
1 -0.011 1.138 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.224 0.000 -0.052 0.000 0.000 0.000 0.000 0.000 0.000
3 -277.171 0.000 0.000 5437.500 0.000 0.000 0.000 0.000 0.000
4 0.088 0.000 0.000 0.000 9.614 0.000 0.000 0.000 0.000
5 0.227 0.000 0.000 0.000 0.000 -0.119 0.000 0.000 0.000
6 0.052 1.003 -0.027 0.000 0.000 0.000 0.000 0.000 0.000
7 -42.636 1.043 0.000 836.046 0.000 0.000 0.000 0.000 0.000
8 -0.005 0.920 0.000 0.000 4.507 0.000 0.000 0.000 0.000
9 0.073 0.820 0.000 0.000 0.000 -0.055 0.000 0.000 0.000
10 -216.621 0.000 -0.031 4251.249 0.000 0.000 0.000 0.000 0.000
11 -0.330 0.000 0.171 0.000 34.999 0.000 0.000 0.000 0.000
12 0.173 0.000 0.105 0.000 0.000 -0.279 0.000 0.000 0.000
13 -175.082 0.000 0.000 3435.088 6.439 0.000 0.000 0.000 0.000
14 -104.626 0.000 0.000 2055.594 0.000 -0.093 0.000 0.000 0.000
15 0.431 0.000 0.000 0.000 -15.647 -0.283 0.000 0.000 0.000
16 1.745 1.007 -0.027 -33.204 0.000 0.000 0.000 0.000 0.000
17 0.003 0.930 -0.003 0.000 3.938 0.000 0.000 0.000 0.000
18 0.047 1.052 -0.034 0.000 0.000 0.015 0.000 0.000 0.000
19 2.772 0.926 0.000 -54.466 4.528 0.000 0.000 0.000 0.000
20 23.004 0.853 0.000 -449.685 0.000 -0.058 0.000 0.000 0.000
21 -0.004 0.920 0.000 0.000 4.477 0.000 0.000 0.000 0.000
22 -100.056 0.000 0.137 1957.297 28.039 0.000 0.000 0.000 0.000
23 139.854 0.000 0.147-2738.823 0.000 -0.380 0.000 0.000 0.000
24 0.042 0.000 0.135 0.000 8.812 -0.234 0.000 0.000 0.000
25 2.676 0.000 0.000 -43.965 -15.792 -0.285 0.000 0.000 0.000
26 2.801 0.936 -0.003 -54.879 3.958 0.000 0.000 0.000 0.000
27 -340.276 3.456 -0.454 6667.283 0.000 0.931 0.000 0.000 0.000
28 -0.002 0.974 -0.010 0.000 3.918 0.013 0.000 0.000 0.000
29 4.442 0.920 0.000 -87.097 4.191 -0.004 0.000 0.000 0.000
30 128.704 0.000 0.163-2521.800 5.531 -0.344 0.000 0.000 0.000
31 -330.208 3.380 -0.440 6469.990 0.240 0.904 0.000 0.000 0.000


112










IWAGAKI AND OTHERS (1974) LABORATORY DATA
n = 23

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.1110 0.0970 1.0000 0.1000 0.00990 1.00514
0.0750 0.0680 1.0000 0.1000 0.00694 1.20049
0.0610 0.0510 1.0000 0.1000 0.00520 1.38621
0.1200 0.1010 1.5000 0.1000 0.00458 1.47755
0.0980 0.0990 1.5000 0.1000 0.00449 1.49241
0.0670 0.0680 1.5000 0.1000 0.00308 1.80074
0.1580 0.1090 1.0000 0.0500 0.01112 0.47410
0.1060 0.0840 1.0000 0.0500 0.00857 0.54006
0.0680 0.0570 1.0000 0.0500 0.00582 0.65561
0.1480 0.1280 1.5000 0.0500 0.00580 0.65625
0.1050 0.0830 1.5000 0.0500 0.00376 0.81496
0.0600 0.0620 1.5000 0.0500 0.00281 0.94293
0.1200 0.0920 2.0000 0.0500 0.00235 1.03209
0.0970 0.0800 2.0000 0.0500 0.00204 1.10680
0.0630 0.0530 2.0000 0.0500 0.00135 1.35980
0.1180 0.0810 1.0000 0.0300 0.00827 0.32998
0.0970 0.0660 1.0000 0.0300 0.00673 0.36556
0.0670 0.0440 1.0000 0.0300 0.00449 0.44772
0.1280 0.1090 1.5000 0.0300 0.00494 0.42669
0.0990 0.0750 1.5000 0.0300 0.00340 0.51439
0.1230 0.0960 2.0000 0.0300 0.00245 0.60622
0.0990 0.0830 2.0000 0.0300 0.00212 0.65197
0.0690 0.0590 2.0000 0.0300 0.00151 0.77328


113











IWAGAKI AND OTHERS (1974) LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 83.59 0.9143
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.0087
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 3.24 0.1801
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 18.87 0.4344
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 15.60 0.3950
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 111.78 -11.78 0.00 0.00 0.00 0.00 0.00 0.00 84.65 0.9200
7 115.30 0.00 -15.30 0.00 0.00 0.00 0.00 0.00 88.85 0.9426
8 91.59 0.00 0.00 8.41 0.00 0.00 0.00 0.00 86.05 0.9276
9 121.51 0.00 0.00 0.00 -21.51 0.00 0.00 0.00 91.23 0.9551
10 0.00 27.60 72.40 0.00 0.00 0.00 0.00 0.00 3.40 0.1844
11 0.00 62.34 0.00 37.66 0.00 0.00 0.00 0.00 64.74 0.8046
12 0.00 -74.64 0.00 0.00 174.64 0.00 0.00 0.00 16.66 0.4082
13 0.00 0.00 -178.30 278.30 0.00 0.00 0.00 0.00 26.56 0.5153
14 0.00 0.00 -130.75 0.00 230.75 0.00 0.00 0.00 22.18 0.4709
15 0.00 0.00 0.00 2720.12-2620.12 0.00 0.00 0.00 26.11 0.5110
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 150.20 -26.70 -23.49 0.00 0.00 0.00 0.00 0.00 91.77 0.9580
17 59.28 23.01 0.00 17.71 0.00 0.00 0.00 0.00 87.32 0.9345
18 126.33 -4.82 0.00 0.00 -21.50 0.00 0.00 0.00 91.35 0.9558
19 104.74 0.00 -17.81 13.07 0.00 0.00 0.00 0.00 93.16 0.9652
20 121.45 0.00 0.69 0.00 -22.14 0.00 0.00 0.00 91.23 0.9552
21 113.34 0.00 0.00 5.46 -18.80 0.00 0.00 0.00 91.83 0.9583
22 0.00 64.81 -5.49 40.68 0.00 0.00 0.00 0.00 67.65 0.8225
23 0.00 128.46 136.07 0.00 -164.53 0.00 0.00 0.00 52.19 0.7224
24 0.00 68.37 0.00 39.43 -7.80 0.00 0.00 0.00 69.57 0.8341
25 0.00 0.00 -180.11 301.56 -21.45 0.00 0.00 0.00 26.56 0.5154
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 81.63 14.61 -14.29 18.05 0.00 0.00 0.00 0.00 93.46 0.9668
27 152.35 -28.81 -25.71 0.00 2.17 0.00 0.00 0.00 91.78 0.9580
28 73.55 24.04 0.00 16.00 -13.59 0.00 0.00 0.00 92.75 0.9631
29 96.09 0.00 -29.80 18.58 15.13 0.00 0.00 0.00 93.54 0.9672
30 0.00 76.01 21.81 32.56 -30.39 0.00 0.00 0.00 72.25 0.8500
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 86.05 8.06 -24.07 19.60 10.36 0.00 0.00 0.00 93.60 0.9675



CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9143 0.0087 -0.1801 0.4344 -0.3950

(2) 83.5941 1.0000 0.1210 0.0534 0.3123 -0.1325

(3) 0.0075 1.4635 1.0000 -0.2617 -0.8359 0.2314

(4) 3.2425 0.2851 6.8481 1.0000 0.2097 0.8239

(5) 18.8663 9.7522 69.8733 4.3989 1.0000 -0.3228

(6) 15.6008 1.7544 5.3556 67.8768 10.4184 1.0000

Numbers in Parentheses are the Variable or Parameter Number











IWAGAKI AND OTHERS (1974) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (88 X8)

B1 B2 B3 B4 B5 B6 B7 B8


EQ BD
NO
1 0.002
2 0.097
3 0.108
4 0.076
5 0.121
6 0.011
7 0.014
8 -0.001
9 0.022
10 0.113
11 -0.099
12 0.112
13 0.089
14 0.118
15 0.097
16 0.032
17 -0.032
18 0.025
19 0.012
20 0.022
21 0.019
22 -0.083
23 0.022
24 -0.077
25 0.090
26 -0.004
27 0.032
28 -0.008
29 0.003
30 -0.073
31 -0.003


0.000 0.000 0.000
0.000 0.000 0.000
0.000 0.000 0.000
4.517 0.000 0.000
0.000 -0.027 0.000
0.000 0.000 0.000
0.000 0.000 0.000
1.715 0.000 0.000
0.000 -0.019 0.000
0.000 0.000 0.000
15.242 0.000 0.000
0.000 -0.028 0.000
5.135 0.000 0.000
0.000 -0.052 0.000
3.562 -0.019 0.000
0.000 0.000 0.000
4.756 0.000 0.000
0.000 -0.018 0.000
2.322 0.000 0.000
0.000 -0.019 0.000
0.890 -0.017 0.000
15.187 0.000 0.000
0.000 -0.144 0.000
14.203 -0.016 0.000
4.977 -0.002 0.000
3.810 0.000 0.000
0.000 0.002 0.000
3.509 -0.017 0.000
3.621 0.016 0.000
11.483 -0.060 0.000
4.086 0.012 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


115


1.199 0.000 0.000
0.000 0.001 0.000
0.000 0.000 -0.181
0.000 0.000 0.000
0.000 0.000 0.000
1.215 -0.007 0.000
1.215 0.000 -0.231
1.132 0.000 0.000
1.151 0.000 0.000
0.000 -0.003 -0.192
0.000 0.085 0.000
0.000 0.007 0.000
0.000 0.000 -0.285
0.000 0.000 0.455
0.000 0.000 0.000
1.247 -0.012 -0.279
0.964 0.021 0.000
1.158 -0.002 0.000
1.127 0.000 -0.274
1.149 0.000 0.009
1.120 0.000 0.000
0.000 0.082 -0.178
0.000 0.068 1.848
0.000 0.083 0.000
0.000 0.000 -0.257
1.043 0.010 -0.261
1.255 -0.013 -0.303
0.977 0.018 0.000
1.134 0.000 -0.503
0.000 0.091 0.666
1.087 0.006 -0.435











WALKER (1974)


LABORATORY DATA
= 15


Bed Equivalent
Slope Wave
Steepness


Water
Depth
(m)

0.0457
0.0396
0.0335
0.0305
0.0518
0.0549
0.0640
0.0610
0.1006
0.0914
0.0732
0.0671
0.0884
0.0914
0.1250


Breaker
Height
(m)

0.0427
0.0366
0.0305
0.0244
0.0518
0.0579
0.0732
0.0762
0.0975
0.0945
0.0914
0.0762
0.0884
0.1036
0.1158


STEPWISE REGRESSION ANALYSIS RESULTS


Eq 100 r*2 r
No


(1) (2) (3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00 0.00 0.00
0.00 100.00 0.00 0.00 0.00
0.00 0.00 100.00 0.00 0.00
0.00 0.00 0.00 100.00 0.00
0.00 0.00 0.00 0.00 100.00
INDEPENDENT VARIABLES


-4.00
0.00
0.00
0.00
-0.03
67.55
2031.31
0.00
0.00
0.00

0.00
9.58
-42.07
0.00
0.00
0.00
-0.40
0.09
126.97
0.00

-0.07
0.10
-52.20
0.00
-4.91

0.10


0.00
100.52
0.00
0.00
100.03
0.00
0.00
99.98
100.04
0.00


0.00
0.00
2.63
0.00
0.00
32.45
0.00
0.02
0.00
-7.50


0.00
0.00
0.00
5.23
0.00
0.00
2131.31
0.00
-0.04
107.50


INDEPENDENT VARIABLES


0.00
5.73
0.00
-0.01
0.00
6.95
-0.18
0.00
31.42
0.00


0.00
0.00
37.48
0.00
-0.03
15.29
0.00
-0.11
-58.39
-0.04


100.55
0.00
0.00
100.69
100.26
0.00
100.58
100.02
0.00
100.04


INDEPENDENT VARIABLES
100.39 -0.03 0.00
100.27 0.00 -0.11
0.00 -2.49 41.64
100.26 -0.02 -0.05
103.89 -1.22 2.24
INDEPENDENT VARIABLES
100.27 0.00 -0.10


(6) (7)
TAKEN ONE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FIVE AT A
0.00 0.00


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


91.82
3.52
5.13
35.94
47.95

91.87
91.95
91.95
91.95
15.17
62.09
67.91
43.48
60.31
48.15

91.95
92.02
93.44
92.02
92.43
92.75
62.50
70.75
80.17
60.31

92.23
94.03
93.46
93.55
80.17


0.9583
0.1877
0.2265
0.5995
0.6925

0.9585
0.9589
0.9589
0.9589
0.3895
0.7880
0.8241
0.6594
0.7766
0.6939

0.9589
0.9593
0.9666
0.9593
0.9614
0.9631
0.7906
0.8411
0.8954
0.7766

0.9604
0.9697
0.9668
0.9672
0.8954


94.03 0.9697


117


Wave
Period
(s)

1.1700
1.6700
2.0000
2.3300
2.3300
2.0000
1.6700
1.1700
1.1700
1.6700
2.0000
2.3300
2.3300
2.0000
1.6700


0.00318
0.00134
0.00078
0.00046
0.00097
0.00148
0.00268
0.00568
0.00727
0.00346
0.00233
0.00143
0.00166
0.00264
0.00424


0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330
0.0330


Surf
Sim
Parm

0.58494
0.90208
1.18345
1.54145
1.05743
0.85856
0.63787
0.43786
0.38702
0.56125
0.68326
0.87198
0.80961
0.64182
0.50693


10


104.00
-0.52
97.37
94.77
0.00
0.00
0.00-
0.00
0.00
0.00

-0.55
84.69
104.60
-0.68
-0.24
77.75
0.00
0.00
0.00
0.00

-0.29
-0.26
113.06
-0.19
0.00

-0.26











WALKER (1974) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9583 -0.1877

(2) 91.8250 1.0000 -0.1747

(3) 3.5242 3.0504 1.0000

(4) 5.1296 7.3585 18.7117

(5) 35.9444 35.4577 65.8554

(6) 47.9538 55.8832 51.7323


0.2265 0.5995 -0.6925

0.2713 0.5955 -0.7476

0.4326 -0.8115 0.7193

1.0000 -0.0786 0.1730

0.6177 1.0000 -0.8301

2.9922 68.9023 1.0000


Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 B2 B3 84 B5 B6 B7 B8


0.907
0.000
0.000
0.000
0.000
0.903
0.916
0.881
0.945
0.000
0.000
0.000
0.000
0.000
0.000
0.915
0.855
1.166
0.894
1.021
0.953
0.000
0.000
0.000
0.000
0.852
1.260
1.204
1.055
0.000
1.252


0.000 0.000
-0.012 0.000
0.000 2373.086
0.000 0.000
0.000 0.000
-0.001 0.000
0.000 -378.369
0.000 0.000
0.000 0.000
-0.022 3966.067
0.056 0.000
0.041 0.000
0.000 2884.630
0.000 3740.098
0.000 0.000
0.000 -355.521
0.004 0.000
-0.018 0.000
0.000 -284.020
0.000 -929.204
0.000 0.000
0.061 -864.320
0.033 2007.118
0.062 0.000
0.000 3736.613
0.008 -623.064
-0.019-1038.754
-0.021 0.000
0.000-1223.032
0.063 -73.803
-0.018-1052.482


0.000
0.000
0.000
8.518
0.000
0.000
0.000
0.637
0.000
0.000
18.608
0.000
8.826
0.000
1.129
0.000
1.548
0.000
0.494
0.000
2.283
19.543
0.000
10.713
0.071
2.269
0.000
-0.710
2.753
10.807
0.173


0.000
0.000
0.000
0.000
-0.061
0.000
0.000
0.000
0.005
0.000
0.000
-0.101
0.000
-0.066
-0.055
0.000
0.000
0.038
0.000
0.011
0.017
0.000
-0.097
-0.068
-0.066
0.000
0.047
0.040
0.028
-0.067
0.046


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


1 0.004
2 0.090
3 -78.242
4 0.045
5 0.115
6 0.006
7 12.489
8 0.004
9 -0.003
10 -130.769
11 -0.084
12 0.072
13 -95.146
14 -123.301
15 0.108
16 11.736
17 -0.004
18 -0.011
19 9.376
20 30.650
21 -0.019
22 28.426
23 -66.151
24 -0.022
25 -123.187
26 20.548
27 34.255
28 -0.007
29 40.323
30 2.412
31 34.707











VAN DORN (1978) LABORATORY DATA


Wave
Period
(s)

1.6500
2.3700
3.4300
4.8000
1.6500
2.3700
3.4300
4.8000
1.6500
2.3700
3.4300
4.8000


n = 12

Bed Equivalent
Slope Wave
Steepness


0.0220
0.0220
0.0220
0.0220
0.0400
0.0400
0.0400
0.0400
0.0830
0.0830
0.0830
0.0830


0.00622
0.00273
0.00113
0.00062
0.00607
0.00262
0.00106
0.00053
0.00585
0.00233
0.00128
0.00048


Surf
Sim
Parm

0.27891
0.42144
0.65518
0.88351
0.51333
0.78206
1.22967
1.74237
1.08546
1.72121
2.31662
3.79508


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
-16.44 0.00 0.00
0.00 9.04 0.00
0.00 0.00 -19.22
39.03 0.00 0.00
0.00 89.69 0.00
0.00 0.00 50.74
690.84 -590.84 0.00
2.18 0.00 97.82
0.00 -430.83 530.83
INDEPENDENT VARIABLES


0.00
10.45
0.00
71.61
0.00
56.32
-309.92
0.00
86.39
-850.68


0.00
0.00
-22.25
0.00
-1.20
-77.07
0.00
-10.36
-74.12
-83.17


INDEPENDENT VARIABLES
-95.82 71.76 0.00
-100.89 0.00 41.87
0.00 40.50 -36.11
-75.53 49.75 14.26
224.58 -149.24 -38.57


(6) (7)
TAKEN ONE AT A T
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A T
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00


INDEPENDENT VARIABLES TAKEN FIVE AT A
31 167.94 -40.18 -122.44 59.49 35.20 0.00 0.00


(8)
IME
0.00
0.00
0.00
0.00
0.00
IME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


64.62
45.33
34.72
59.93
63.69

65.29
81.41
68.26
76.15
80.04
59.97
69.99
91.59
63.70
80.76

85.98
68.69
76.30
92.28
81.44
81.15
91.61
80.78
83.53
91.69

92.28
88.50
83.90
92.78
91.93

93.16


0.8039
0.6733
0.5892
0.7742
0.7981

0.8080
0.9023
0.8262
0.8727
0.8947
0.7744
0.8366
0.9570
0.7981
0.8987

0.9273
0.8288
0.8735
0.9606
0.9024
0.9008
0.9571
0.8988
0.9139
0.9575

0.9606
0.9407
0.9160
0.9632
0.9588

0.9652


Water
Depth
(m)

0.2080
0.1870
0.1380
0.1500
0.2170
0.1690
0.1110
0.1110
0.1540
0.1080
0.0930
0.0930


Breaker
Height

0.1660
0.1500
0.1300
0.1400
0.1620
0.1440
0.1220
0.1190
0.1560
0.1280
0.1480
0.1080


(1)

100.00
0.00
0.00
0.00
0.00

106.27
116.44
90.96
119.22
0.00
0.00
0.00
0.00
0.00
0.00

223.87
82.61
127.87
123.95
117.29
120.75
0.00
0.00
0.00
0.00

124.21
255.62
49.60
111.52
0.00


(2)

0.00
100.00
0.00
0.00
0.00

-6.27
0.00
0.00
0.00
60.97
10.31
49.26
0.00
0.00
0.00

-61.43
6.95
-5.63
0.00
0.00
0.00
31.60
65.37
87.73
0.00

-0.14
-96.59
46.00
0.00
63.23


-62.44
0.00
0.00
-95.56
-16.09
0.00
378.32
44.98
0.00
1033.85


1
]
1











VAN DORN (1978) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8039 -0.6733 -0.5892 0.7742 -0.7981

(2) 64.6176 1.0000 -0.7729 -0.2379 0.8309 -0.6849

(3) 45.3275 59.7383 1.0000 0.0000 -0.8822 0.5894

(4) 34.7164 5.6589 0.0000 1.0000 -0.0348 0.7310

(5) 59.9328 69.0379 77.8190 0.1211 1.0000 -0.5315

(6) 63.6885 46.9095 34.7433 53.4369 28.2442 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (82 X2) + (83 X3) + ... + (B8 X8)

BO B1 B2 B 3 B4 B5 B6 B7 B8


1.923 0.000 0.000
0.000 -0.024 0.000
0.000 0.000 -0.957
0.000 0.000 0.000
0.000 0.000 0.000
1.684 -0.005 0.000
1.683 0.000 -0.685
1.241 0.000 0.000
1.159 0.000 0.000
0.000 -0.024 -0.957
0.000 0.002 0.000
0.000 -0.011 0.000
0.000 0.000 -0.914
0.000 0.000 -0.020
0.000 0.000 0.000
0.991 -0.012 -0.797
1.284 0.005 0.000
1.058 -0.002 0.000
0.387 0.000 -0.860
1.629 0.000 -0.645
0.311 0.000 0.000
0.000 -0.001 -0.916
0.000 -0.029 -1.254
0.000 0.013 0.000
0.000 0.000 -0.995
0.386 0.000 -0.860
1.175 -0.020 -1.337
0.307 0.013 0.000
0.530 0.000 -1.035
0.000 -0.005 -1.113
0.563 -0.006 -1.184


0.000 0.000
0.000 0.000
0.000 0.000
15.060 0.000
0.000 -0.035
0.000 0.000
0.000 0.000
6.676 0.000
0.000 -0.020
0.000 0.000
15.809 0.000
0.000 -0.027
14.679 0.000
0.000 -0.034
9.490 -0.023
0.000 0.000
8.790 0.000
0.000 -0.020
12.090 0.000
0.000 -0.002
7.846 -0.022
14.087 0.000
0.000 0.011
15.203 -0.026
15.361 0.003
12.082 0.000
0.000 0.021
13.569 -0.024
12.796 0.007
13.883 0.007
10.801 0.013


-0.123
0.217
0.191
0.106
0.190
-0.076
-0.057
-0.045
0.009
0.263
0.100
0.213
0.151
0.190
0.151
0.083
-0.072
0.030
0.101
-0.049
0.109
0.157
0.280
0.100
0.150
0.102
0.081
0.059
0.079
0.169
0.098


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











HANSEN AND SVENDSEN (1979) LABORATORY DATA
n = 16

Water Breaker Wave Bed Equivalent Surf
Depth Height Period SLope Wave Sim
(m) (m) (s) Steepness Parm

0.0470 0.0430 0.8300 0.0292 0.00637 0.36576
0.1430 0.1070 1.0000 0.0292 0.01092 0.27945
0.0950 0.0780 1.0000 0.0292 0.00796 0.32730
0.0570 0.0510 1.0000 0.0292 0.00520 0.40477
0.1420 0.1220 1.2500 0.0292 0.00797 0.32713
0.1050 0.0910 1.2500 0.0292 0.00594 0.37878
0.0640 0.0570 1.2500 0.0292 0.00372 0.47860
0.1490 0.1400 1.6700 0.0292 0.00512 0.40799
0.1440 0.1300 1.6700 0.0292 0.00476 0.42339
0.1370 0.1180 1.6700 0.0292 0.00432 0.44440
0.1190 0.1040 1.6700 0.0292 0.00381 0.47336
0.1110 0.1090 2.0000 0.0292 0.00278 0.55375
0.0730 0.0720 2.0000 0.0292 0.00184 0.68133
0.1320 0.1290 2.5000 0.0292 0.00211 0.63627
0.0890 0.0880 2.5000 0.0292 0.00144 0.77036
0.1000 0.0950 3.3300 0.0292 0.00087 0.98759


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 93.02 0.9645
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 2.69 0.1641
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 22.40 0.4733
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 5.50 0.2346
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 3.37 0.1836
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 115.73 -15.73 0.00 0.00 0.00 0.00 0.00 0.00 96.07 0.9802
7 1.67 0.00 98.33 0.00 0.00 0.00 0.00 0.00 93.04 0.9646
8 88.82 0.00 0.00 11.18 0.00 0.00 0.00 0.00 98.21 0.9910
9 117.07 0.00 0.00 0.00 -17.07 0.00 0.00 0.00 96.03 0.9800
10 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 22.41 0.4734
11 0.00 57.65 0.00 42.35 0.00 0.00 0.00 0.00 41.53 0.6444
12 0.00-1245.15 0.00 0.00 1345.15 0.00 0.00 0.00 91.68 0.9575
13 0.00 0.00 99.99 0.01 0.00 0.00 0.00 0.00 32.44 0.5695
14 0.00 0.00 100.01 0.00 -0.01 0.00 0.00 0.00 30.00 0.5477
15 0.00 0.00 0.00 72.83 27.17 0.00 0.00 0.00 5.64 0.2375
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 0.49 -0.07 99.59 0.00 0.00 0.00 0.00 0.00 96.32 0.9814
17 80.42 5.80 0.00 13.78 0.00 0.00 0.00 0.00 98.39 0.9919
18 114.64 -26.87 0.00 0.00 12.23 0.00 0.00 0.00 96.08 0.9802
19 0.38 0.00 99.57 0.05 0.00 0.00 0.00 0.00 98.55 0.9927
20 0.42 0.00 99.64 0.00 -0.07 0.00 0.00 0.00 96.31 0.9814
21 80.41 0.00 0.00 13.54 6.05 0.00 0.00 0.00 98.42 0.9920
22 0.00 0.05 99.91 0.04 0.00 0.00 0.00 0.00 52.99 0.7279
23 0.00 0.40 100.04 0.00 -0.43 0.00 0.00 0.00 93.01 0.9644
24 0.00 829.84 0.00 72.02 -801.86 0.00 0.00 0.00 94.36 0.9714
25 0.00 0.00 99.99 0.01 0.00 0.00 0.00 0.00 32.44 0.5696
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 0.39 0.03 99.52 0.07 0.00 0.00 0.00 0.00 98.70 0.9935
27 0.48 -0.06 99.60 0.00 -0.01 0.00 0.00 0.00 96.32 0.9814
28 81.63 -13.03 0.00 12.66 18.75 0.00 0.00 0.00 98.44 0.9922
29 0.41 0.00 99.49 0.07 0.03 0.00 0.00 0.00 98.71 0.9935
30 0.00 0.39 99.95 0.04 -0.38 0.00 0.00 0.00 95.71 0.9783
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 0.42 -0.01 99.48 0.07 0.04 0.00 0.00 0.00 98.71 0.9935











HANSEN AND SVENDSEN (1979) LABORATORY DATA (CONT)

CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9645 0.1641

(2) 93.0208 1.0000 0.3405

(3) 2.6927 11.5928 1.0000

(4) 22.4017 22.8697 10.6490

(5) 5.5035 0.0050 65.2841

(6) 3.3702 0.0110 87.2480


0.4733 0.2346 -0.1836

0.4782 0.0070 -0.0105

0.3263 -0.8080 0.9341

1.0000 -0.1645 0.1845

2.7052 1.0000 -0.8623

3.4043 74.3567 1.0000


Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 B2 B3 B4 B5 B6 B7 B8


1.100
0.000
0.000
0.000
0.000
1.172
1.092
1.098
1.098
0.000
0.000
0.000
0.000
0.000
0.000
1.145
1.063
1.233
1.061
1.064
1.099
0.000
0.000
0.000
0.000
1.031
1.132
1.187
1.064
0.000
1.079


0.000 0.000
0.008 0.000
0.000 6366.667
0.000 0.000
0.000 0.000
-0.009 0.000
0.000 210.534
0.000 0.000
0.000 0.000
0.001 6319.292
0.050 0.000
0.130 0.000
0.000 7077.138
0.000 7062.731
0.000 0.000
-0.010 770.643
0.004 0.000
-0.017 0.000
0.000 910.639
0.000 822.247
0.000 0.000
0.040 4895.088
0.122 1758.246
0.130 0.000
0.000 7085.818
0.004 870.062
-0.008 779.125
-0.011 0.000
0.000 846.972
0.123 1771.181
-0.002 837.561


0.000
0.000
0.000
2.906
0.000
0.000
0.000
2.822
0.000
0.000
13.103
0.000
3.978
0.000
3.686
0.000
3.720
0.000
2.963
0.000
3.778
11.815
0.000
4.009
3.822
3.776
0.000
3.759
3.792
4.024
3.789


0.000
0.000
0.000
0.000
-0.033
0.000
0.000
0.000
-0.031
0.000
0.000
-0.471
0.000
-0.050
0.013
0.000
0.000
0.025
0.000
-0.033
0.016
0.000
-0.449
-0.422
-0.003
0.000
-0.005
0.053
0.014
-0.400
0.020


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


122


1 0.001
2 0.093
3 -185.796
4 0.093
5 0.123
6 0.010
7 -6.145
8 -0.012
9 0.017
10 -184.414
11 -0.039
12 0.124
13 -206.560
14 -206.096
15 0.083
16 -22.489
17 -0.020
18 0.004
19 -26.599
20 -23.988
21 -0.024
22 -142.949
23 -51.214
24 0.081
25 -206.811
26 -25.422
27 -22.736
28 -0.033
29 -24.751
30 -51.635
31 -24.478











SINGAMSETTI AND WIND (1980) LABORATORY DATA
n =95

Water Breaker Wave Bed Equivalent Surf
Depth Height Period SLope Wave Sim
(m) (m) (s) Steepness Parm

0.1310 0.1170 1.5500 0.2000 0.00497 2.83715
0.1600 0.1930 1.5500 0.2000 0.00820 2.20900
0.1240 0.1560 1.2800 0.2000 0.00972 2.02904
0.1030 0.0970 1.2800 0.2000 0.00604 2.57316
0.0780 0.0950 1.5500 0.2000 0.00403 3.14857
0.0820 0.1060 1.0400 0.2000 0.01000 1.99997
0.1340 0.1600 1.2800 0.2000 0.00996 2.00352
0.0790 0.0910 1.0400 0.2000 0.00859 2.15852
0.0990 0.1170 1.0400 0.2000 0.01104 1.90363
0.1950 0.1840 1.7200 0.2000 0.00635 2.51051
0.1170 0.1250 1.7200 0.2000 0.00431 3.04591
0.1390 0.1620 1.2800 0.2000 0.01009 1.99111
0.1040 0.1210 1.2800 0.2000 0.00754 2.30388
0.0830 0.0890 1.2800 0.2000 0.00554 2.68632
0.0980 0.0930 1.5500 0.2000 0.00395 3.18224
0.0820 0.0870 1.0400 0.2000 0.00821 2.20758
0.1340 0.1500 1.2800 0.2000 0.00934 2.06922
0.0990 0.0770 1.0400 0.2000 0.00726 2.34656
0.0990 0.1180 1.0400 0.2000 0.01113 1.89555
0.1950 0.1840 1.7200 0.2000 0.00635 2.51051
0.1770 0.1240 1.7200 0.2000 0.00428 3.05816
0.1290 0.1370 1.5500 0.1000 0.00582 1.31094
0.2000 0.1690 1.5500 0.1000 0.00718 1.18032
0.1290 0.1180 1.2800 0.1000 0.00735 1.16649
0.1080 0.0860 1.2800 0.1000 0.00536 1.36639
0.1030 0.1110 1.5500 0.1000 0.00471 1.45641
0.1130 0.0910 1.0400 0.1000 0.00859 1.07926
0.1460 0.1350 1.2800 0.1000 0.00841 1.09058
0.0970 0.0730 1.0400 0.1000 0.00689 1.20499
0.1290 0.1060 1.0400 0.1000 0.01000 0.99998
0.1860 0.1690 1.7200 0.1000 0.00583 1.30978
0.1170 0.1410 1.7200 0.1000 0.00486 1.43394
0.1840 0.1500 1.2800 0.1000 0.00934 1.03461
0.1300 0.1410 1.5500 0.1000 0.00599 1.29222
0.1880 0.1700 1.5500 0.1000 0.00722 1.17685
0.1310 0.1180 1.2800 0.1000 0.00735 1.16649
0.0900 0.1010 1.2800 0.1000 0.00629 1.26085
0.1080 0.1190 1.5500 0.1000 0.00505 1.40660
0.0900 0.0860 1.0300 0.1000 0.00827 1.09951
0.1730 0.1430 1.2800 0.1000 0.00891 1.05963
0.0780 0.0780 1.0300 0.1000 0.00750 1.15452
0.1350 0.1120 1.0300 0.1000 0.01077 0.96348
0.1850 0.1750 1.7100 0.1000 0.00611 1.27965
0.1240 0.1400 1.7100 0.1000 0.00489 1.43069
0.1880 0.1560 1.2800 0.1000 0.00972 1.01452
0.1700 0.1400 1.5500 0.0500 0.00595 0.64841
0.2020 0.1740 1.5500 0.0500 0.00739 0.58162
0.1270 0.1150 1.2800 0.0500 0.00716 0.59080
0.1020 0.0970 1.2800 0.0500 0.00604 0.64329
0.1030 0.1060 1.5500 0.0500 0.00450 0.74518
0.1080 0.0880 1.0400 0.0500 0.00830 0.54875
0.1740 0.1350 1.2800 0.0500 0.00841 0.54529
0.0930 0.0790 1.0400 0.0500 0.00745 0.57917
0.1300 0.1010 1.0400 0.0500 0.00953 0.51222
0.2020 0.1760 1.7300 0.0500 0.00600 0.64547
0.1250 0.1330 1.7300 0.0500 0.00453 0.74251
0.2030 0.1630 1.2800 0.0500 0.01015 0.49625
0.1600 0.1420 1.5500 0.0500 0.00603 0.64383
0.2130 0.1810 1.5500 0.0500 0.00769 0.57026
0.1350 0.1190 1.2800 0.0500 0.00741 0.58079
0.1060 0.1010 1.2800 0.0500 0.00629 0.63042
0.1040 0.1060 1.5500 0.0500 0.00450 0.74518
0.1000 0.0920 1.0400 0.0500 0.00868 0.53669
0.1650 0.1330 1.2800 0.0500 0.00828 0.54937
0.0830 0.0770 1.0400 0.0500 0.00726 0.58664
0.1350 0.1010 1.0400 0.0500 0.00953 0.51222


123











SINGAMSETTI AND WIND (1980) LABORATORY DATA COUNT )


Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.1810 0.1710 1.7300 0.0500 0.00583 0.65483
0.1280 0.1320 1.7300 0.0500 0.00450 0.74532
0.1930 0.1650 1.2800 0.0500 0.01028 0.49323
0.1450 0.1360 1.5500 0.0250 0.00578 0.32894
0.2030 0.1700 1.5500 0.0250 0.00722 0.29421
0.1400 0.1190 1.2800 0.0250 0.00741 0.29040
0.1110 0.0930 1.2800 0.0250 0.00579 0.32849
0.1180 0.1120 1.5500 0.0250 0.00476 0.36247
0.1170 0.0960 1.0400 0.0250 0.00906 0.26269
0.0930 0.0790 1.0400 0.0250 0.00745 0.28958
0.2200 0.1590 1.2800 0.0250 0.00990 0.25123
0.1510 0.1370 1.2800 0.0250 0.00853 0.27065
0.1530 0.1100 1.0400 0.0250 0.01038 0.24541
0.1690 0.1240 1.0400 0.0250 0.01170 0.23114
0.2050 0.1330 1.0400 0.0250 0.01255 0.22318
0.1500 0.1320 1.7200 0.0250 0.00455 0.37051
0.1450 0.1360 1.5500 0.0250 0.00578 0.32894
0.2130 0.1690 1.5500 0.0250 0.00718 0.29508
0.1300 0.1180 1.2800 0.0250 0.00735 0.29162
0.1010 0.0920 1.2800 0.0250 0.00573 0.33027
0.1180 0.1110 1.5500 0.0250 0.00471 0.36410
0.1270 0.0950 1.0400 0.0250 0.00896 0.26407
0.0930 0.0780 1.0400 0.0250 0.00736 0.29143
0.1950 0.1600 1.2800 0.0250 0.00996 0.25044
0.1660 0.1320 1.2800 0.0250 0.00822 0.27572
0.1430 0.1090 1.0400 0.0250 0.01028 0.24653
0.1640 0.1250 1.0400 0.0250 0.01179 0.23021
0.1950 0.1370 1.0400 0.0250 0.01292 0.21990
0.1550 0.1400 1.7200 0.0250 0.00483 0.35976


124











SINGAMSETTI AND WIND (1980) LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(1)

100.00
0.00
0.00
0.00
0.00

141.68
113.80
83.97
112.30
0.00
0.00
0.00
0.00
0.00
0.00

158.64
63.50
138.61
95.20
113.76
97.74
0.00
0.00
0.00
0.00

73.21
351.25
60.29
75.97
0.00


(2)

0.00
100.00
0.00
0.00
0.00

-41.68
0.00
0.00
0.00
126.05
62.24
126.74
0.00
0.00
0.00

-42.16
15.27
-26.01
0.00
0.00
0.00
65.23
115.17
66.14
0.00

15.12
-224.00
25.40
0.00
61.77


31 217.72 -116.40


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
-13.80
0.00
0.00
-26.05
0.00
0.00
-83.03
16.10
0.00


0.00
0.00
16.03
0.00
0.00
37.76
0.00
183.03
0.00
195.64


0.00
0.00
0.00
-12.30
0.00
0.00
-26.74
0.00
83.90
-95.64


INDEPENDENT VARIABLES


-16.47
0.00
0.00
-11.34
-20.20
0.00
-4.09
54.06
0.00
-143.90


0.00
21.23
0.00
16.14
0.00
11.96
38.86
0.00
37.54
133.96


0.00
0.00
-12.60
0.00
6.44
-9.70
0.00
-69.23
-3.67
109.93


INDEPENDENT VARIABLES
-9.61 21.28 0.00
-169.82 0.00 142.57
0.00 21.62 -7.31
-47.15 32.66 38.52
-13.78 42.06 9.94
INDEPENDENT VARIABLES
-111.81 16.81 93.68


(6) (7) (8)
TAKEN ONE AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN TWO AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN THREE AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN FOUR AT A TIME
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
0.00 0.00 0.00
TAKEN FIVE AT A TIME
0.00 0.00 0.00


70.53
12.79
8.40
6.39
8.83

73.50
80.17
73.92
78.90
22.55
69.54
27.12
13.96
8.85
12.20

82.56
73.99
79.91
82.90
80.33
80.22
78.26
33.47
76.56
16.58

82.96
89.36
80.40
88.71
80.21


0.8398
0.3576
0.2899
0.2529
0.2972

0.8573
0.8954
0.8598
0.8883
0.4749
0.8339
0.5207
0.3737
0.2974
0.3493

0.9087
0.8602
0.8939
0.9105
0.8963
0.8956
0.8847
0.5785
0.8750
0.4072

0.9108
0.9453
0.8966
0.9419
0.8956


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8398 0.3576 -0.2899 0.2529 -0.2972

(2) 70.5252 1.0000 0.5914 0.0245 0.0823 -0.0093

(3) 12.7879 34.9789 1.0000 0.0614 -0.7308 0.2050

(4) 8.4034 0.0603 0.3767 1.0000 -0.0603 0.9646

(5) 6.3941 0.6772 53.4070 0.3633 1.0000 -0.2534

(6) 8.8309 0.0086 4.2044 93.0547 6.4189 1.0000

Numbers in Parentheses are the Variable or Parameter Number


89.42 0.9456











SINGAMSETTI AND WIND (1980) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 XI) + (B2 X2) + (B3 X3) + ... + (B8 X8)

BO B1 B2 B3 B4 B5 B6 B7 B8


0.000 0.000 0.000 0.000 0.000
0.057 0.000 0.000 0.000 0.000
0.000 -0.172 0.000 0.000 0.000
0.000 0.000 4.593 0.000 0.000
0.000 0.000 0.000 -0.014 0.000
-0.034 0.000 0.000 0.000 0.000
0.000 -0.185 0.000 0.000 0.000
0.000 0.000 3.361 0.000 0.000
0.000 0.000 0.000 -0.014 0.000
0.060 -0.186 0.000 0.000 0.000
0.185 0.000 20.047 0.000 0.000
0.069 0.000 0.000 -0.018 0.000
0.000 -0.164 4.292 0.000 0.000
0.000 -0.028 0.000 -0.012 0.000
0.000 0.000 3.447 -0.012 0.000
-0.031 -0.179 0.000 0.000 0.000
0.021 0.000 5.262 0.000 0.000
-0.021 0.000 0.000 -0.012 0.000
0.000 -0.179 3.019 0.000 0.000
0.000 -0.272 0.000 0.007 0.000
0.000 0.000 2.163 -0.012 0.000
0.187 -0.176 19.869 0.000 0.000
0.095 0.670 0.000 -0.071 0.000
0.187 0.000 18.941 -0.013 0.000
0.000 -0.683 7.552 0.043 0.000
0.019 -0.178 4.695 0.000 0.000
-0.096 -1.092 0.000 0.076 0.000
0.035 0.000 5.246 -0.012 0.000
0.000 -0.955 7.862 0.065 0.000
0.186 -0.623 22.603 0.037 0.000
-0.075 -1.078 1.926 0.075 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


126


1 0.004
2 0.062
3 0.154
4 0.104
5 0.153
6 0.029
7 0.019
8 -0.019
9 0.019
10 0.075
11 -0.259
12 0.065
13 0.121
14 0.153
15 0.125
16 0.041
17 -0.047
18 0.033
19 -0.002
20 0.018
21 0.003
22 -0.245
23 0.028
24 -0.239
25 0.096
26 -0.027
27 0.081
28 -0.043
29 -0.042
30 -0.264
31 0.052


1.072
0.000
0.000
0.000
0.000
1.233
1.082
1.052
1.068
0.000
0.000
0.000
0.000
0.000
0.000
1.226
0.941
1.166
1.064
1.088
1.056
0.000
0.000
0.000
0.000
0.965
1.604
0.874
1.093
0.000
1.491











NADAOKA (1986) LABORATORY DATA


Wave
Period
(s)

1.3200
1.5100
1.7000
1.9100
2.3400
2.9900
1.2100
1.0700
0.9200
2.1300
1.8900
1.6200


n = 12

Bed Equivalent
Slope Wave
Steepness


0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500
0.0500


0.02981
0.01025
0.00812
0.00682
0.00464
0.00298
0.01310
0.01301
0.01242
0.00553
0.00574
0.00587


Surf
Sim
Parm

0.28954
0.49390
0.55484
0.60523
0.73401
0.91608
0.43681
0.43832
0.44870
0.67219
0.65985
0.65254


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(1)

100.00
0.00
0.00
0.00
0.00

153.92
-0.02
79.46
191.53
0.00
0.00
0.00
0.00
0.00
0.00

0.07
154.73
280.09
0.03
0.12
214.12
0.00
0.00
0.00
0.00

0.04
27.04
231.64
0.05
0.00


(2)

0.00
100.00
0.00
0.00
0.00

-53.92
0.00
0.00
0.00
0.00
54.19
-294.27
0.00
0.00
0.00

-0.03
-54.49
145.59
0.00
0.00
0.00
-0.04
-0.06
1721.78
0.00

-0.01
14.01
155.74
0.00
-0.04


31 0.03 -0.02


(3) (4) (5)
INDEPENDENT VARIABLES
0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
0.00 0.00 100.00
INDEPENDENT VARIABLES


0.00
100.02
0.00
0.00
100.00
0.00
0.00
100.00
100.00
0.00


0.00
0.00
20.54
0.00
0.00
45.81
0.00
0.00
0.00
43.51


0.00
0.00
0.00
-91.53
0.00
0.00
394.27
0.00
0.00
56.49


INDEPENDENT VARIABLES


0.00
-0.24
0.00
0.01
0.00
-4.65
-0.03
0.00


0.00
0.00
-325.68
0.00
-0.06
-109.47
0.00
0.07


262.67-1884.45
0.00 0.00


99.96
0.00
0.00
99.96
99.94
0.00
100.06
99.98
0.00
99.99


INDEPENDENT VARIABLES
99.96 0.01 0.00
90.33 0.00 -31.39
0.00 11.24 -298.62
99.96 0.01 -0.01
99.98 0.00 0.06


(6) (7)
TAKEN ONE AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN FOUR AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00


INDEPENDENT VARIABLES TAKEN FIVE AT A
99.97 0.01 0.02 0.00 0.00


127


Water
Depth
(m)

0.6050
0.2840
0.2480
0.2570
0.2480
0.2210
0.2590
0.1850
0.1400
0.2500
0.2090
0.1440


Breaker
Height
(m)

0.5090
0.2290
0.2300
0.2440
0.2490
0.2610
0.1880
0.1460
0.1030
0.2460
0.2010
0.1510


(8)
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


91.61
0.54
85.42
62.19
22.48

98.90
94.92
98.22
99.32
87.04
89.66
94.19
85.64
85.49
76.88

99.31
98.90
99.48
99.30
99.45
99.33
90.02
99.04
97.51
85.93

99.62
99.48
99.50
99.59
99.19

99.69


0.9571
0.0733
0.9242
0.7886
0.4741

0.9945
0.9743
0.9910
0.9966
0.9330
0.9469
0.9705
0.9254
0.9246
0.8768

0.9966
0.9945
0.9974
0.9965
0.9972
0.9966
0.9488
0.9952
0.9875
0.9270

0.9981
0.9974
0.9975
0.9980
0.9959

0.9984











NADAOKA (1986) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9571 -0.0733 -0.9242 0.7886 -0.4741

(2) 91.6127 1.0000 0.1998 -0.8729 0.6114 -0.2119

(3) 0.5376 3.9909 1.0000 0.2139 -0.6164 0.9069

(4) 85.4191 76.1957 4.5753 1.0000 -0.8778 0.5381

(5) 62.1852 37.3852 37.9972 77.0511 1.0000 -0.8541

(6) 22.4811 4.4887 82.2557 28.9528 72.9467 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 83 B4 B5 B6 B7 B8
NO
1 -0.007 1.136 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.280 0.000 -0.015 0.000 0.000 0.000 0.000 0.000 0.000
3 1913.858 0.000 0.000********* 0.000 0.000 0.000 0.000 0.000
4 0.124 0.000 0.000 0.000 13.171 0.000 0.000 0.000 0.000
5 0.449 0.000 0.000 0.000 0.000 -0.339 0.000 0.000 0.000
6 0.075 1.201 -0.056 0.000 0.000 0.000 0.000 0.000 0.000
7 771.863 0.750 0.000********* 0.000 0.000 0.000 0.000 0.000
8 -0.006 0.900 0.000 0.000 5.424 0.000 0.000 0.000 0.000
9 0.126 1.064 0.000 0.000 0.000 -0.203 0.000 0.000 0.000
10 1971.534 0.000 0.027********* 0.000 0.000 0.000 0.000 0.000
11 -0.177 0.000 0.136 0.000 20.024 0.000 0.000 0.000 0.000
12 0.494 0.000 0.411 0.000 0.000 -1.643 0.000 0.000 0.000
13 2093.625 0.000 0.000********* -1.652 0.000 0.000 0.000 0.000
14 1950.166 0.000 0.000********* 0.000 0.023 0.000 0.000 0.000
15 -0.282 0.000 0.000 0.000 23.683 0.527 0.000 0.000 0.000
16 -466.415 1.456 -0.075 9329.416 0.000 0.000 0.000 0.000 0.000
17 0.075 1.203 -0.057 0.000 -0.044 0.000 0.000 0.000 0.000
18 0.179 0.913 0.063 0.000 0.000 -0.424 0.000 0.000 0.000
19 -972.631 1.199 0.00019450.603 9.735 0.000 0.000 0.000 0.000
20 -223.350 1.166 0.000 4469.456 0.000 -0.232 0.000 0.000 0.000
21 0.138 1.082 0.000 0.000 -0.547 -0.221 0.000 0.000 0.000
22 517.408 0.000 0.109********* 15.002 0.000 0.000 0.000 0.000
23 852.104 0.000 0.281********* 0.000 -1.072 0.000 0.000 0.000
24 0.222 0.000 0.318 0.000 8.461 -1.040 0.000 0.000 0.000
25 2463.491 0.000 0.000********* -7.032 -0.138 0.000 0.000 0.000
26 -824.948 1.371 -0.04116498.396 5.162 0.000 0.000 0.000 0.000
27 -0.522 0.914 0.063 14.025 0.000 -0.424 0.000 0.000 0.000
28 0.170 0.851 0.077 0.000 0.962 -0.438 0.000 0.000 0.000
29 -595.920 1.200 0.00011918.957 4.239 -0.142 0.000 0.000 0.000
30 1158.185 0.000 0.279********* -4.075 -1.158 0.000 0.000 0.000
31-1902.868 2.108 -0.21538051.320 10.486 0.644 0.000 0.000 0.000


128











SMITH AND KRAUS (1990) LABORATORY DATA
n = 82

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.1829 0.1311 1.0200 0.0873 0.01285 0.76970
0.1463 0.1372 1.0200 0.1658 0.01345 1.42955
0.1341 0.1128 1.0200 0.2409 0.01106 2.29014
0.1372 0.1189 1.0200 0.3403 0.01166 3.15199
0.1585 0.1250 1.0200 0.0873 0.01226 0.78824
0.1829 0.1189 1.0200 0.1012 0.01166 0.93752
0.1280 0.1250 1.0200 0.2059 0.01226 1.86026
0.1311 0.1189 1.0200 0.2094 0.01166 1.93969
0.1402 0.1158 1.0200 0.3316 0.01136 3.11132
0.1768 0.1189 1.0200 0.0925 0.01166 0.85670
0.1433 0.1311 1.0200 0.1658 0.01285 1.46242
0.1280 0.1158 1.0200 0.2566 0.01136 2.40718
0.1402 0.1158 1.0200 0.3595 0.01136 3.37333
0.1768 0.1189 1.0200 0.0925 0.01166 0.85670
0.1128 0.1189 1.0200 0.1710 0.01166 1.58408
0.1189 0.1189 1.0200 0.2583 0.01166 2.39228
0.1280 0.1189 1.0200 0.3316 0.01166 3.07117
0.1219 0.0914 1.0200 0.0838 0.00897 0.88463
0.0914 0.0975 1.0200 0.1710 0.00957 1.74878
0.0945 0.0914 1.0200 0.2461 0.00897 2.59861
0.1402 0.0914 1.0200 0.3316 0.00897 3.50168
0.1311 0.1097 1.0200 0.0908 0.01076 0.87485
0.0945 0.1006 1.0200 0.1990 0.00987 2.00323
0.1067 0.0975 1.0200 0.2496 0.00957 2.55179
0.1646 0.1067 1.0200 0.3316 0.01046 3.24192
0.1341 0.0945 1.0200 0.0803 0.00927 0.83399
0.0975 0.0975 1.0200 0.1815 0.00957 1.85584
0.1067 0.0884 1.0200 0.2304 0.00867 2.47433
0.1341 0.0914 1.0200 0.3037 0.00897 3.20680
0.1128 0.0914 1.0200 0.0960 0.00897 1.01364
0.0945 0.1006 1.0200 0.0960 0.00987 0.96647
0.1219 0.0914 1.0200 0.2251 0.00897 2.37745
0.1554 0.0914 1.0200 0.3595 0.00897 3.79655
0.1829 0.1554 1.5000 0.0942 0.00705 1.12249
0.1341 0.1859 1.5000 0.1693 0.00843 1.84366
0.1554 0.1890 1.5000 0.2374 0.00857 2.56400
0.1890 0.2164 1.5000 0.3770 0.00981 3.80539
0.1920 0.1402 1.5000 0.0908 0.00636 1.13815
0.1372 0.1737 1.5000 0.1693 0.00788 1.90725
0.1585 0.1920 1.5000 0.2478 0.00871 2.65578
0.2347 0.1829 1.5000 0.3595 0.00829 3.94790
0.1951 0.1707 1.5000 0.0855 0.00774 0.97202
0.1372 0.1829 1.5000 0.1850 0.00829 2.03144
0.1768 0.1737 1.5000 0.2740 0.00788 3.08700
0.2286 0.1737 1.5000 0.3316 0.00788 3.73586
0.1707 0.1554 1.5000 0.0925 0.00705 1.10170
0.1372 0.1676 1.5000 0.1815 0.00760 2.08174
0.1615 0.1798 1.5000 0.2443 0.00816 2.70568
0.1981 0.1646 1.5000 0.3595 0.00746 4.16145
0.1646 0.1646 1.7400 0.0977 0.00555 1.31227
0.1433 0.1707 1.7400 0.1763 0.00575 2.32413
0.1707 0.2012 1.7400 0.2443 0.00678 2.96748
0.2012 0.2012 1.7400 0.3316 0.00678 4.02730
0.1707 0.1463 1.7400 0.0925 0.00493 1.31731
0.1554 0.1951 1.7400 0.1833 0.00657 2.26012
0.1494 0.1646 1.7400 0.2443 0.00555 3.28068
0.1859 0.1768 1.7400 0.4119 0.00596 5.33618
0.1768 0.1524 1.7400 0.0873 0.00514 1.21764
0.1615 0.1829 1.7400 0.1920 0.00616 2.44540
0.1463 0.1707 1.7400 0.2880 0.00575 3.79684
0.2103 0.1707 1.7400 0.3857 0.00575 5.08546
0.1737 0.1494 1.7400 0.0977 0.00503 1.37760
0.1554 0.1707 1.7400 0.1955 0.00575 2.57725
0.1463 0.1646 1.7400 0.2880 0.00555 3.86651
0.1981 0.1646 1.7400 0.3595 0.00555 4.82728
0.1128 0.1341 2.4900 0.0942 0.00221 2.00609


129











SMITH AND KRAUS (1990) LABORATORY DATA (CONT)


Wave
Period
(s)

2.4900
2.4900
2.4900
2.4900
2.4900
2.4900
2.4900
2.4900
2.4900
2.4900
2.4900
1.0200
1.0200
1.4900
1.7400
2.4900


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


Water
Depth
(m)

0.1280
0.1524
0.1128
0.1189
0.1707
0.1097
0.1189
0.1646
0.1158
0.1189
0.1585
0.2042
0.1311
0.2164
0.2164
0.1494


Breaker
Height
(m)

0.1341
0.1219
0.1219
0.1128
0.1250
0.1250
0.1097
0.1219
0.1280
0.1128
0.1158
0.1219
0.0823
0.1554
0.1646
0.1280


3.41778
5.02623
2.18193
4.82090
6.69637
1.92424
5.05168
6.11719
1.97725
3.97016
6.27610
0.30178
0.36731
0.39041
0.44307
0.71894


STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2


(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 111.83 -11.83 0.00 0.00 0.00 0.00 0.00 0.00
7 98.80 0.00 1.20 0.00 0.00 0.00 0.00 0.00
8 93.14 0.00 0.00 6.86 0.00 0.00 0.00 0.00
9 98.73 0.00 0.00 0.00 1.27 0.00 0.00 0.00
10 0.00 54.71 45.29 0.00 0.00 0.00 0.00 0.00
11 0.00 58.62 0.00 41.38 0.00 0.00 0.00 0.00
12 0.00 61.39 0.00 0.00 38.61 0.00 0.00 0.00
13 0.00 0.00 190.46 -90.46 0.00 0.00 0.00 0.00
14 0.00 0.00 97.07 0.00 2.93 0.00 0.00 0.00
15 0.00 0.00 0.00 4.52 95.48 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 111.27 -11.72 0.45 0.00 0.00 0.00 0.00 0.00
17 133.79 -24.81 0.00 -8.97 0.00 0.00 0.00 0.00
18 110.61 -16.02 0.00 0.00 5.41 0.00 0.00 0.00
19 92.73 0.00 0.49 6.78 0.00 0.00 0.00 0.00
20 98.70 0.00 0.07 0.00 1.22 0.00 0.00 0.00
21 87.58 0.00 0.00 8.69 3.73 0.00 0.00 0.00
22 0.00 54.42 7.95 37.63 0.00 0.00 0.00 0.00
23 0.00 83.71 85.37 0.00 -69.08 0.00 0.00 0.00
24 0.00 54.85 0.00 41.01 4.14 0.00 0.00 0.00
25 0.00 0.00 -461.68 283.12 278.56 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 133.07 -24.67 0.56 -8.97 0.00 0.00 0.00 0.00
27 195.08 -96.36 -81.16 0.00 82.45 0.00 0.00 0.00
28 133.18 -30.33 0.00 -9.32 6.47 0.00 0.00 0.00
29 78.03 0.00 -21.67 21.31 22.32 0.00 0.00 0.00
30 0.00 64.68 30.38 29.21 -24.27 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 259.65 -145.81 -106.96 -15.58 108.71 0.00 0.00 0.00


35.39 0.5949
0.93 0.0966
1.48 0.1217
0.16 0.0406
0.98 0.0992

36.18 0.6015
35.42 0.5951
35.98 0.5998
35.43 0.5952
2.54 0.1592
3.38 0.1838
1.31 0.1144
1.72 0.1313
1.48 0.1217
0.98 0.0992

36.18 0.6015
36.23 0.6019
36.60 0.6050
35.99 0.5999
35.43 0.5952
36.29 0.6024
4.71 0.2170
5.10 0.2258
3.73 0.1931
2.19 0.1482

36.23 0.6019
40.35 0.6352
36.65 0.6054
38.02 0.6166
6.63 0.2576

40.42 0.6357


130


0.1606
0.2251
0.0977
0.2077
0.3037
0.0873
0.2147
0.2740
0.0908
0.1710
0.2740
0.0330
0.0330
0.0330
0.0330
0.0330


0.00221
0.00201
0.00201
0.00186
0.00206
0.00206
0.00181
0.00201
0.00211
0.00186
0.00191
0.01196
0.00807
0.00714
0.00555
0.00211











SMITH AND KRAUS (1990) LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.5949 0.0966 0.1217 -0.0406 0.0992

(2) 35.3935 1.0000 0.3040 0.1800 -0.1946 0.1364

(3) 0.9334 9.2426 1.0000 -0.0488 -0.9457 0.4662

(4) 1.4817 3.2405 0.2377 1.0000 0.0699 0.8028

(5) 0.1646 3.7886 89.4418 0.4889 1.0000 -0.4371

(6) 0.9836 1.8600 21.7336 64.4494 19.1099 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (88 X8)

EQ BO B1 B2 B3 B4 85 B6 B7 B8


0.590 0.000
0.000 0.006
0.000 0.000
0.000 0.000
0.000 0.000
0.618 -0.006
0.587 0.000
0.605 0.000
0.587 0.000
0.000 0.007
0.000 0.036
0.000 0.004
0.000 0.000
0.000 0.000
0.000 0.000
0.617 -0.006
0.625 -0.011
0.618 -0.008
0.603 0.000
0.587 0.000
0.602 0.000
0.000 0.034
0.000 0.023
0.000 0.033
0.000 0.000
0.624 -0.011
0.766 -0.035
0.626 -0.013
0.658 0.000
0.000 0.045
0.774 -0.040


0.000
0.000
0.040
0.000
0.000
0.000
0.005
0.000
0.000
0.041
0.000
0.000
0.041
0.039
0.000
0.002
0.000
0.000
0.002
0.000
0.000
0.038
0.178
0.000
0.097
0.002
-0.220
0.000
-0.126
0.158
-0.220


0.000 0.000
0.000 0.000
0.000 0.000
-0.412 0.000
0.000 0.002
0.000 0.000
0.000 0.000
0.795 0.000
0.000 0.000
0.000 0.000
4.891 0.000
0.000 0.002
-0.501 0.000
0.000 0.000
0.035 0.002
0.000 0.000
-0.748 0.000
0.000 0.002
0.787 0.000
0.000 0.000
1.065 0.001
4.625 0.000
0.000 -0.011
4.867 0.002
-1.539 -0.005
-0.750 0.000
0.000 0.018
-0.782 0.002
3.204 0.010
3.938 -0.010
-0.829 0.018


131


1 0.070
2 0.141
3 0.143
4 0.154
5 0.145
6 0.075
7 0.069
8 0.062
9 0.069
10 0.133
11 0.060
12 0.141
13 0.147
14 0.143
15 0.145
16 0.075
17 0.087
18 0.074
19 0.061
20 0.069
21 0.056
22 0.056
23 0.109
24 0.059
25 0.155
26 0.086
27 0.097
28 0.086
29 0.035
30 0.047
31 0.110


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











SMALL LABORATORY DATA SETS COMBINED
n = 17


Water Breaker Wave Bed
Depth Height Period Slope


Equivalent
Wave


(m) (s) Steepness
SAEKI AND SASAKI (1973)
0.1060 1.3000 0.0200 0.00640
0.0990 2.5000 0.0200 0.00162
MIZUGAUCHI (1981)
0.1000 1.2000 0.1000 0.00709
VISSER (1982)
0.1050 2.0100 0.1000 0.00265
0.1000 1.0000 0.1000 0.01020
0.0970 1.0000 0.1000 0.00990
0.0910 1.0200 0.0500 0.00893
0.1080 1.8500 0.0500 0.00322
0.0580 0.7000 0.0500 0.01208
0.0900 1.0200 0.0500 0.00883
MARUYAMA AND OTHERS (1983)
1.2900 3.1000 0.0340 0.01370
STIVE (1985)
0.1800 1.8000 0.0250 0.00567
1.5000 5.0000 0.0250 0.00612
WATANBE AND DIBAJNIA (1988)
0.0770 1.1900 0.0500 0.00555
0.0820 1.1800 0.0500 0.00601


Farm

0.25000
0.49747

1.18794

1.94184
0.98995
1.00514
0.52925
0.88114
0.45495
0.53218

0.29051

0.33204
0.31950

0.67125
0.64500


0.1000 0.0750 0.9400 0.0500 0.00866 0.53725


TAKIKAMA AND OTHERS (1997)
0.2320 0.2150 2.0800 0.0500 0.00507


0.70215


(m)

0.1640
0.0970

0.0830

0.1040
0.1090
0.1140
0.1100
0.1160
0.0880
0.1220

2.0000

0.2000
1.9000

0.1000
0.1100











SMALL LABORATORY DATA SETS COMBINED (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 98.53 0.9926
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 68.96 0.8304
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 12.21 0.3494
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 10.05 0.3171
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 13.31 0.3648
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 160.35 -60.35 0.00 0.00 0.00 0.00 0.00 0.00 99.13 0.9956
7 102.86 0.00 -2.86 0.00 0.00 0.00 0.00 0.00 98.53 0.9926
8 78.89 0.00 0.00 21.11 0.00 0.00 0.00 0.00 99.00 0.9950
9 107.24 0.00 0.00 0.00 -7.24 0.00 0.00 0.00 98.58 0.9929
10 0.00 101.69 -1.69 0.00 0.00 0.00 0.00 0.00 68.97 0.8305
11 0.00 56.67 0.00 43.33 0.00 0.00 0.00 0.00 95.32 0.9763
12 0.00 136.68 0.00 0.00 -36.68 0.00 0.00 0.00 72.99 0.8543
13 0.00 0.00-1475.44 1575.44 0.00 0.00 0.00 0.00 25.86 0.5085
14 0.00 0.00 40.77 0.00 59.23 0.00 0.00 0.00 13.91 0.3730
15 0.00 0.00 0.00 5499.40-5399.40 0.00 0.00 0.00 17.90 0.4231
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 196.95 -80.37 -16.57 0.00 0.00 0.00 0.00 0.00 99.20 0.9960
17 268.01 -137.09 0.00 -30.92 0.00 0.00 0.00 0.00 99.14 0.9957
18 163.10 -60.60 0.00 0.00 -2.50 0.00 0.00 0.00 99.13 0.9956
19 83.18 0.00 -7.91 24.72 0.00 0.00 0.00 0.00 99.07 0.9953
20 103.61 0.00 11.63 0.00 -15.24 0.00 0.00 0.00 98.61 0.9930
21 80.39 0.00 0.00 21.08 -1.48 0.00 0.00 0.00 99.00 0.9950
22 0.00 57.91 -3.18 45.28 0.00 0.00 0.00 0.00 95.47 0.9771
23 0.00 96.97 78.25 0.00 -75.23 0.00 0.00 0.00 82.34 0.9074
24 0.00 57.19 0.00 43.52 -0.71 0.00 0.00 0.00 95.33 0.9764
25 0.00 0.00 -276.23 200.49 175.74 0.00 0.00 0.00 33.84 0.5818
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 325.66 -168.94 -25.27 -31.46 0.00 0.00 0.00 0.00 99.21 0.9960
27 374.63 -237.18 -156.26 0.00 118.81 0.00 0.00 0.00 99.45 0.9973
28 291.64 -150.28 0.00 -36.03 -5.33 0.00 0.00 0.00 99.15 0.9957
29 67.35 0.00 -45.23 42.41 35.47 0.00 0.00 0.00 99.37 0.9969
30 0.00 53.17 -19.56 51.16 15.23 0.00 0.00 0.00 96.07 0.9802
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 165.93 -78.86 -87.78 32.51 68.20 0.00 0.00 0.00 99.48 0.9974



CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9926 0.8304 -0.3494 0.3171 -0.3648

(2) 98.5280 1.0000 0.8744 -0.3444 0.2525 -0.3462

(3) 68.9610 76.4573 1.0000 -0.4092 -0.2211 -0.2026

(4) 12.2069 11.8585 16.7471 1.0000 0.1397 0.8436

(5) 10.0525 6.3771 4.8905 1.9511 1.0000 -0.3107

(6) 13.3092 11.9849 4.1056 71.1658 9.6535 1.0000

Numbers in Parentheses are the Variable or Parameter Number











TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (88 X8)

BO B1 B2 B3 B4 B5 B6


1 -0.022
2 -0.467
3 0.744
4 -0.085
5 0.703
6 0.082
7 -0.011
8 -0.110
9 0.005
10 -0.449
11 -1.282
12 -0.221
13 0.307
14 0.749
15 0.333
16 0.129
17 0.175
18 0.088
19 -0.082
20 -0.007
21 -0.101
22 -1.219
23 -0.609
24 -1.262
25 -0.168
26 0.205
27 0.218
28 0.189
29 -0.171
30 -1.317
31 0.087


1.399
0.000
0.000
0.000
0.000
1.596
1.395
1.374
1.387
0.000
0o.obo
0.000
0.000
0.000
0.000
1.598
1.696
1.589
1.358
1.390
1.371
0.000
0.000
0.000
0.000
1.682
1.753
1.695
1.333
0.000
1.620


0.000 0.000 0.000 0.000 0.000
0.474 0.000 0.000 0.000 0.000
0.000 -7.460 0.000 0.000 0.000
0.000 0.000 59.059 0.000 0.000
0.000 0.000 0.000 -0.528 0.000
-0.091 0.000 0.000 0.000 0.000
0.000 -0.183 0.000 0.000 0.000
0.000 0.000 13.210 0.000 0.000
0.000 0.000 0.000 -0.035 0.000
0.471 -0.245 0.000 0.000 0.000
0.540 0.000 98.062 0.000 0.000
0.450 0.000 0.000 -0.297 0.000
0.000 -8.572 69.505 0.000 0.000
0.000 -3.082 0.000 -0.352 0.000
0.000 0.000 41.999 -0.426 0.000
-0.099 -0.636 0.000 0.000 0.000
-0.131 0.000 -7.034 0.000 0.000
-0.089 0.000 0.000 -0.009 0.000
0.000 -0.611 14.499 0.000 0.000
0.000 0.739 0.000 -0.076 0.000
0.000 0.000 12.922 -0.009 0.000
0.530 -0.912 98.473 0.000 0.000
0.541 13.639 0.000 -1.030 0.000
0.538 0.000 97.270 -0.016 0.000
0.000 -25.743 141.877 1.286 0.000
-0.132 -0.617 -5.837 0.000 0.000
-0.168 -3.461 0.000 0.207 0.000
-0.132 0.000 -7.526 -0.012 0.000
0.000 -4.237 30.170 0.261 0.000
0.518 -5.957 118.278 0.364 0.000
-0.117 -4.057 11.411 0.248 0.000


B7 B8

0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000












APPENDIX IV

Stepwise regression results for a special data set



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (11) of the text.

100 r*2 = 100 r2.

The dependent variable, db, is the water depth at the shore-breaking position.

Independent variables:

(1) Shore-breaking wave height, Hb
(2) Wave period, T
(3) Bed slope leading to shore-breaking, tan ab
(4) Equivalent wave steepness, Hb/g T2
(5) Surf similarity parameter, tan ab/[Hb/(g T2)]05

Correlation Matrix.

Correlation matrix treatment is given by Table 5 of the text.

Variables:

(1) db (2) Hb (3) T (4) tan ab

(5) Hbl(g T2) (6) tan ab/[Hb/(g T2)]0'5











A SPECIAL DATA SET
n = 171
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 88.23 0.9393
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.47 0.0688
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 3.91 0.1976
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 34.49 0.5873
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 14.78 0.3845
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 126.17 -26.17 0.00 0.00 0.00 0.00 0.00 0.00 90.84 0.9531
7 94.30 0.00 5.70 0.00 0.00 0.00 0.00 0.00 88.69 0.9417
8 82.29 0.00 0.00 17.71 0.00 0.00 0.00 0.00 93.05 0.9646
9 102.84 0.00 0.00 0.00 -2.84 0.00 0.00 0.00 88.33 0.9399
10 0.00 -13.96 113.96 0.00 0.00 0.00 0.00 0.00 3.93 0.1982
11 0.00 52.18 0.00 47.82 0.00 0.00 0.00 0.00 62.18 0.7886
12 0.00 -151.27 0.00 0.00 251.27 0.00 0.00 0.00 17.05 0.4129
13 0.00 0.00 -112.86 212.86 0.00 0.00 0.00 0.00 43.71 0.6611
14 0.00 0.00 -66.76 0.00 166.76 0.00 0.00 0.00 16.44 0.4054
15 0.00 0.00 0.00 166.92 -66.92 0.00 0.00 0.00 38.26 0.6186
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 120.11 -23.59 3.48 0.00 0.00 0.00 0.00 0.00 90.94 0.9536
17 77.22 3.80 0.00 18.98 0.00 0.00 0.00 0.00 93.09 0.9648
18 124.28 -26.72 0.00 0.00 2.45 0.00 0.00 0.00 90.89 0.9534
19 82.51 0.00 -0.47 17.96 0.00 0.00 0.00 0.00 93.05 0.9646
20 97.27 0.00 16.51 0.00 -13.78 0.00 0.00 0.00 89.97 0.9485
21 80.95 0.00 0.00 17.63 1.42 0.00 0.00 0.00 93.08 0.9648
22 0.00 56.38 -12.81 56.42 0.00 0.00 0.00 0.00 65.99 0.8123
23 0.00 164.90 156.76 0.00 -221.66 0.00 0.00 0.00 25.74 0.5073
24 0.00 60.28 0.00 51.29 -11.57 0.00 0.00 0.00 65.90 0.8118
25 0.00 0.00 -128.35 167.04 61.31 0.00 0.00 0.00 45.54 0.6748
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 77.34 3.91 -0.57 19.31 0.00 0.00 0.00 0.00 93.09 0.9648
27 118.73 -21.96 5.12 0.00 -1.89 0.00 0.00 0.00 90.95 0.9537
28 77.16 3.09 0.00 18.68 1.07 0.00 0.00 0.00 93.11 0.9649
29 79.23 0.00 -6.61 20.81 6.57 0.00 0.00 0.00 93.24 0.9656
30 0.00 58.60 -7.43 54.52 -5.69 0.00 0.00 0.00 66.18 0.8135
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 78.26 0.85 -6.35 20.97 6.27 0.00 0.00 0.00 93.24 0.9656


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9393 0.0688 -0.1976 0.5873 -0.3845

(2) 88.2336 1.0000 0.2399 -0.2791 0.4124 -0.3782

(3) 0.4727 5.7561 1.0000 -0.2729 -0.5998 0.2045

(4) 3.9052 7.7921 7.4486 1.0000 0.1727 0.7394

(5) 34.4926 17.0087 35.9793 2.9824 1.0000 -0.3442

(6) 14.7802 14.3019 4.1837 54.6747 11.8502 1.0000

Numbers in Parentheses are the Variable or Parameter Number


139











A SPECIAL DATA SET (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 0.007 1.072 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.107 0.000 0.006 0.000 0.000 0.000 0.000 0.000 0.000
3 0.136 0.000 0.000 -0.239 0.000 0.000 0.000 0.000 0.000
4 0.065 0.000 0.000 0.000 9.495 0.000 0.000 0.000 0.000
5 0.150 0.000 0.000 0.000 0.000 -0.026 0.000 0.000 0.000
6 0.026 1.118 -0.015 0.000 0.000 0.000 0.000 0.000 0.000
7 -0.002 1.094 0.000 0.085 0.000 0.000 0.000 0.000 0.000
8 -0.002 0.959 0.000 0.000 3.894 0.000 0.000 0.000 0.000
9 0.012 1.057 0.000 0.000 0.000 -0.002 0.000 0.000 0.000
10 0.133 0.000 0.001 -0.234 0.000 0.000 0.000 0.000 0.000
11 -0.063 0.000 0.058 0.000 15.873 0.000 0.000 0.000 0.000
12 0.131 0.000 0.014 0.000 0.000 -0.028 0.000 0.000 0.000
13 0.090 0.000 0.000 -0.373 10.356 0.000 0.000 0.000 0.000
14 0.144 0.000 0.000 0.231 0.000 -0.036 0.000 0.000 0.000
15 0.090 0.000 0.000 0.000 8.344 -0.014 0.000 0.000 0.000
16 0.021 1.127 -0.014 0.042 0.000 0.000 0.000 0.000 0.000
17 -0.007 0.937 0.003 0.000 4.344 0.000 0.000 0.000 0.000
18 0.024 1.131 -0.015 0.000 0.000 0.002 0.000 0.000 0.000
19 -0.001 0.956 0.000 -0.007 3.927 0.000 0.000 0.000 0.000
20 0.006 1.057 0.000 0.230 0.000 -0.012 0.000 0.000 0.000
21 -0.005 0.965 0.000 0.000 3.967 0.001 0.000 0.000 0.000
22 -0.036 0.000 0.053 -0.245 15.919 0.000 0.000 0.000 0.000
23 0.082 0.000 0.036 0.693 0.000 -0.061 0.000 0.000 0.000
24 -0.038 0.000 0.058 0.000 14.724 -0.014 0.000 0.000 0.000
25 0.076 0.000 0.000 -0.658 12.605 0.019 0.000 0.000 0.000
26 -0.006 0.933 0.003 -0.009 4.395 0.000 0.000 0.000 0.000
27 0.020 1.120 -0.013 0.062 0.000 -0.001 0.000 0.000 0.000
28 -0.008 0.945 0.002 0.000 4.320 0.001 0.000 0.000 0.000
29 -0.005 0.948 0.000 -0.101 4.701 0.006 0.000 0.000 0.000
30 -0.035 0.000 0.055 -0.142 15.345 -0.007 0.000 0.000 0.000
31 -0.006 0.943 0.001 -0.098 4.772 0.006 0.000 0.000 0.000


140












APPENDIX V

Stepwise regression results where dbf/H is the independent variable



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (12) of the text.

100r*2 = 100r2.

The dependent variable is db/Hb, where db is the water depth at the
shore-breaking position, and Hb is the shore-breaking wave height..

Independent variables:

(1) Wave period, T
(2) Bed slope leading to shore-breaking, tan ab
(3) Equivalent wave steepness, Hb/g T2
(4) Surf similarity parameter, tan ab/[Hb/(g T2)]0.5
(5) tanh 0.4 b


Correlation



(1)

(4)


Correlation Matrix.

matrix treatment is given by Table 5 of the text.

Variables:

dh/Hb (2) T (3) tan a


Hb/(g T2)


h


(5) tan a/[Hb/(g T2)]0.5

tanh 0.4 !b











db/Hb VERSUS T, tan ab, Hb/(g T2), fb, tanh 0.4 fb


ALL DATA
n = 771

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1) (2) (3) (4) (5) (6)
INDEPENDENT VARIABLES TAKEN
10.00 0.00 0.00 0.00 0.00 0.00
0.00 100.00 0.00 0.00 0.00 0.00
0.00 0.00 100.00 0.00 0.00 0.00
0.00 0.00 0.00 100.00 0.00 0.00
0.00 0.00 0.00 0.00 100.00 0.00
INDEPENDENT VARIABLES TAKEN
0.69 620.69 0.00 0.00 0.00 0.00
5.38 0.00 54.62 0.00 0.00 0.00
15.77 0.00 0.00 605.77 0.00 0.00
9.17 0.00 0.00 0.00 149.17 0.00
0.00 1177.40-1077.40 0.00 0.00 0.00
0.00 36.39 0.00 63.61 0.00 0.00
0.00 -38.08 0.00 0.00 138.08 0.00
0.00 0.00 -71.20 171.20 0.00 0.00
0.00 0.00 -20.49 0.00 120.49 0.00
0.00 0.00 0.00 -53.91 153.91 0.00
INDEPENDENT VARIABLES TAKEN
.7.33 -50.62 93.29 0.00 0.00 0.00
4.06 -97.57 0.00 921.63 0.00 0.00
7.94 -90.59 0.00 0.00 248.53 0.00
'0.05 0.00 78.33 -48.38 0.00 0.00
3.52 0.00 345.26 0.00 -548.78 0.00
1.97 0.00 0.00 -67.55 199.52 0.00
0.00 2314.24-2079.59 -134.65 0.00 0.00
0.00 -40.93 8.78 0.00 132.14 0.00
0.00 -22.33 0.00 -45.41 167.74 0.00
0.00 0.00 -20.15 -67.67 187.82 0.00
INDEPENDENT VARIABLES TAKEN F
4.39 -55.53 93.10 8.04 0.00 0.00
6.92 -193.25 -155.69 0.00 675.86 0.00
2.34 -59.33 0.00 -49.38 251.05 0.00
7.71 0.00 -700.63 -774.47 2142.81 0.00
0.00 -17.23 -7.90 -52.75 177.87 0.00
INDEPENDENT VARIABLES TAKEN F
1.61 -66.99 -298.08 -347.79 1084.48 0.00


ONE





TWO










THR










FOUI





FIVE


(7) (8)
AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
EE AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
R AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
E AT A TIME
0.00 0.00


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.2113 -0.2225 0.0919 -0.2356 -0.3339

(2) 4.4630 1.0000 -0.2759 -0.4391 -0.1031 -0.1826

(3) 4.9486 7.6131 1.0000 0.2593 0.8204 0.8388

(4) 0.8439 19.2788 6.7219 1.0000 -0.1387 -0.1203

(5) 5.5485 1.0629 67.3045 1.9247 1.0000 0.8956

(6) 11.1478 3.3338 70.3648 1.4477 80.2042 1.0000

Numbers in Parentheses are the Variable or Parameter Number


10





-52
4
-50
-4







5
-72
-5
7
30
-3





5
-22
-4
-56


-27


4.46
4.95
0.84
5.55
11.15

7.38
8.69
9.08
13.48
7.35
5.81
12.27
5.91
11.42
13.18

12.94
9.10
15.58
11.80
15.47
14.99
7.35
12.33
13.64
13.57

12.97
15.96
16.21
17.00
13.68

17.03


0.2113
0.2225
0.0919
0.2356
0.3339

0.2717
0.2947
0.3014
0.3672
0.2710
0.2410
0.3503
0.2430
0.3379
0.3631

0.3598
0.3016
0.3947
0.3435
0.3934
0.3872
0.2711
0.3511
0.3694
0.3684

0.3601
0.3995
0.4026
0.4123
0.3699

0.4127











db/Hb VERSUS T, tan ab, Hb/(g T2), fb, tanh 0.4 [b (CONT)


ALL DATA

TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 85 B6 B7 B8


0.021 0.000 0.000
0.000 -0.801 0.000
0.000 0.000 7.260
0.000 0.000 0.000
0.000 0.000 0.000
0.016 -0.640 0.000
0.031 0.000 18.075
0.019 0.000 0.000
0.015 0.000 0.000
0.000 -0.951 12.670
0.000 -0.322 0.000
0.000 0.700 0.000
0.000 0.000 4.769
0.000 0.000 4.145
0.000 0.000 0.000
0.027 -0.783 21.016
0.019 0.086 0.000
0.019 0.984 0.000
0.027 0.000 14.811
0.023 0.000 12.735
0.014 0.000 0.000
0.000 -1.006 13.173
0.000 0.847 -2.647
0.000 0.469 0.000
0.000 0.000 4.966
0.027 -0.908 22.168
0.022 0.625 7.332
0.017 0.795 0.000
0.021 0.000 12.797
0.000 0.331 2.213
0.021 0.174 11.287


0.000 0.000
0.000 0.000
0.000 0.000
-0.053 0.000
0.000 -0.347
0.000 0.000
0.000 0.000
-0.049 0.000
0.000 -0.318
0.000 0.000
-0.037 0.000
0.000 -0.517
-0.051 0.000
0.000 -0.340
0.072 -0.645
0.000 0.000
-0.053 0.000
0.000 -0.549
-0.041 0.000
0.000 -0.283
0.063 -0.580
0.004 0.000
0.000 -0.556
0.062 -0.717
0.075 -0.647
0.009 0.000
0.000 -0.445
0.043 -0.685
0.063 -0.547
0.066 -0.696
0.059 -0.574


1.146
1.262
1.162
1.263
1.332
1.208
1.025
1.210
1.281
1.206
1.268
1.342
1.236
1.307
1.359
1.081
1.207
1.284
1.100
1.181
1.311
1.204
1.360
1.362
1.331
1.074
1.225
1.304
1.210
1.349
1.220


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000












APPENDIX VI

Stepwise regression results where tan ab is the independent variable



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (13) of the text.

100r*2 = 100r2.

The dependent variable is tan ab, the bed slope in the
vicinity leading up to shore-breaking.

Independent variables:

(1) Water depth at shore-breaking, db
(2) Shore-breaking wave height, Hb
(3) Wave period, T
(4) Equivalent wave steepness, Hb/g T2

Correlation Matrix.

Correlation matrix treatment is given by Table 5 of the text.

Variables:

(1) tan ab (2) db (3) Hb

(4) T (5) Hb/(g T2)


145











tan ab VERSUS db, Hb, T, and Hb/(g T2)

ALL DATA
n = 771

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1) (2) (3) (4) (5)
INDEPENDENT VARIABLES TAKEN ON


0.00
100.00
0.00
0.00

67.77
0.00
0.00
18.20
-51.15
0.00

-5.15
-25.16
0.00
-49.61

-32.40


0.00 0.00 0.00
0.00 0.00 0.00
100.00 0.00 0.00
0.00 100.00 0.00
INDEPENDENT VARIABLES
0.00 0.00 0.00
81.54 0.00 0.00
0.00 145.54 0.00
81.80 0.00 0.00
0.00 151.15 0.00
-251.50 351.50 0.00
INDEPENDENT VARIABLES
83.03 0.00 0.00
0.00 149.56 0.00
15.62 128.83 0.00
31.35 118.26 0.00


(6) (7)
E AT A TIME
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN TWO AT A
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00
TAKEN THREE AT
0.00 0.00
0.00 0.00
0.00 0.00
0.00 0.00


INDEPENDENT VARIABLES TAKEN FOUR AT A
30.53 118.06 0.00 0.00 0.00


(8)

0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
TIME
0.00


6.79
6.92
7.61
6.72

6.97
7.82
11.34
7.75
11.34
9.98

7.82
11.41
11.38
11.49

11.55


0.2606
0.2631
0.2759
0.2593

0.2640
0.2796
0.3368
0.2784
0.3367
0.3159

0.2796
0.3378
0.3373
0.3389

0.3399


CORRELATION MATRIX

(1) (2) (3) (4) (5)


(1) 1.0000 -0.2606 -0.2631 -0.2759

(2) 6.7908 1.0000 0.9709 0.8617

(3) 6.9235 94.2587 1.0000 0.8926

(4) 7.6131 74.2542 79.6819 1.0000

(5) 6.7219 3.6632 4.1475 19.2788


0.2593

-0.1914

-0.2037

-0.4391

1.0000


100.00
0.00
0.00
0.00

32.23
18.46
-45.54
0.00
0.00
0.00

22.11
-24.40
-44.45
0.00

-16.19










tan ab VERSUS db, Hb, T, and Hb/(g T2) (CONT)


ALL DATA

TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (81 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 82 B3 B4 B5 B6 B7 B8
NO
1 0.087 -0.019 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.087 0.000 -0.027 0.000 0.000 0.000 0.000 0.000 0.000
3 0.097 0.000 0.000 -0.008 0.000 0.000 0.000 0.000 0.000
4 0.047 0.000 0.000 0.000 5.691 0.000 0.000 0.000 0.000
5 0.087 -0.007 -0.018 0.000 0.000 0.000 0.000 0.000 0.000
6 0.095 -0.007 0.000 -0.005 0.000 0.000 0.000 0.000 0.000
7 0.060 -0.016 0.000 0.000 4.771 0.000 0.000 0.000 0.000
8 0.095 0.000 -0.009 -0.006 0.000 0.000 0.000 0.000 0.000
9 0.061 0.000 -0.023 0.000 4.710 0.000 0.000 0.000 0.000
10 0.072 0.000 0.000 -0.006 3.756 0.000 0.000 0.000 0.000
11 0.095 -0.008 0.002 -0.006 0.000 0.000 0.000 0.000 0.000
12 0.060 -0.008 -0.011 0.000 4.726 0.000 0.000 0.000 0.000
13 0.056 -0.019 0.000 0.001 5.057 0.000 0.000 0.000 0.000
14 0.053 0.000 -0.032 0.003 5.332 0.000 0.000 0.000 0.000
15 0.053 -0.008 -0.021 0.003 5.332 0.000 0.000 0.000 0.000


148
















APPENDIX VII

Additional data needed to improve the statistical fit of Figure 5 of the main text.


149









Additional Data Needed to Improve the
Statistical Fit of Figure 5 of the Main Text


The purpose of this exercise is to
determine the number of data points, in
addition to those data already existing, that
would be required to improve the statistically
fitted relationship.

We shall employ the data of Figure 5
of the main text wherein the water depth at
the shore-breaking position, db, is plotted
against the shore-breaking wave height, Hb.
A total of 812 data points were plotted
covering a domain of over 2.5 orders of
magnitude. The fitted equation is that given
by McCowan (1894) as:

d, = 1.28 H, (VII-1)

The question is how much more data is
required, in addition to the 81 2 existing data
points, to enhance the power of the fitted
relationship? This may be assessed by
restructuring equation (VII-1) as:

d
-1.28 + v (VII-2)
H4
in which v is the percent variability
associated with the relating coefficient 1.28.
The value of v will diminish in magnitude as
sample size, n, increases, and the correlation
will increase in strength (even though the
existing correlation of r = 0.9670 is
significantly strong).

It is assumed for purposes of this
treatment that researchers obtaining the
additional data not only make precise
measurements, but that they precisely apply
the surrogate definitions as to where or
when waves shore-break such that the
coefficient 1.28 of equation (VII-1) is exactly
obtained for each additional data point.

A computer program was written to
assess the resulting effect. Resulting


outcomes are plotted in Figure VII from
which values are visually determined and
listed in Table VII. Column (1) of Table VII
gives + v values from the abscissa of Figure
VII at even integer per cent increments from
1 to 10. Column (3) provides the estimated
average number of observations (in addition
to the 812 existing data points) required to
result in its associated value of v (which
represents both the additional data plus the
812 existing data points of Figure 5).
Column (2) gives the applied domain of
relative depths where the standard value of
db/Hb is 1.28.

Noting that it has required 93 years to
collect the 812 data points plotted in Figure
5, the number of years required to collect
the additional data is listed in column (4). It
is apparent that the number of needed
additional data points is not small, nor is the
prognosticated amount of time required for
their obtention. This would not seem to
present a probable programmatic situation
for strengthening the statistical fit of Figure
5.

Moreover, it is to be noted that the
required number of additional data points of
column (3) of Table VII represents a
conservative minimum. It is highly unlikely
that a constant value for db/Hb will be
forthcoming because of the surrogate
method of determining where shore-breaking
occurs. Such associated variability will,
therefore, require many more observations
than those listed in column (3) of Table VII.
It is to be concluded that the amount of data
of Figure 5 is so significantly large and
occupying a significantly large domain of
magnitude, that future attempts (using the
surrogate method) will not greatly contribute
to refine the fitted results. The answer is to
find an alternative method for determining
when or where shore-breaking occurs.










"I UUI



10000


Additional
Data 1000 -=
Required -

100. . .I

!- i'^^ -^^ ---, -- g-


0 1 2 4 5 6 7 10 11 12 141
V (%)
Figure VII. Additional data points required in the presence of the
812 data points of Figure 5 of text versus the associated
variability for both, v, as given by equation (VII-2).


Table VII. Table of number of additional data points required to produce
a better fit for data of Figure 5 of the main text.


(1) (2) (3) (4)
% Estimated Predicted
Variability v Number of Number of
Attained Additional Years
for Additional Domain Data Needed
Data Points Plus Points to Collect
Existing 812 Data Required Additional
Points Data

1% 1.267 < db/Hb < 1.293 11,000 1260

2% 1.254 < db/Hb < 1.307 4,600 527

3% 1.242 < db/Hb < 1.318 2,800 321

+4% 1.229 < db/Hb < 1.331 1,950 223

5% 1.216 < db/Hb < 1.344 1,400 160

+6% 1.203 < db/Hb < 1.357 1,050 120

+7% 1.190 < db/Hb < 1.370 800 92

8% 1.178 < db/Hb < 1.382 600 69

+9% 1.165 < db/Hb < 1.395 450 52

10% 1.152 < db/Hb < 1.408 320 37














APPENDIX VIII

Field data for determination of additional values of Hb"/Hb and b for Figure 3










Table of data for determination of Hb"/Hb and b from field data
collected at Alligator Point, Franklin County, Florida (July and
August, 1998), where all waves were spilling shore-breakers.


Exp H Hb Hb" T
No. () (m) (m) (s) tan b
1 0.159(30) 0.306(14) 0.028(14) 5.4 (30) 0.00703
2 0.185 (30) 0.287 (15) 0.033 (15) 4.0 (40) 0.00917
3 0.286 (15) 0.354(16) 0.039 (16) 5.0 (30) 0.01234
Values in ( ) indicate number of waves measured from which averages were
calculated. Participants were James H. Balsillie, Paulette Bond, Don L. Hargrove,
Jacqueline M. Lloyd, and William C. Parker.










SHORE-BREAKING WAVE ENERGETIC

by

James H. Balsillie, P. G. No. 167


ABSTRACT

Wave energy at breaking is investigated using "classical" theory (Airy), and empirically
replicated horizontally and vertically distorted waves. The goal is to determine how wave energy
is distributed across the wavelength, e.g. wave crest energy versus wave trough energy.

The total energy contained in the Airy wave crest at the shore-breaking position
comprises 83% of the energy of the entire wave (i.e., across its entire wavelength). Moreover,
the wave crest contains 5.0 times the energy of the wave trough.

Empirically evaluated distorted shore-breaking waves occurring on a horizontal bed
resulted in wave crest energy densities close (0.933) to the wave crest energy density for the
Airy shore-breakers, and 1.56 times the energy density for the entire Airy wave (i.e., averaged
across its wavelength). Total wave crest energy of the distorted shore-breakers is 83% of the
energy assessed across the wavelength for distorted shore-breakers, and 5.0 times the energy
contained in the wave trough; both similar to Airy results on a horizontal bed.

Empirically evaluated distorted shore-breaking waves occurring on non-horizontal bed
slopes yield quite different outcomes. Distorted wave crest total energy ranges from 83% to
93% of the wavelength total energy, increasing with an increase in the bed slope and surf
similarity parameter. Moreover, distorted total wave crest energy ranges from 5.0 to 14.0 times
the total wave trough energy, increasing with increases in bed slope and surf similarity parameter
values! Wave crest energy density is 14 times the wave trough energy density, and independent
of bed slope! These results invoke significant questions as to the veracity of determining energy
density across the wavelength. (Balsillie, 1997).


INTRODUCTION

Alaska, California, Florida, and Texas
have the longest ocean-fronting shorelines in
the United States. Florida has approximately
1,253 miles of shoreline that front directly
upon the Atlantic Ocean, Gulf of Mexico,
and Straits of Florida. Moreover, annual
average wave energy levels range from near
zero for Florida's Big Bend Gulf Coast, low to
moderate for the remainder of the Gulf
Coast, to high along Florida's Atlantic shores
(Tanner, 1960). Only Alaska experiences
such wave energy variability, not only
because of its large waves but because of
zero-energy occurring along ice-wedged
shores during a significant portion of the


year. Alaska does not, however, have a
coastal population of any significant
proportions (i.e., socio-economic
constraints). California has a significant
coastal population and large Pacific Ocean
waves, but not the wave height variability.
Coastal Florida and Texas also experience
tropical storm and hurricane wave impacts
that far exceed annual average wave energy
levels. Texas does not, however, have
Florida's wave energy variability. Florida,
then, has perhaps the longest shoreline in
terms of wave energy variability and socio-
economic characteristics. Such wave energy
variability requires those of us in Florida to
be sensitively precise in wave energy
assessment and application.


157









This work addresses Earth's ocean
propagating surface gravity water waves.
Specifically, this account addresses energy
of such waves. Its importance is manifest
primarily in sediment transport mechanics
and in assessing horizontal and vertical
dynamic and impact wave pressures. Both
are required in order to realize responsible
coastal engineering design solutions.

It has become apparent to the author
that, to varying extent, depending on the
individual practitioner, wave energy has
become elevated to the status of a
"fundamental property" of ocean waves.
Such a view is far from reasonable. From a
mechanical or kinematic viewpoint (for this
paper) mass, length, and time constitute the
fundamental properties, and energy is a
derived parameter. Other appropriate
parameters used in determining wave
energy, namely the acceleration of gravity,
g, and the mass density of sea water, pf,
are also derived parameters. Both g and pf
can vary in value depending on site and
steric conditions. Nevertheless, the reader
might be surprised at the number of
practioners who regard g and pf to be
constants. While it is true that variations
associated with g and pf are so small that
they are normally considered to be
inconsequential for Earth-bound ocean wave
applications they are, nevertheless, real.
The author should like to posit that when
considering wave energy, similar
assumptions have been taken for granted
which may, at least in part, explain why final
energy computations result in but
approximating or unsatisfying outcomes.
We have become comfortable, for instance,
in viewing wave energy as directly
proportional to the square of a vertical
length, namely, the height of the wave.
Moreover, we use this approach whether the
wave is in deep, intermediate, or shallow
water. The approach is not without
justification. However, there is another
fundamental property which deserves


concerted attention, one which the author
believes has been neglected.

In this work wave energy and how it
is distributed across the wave (e.g., the
wavelength) are investigated at breaking,
and specifically at the shore-breaking
position. Wave energy is virtually always
expressed and applied as an average per unit
length along the wavelength for a unit width
(termed energy density or specific energy).
It is the wave-normal length that we shall
focus on in this work, since it is a primary
factor controlling the value of energy
density.

The paper is presented in three
primary sections. The first deals with Airy
wave theory to provide a fundamental
development, in which Airy waves are shore-
broken on a horizontal bed. The second
section is developed using empirical data to
describe the vertical and horizontal distorted
nature of waves (termed distorted waves) at
the shore-breaking position occurring on a
horizontal bed. The third section describes
wave energetic for distorted shore-breakers
on non-horizontal bed slopes. The waves
are assessed in the Eulerian sense, where
assessment occurs at the point of breaking.

AIRY WAVE THEORY

Analysis of wave energy conditions
begins using classical theory, in this case
Airy wave theory. In this approach, where
the Airy waves are assumed to be shore-
breaking, some conditions require definition.
These are: 1) the depth of water at the
shore-breaking position is given (McCowan,
1892; Munk, 1949; Balsillie, 1983a, 1999;
Balsillie and Tanner, 1989) by:


d, = 1.28 Hb


in which db is the water depth at the
breaking position and Hb is the mean shore-
breaking wave height both referenced to the


158










Direction of Wave Propagation


1.4-
1.2-
1-
g 0.8
0.6a
E0.4
So.2
o 0-
> -0.2
ll -0.4
-0.6
-0.8
-1
-1.2
AA
-1.4 -
-1.6
-2


Hbn 2t



SWL






Bed

6 4 A 10 1'214 1' 1'8 20 22 24 26 28 30
Distance (m)


Figure 1. Definition sketch of wave conditions for an Airy sine wave at


the shore-breaking position; Hb =
definition of terms).

still water level (SWL of Figure 1) as defined
by Galvin (1969), 2) the wave phase speed
at the shore-breaking point is given (Balsillie,
1984) by:


--L, gl H
T 2


where cb is the wave celerity and Lb is the
wavelength of the breaking wave, T is the
wave period, and g is the acceleration of
gravity, and 3) the bed slope is horizontal.
Such theoretical waves at shore-breaking, of
course, are not realistic since they are
perfectly symmetrical in both vertical and
horizontal directions (Figure 1), while natural
shore-breaking waves are highly distorted in
both horizontal and vertical directions. The
bed is seldom horizontal, having more nearly
a shoaling slope even if locally, such as
those associated with longshore bars. Initial
consideration of Airy theoretical treatment


1.0 m, T = 10 s (see text for



does, however, provide a place to begin,
principally because it is the basis upon which
wave energy constraints, including base
conventions, are developed (e.g., Eagleson
and Dean, 1966; U. S. Army, 1984).

Potential Energy, EP

The Newtonian representation of
potential energy, EP, is given by:


EP = mg h


where m is the mass, g is the acceleration of
gravity, and h is the elevation of the centroid
of the mass.

Classical Derivation

For Airy waves, the classical
approach is to determine the potential energy
associated with the wave as the difference
between the potential energy with and


159









without the waves present. The total
potential wave energy averaged over one
wavelength for a progressive wave height H
is given by Dean and Dalrymple (1984) as:


EP, = p,g d, dx
2


in which db is the water depth at shore-
breaking and rq is the local wave height
departure from the SWL. Integration results
in:

~EPLLH22 bL (5)
EP(L 4 2 ) L+ (p,g -
2 16

in which the first term on the right hand side
of the equation (i.e., pf g (db2/2)) represents
the potential energy in the absence of
waves, and the second term is the potential
energy in their presence. Hence, by
subtraction the total potential energy due to
the presence of waves becomes:


1
EP = 16
16


P g H,2 L,


and the total average potential wave ener
per unit surface area, EPL, termed t
energy density (or specific energy) become<


The volume of water of the wave crest
above the SWL, vc, is given by:


where Lb is the wavelength, and the wave
crest mass, m, lying above the SWL
becomes:


Hb L,
m 2w


(10)


Combining equations (3), (8), and (10) yields
the total potential energy as:


H1, Lb w Hb
EP, = p w 8g
27r 8

or when simplified:


EPC 16 p, g HL
16


(11)


(12)


which occurs over one-half of the
(6) wavelength. Therefore, dividing by Lb/2
yields the potential wave energy density
gy given by:


he
es:


EPL = 1 pfg H,


A Different Approach

This section approaches the subject
from the point of view that the still water
level (SWL) is the pertinent reference (see
Figure 1). The vertical distance to which the
center of wave crest mass is elevated from
the preceding trough, h, is given (see Dean
and Dalrymple, 1984, p. 96) by:


wn Hi


1
EP,= pr g H,2
8


(13)


So, we have obtained the identical
outcome as the classical approach (equation
(6)), except that the last result (equation
(13)) clearly identifies that we are dealing
with only the Airy wave crest portion of the
wavelength which occupies one-half of the
wavelength. That is, the trough portion of
the wavelength contains no potential energy,
because there is nothing (i.e., no mass)
present above the SWL.

Kinetic Energy, EK

The Newtonian representation of
kinetic energy, EK, is given by:


160


Hb Lb










EK= 1 m v2 (14)
2

in which m is the mass, and v is a velocity,
in this case, the velocity of internal water
particles.

Classical Derivation

Kinetic energy determination is
somewhat more involved than that for
determination of the potential energy. To
find the total wave energy under one
wavelength of an Airy wave, EKL, the
change in kinetic energy is integrated over
depth for the wavelength (Dean and
Dalrymple, 1984) according to:


EK = p U2 dz dx (15)
o E 4J, 2
0 d'

Using known transcendental
hyperbolic solutions for water particle
velocities in a progressive wave and carrying
out the integration and simplification (Dean
and Dalrymple, 1984) yields:

1
EK, = pfg H,2 L (16)
16

In this work, the horizontal water
particle speed, u, is given (U. S. Army,
1984) by:


(17)


u = cos 0
2 '4b


where 0 is the phase angle. The vertical
water particle speed, w, is given by:


w- 1 +f sin
T' de


(18)


Equations (16), (17), and (18) were
evaluated for many input values of Hb and T,
the results of which are listed in Table 1 for
some typical examples. Each wave was
divided into 30,000 equivalently sized cells
representing water particles. An example
of horizontal and vertical water particle
velocity fields are illustrated in Figure 2. In
addition to Eulerian kinetic energies being
determined across the entire wavelength,
wave crest and wave trough total kinetic
energies were simultaneously calculated, to
provide some interesting results.

Kinetic energy results were generally
equivalent to those forthcoming from
equation (16), although the intergrally
derived results slightly underestimate those
for equation (16) for steeper waves. This
occurs because integrally derived results are
critically sensitive to water particle velocities
determined from equations (17) and (18). It
is of especial importance to record that
transcendental hyperbolic equations (e.g.,
Dean and Dalrymple, 1984, p. 86; U. S.
Army, 1984, p. 2-32) produce different and
somewhat less satisfactory results for u and
w than do equations (17) and (18) at the
shore-breaking position. Useful equations
describing particle velocities have been
determined from this study (Figure 3) where
the ratio of integrally derived and theoretical
results have a ratio, )p, near unity (see Table
1). Exact equations from the data of Table
1 are:


(U2 )L

(u2 + w1)
and:


(U2 + W2 )


0.0624 c 2

0.0676 c


0.0538 c0


(19)

(20)


(21)


in which z is the vertical departure from the
SWL.


where the phase speed at the shore-breaking
position is given by equation (2).









Table 1. Some examples of total kinetic wave energies for Airy shallow water
waves assumed to be shore-breaking.

Total Wave Energy u2 + 2
Hb T Lb (Joules)
P (m2/s2)
(ml (s) (m)
Entire Wave Wave Entire Wave Wave
Wave Crest Trough Wave Crest Trough
0.25 6.0 11.9 466 315 151 0.9997 0.2391 0.2589 0.2061
0.50 8.0 22.4 3,523 2,380 1,143 0.9980 0.4790 0.4987 0.4128
0.75 9.5 32.6 11,539 7,797 3,742 0.9971 0.7192 0.7789 0.6198
1.00 10.0 39.6 25,014 16,906 8,108 0.9938 0.9621 1.0421 0.8287
1.25 11.5 50.9 50,201 33,926 16,276 0.9949 1.2013 1.3011 1.0349
1.50 12.5 60.6 86,099 58,188 27,912 0.9946 1.4420 1.5619 1.2422
1.75 13.5 70.7 136,707 92,390 44,319 0.9946 1.6823 1.8222 1.4493
2.00 15.0 84.0 211,804 143,129 68,678 0.9959 1.9200 2.0795 1.6543

1 pg H' L
16
n U + WK dzd

0 d
where q is the correction factor for the entire (i.e., over the wavelength) kinetic wave energy column values.


For later applications, it is deel
useful here to identify the contribution of
vertical velocity field. Equations for dat
Table 1 are:


(w2)L = 0.0170 (U2 + w2)L
where the subscript L denotes the equa
applies to the entire wave length,


(w2) = 0.0181 (U2 + w2)0
where the subscript c denotes the equal
applies to the wave crest, and


(w2)t = 0.0147 (U2 + w2),


where the subcript t denotes the equation
applies to the wave trough. These
equations, whose relating coefficients
represent average values, testify to the
minimal contribution of vertical particle
velocity fields for wavelength, crest, and


trough. In fact, that part of the kinetic
energy due to vertical particle motion is less
than 2% of the total kinetic energy for the
wavelength, less than 3% of the total kinetic
energy for the wave crest, and less than
1.7% of the total kinetic energy of the wave
trough, for the data of Table 1.

Where EKL is the total kinetic energy
over the wavelength, EK, is the total kinetic
energy over the crest, and EKt is the total
kinetic energy over the wave trough, then
for the data of Table 1 and Figure 4:


2 E
EK = 0.6743 EK, EK
(24) 3


1
EKf = 0.3242 EKL EKL
3

and
1
EKt = 0.4809 EK, 1 EK,
2


162


(25)


(26)


(27)











Direction of Wave Travel


- I II 1 1i l 111


SWL
A~t


-12 -12 f


Direction of Wave Travel


" Bed "


Figure 2. Eulerian horizontal (top) and vertical (bottom) water particle velocity (m/s) fields for
an Airy shore-breaker where Hb = 1.0 m, T = 8.0 s, and Lb = 39.68 m (Note: the
terminology "bed" is used since this work does not extend into the bottom eddy layer (BEL)
nor is it concerned at this time with BEL kinematics).











163


f1


T


















u2 + w2
(m2 /s2)


0 5 10 15 20 25
c2 (m21s2)


30 35 40


Figure 3. The average of the sum of horizontal and vertical water particle
velocities squared (from equations (17) and (18), respectively) versus the wave
celerity squared for Airy waves at the shore-breaking position.


160000
o A Wave crest
2 140000
S140000 Wave trough
S120000
SEKt = 0.6743 EK L
U 100000

S80000

S60000

S40000
S 0 EKt = 0.3242 EK
o 20000
I.-
0 50000 100000 150000 200000 250000
Total Kinetic Energy Over Wavelength (Joules)

Figure 4. Tabulated (Table 1) values of wave crest and wave trough kinetic
energies versus total kinetic energy for one wavelength.


164








Total and Specific Wave Energy
Over a Wavelength

The total energy contained in a wave
over one wavelength, EL is the sum of the
total potential (equation (6)) and total kinetic
(equation (16)) energies to yield:


1
E, 1 Pr g H2 Lb
8
and the energy density becomes;


1
E,= p, g H6


(28)


in the wave crest (equation (12)) and the
total kinetic energy given by the combination
of equations (16) and (25) yields:


E 1 6 p, gH,, L,,
+ -2 pH,2 L,

3 16
1 pg H2 L,
9.6 _


(32)


and the crest energy density becomes:


(29) E= 4 p, 9 H,2
4.8


Total and Specific Wave Trough Energy

The wave trough is, by definition,
devoid of potential energy. Therefore, we
deal only with kinetic energy. The total
energy of the wave trough is given by the
combination of equations (16) and (26) to
yield:


E. ( p, gH,2 L1)
(30)
1
48 p g H& L1
48


and the wave trough
becomes:


energy density


1
S p, gH H (31)

which was derived by dividing equation (30)
by Lb/2 since for an Airy wave, the trough
occupies one-half of the wavelength.
Comparison of these equations with
equations (28) and (29) yields the result that
total energies associated with the wave
trough comprise 16 2/% of the wave energy
for the entire wave.

Total and Specific Wave Crest Energy

The sum of the total potential energy


(33)


which was derived by dividing equation (32)
by L/2 because for an Airy wave the wave
crest occupies one-half of the wavelength.

Discussion of Airy Results

From the last section it should be
apparent that the energy contained in the
Airy wave crest at shore-breaking comprises
83.3% of the energy of the entire wave (i.e.,
across its wavelength). Moreover, the wave
crest energy is 5 times that of the wave
trough energy. The last result is significant
enough that one is compelled to posit certain
questions. Was the original convention of
using the wavelength as the "disciplinary"
or, pragmatically, the "industry" standard
over which to assess energy density in error,
and oversight, or an misnomer? Was it a
matter of convenience relative to empirical
knowledge and computational technologies
available today? Do we need to treat such a
method for determining average energy (i.e.,
specific energy or energy density) as a
paradigm? Perhaps we should next look at
highly distorted waves which break upon our
shores.

DISTORTED SHORE-BREAKING
WA VES

Waves at the shore-breaking position
are, as we know, highly distorted. The


165








1.6-
1.5
1.4-
1.3
12-
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1 -
0-
-0.1
-0.2
-0.3-
-0.4-
-0.5-
-0.6-
-0.7-
-0.8-
-0.9
-2


Distance (m)


Figure 5. Definition sketch of distorted waves at the shore-breaker position.


design, or "industry" literature, however,
while variously presenting some complicated
mathematical representations of distorted
shore-breakers, is conspicuously devoid of
information related to straightforward
prediction or applied solutions. Given the
available empirical information, this condition
is simply not necessary. Regarding the
approach taken here to represent distorted
shore-breaking waves (Figure 5), it is
certainly not as sophisticated as the author
wishes it to be. In fact, from time-to-time,
these distorted shore-breakers might be
referred to as "stick waves" (because of the
linear segments describing the wave form).
There is no need, however, to apologize for
the approach. It probably has more veracity
than its simplicity might suggest, and shall
form a rudimentary approach as a
prerequisite for future, more precise work.

Shore-breaking waves considered in
this work (to include nearshore bar-breaking
waves) are depth limited waves, so named


to distinguish them from fully forced waves
in deeper water (Balsillie and others, 1976;
Mooers, 1976) which break due to critically
high wind stresses. Shore-breaking occurs
when internal horizontal water particle
velocities in the crest exceed the wave
phase speed. The shore-breaking position is
defined to be reached when conditions of
equation (1) occur at which point:


= 1.0
C,


(34)


termed the kinematic stability parameter
(Kinsman, 1964; Dean, 1968), where Ub is
the horizontal water particle speed in the
breaking wave crest and cb is the breaker
phase speed. This work also recognizes the
various shore-breaking wave types where it
has been demonstrated (Balsillie, 1985)that:


tan ab

,W1(0 r2)


< 0.64, Spillng
0.64-5.0, Plunging
5.0- ?, Surging
> ?, Collapsing


(35)


Lb Lb

I c It
^ Icf Icb- itf --- itb
SDirection of
Wave Travel



Hb Hb



db



Bed

0 2 4 6 8 10 12 14 16 18 20 22 24









in which fb is the modified surf similarity
parameter of Irribarren and Nogales (1949),
tan ab is the bed slope, and which suggests
a slight modification from that reported by
Balsillie (1985, 1999). These shore-breaker
types may be incorporated in a continuum
based on the amount of the wave involved in
breaking, measured from the wave crest
apex downward, Hb", according to:


H, = tanh 8 (36)
8

Spilling occurs where the top of the
unstable wave crest results in aerated and
turbulent water slipping down the front face
of the wave; up to 25% of the upper part of
the wave crest is considered to be involved
in spilling. Plunging occurs where the upper
portion of the wave crest curls over, forms
an air pocket and the curling crest eventually
falls onto the trough fronting the wave crest.
Breaking is defined to occur for a plunger
when the upper portion of the wave crest
front becomes vertical; greater than 25% of
the wave crest top is involved in plunging.
Surging occurs when the basal portion of the
wave crest rushes forward out from under
the wave, sliding up the beach face with a
minimum of bubble production. Collapsing
occurs when the wave crest advance
terminates and the wave crest collapses, and
is at variance with the definition proposed by
Galvin (1968).

The following equations have been
developed from empirical data to represent
the distorted geometry (see Figure 5) of
waves at the shore-breaking position. The
vertical distortion of the wave crest lying
above the still water level (SWL) or design
water level (DWL), Hb' is given (Balsillie,
1983b) by:


Horizontal distortion of the shore-
breaking wave is given by a series of
equations. As the wave, represented by its
wavelength, Lb approaches shore-breaking,
the wave crest length, IC attenuates (i.e.,
becomes < 0.5 Lb), while the following
wave trough length, It increases (i.e.,
becomes > 0.5 Lb). This condition is well
represented by the data of Hansen and
Svendsen (1979) plotted in Figure 6. The
wave crest length (see Figures 5 and 7) is
given by:


/= 0.12+ (0.14 e 0.s5)
Lb
and the trough length, It becomes:


t = Lb -


(38)


(39)


Horizontal distortion of the wave
crest is evaluated by first determining the
length of the wave crest front, Icf, according
to:


0.84 Hb
tan 8,


(40a)


which is applicable where Icf 5 0.5 Ic and
tan Of the angle of the crest front face
measured from the horizontal, is evaluated
as:


tan = 0.52 e0.48 O


(40b)


illustrated in Figure 8. Where Ic > 0.5 1,
then:


/ = 0.5 /


(40c)


Then the crest back or crest stoss length,
Icb, becomes:


H = 0.84 H,


(37)


and, therefore, the amount of the distorted
wave lying below the SWL becomes 0.16
Hb'.


/c l/o


(41)


There is a paucity of information
regarding horizontal wave trough geometry.
The wave trough free surface is, however,
relatively smooth and gentle becoming


167

























0.2 .- "P 2.00 -
-1 PI 0 1041 .
S. = 1.0 + 3p 41071 2.0
0.1 t pl ,41,41
PI 031041 3.33

0 02 0.4 0.6 0.8 1.0

It

Figure 6. Laboratory data of Hansen and Svendsen (1979) illustrating the
attenuation of wave crest length, I during the shore-breaking process.






Snillin- --Plunnina Su ? Co--


A. I I -II I 1 1 i i
tan a Su= Surging
S* 0.0292 Hansen and Svendsen (1979) Co = Collapsing
0.0200 Iverson (1952)
A 0.0500 **
0.1000
o 0.0250 Singamsetti and Wind (1980)
0 0.0500
-A 0.1000
0 0.2000

0.4 ----- --- -.^ ^ - --
C 0.3 2
Lb 0.2 L- = 0.12 + (0.14 etanh 0-5 b)

0.1 r0r = 0.8629
0.0
0.1 1 10
b
Figure 7. Wave crest length at shore-breaking, Ic, measured at the SWL (L =
wavelength at shore-breaking) versus the surf similarity parameter, b (values following
symbols are bed slopes; numbers associated with plotted points are number of waves
representing plotted averages)


168


----,---,


-----r ...










Plunging i Surging


ton of
tane, 0.526b

tan b

0 0.05
A 0.1

101 1 10
tb


Figure 8. Determination of the front face angle, tan ab of the shore-
breaking wave crest; data from Iverson (1952).


depressed near the back part of the trough.
Specific details of its shape do not appear to
be as important (unlike the breaking wave
crest) and it would appear the some
quantitative representation of its general
shape will suffice. Adyemo (1970) provides
some insight (Figure 9), the results of which
are generally confirmed by the results of
Morison and Crooke (1953), Inman and Nasu
(1956), Kemp (1975), Hwang (1984), etc.
The front face length of the trough, It may
be quantified by:


1.0 6.35


fa = t /f


(43)


Potential Energy, EP

Following the Newtonian approach of
equation (3), the potential energy of the
distorted shore-breaking wave is determined
as for the Airy wave approach. The vertical
distance to which the center of wave crest
mass is elevated from the preceding wave
trough, h, is closely given by:


(42a) h = 1 0.84 H, + 1 0.16 Hb=- 1 H
3 3 3


(44)


which is applicable where db /Lb < 0.137.
Where db /Lb 0.137, then:


/ = 0.5 /


The volume of the crest above the SWL
becomes:


(42b) vo = 0.5 (0.84 Hb/ 4)


(45)


and the wave trough back or trough stoss
length, Itb becomes:


and the total potential energies, EPL and EP,,
are assessed as:


169










1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45


0 0.025 0.05 0.075 0,1 0.125 0.15 0.175 0.2
dbl Lb
Figure 9. Data from Adeymo (1970) for determination of the length of
the breaking wave trough front, I where I, = length of the wave
trough, dd = water depth at shore-breaking, and Lb = wavelength at
shore-breaking.


EP, = EP,


0.14 p, g H'


(46)


If we further assess this representation over
the wave crest length, the energy density
becomes:

EP = EP, = 0.14 p, g H2


(47)


7.1429 p H

= 1.12 pg H


which is not far removed from equation (13).

Based on additional analytical
considerations, two more exact and useful
expressions quantifying equation (46) are
given by:

PL = H, Lb H 02--
EP, = H .00 + 2.24 (48a)
16 L,
and:


EP g H26 (224 + 0.00)
16 L b


(48b)


both of which reduce to equation (46) and
also represent EPc.

Kinetic Energy, EK

While considerable investigation on
the internal water particle velocity field has
been conducted for near-breaking or breaking
waves, our precise quantifying capability
beneath the entire wavelength is incomplete.
We have more knowledge about the
horizontal particle velocity field beneath the
breaking wave crest, less about its behavior
beneath the wave trough. Maximum
horizontal particle velocities for the water
column directly beneath the breaker crest
apex, uc max, are given by Van Dorn (1978)
as an invariant vertical distribution according
to:


170


A Spilling shore-breakers-
Plunging shore-breakers-





It"
Stf db t 0.5
= 1.0 3.65
It Lb
I=5I I =









max 0.2 +
Cb


o.1 (yl/t)
1.125 (y/yt)


Eventually, the vertical velocity field needs to
(49) be incorporated.


in which yt is the total water depth at
breaking and y is the local water depth
measured upward from the bed. Equation
(49) appears applicable to results of other
investigators (e.g., Adeymo, 1970; Divoky
and others, 1970; Iwagaki and others, 1974;
Sakai and Iwagaki, 1978; Easson and others,
1988; etc.).

Iwagaki and others (1974) provide a
representative assessment of maximum
horizontal particle velocities for the water
column, utma occurring beneath the lowest
part of the wave trough. It is given by:


utxi = -0.2 0.05 (1 (50)

It is of importance to note that equation (50)
is at significant variance with Airy theory
(equation (17)) which, for the wave trough,
results in absolute magnitudes equivalent to
those occurring beneath the wave crest
(when measured upward from the bed).
Equation (50) results in significantly smaller
values than those occurring beneath the
wave crest given by equation (49).

Little is known about the vertical
velocity field in the vicinity of shore-
breaking. It is necessary to note that vertical
water particle velocities are not considered in
this work for distorted shore-breakers
because of the lack of information. Their
exclusion has been justified on the basis of
their minimal contribution toward
determination of Airy kinetic energy levels
(see equations (22), (23), and (24) and
kinetic energy discussion), from which it is
assumed they are relatively unimportant for
the distorted shore-breaking wave
application. It is recognized, however, that
the wave crest cannot be maintained without
a vertical component as explained by the
dynamic stability parameter (Dean, 1968).


The horizontal water particle field
beneath the breaker crest was determined as
a horizontal linear field with a maximum
directly beneath the wave crest apex
(equation (49)) going to zero at the forward
and stoss stagnation point (point b of Figure
5) and forward stagnation point of the
following crest (point c of Figure 5). An
example of the horizontal water particle field
for a distorted shore-breaker is given by
Figure 10. Note the differences when
compared to those forthcoming from Airy
theory of Figure 2.

Each wave evaluated was divided into
30,000 equivalently sized cells representing
water particle velocity behavior. Velocities
are Eulerian, with the assessment point (i.e.,
"measurement location") occurring at the
shore-breaking position (i.e., at db = 1.28
Hb). Integrally derived results were
determined as for equation (15).

For energies so obtained all are
directly related to IC/Lb which, in turn is
directly related to Fb (equation (38)). The
following equations represent exact
solutions.

Total and specific (energy density)
kinetic energies of distorted shore-breaking
waves for the wavelength are given by:


(51)
LJ


Pr g H2 L. 0
EKL = 6 b (0.225 + 0.64
and, the energy density becomes:

and, the energy density becomes:


P- pg Hb
EKL
16


0.225 + 0.64 ) -
Lb


(52)


Total wave trough kinetic energy is
quantified by:









Direction of Wave Travel


3.92
.00
200
.50
1 25


w---


0.8 0.6 A 0.2 0.0 -1 -0-2 -0.3 -OA -0.5 4. -017 -08
0.7 0. 0.3 0.1


" Bed"


Figure 10. Eulerian horizontal water particle velocity (m/s) field for a distorted shore-
breaker where Hb = 1.0 m, T = 8.0 s, tan ab = 0.02, cb = 3.92 m/s, Lb = 31.39 m,
IC = 9.73 m, 1t = 21.66 m (Note: the terminology "bed" is used since this work does
not extend into the bottom eddy layer (BEL) nor is it concerned at this time with BEL
kinematics).


EKt = p, H2 Lb 0.222 0.222 /c (53a)
or more straightforwardly by:
or more straightforwardly by:


EK = p, g Hb Lb 0.003 + 0.862 Jb (55a)
S16 L,
or more simply by:


EKt p H 0.222 + 0.00-
16 [L 4

0.222 g H:
= 0.222


(53b)


= 0.014 p gH2 /I

and the wave trough energy density
becomes:


EK,


0.222 pg


(54)


1 P g H2
3 24


Total kinetic energy for the wave
crest is given by:


p 'b2 /4
E = _0. 0.871
S16

= 0.871 p, g H,2
16
= 0.054 p, g H /


0.00 /)
i-


(55b)


and wave crest energy density becomes:


P, g H2
EK, = 0.871 16 (56)

= 0.054 p, g H2

Total and Specific Wave Energy
Over a Wavelength

The total energy contained in a wave
over one wavelength, EL is the sum of the


0.0


0,0








total potential and kinetic energies.
Combining equations (48a), and (51) yields:


E 8 0.114 + 1.44 (57)


Dividing by Lb straightforwardly yields the
energy density according to:

E = 0. H2 114+ 1.44 (58)
8 58

Total and Specific Wave Trough Energy

The wave trough is, by definition,
devoid of potential energy which is assessed
above the SWL. Therefore, we deal only
with kinetic energy. The total energy of the
wave trough is given by equations (53a) and
(53b) according to:


E,= 0p,g H .111 0.111 /
8 L,

or by:


Et= H It111 + 0.00
8 L 0
pg a,2 /t
= 0.111 p, H
8
= 0.014 p, g H,2 /I
with an energy density given by:

Prg H,2b
Et = 0.111
8


(59a)


(59b)


1 P g H,2
3 24

which is a small fraction of the Airy trough
kinetic energy (see equation (31)) and is not
dependent on either Ic/Lb or bb.


Total and Specific Wave Crest Energy

Combination of equation (48a), (48b),
(55a), and (55b) yields the total wave crest
energy given by:


E, = P02 .0014 + 1.551 (61a)
8 L,)
or by:


E=, p 1.555 + 0.00

= 1.555 pH2


(61b)


=0.194 p, g Hb2

and was found to result in an energy density
given by:
S1.555 g H
Ec = 1.555


(62)


= 0.194 p, g H,

= 0.933 P
4.8


which is close (6.7% less) to that derived for
Airy crest energy (see equation (33)), and is
not dependent on either Ic/Lb or fb.

DISTORTED SHORE-BREAKING
WAVES ON A HORIZONTAL BED


This investigation was begun by
considering Airy waves shore-breaking on a
horizontal slope in order to identify the
"industry standard" for determination of
wave energy constraints (i.e., across the
entire wavelength). This "standardized"
methodology was then followed to determine
wave energies for distorted shore-breaking
waves. It is deemed useful and, therefore,
necessary to compare Airy results with those
from distorted shore-breakers evaluated on a
horizontal bed. The domain of wavelengths
considered in this work range from near 8 to
70 m. A slope of tan ab = 0.001 for such
wavelengths is essentially a horizontal bed.









Alternatively, one can set 1b equal to zero in
the appropriate kinetic energy equations.
Which one is done does not matter, the
former is simply easier for computer
functions written for the investigation.

It is of tacit import to note that for a
horizontal bed, all distorted shore-breakers
will be of the spilling variety. Greater bed
slopes are required to produce plunging,
surging, or collapsing shore-breakers.

Concerning water particle velocities,
it would seem of consequence to report
distorted shore-breaking results so that they
can be compared to those resulting from the
Airy treatment. Airy results included vertical
water particle velocities, which were deemed
inconsequential to the total contribution. For
distorted shore-breakers horizontal average
water particle velocities were found to be
related to the wave speed according to:


(u2)L = 0.015 c (63)
across the wavelength,


(64)


(u2) 0.033 2
for the wave crest, and


(y2) = 0.0083 c,


for the wave trough.

Results from Airy and distorted shore-
breakers for a horizontal bed are listed in
Table 2 for a relatively wide range of
breaking wave conditions. It is to be noted
that distorted shore-breaker wavelength,
wave crest, and wave trough total energies
do not attain magnitudes forthcoming from
the Airy treatment. This occurs because
Airy wave water particle velocities beneath
the wave trough far exceed those resulting
from the distorted shore-breaker analysis.
The result is interesting even though the
distorted shore-breaker wave trough can
occupy up to 72% of the wavelength, while
the Airy trough is always 50% of the


wavelength.


On a horizontal bed, distorted shore-
breaking waves result in wave crest total
energies 5.00 times that contained in the
wave trough, and 0.833 times that
contained across the wavelength. These
values are similar to those found for shore-
breaking Airy waves. We seldom consider
total wave energies, however, almost
singularly applying energy densities that
have been calculated across the entire
wavelength. What is most significant
appears when comparing energy density
results for Airy and distorted breakers.
Wavelength and wave trough energy
densities of distorted shore-breakers do not
reach Airy energy density magnitudes.
However, distorted shore-breaker crest
energy densities are close to Airy crest
energy density results (equation (62)), and
exceed Airy wavelength energy density by
155%. This result should raise a "red flag"
concerning determination of wave energy
across the wavelength because of crest
versus trough energy density insensitivities
that result.


DISTORTED SHORE-BREAKING
WA VES ON SLOPING BEDS

When assessing shore-breaking wave
energy on a horizontal bed, we have
introduced a significant and fortuitous
condition. The water depth beneath the
wave is constant, and we are assessing
water particle velocities as if they are locally
passing the point where the wave crest apex
shore-breaks. If we, however, introduce a
slope to the bed, conditions become much
more complex. The problem is solved by
assessing water particle velocities using an
Eulerian approach as different parts of the
wavelength progressively pass the point of
shore-breaking at db.

For sloping beds, the relationship
between the average horizontal water
particle velocity and phase speed at breaking


(65)











Table 2. Comparison of distorted shore-breaking wave and Airy wave energies on a
horizontal bed.

TOTAL ENERGY ENERGY DENSITY

WAVE CREST TROUGH
Hb T Lb lb WAVE CREST TROUGH WAVE CREST TROUGH

(i) Is) Fm) (mlb EA sr E0 ar E Iwr E.EtDS
ELAIHY A cAMY Ee IYAIY



0.25 4 7.86 2.05 0.0250 0.490 0.489 0.492 0.490 0.934 0.333
0.25 6 11.77 3.09 0.0380 0.491 0.490 0.491 0.491 0.933 0.333
0.25 8 15.69 4.14 0.0500 0.491 0.491 0.491 0.491 0.932 0.333
0.25 10 19.62 5.19 0.0630 0.493 0.494 0.490 0.493 0.934 0.333
0.25 12 23.54 6.25 0.0750 0.494 0.495 0.490 0.494 0.934 0.333

0.50 4 11.10 2.90 0.0180 0.489 0.489 0.492 0.489 0.936 0.333
0.50 6 16.65 4.36 0.0270 0.489 0.489 0.492 0.489 0.933 0.333
0.50 8 22.19 5.83 0.0350 0.490 0.490 0.491 0.490 0.933 0.333
0.50 10 27.74 7.30 0.0440 0.492 0.492 0.491 0.492 0.935 0.333
0.50 12 33.29 8.78 0.0530 0.493 0.493 0.491 0.493 0.934 0.333

0.75 4 13.59 3.55 0.0140 0.488 0.487 0.493 0.488 0.932 0.333
0.75 6 20.39 5.33 0.0220 0.488 0.487 0.492 0.488 0.931 0.333
0.75 8 27.18 7.12 0.0290 0.490 0.489 0.442 0.490 0.934 0.333
0.75 10 33.98 8.92 0.0360 0.490 0.490 0.491 0.490 0.933 0.333
0.75 12 40.77 10.73 0.0430 0.491 0.492 0.491 0.491 0.934 0.333

1.00 4 15.69 4.09 0.0130 0.488 0.487 0.493 0.488 0.934 0.333
1.00 6 23.54 6.15 0.0190 0.489 0.489 0.492 0.489 0.935 0.333
1.00 8 31.39 8.22 0.0250 0.490 0.489 0.492 0.490 0.935 0.333
1.00 10 39.23 10.29 0.0310 0.490 0.489 0.492 0.490 0.933 0.333
1.00 12 47.08 12.37 0.0380 0.491 0.490 0.491 0.491 0.934 0.333

1.25 4 17.55 4.58 0.0110 0.488 0.487 0.492 0.488 0.934 0.333
1.25 6 26.32 6.87 0.0170 0.489 0.488 0.492 0.489 0.934 0.333
1.25 8 35.09 9.18 0.0220 0.489 0.489 0.492 0.489 0.934 0.333
1.25 10 43.87 11.49 0.0280 0.490 0.489 0.492 0.490 0.933 0.333
1.25 12 52.64 13.81 0.0340 0.490 0.490 0.492 0.490 0.934 0.333

1.50 4 19.22 5.01 0.0100 0.488 0.487 0.493 0.488 0.934 0.333
1.50 6 28.83 7.53 0.0150 0.488 0.487 0.492 0.488 0.933 0.333
1.50 8 38.44 10.05 0.0200 0.490 0.489 0.492 0.490 0.935 0.333
1.50 10 48.05 12.58 0.0260 0.489 0.489 0.492 0.489 0.933 0.333
1.50 12 57.66 15.12 0.0310 0.490 0.490 0.492 0.490 0.934 0.333

1.75 4 20.76 5.41 0.0090 0.488 0.487 0.493 0.488 0.934 0.333
1.75 6 31.14 8.13 0.0140 0.488 0.487 0.493 0.488 0.933 0.333
1.75 8 41.52 10.85 0.0190 0.489 0.489 0,492 0.489 0.935 0.333
1.75 10 51.90 13.58 0.0240 0.488 0.487 0.492 0.488 0.931 0.333
1.75 12 62.28 16.32 0.0280 0.489 0.489 0.492 0.489 0.933 0.333

2.00 4 22.19 6.23 0.0090 0.488 0.487 0.493 0.488 0.934 0.333
2.00 6 33.29 9.35 0.0130 0.488 0.487 0.493 0.488 0.934 0.333
2.00 8 44.39 12.48 0.0180 0.489 0.489 0.492 0.489 0.936 0.333
2.00 10 55.49 15.61 0.0220 0.489 0.489 0.492 0.489 0.934 0.333
2.00 12 66.58 18.75 0.0270 0.489 0.489 0.492 0.489 0.933 0.333

Averages 0.490 0.489 0.492 0.490 0.933 0.333


175










is slightly dependent on bed slope when
assessed across the entire wavelength,
according to:


(u2)L = (0.015 + 0.028 tan ah) c2 (66)

Average horizontal water particle speeds for
the wave crest and trough are independent
of the bed slope and equations (64) and (65)
apply.

Shore-breaker wavelength, wave
crest, and wave trough energy densities for
vertically and horizontally distorted waves
shore-breaking on non-horizontal bed slopes
are compared to their Airy counterparts
(horizontal bed only for the latter) in Figure
11. Distorted wavelength energy densities
remain less than for Airy waves at shore-
breaking (49% to 83%). Distorted trough
energy densities are constant at 33.3% of
Airy trough energy densities. Distorted
breaker crest energy densities are only
slightly less than Airy crest energy densities,
and 1.56 times larger than the wavelength
energy density of Airy waves.

When considering distorted shore-
breakers only, the distorted wave crest
contains from 83% to 93% of the total
energy measured across the entire
wavelength. The trough, therefore, accounts
for but 7% to 17% of the wavelength total
energy. However, distorted wave crest
energies range from 5.0 to 14.0 times the
wave trough energies! Both outcomes are
illustrated by Figure 12. Wave crest energy
density is 14 times wave crest energy
density, and is independent of bed slope (see
equations (60) and 62)).

Another reason for addressing non-
horizontal bed slopes is to include shore-
breaker types other than the spilling breaker
which is the only breaker type that can occur
on a horizontal bed. It is the author's
opinion that the overwhelming reason waves


shore-break is because they are depth
limited. Other researchers (e.g., Galvin,
1969; Collins and Weir, 1969; Weggel,
1972a, 1972b; Mallard, 1978) include, in
addition to the water depth, wave steepness
and/or bed slope. It is the author's
contention (Balsillie, 1983a, 1985, 1999)
that these factors contribute very little, if
anything, toward wave instability. They are,
rather, influential in determining the type of
shore-breaking wave (see equation (35)).
Results of this study (Figure 11) suggest that
energy is transferred from the trough to the
crest, thereby, affecting the shore-breaker
type.

DISCUSSION AND CONCLUSIONS

This work has investigated how
energy is distributed across the wavelength
of ocean-propagating waves. It has included
assessment of both total energy and energy
density. The latter, also termed specific
energy, provides a measure for assessing
inherent variability. It does so because it is
an average quantity representing some
length such as wavelength, crest length,
trough length, etc. When we determine
averaged quantities, it is desirable that the
variability associated with the average
outcome is minimized. This work, however,
demonstrates that for shore-breaking waves,
the associated variability for currently applied
theory across the wavelength is not only not
minimized, it is very large. Moreover, this
work demonstrates that such a condition
would not appear to be necessary.

The proper statistical tool to use
when assessing variability is the relative
dispersion (alternatively named the
coefficient of variability or coefficient of
dispersion) in which the standard deviation is
divided by the mean. The mean in our case
is the energy density, and the standard
deviation is assessed using energy
associated with each water particle column.
The relative dispersion analysis unequivocally


176














Ec DIST
Ec AIRY

EL DIST

L AIRY
Et DIST
t AIRY


----- SpillingPling ungng -4 Surging?Collapsing-
2

9 E_ 1162) I _-, j I & tan ao
S0.001
9 M 0.010
8-
8 : : 0.020
7 Eq. 58 0.030
. 0 050
S- --- 0.100
4 1 0.200

2 -- Eq. (60)/I

0
0.01 0.1 1 10 1C


Figure 11. Eulerian wave energy density ratios for distorted and Airy waves
(the latter on a horizontal bed) at shore-breaking, illustrating the effect of bed
slope, tan ab and the surf similarity parameter, b ; bed slope wave groups
represent breaker heights ranging from 0.25 to 2.0 m with periods ranging from
4 to 12s.


-- Spilling -Plunging- -Surging?Collapsing--
16 -

14 lltan b
E 12 :- 0.001
U 0.010
Et 10 -111 0 0 020

8 I I I T I 0.030
S0.050
6 To ooo
4 .... _. ._. IO 0.200

E 2
F n --7 ,- _


0.01


Figure 12. Ratios of Eulerian distorted shore-breaker crest to trough, and crest
to wavelength total energies, illustrating the effect of bed slope, tan ab and
surf similarity parameter, b ; bed slope wave groups represent breaker heights
ranging from 0.25 to 2.0 m with periods ranging from 4 to 12 s.


177










Spilling Plunging- -Surging?Collapsing-
2.0
1.9 L l t-1 1
1.8 "IFI4ll tanab
1.7 Wavelen thA 0001
1.6 1 1 0.010
S1.5 -0-,,020
S1.4 0,030
0 1.2 & 0.050
S1 1Ill0.100
A 1.0 Wave Trougho 0200

0.8
0.7
0.6 Wave Crest --
0.5 ,- I lllllii 1 1
0.5
0.01 0.1 1 10 100
b
Figure 13. Energy density relative dispersion for energy associated with water
particle columns for distorted shore-breakers.


demonstrates that the energy densities
separately determined for wave crest and
wave trough are far superior to determining
energy density content across the entire
wavelength (see Figure 13 and Table 3).
Further subdivision of the wave (e.g.,"
quarter-zoning" the wave into wave crest
front, wave crest back, wave trough front,
and wave trough back) results in but only
very slightly different values of the relative
dispersion for the associated crest and
trough. The "bottom line" is that, in terms
of wave energy density statistical variability
or statistical dispersion, wave trough energy
density is better than wavelength energy
density assessment by a factor ranging from
1.5 to 2.1, and wave crest energy density
assessment is better by a factor ranging
from 1.7 to 2.4. The ramifications of these
conclusions may be of rudimentary
importance in the way that one can view and
understand wave energy. They may,
perhaps, be why when dealing with energy
applications (e.g., in sediment transport) we
achieve either confounding results or
unsatisfactory results which require the use
of unexplained proportionality constants.


When considering energy
applications, there are two highly visible
issues that require discussion. These are:
1) wave pressures and 2) sediment
transport.

Pressures associated with waves are
described to consist of a first extremely high
pressure of very short duration termed


Table 3.
dispersion


Energy density relative
analysis results for distorted


shore-breakers.
General Relative
Wave Segment
S Description Dispersion
1.333 to
Wave Length Variable .1
1.811
Wave Crest Constant 0.768
Wave Crest Front 0.788
Wave Crest Back 0.767
Wave Trough Constant 0.870
Wave Trough Front "0.869
Wave Trough Back "0.879


178









Table 4. Summary of wavelength, wave crest, and wave trough energy constraints for Airy
and distorted breaking waves.
Total Energy Energy Density2

Description of Breaking tan E E EL /Sr Ec ,ST Et asr
Condition ELb jE Y CA R
L Et E AIRY AIRY Et AIRY


AIRY: shore-breaking at 0 0.833 5.00 NA NA NA
db = 1.28 Hb
horizontal bed.

DIST: JHB Distorted shore- 0 0.833 5.00 0.490 0.933 0.333
breaking waves at
db = 1.28 Hb
horizontal bed.

0.001 0.83 5.00- 5.06 0.49 0.933 0.333
0.010 0.84- 0.87 5.07- 6.60 0.49- 0.57 0.933 0.333
DIST: JHB Distorted shore-
SJ Distorted shre 0.020 0.85 0.89 5.26- 8.74 0.51 -0.67 0.933 0.333
0.030 0.85 0.91 5.44- 10.76 0.52 -0.74 0.933 0.333
db = 1.28 Hb 0.050 0.86- 0.93 5.83- 13.10 0.54 0.81 0.933 0.333
sloping beds. 0.100 0.87 -0.93 6.98- 13.92 0.60- 0.83 0.933 0.333
0.200 0.90- 0.93 9.51 14.00 0.71 -0.83 0.933 0.333

NOTES:
NA = not applicable; Subscripts: L = wavelength, c = wave crest, t = wave trough; Hb = shore-breaking
wave height, db = water depth at shore-breaking position; 1 Energy results were calculated for each bed
slope with shore-breaking wave heights ranging from 0.25 to 2.0 m; 2 Wavelength, crest, and trough
energy densities for AIRY waves conform to the following equations,
all assessed at shore-breaking on a horizontal bed:


1 p H2
EL AIRY P g H,,b


Am -18 p gH
E. ANY = 4.8 p, g H,,2


E AIRY 4 pgH,


"gifle" or impact pressure, followed by a
second pressure less in magnitude and
longer in duration termed "bourrage" or
dynamic pressure (Larras, 1937). These
pressures are commonly applied as horizontal
or vertical components. Field data (Miller
and others, 1974a, 1974b; Miller, 1976)
indicate that horizontal impact pressures
from shore-breaking and broken waves
significantly exceed those from non-breaking
waves. Highest impact pressures occur in
post-breaking bores. Shore-breaking waves
produced next highest horizontal impact
pressures, with greater pressures
forthcoming from plunging rather than


spilling breakers. The difference between
breaking and post-breaking pressures is the
elevation at which they occur, being higher
for the shore-breaking waves. (Balsillie,
1985) Clearly, from the viewpoint of using
the higher elevation associated with waves,
we are interested in pressures forthcoming
from the wave crest and not the trough.
This work, for instance, found that the
energy density of the shore-breaking wave
crest is close to that given by Airy theory for
the crest, and is 1.555 times Airy energy
density assessed across the entire
wavelength (equation (62)).

Empirically evaluated distorted shore-









breaking waves occurring on non-horizontal
bed slopes yield quite different outcomes.
Distorted wave crest total energy ranges
from 83% to 93% of the wavelength total
energy, increasing with an increase in the
bed slope and surf similarity parameter.
Moreover, distorted total wave crest energy
ranges from 5.0 to 14.0 times the total
wave trough energy, increasing with
increases in bed slope and surf similarity
parameter values! Wave crest energy
density is 14 times the wave trough energy
density, and independent of bed slope
(equations (60) and (62))! These results
invoke significant questions as to the
veracity of determining energy density
across the wavelength.

Sediment transport does require,
however, knowledge of crest and trough
kinematics since net drift beneath the crest
is in the direction of wave propagation, and
opposite the direction of wave propagation
beneath the trough. Energy densities for
distorted shore-breaker crests are large (5 to
14 times trough energy densities).

This paper has investigated the lateral
distribution of energy across a wave at the
shore-breaking position in three sections: 1)
symmetrical Airy waves on a horizontal bed,
2) horizontally and vertically distorted shore-
breakers on a horizontal bed, and 3)
horizontally and vertically distorted shore-
breakers on sloping beds. The latter two are
based on empirical data. Each section
contains pertinent summary information,
which is summarized in Table 4. Results
indicate that at the shore-breaker position,
the wavelength is not the appropriate length
over which to determine wave energy
density. It is proffered that it would be more
appropriate to determine energy densities
separately for wave crest and wave trough.

ACKNOWLEDGEMENTS

This work reflects, in part,
investigations initiated by the author while


he was chief of the Analysis/Research
Section of the Division of Beaches and
Shores (now the Bureau of Beaches and
Coastal Systems) of the Florida Department
of Natural Resources (now the Department
of Environmental Protection), as encouraged
by Robert G. Dean then Director of the
Division of Beaches and Shores and
Professor, Department of Coastal and
Oceanographic Engineering, University of
Florida, Gainesville, FL. The final form of the
work, as it appears here, recognizes the
concerted interest, countless hours of
discussion, and encouragement of William F.
Tanner, Regents Professor, Department of
Geology, Florida State University,
Tallahassee, FL.

Editorial comments of the FGS staff
from Jon Arthur, Kenneth Campbell, Joel
Duncan, Ted Kiper, Ed Lane, Frank Rupert,
Thomas Scott, and Steven Spencer are
acknowledged.

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On the breaking of nearshore waves
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 Material Information
Title: On the breaking of nearshore waves
Added title page title: Shore-breaking wave energetics
Physical Description: xii, 183 p. : ill. ; 28 cm.
Language: English
Creator: Balsillie, James H
Publisher: Florida Geological Survey
Place of Publication: Tallahassee, Fla.
Publication Date: 1999
 Subjects
Subjects / Keywords: Water waves -- Mathematical models   ( lcsh )
Ocean waves -- Research   ( lcsh )
Genre: bibliography   ( marcgt )
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Table of Contents
    Front Cover
        Front Cover
    Errata
        Errata
    Title Page
        Page i
        Page ii
    Letter of transmittal
        Page iii
        Page iv
    Foreword
        Page v
        Page vi
    Table of Contents
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
    On the breaking of nearshore waves
        Page 1
        Page 2
        Page 3
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    Shore-breaking wave energetics
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    Back Cover
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Full Text





State of Florida
Department of Environmental Protection
David B. Struhs, Secretary


Division of Resource Assessment and Management
Ed Conklin, Director


.: --
lorida Geological Survey.
Walter SchrmidtrStaWGe6ologist and Chief


Special Publication No. 45-


On the Breaking of Nearshore Waves


by,.

- aiames H. Balsillie
SL"""~~~~~"""3


.-:._


and


Shore-Breaking Wave Energetics




James H. Balsillie


Florida Geological Survey
Tallahassee, Florida
1999


QE
99
.A341
no.45


--






















ERRATA


First paragraph on page 24, last sentence should read equation(20) instead of equation


(18).


The first paragraph of the DISCUSSION on page 27, "Figure 4" should read
Figure 5.



On page 34 add the following reference:

Mallard, W. W., 1978. Investigation of the effect of beach slope on the breaking height to
depth ratio: Ms Thesis, Department of Civil Engineering, Newark, DE, University of
Delaware, 168 p,











State of Florida
Department of Environmental Protection
David B. Struhs, Secretary




Division of Resource Assessment and Management
Ed Conklin, Director



Florida Geological Survey
Walter Schmidt, State Geologist and Chief





Special Publication No. 45


On the Breaking of Nearshore Waves

by

James H. Balsillie

and


Shore-Breaking Wave Energetics

by

James H. Balsillie


Florida Geological Survey
Tallahassee, Florida
1999






















cq























Printed for the
Florida Geological Survey

Tallahassee, Florida
1999

ISSN 0085-0640







LETTER OF TRANSMITTAL


Florida Geological Survey
Tallahassee

Govemor Jeb Bush
Florida Department of Environmental Protection
Tallahassee, Florida 32304-7700

Dear Governor Bush:

The Florida Geological Survey, Division of Administrative and Technical Services,
Department of Environmental Protection is publishing two papers in Special Publication No. 45:
"On the breaking of shore-breaking waves" and "Shore-breaking wave energetics.

The first paper identifies where waves shore-break. When or where waves shore-break
has been a controversial topic for over a century. Where they break, significantly affects wave
energy constraints. This work has finally settled the issue. It is found that nearshore waves are
water depth limited and that the water depth in which they break is 1.28 times the mean shore-
breaker height. The second paper provides a numerical methodology for determining how energy
is distributed across the wavelength for waves at the shore-breaking position. It was found that
energy density of the breaker crest is 14 times that of the breaking wave trough.

Practical uses for the results presented in these works include the determination of
dynamic impact pressures necessary in the design of coastal structures, increased precision in
identifying breaking wave parameters leading to increased precision in predicting sediment
transport, sediment budgets, beach and coast erosion during storm/hurricane impacts, etc.

Respectfully yours,


Walter Schmidt, Ph.D., P.G.
State Geologist and Chief
Florida Geological Survey





























































iv












FOREWORD


Whether rocks of our planet (or others for that matter) are igneous, metamorphic, or
sedimentary, the final form of the rock deposit is dependent upon forces which led to its
formation. The former we refer to as a response element (i.e., the final form of the deposit),
the latter as force elements (i.e., forces which led to deposition and induration). Force
elements include wind, hydraulic forces, gravity, pressure, temperature, chemical reactions,
etc.

This work is concerned with sedimentary deposits. The majority of geologists have
been involved in studying and describing insitu unconsolidated or lithified sedimentary
deposits. Unless fossils are present, they will have little idea of the conditions leading to
deposition. Moreover, fossils may not indicate specific conditions of transport and deposition.
While relatively small in number, there are, however, geologists who have adopted an
expanded Earth Science perspective, and have sought to study first currently occurring forces
(e.g., water waves, water currents, wind, gravity, etc.) and then describe the resulting
sedimentary deposits (e.g., based on their granulometry and/or bedding characteristics). This
does, after all, constitute an underlying and basic Geological or Earth Science concept
proposed in 1785 by James Hutton, termed the Principle of Uniformitarianism. It states: "The
present is the key to the past."

It is from the latter school of geology that this work surfaces. It has always been an
errand of keen professional interest to the author for two reasons: 1) it allows for the study
of natural environmental processes using the robust scientific method that, in combination,
2) provides results that can not only be used to interpret ancient sedimentary deposits and
rocks, but provides information that can have significant value for current and future
environmental concerns. After all, if Hutton's principle is true, then the corollary must also
be true that ... the present is the key to the future.

The work presented in this study is concerned with numerically quantifying where
waves break and wave energy. Alaska, California, Florida, and Texas have the longest ocean-
fronting shorelines in the United States. Florida has approximately 1,253 miles of shoreline
that front directly upon the Atlantic Ocean, Gulf of Mexico, and Straits of Florida. Moreover,
annual average wave energy levels range from near zero for Florida's Big Bend Gulf Coast, low
to moderate for the remainder of the Gulf Coast, to high along Florida's Atlantic shores. Only
Alaska experiences such wave energy variability, not only because of its large waves but
because of zero-energy occurring along ice-wedged shores during a significant portion of the
year. Alaska does not, however, have a coastal population of any significant proportions.
California has a significant coastal population and large Pacific Ocean waves, but not the wave
height variability. Coastal Florida and Texas also experience tropical storm and hurricane wave
impacts that far exceed annual average wave energy levels. Texas does not, however, have
Florida's wave energy variability. Florida, then, has perhaps the longest shoreline in terms of
wave energy variability and socio-economic characteristics. Such wave energy variability
requires those of us in Florida to be sensitively precise in wave characteristics and wave
energy assessment and application. We must remind ourselves that it was and is marine
forces (i.e., primarily waves) that, for the most part, formed and form Florida's surficial
sediment configurations.









It is with this explanation that I submit the content of this work to the record

James H. Balsillie
May 1999









TABLE OF CONTENTS


ON THE BREAKING OF NEARSHORE WAVES

ABSTRACT ....................................................
INTRODUCTION ................................................
PREVIOUS W ORK ..............................................
THE DA TA ...................................................
Laboratory Data ...........................................
Field D ata ... .. .. .... .. ... .. ... ... ... .. ... .. .. ... .. ...
G aillard Data ........................................
Scripps Leica Data ....................................
Scripps Special Measurements ...........................
Balsillie and Carter Data ................................
W eishar Data .......................................
ANALYTICAL PROCEDURE ........................................
Occam 's Razor ......................................
Doyle's Principle .....................................
Procedure ..........................................
STEPW ISE REGRESSION ..........................................
Assessment of Equation (11) ..................................
Data Redundancy .....................................
Net Contributions .....................................
Some Observations ...................................
Data Set Listings and Stepwise Regression Results .............
Stepwise Assessment of Individual Data Sets .................
Evaluation of a Special Data Set ..........................
Assessment of Equations (12) and (13) ..........................
A Note on Matrix Algebra ....................................
COMPARATIVE ANALYSIS OF PREDICTIVE METHODS ....................
A Logarithmic Relationship .................................. .
Variability Analysis .........................................
Functional Regression Variability Analysis ................... .
Natural Variability Analysis ..............................
DISCUSSIO N .......................... .......................
The Nelson-Gourlay Horizontal Bed Slope Breaking Anomaly ............
CONCLUSIONS ................................................
ACKNOW LEDGEMENTS ..........................................
REFERENCES ..................................................

LIST OF FIGURES
Figure 1. Water depth in which shore-breaking occurs, db, given by db = c Hb (Hb
= shore-breaking wave height), versus the relative breaker energy level.. ..

Figure 2. Three basic types of shore-breaking waves. .....................


Page










Figure 3. Relationship between the percentage of the wave crest that is involved in
shore-breaking, Hb' (measured from the crest apex downward) where Hb is
the mean shore-breaking wave height, as a function of the modified surf
similarity parameter, Jb. Numbers next to symbols represent number of
values from which averages were determined. (After Balsillie, 1985). .......... 4

Figure 4. Comparison of results of predicted d/Hb using the relationships of Galvin
(1969), Collins and Weir (1969), and Mallard (1978) which consider bed
slope, tan ab, only, described in text.................................. 8

Figure 5. The McCowan equation as it represents the data of Table 3 for db and
Hb. Data include 810 total points, 156 field data points, and 16 laboratory
prototype data pairs, and 624 small wave laboratory data pairs; + 3 sb
indicate the limits of natural shore-breaker wave height variability within a
shore-breaking wave train. ...................................... 25

Figure 6. Relationship between the mean shore-breaking wave height, Hb, for a
single wave train and its associated standard deviation, sb, (after Balsillie and
Carter, 1984a, 1984b). ........................................ 27

LIST OF TABLES

Table 1. Some early general observations reported by Gailliard (1904) and Scripps
Institution of Oceanography (1945). ................................ 6

Table 2. Breaker depth studies resulting in quantifying relationships. ............. .. 7

Table 3. List of general characteristics of field and laboratory data used in
this w ork. .................................................. 10

Table 4. Stepwise regression results for equation (11). ........................ 15

Table 5. Correlation matrix for results forthcoming from equation (11). Numerical
values above and to the right of the zig-zag diagonal line are Pearson
product-moment correlation coefficients (r), those below and to the left are
100 r2. See text for explanation. .................................. 16

Table 6. Net contributions of independent variables ................. ........ 17

Table 7. Net contributions of independent variables in predicting db from stepwise
regression analysis for equation (11). Results are listed for each
investigation and/or data groupings and averages associated with data
groupings .. ................................................. 19

Table 8. Characteristics of data and assessments for a special data set ........... 20

Table 9. Net contributions of independent variables for special data set ........... 21

Table 10. Determination of final regression coefficient relating db and Hb ........... 22










Table 11. Comparative analysis results for three prediction methods of db. .......

Table 12. Determination of corrective regression coefficients for Mallard's and
Weggel's equations. .........................................

Table 13. Examples of the degree on which shore-breaking wave energy levels are
dependent on the depth in which the waves break. Assessment is made in
terms of sb given by equation (22), where the wave energy is directly
proportional to the wave height squared ..........................

APPENDICES

APPENDIX I. Example calculations of the effect of shore-breaking water depth
on wave energy constraints. ...................................


APPENDIX II. Stepwise regression results for data sets and subsets. ..........
All data. ...................................... ..........
All data minus Gaillard's (1904) field data. .........................
All data minus Gaillard's (1904), Scripps Leica Type I and II (1944a, 1944b,
1945), and Weishar's (1976)field data............................
All Laboratory data. .........................................
Laboratory data minus prototype laboratory data. ...................
Prototype laboratory data. ...................................
A ll Field data. ............................................


.. 23


. 24





. 28


..37


. 41
... 43
... 45

... 47
... 49
... 51
... 53
... 55


APPENDIX III. Data listings and stepwise regression
Gaillard (1904) field data. ..............
Scripps (1944a, 1944b, 1945) Leica Type I fie
Scripps (1944a, 1944b, 1945) Leica Type II fie
Balsillie and Carter (1980)field data. ......
Weishar (1976) original field data. ........
Munk (1949) Beach Erosion Board laboratory d


results for each investigation. ..


Id data.. .
ld data. .


ata. . .


Munk (1949) Berkeley laboratory data..............
Putnam and others (1949) laboratory data. ...........
Iversen (1952) labooratory data. ..................
Morison and Crooke (1953) laboratory data. .........
Galvin and Eagelson (1965) laboratory data. .........
Eagleson (1965) labooratory data. ..................
Horikawa and Kuo (1967) laboratory data. ..........
Bowen and others (1968) laboratory data. ..........
Komar and Simmons (1968)laboratory data..........
Galvin (1969) laboratory data.....................
Weggel and Maxwell (1970) laboratory data. ........
Iwagaki and others (1974) laboratory data...........
Walker (1974) labooratory data. ..................
Van Dor 1978) labotoatry data ..................
Hansen and Svendsen (1979) laboratory data .......
Singamsetti and Wind (1980) laboratory data. ........
Nadaoka (1986) labotoatory data. .................
Smith and Kraus (1990) laboratory data ............


. 57


. . . . . . . . . 59
.. .. .. ... .. .. .. 63
. . . . . . . . . 67
. ... .. .. .. .. .. .. 69
. .. .. .. .. .. .. .. 73
. . . . . . . . . 77
. . . . . . . . . 8 1
.. .. .. .. .. .. .. 83
. .. ... .. .. .. .. 87
. . . . . . . . . 9 1
.. .. .. .. .. .. .. .. 93
. . . . . . . . . 97
. .. .. .. .. .. .. .. 99
................ 103
................ 105
................ 109
. . . . . . . . 1 1 1
. . . . . . . . 1 13
. ... ... .. .. .. 117
. . . . . . . . 1 19
. . . . . . . . 12 1
................ 123
................ 127
................ 129









Small laboratory wave sets combined. .............................. 133


APPENDIX IV. Stepwise regression results for a special data set ............


. 137


APPENDIX V. Stepwise regression results where db\Hb is the independent
param eter. ................................................ 14 1

APPENDIX VI. Stepwise regression results where tan ab is the independent
variable. ................................................... 145

APPENDIX VII. Additional data needed to improve the statistical fit of Figure 5
of the maintext. .................. ........................... 149


APPENDIX VIII. Field data for determination of additional values of Hb"/Hb and
b for Figure 3. ...........................................


153


SHORE-BREAKING WAVE ENERGETIC


ABSTRACT ..............................................
INTRODUCTION ...........................................
AIRY WAVE THEORY ......................................
Potential Energy, EP ..................................
Classical Derivation .............................
A Different Approach ............................
Kinetic Energy, EK ...................................
Classical Derivation .............................
Total and Specific Wave Energy Over a Wavelength ..........
Total and Specific Wave Trough Energy ...................
Total and Specific Wave Crest Energy ....................
Discussion of Airy Results .............................
DISTORTED SHORE-BREAKING WAVES ........................
Potential Energy, EP ..................................
Kinetic Energy, EK ...................................
Total and Specific Wave Energy Over a Wavelength ..........
Total and Specific Wave Trough Energy ...................
Total and Specific Wave Crest Energy ....................
DISTORTED SHORE-BREAKING WAVES ON A HORIZONTAL BED ....
DISTORTED SHORE-BREAKING WAVES ON SLOPING BEDS ........
DISCUSSION AND CONCLUSIONS ............................
ACKNOWLEDGEMENTS ....................................
REFERENCES ............................................


.......... 157
.......... 157
.......... 158
.......... 159
.......... 159
.......... 160
.......... 160
.......... 161
.......... 165
.......... 165
.......... 165
.......... 165
.......... 165
.......... 169
.......... 170
.......... 171
.......... 173
.......... 173
. .. ... 173
.......... 174
.......... 176
.......... 180
.......... 180


LIST OF FIGURES

Figure 1. Definition sketch of wave conditions for an Airy sine wave at the shore-
breaking position; Hb = 1.0 m, T = 10 s. ............................. 159









Figure 2. Figure 2. Eulerian horizontal (top) and vertical (bottom) water particle
velocity (m/s) fields for an Airy shore-breaker where Hb = 1.0 m, T = 8.0 s,
and Lb = 39.68 m (Note: the terminology "bed" is used since this work does
not extend into the bottom eddy layer (BEL) nor is it concerned at this time
with BEL kinem atics). .......................................... 163

Figure 3. The average of the sum of horizontal and vertical water particle
velocities squared (from equations (17) and (18), respectively) versus the
wave celerity squared for Airy waves at the shore-breaking position ....... 164

Figure 4. Tabulated (Table 1) values of wave crest and wave trough kinetic
energies versus total kinetic energy for one wavelength ................. 164

Figure 5. Definition sketch of distorted waves at the shore-breaker position ....... 166

Figure 6. Laboratory data of Hansen and Svendsen (1979) illustrating the
attenuation of wave crest length, Ic, at the shore-breaking position ........ 168

Figure 7. Wave crest length at shore-breaking, Ic (Lb = wave length at shore-
breaking) versus the surf similarity parameter, b (values following
symbols are bed slopes; numbers associated with plotted points are
number of waves representing plotted averages). ................... .. 168

Figure 8. Determination of the front face angle, tan ab of the shore-breaking
wave crest; data from Iverson (1952). ................................ 169

Figure 9. Data from Adeymo (1970) for determination of the length of the breaking
wave trough front, It where It = length of the wave trough, db = water
depth at shore-breaking, and Lb = wave length at shore-breaking ......... 170

Figure 10. Eulerian horizontal water particle velocity (m/s) field for a distorted
shore-breaker where Hb = 1.0 m, T = 8.0 s, tan ab = 0.02, cb = 3.92 m/s, Lb
=31.39 m, Ic = 9.73 m, t = 21.66 m (Note: the terminology "bed" is used
since this work does not extend into the bottom eddy layer (BEL) nor is it
concerned at this time with BEL kinematics). ....................... 172

Figure 11. Eulerian wave energy density ratios for distorted and Airy waves (the
latter on a horizontal bed) at shore-breaking, illustrating the effect of bed
slope, tan ab and the surf similarity parameter, ; bed slope wave
groups represent breaking heights ranging from 0.5 to 2.0 m with periods
ranging from 4 to 12 s. ......................................... 177

Figure 12. Ratios of Eulerian distorted shore-breaker crest to trough, and crest to
wavelength total energies, illustrating the effect of bed slope, tan ab and
surf similarity parameter, &b ; bed slope wave groups represent breaker
heights ranging from 0.25 to 2.0 m with periods ranging from 4 to 12 s .... 177

Figure 13. Energy density relative dispersion for energy associated with water
particle columns for distorted shore-breakers. .................. ..... 178









LIST OF TABLES

Table 1. Some examples of total kinetic wave energies for Airy shallow water waves
assumed to be shore-breaking. ................................... 162

Table 2. Comparison of distorted shore-breaking wave and Airy wave energies on a
horizontal bed. .............................................. 175

Table 3. Energy density relative dispersion analysis results for distorted
shore-breakers. ............................................... 178

Table 4. Summary of wavelength, wave crest, and wave trough energy constraints
for Airy and distorted breaking waves. .............................. 179











ON THE BREAKING OF NEARSHORE WAVES

by

James H. Balsillie, P. G. No. 167

Coastal Engineering Geologist, Geologic Investigations Section
The Florida Geological Survey

ABSTRACT

This investigation considers data for 624 small laboratory shore-breaking waves,
16 prototype (large) laboratory wave tank shore-breaking waves, and from 131 to 172
(depending on the number of variables available for analysis) field shore-breaking
waves, for the determination as to where shore-breaking occurs. The original, formal
definition of McCowan (1894) suggested that nearshore waves are depth limited (i.e.,
related to water depth only). Subsequent investigators, feeling the answer must be
more complicated have, in addition, included bed slope, tan ab, equivalent wave
steepness, Hb/(g T2), the surf similarity parameter, fb, etc., in endeavors to "refine"
predictive power. Results of this study, to a highly significant level, however, confirm
that nearshore waves are depth limited. It was found that db = 1.277 Hb, where db
is the water depth at the shore-breaking position and Hb is the average shore-breaking
wave height. This result is so closely related to McCowan's original result of db = 1.28
Hb, that McCowan's relationship remains as the standard instrument for prediction.


INTRODUCTION

Where or when waves shore-break
affects our ability to predict longshore
currents, water levelss, sediment transport,
wave impact pressures, longshore bar
dynamics, wave reformation, etc. It is, in
fact, an issue of fundamental importance in
coastal applications, even when rated in the
context of the most basic of Earth-bound
Newtonian physics. It is, perhaps, no where
more importantly sensitive than when we
consider wave energy. Following are
some examples concerning its importance.
Golob and Brus (1993) state that at any
moment the "... total amount of power in
waves breaking along the world's shorelines
has been estimated at about 2 million to 3
million megawatts, which is about equal to
the generating capacity of 3,000 large power
plants". They also note that at any moment
"... the total wave-power potential along


Europe's Atlantic Coast is about 111,000
megawatts, or about 85 percent of the
European Community's current demand for
electricity".

In Florida (1990-1992 data; Florida
Department of Community Affairs (1995), U.
S. Department of Energy, 1997)), the
average residential structure uses an average
of 39 kilowatt-hours in one day or 1188
kilowatt-hours in one average month (30.42
days). Suppose that for your beach-front
community a specialized power plant placed
in the surf could precisely harness all the
breaking wave energy along but a 10 m
length of beach. For a 0.25 m breaker
height with a 4 s wave period, enough
energy would be harnessed in one day
(133,013,388 joules or 36.95
kilowatt-hours) to power your home for
almost one day (22 .7 hours). For a 0.5 m
breaker height with a period of 6 s, enough









energy would be
harnessed in one day 20
(752,439,744 joules or 10-
209.1 kilowatt-hours) to
5 -
provide power for more
RELATIVE -
than 5 homes (5.36 BREAKER 2-
homes to be precise). For ENERGY
a 1.0 m breaker height LEVEL
with a period of 8 s, o.5
enough energy would be
harnessed in one day 0
(4,256,442,000 joules or 0.1-
1182 kilowatt-hours) to
provide power for more Figure 1. Wat
than 30 homes (30.28 given by db
homes to be precise). height), e
Several breaker rsu
height/period examples are
given since the ocean-generated wave field
is almost always comprised of multiple wave
trains. Hence, if all the above waves were
present, then enough energy would be
available to power over 36 homes (36.3
homes to be precise).

In addition, where waves shore-break
has remarkable influence on wave energy
constraints. Let us assume, for the moment,
that nearshore waves are simply water depth
limited. That is, water depth only
determines where they shore-break. The
criterion of McCowan (1894) is used to
illustrate this simple example which is
plotted in Figure 1, where wave energy is
generally considered to be proportional to
the breaker height squared. Hence, a wave
with a height of 2 m will contain 4 times the
energy of a 1 m wave, a wave height of 3 m
has 9 times the energy of a 1 m wave, a
wave with a height of 4 m has 16 times the
energy of a 1 m wave, and so on. The value
of c (Figure 1) as given by McCowan (1894)
is 1.28 (although its value has been
questioned by many investigators). Let us
assume it is valid for the following examples.
If a value of c = 0.5 is used, the breaker
energy level becomes 6.55 times that for c
= 1.28. If the value of c = 3, the wave
energy level becomes 0.182 times the


0.5 1.0


1.5


2.5 3.0


er depth in which shore-breaking occurs, db
= c Hb (Hb = mean shore-breaking wave
s the relative breaker energy level.

energy level where c = 1.28 (i.e., 5.5 times
less). One can easily see from Figure 1 that
wave energy constraints are sensitively
dependent upon the water depth in which
breaking occurs (see Appendix I for example
calculations).

In Florida, out of necessity, it
becomes an important issue because wave
conditions range from zero energy to high
energy (Tanner, 1960), and so we must be
sensitively precise in our assessment of
wave breaking. Almost every coastal
application, in one way or another, is
dependent on where or when waves shore-
break. It is an issue that is of vital
importance to every professional coastal
practitioner.

The breaking of ocean waves (i.e.,
gravity waves) is defined to occur when the
internal horizontal water particle velocities,
ub, in any part of the wave crest exceed the
wave phase speed, cb (e.g., McCowan,
1894; Munk, 1949; Kinsman, 1965), or:

> 1.0 (1)

termed the kinematic stability parameter
(Dean, 1968). There is also involved the
dynamic stability parameter (Dean, 1968)


Maximum natural varability
associated with a shore-beaker
S for a single wave train
I : --1 I


I










which involves vertical water particle
accelerations necessary to conserve the
integrity of the wave form as regards its
height. Its treatment is beyond the scope of
this work, nor is it necessary (i.e., we shall
assume wave height integrity is maintained,
which in reality it is, at least until shore-
breaking is final).

The author is careful to use the
terminology shore-breaking waves (including
bar-breaking waves) which are produced due
to nearshore shoaling conditions (i.e., depth
limitations), so named to distinguish them
from fully forced waves breaking in deeper
water (Balsillie and others, 1976; Mooers,
1976) which may break due to critically high
wind stresses (e.g., "white caps" or
"horses"), seismic impulses, explosive
events of anthropic origin, or even, perhaps,
in the laboratory by improperly sited wave
generators.

For the bulk of the history (- 1859 to
present) in endeavors to determine the
causes) of shore-breaking, technical means
were not available to measure the criterion
of equation (1). Nor are they available to
many of us today. We have had to or do,
therefore, rely upon an approximating
surrogate set of visual definitions, which
more nearly identify when waves shore-
break.

By 1946, Dean M. P. O'Brien had
identified and suggested conditions required
to produce spilling and plunging type shore-
breakers (Beach Erosion Board, 1949).
Subsequent evaluations (e.g., Galvin, 1968)
led to a set of "standardized" visual
definitions of shore-breakers in profile view.
The author (JHB), however, for the principal
types of shore-breakers (Balsillie, 1985;
1999), prefers the following.

Spilling occurs where the top of the
unstable wave crest results in aerated,
turbulent water slipping down the front face
of the wave (Figure 2). Up to 25% of the


Figure 2. Three basic types of shore-
breaking waves.

upper portion of the breaking wave crest is
considered to be involved in spilling.
Aerated water at the crest apex signifies that
the breaking position has been reached.

Plunging occurs (Figure 2) where the
upper portion of the wave crest (> 25%)
curls over, forms an air pocket and the
curling crest eventually falls into the trough
fronting the breaking wave crest. Shore-
breaking for a plunger is defined to occur
when the front face of the crest becomes
vertical.

Surging occurs at the point when the
basal portion of the wave crest rushes
forward from beneath the crest, sliding up
the foreshore slope with a minimum of
bubble production (Figure 2).

Collapsing occurs when wave crest











SPILLING
Impact Pressure Data


H; 60 10.10 arcia(1) -- f -
100 1 ljt,'4
b10 Hb 0.0609 Weggel (1968)
0 --
0 0.0667
(% 40) 0.102
40 0.102 Kirkgoz (1982) Direct Wave Measurements
t 0.1404 26 tanab
30-- 1 0.2247 --f
0.065 Ippen & Kulin (1964)
20 V V0.10 Wiegel & Skjei (1958)
0.0509 Weggel (1968)
10- --------- 0.0070,
1 1 0.0092, This study
0.0123
0 I I
0.1 1 10

b
Figure 3. Relationship between the percentage of the wave crest that is
involved in shore-breaking, Hb" (measured from the crest apex downward)
where Hb is the mean shore-breaking wave height, as a function of the modified
surf similarity parameter, Fb. Numbers next to symbols identify the number of
values from which averages were determined. (After Balsillie, 1985).


advance suddenly terminates, and the wave
crest simply collapses. In all his years of
observing shore-breaking waves on sandy
shores in Florida, the author has seen this
occur but once.

A shore-breaking continuum (Figure
3) has been proposed (Balsillie, 1985)
according to:

Hb"
S- tanh 7 (2)
Hb 8
where Hb" is the amount of the shore-
breaking wave crest, measured from the
crest apex downward, involved in translator


shore-breaking, Hb is the mean breaker
height, and fb is the modified surf similarity
parameter of Iribarren and Nogales (1949).
Surrogate definitions fit into the continuum
of equation (2) according to:
< 0.64, Spilling
tan a, 0.64- 5.0, Plunging (3)
6H=(g ) 5.0- ?, surging
& r) I> ?, Collapsing
in which tan ab is the bed slope, g is the
acceleration of gravity, and T is the wave
period. It was determined that the breaker
with the most destructive potential (i.e., in
terms of the horizontal impact pressure) at
the highest elevation occurs where fb = 1.0.
This correlates with a plunging shore-breaker










where 38% of the breaker crest top is
involved in translator shore-breaking.

Scripps Institution of Oceanography
(1945) in its World War II field experiments
on shore-breaking was careful to note that it
is not the bed slope that appears to be so
important in influencing breaking but, rather,
a sudden change in the bed slope. The
author posits that not even that is important
as it concerns the physical processess.
More nearly, a wave-encountered sudden
change in the bed slope induces a change in
wave profile shape which aids the observers'
cognitive abilities to more readily identify a
change in the wave shape that he perceives
to have reached the shore-breaking position.
In short, the surrogate definitions provide but
discontinuous, albeit useful, clues as to
where shore-breaking occurs. In fact, for
such conditions, the author can provide a
precise definition for the equilibrium
nearshore profile, when the shore-breaker is
of the spilling type. This occurs for a profile
upon which the spilling shore-breaker
continuously breaks until, at the shoreline
it's energy is completely dissipated and the
wave crest form is no longer evident.
Moreover, because uc is only very slightly
greater than cb, the observer may be unable
to detect according to definition that the
wave has reached shore-breaking conditions
anywhere along its entire shore-breaking
journey.

PREVIOUS WORK

Interest in where waves shore-break,
at least in terms of published work on the
subject, has been a subject of serious
interest for well over a century. The first
published formal account known to the
author was the theoretical work of
McCowan (1881) which he subsequently
modified (McCowan, 1894), resulting in:

db = 1.28 Hb (4)
where db is the water depth at shore-
breaking (measured as the vertical distance


from the still water level (SWL) to the bed),
and Hb is the mean shore-breaking wave
height.

The first comprehensive field data
collection effort was compiled by Gaillard
(1904). He also identified an earlier wave
tank study conducted by Henri Bazin in
1859, predating McCowan's work by 25
years. Other earlier studies, both in the field
and (perhaps) laboratory, were also identified
by Gaillard. These and some other early
studies identified by the Scripps Institution of
Oceanography (1945), leading to some
general observations, are listed in Table 1.

Interest in the issue accelerated
during World War II when landing craft
operations became of serious concern. The
U. S. Government initiated a program of
research through the Joint Army-Navy
Intelligence Service (JANIS) with the U. S.
Army, Beach Erosion Board (BEB) assigned
as the lead agency (Quinn, 1977). While the
BEB conducted much of its own research, it
also contracted with such institutions as
Wood's Hole Oceanographic Institution and
Scripps Institution of Oceanography (Scripps
Institution of Oceanography, 1945). A
considerable number of confidential works
were completed. They remained virtually
unknown to the public until the summer of
1976 when the author, then on the staff of
the U. S. Army, Coastal Engineering
Research Center (CERC, successor to the
Beach Erosion Board), found them in a
secured section of the agency and had them
released (ironically, these documents had
been declassified in the summer of 1950,
but were apparently forgotten and not made
available to the public). The Scripps (1945)
Leica Type I and II data, and special
measurements listed in these secured
documents were published by Munk
(1949). In both the Scripps (1945) and
Munk's (1949) works the special
measurements data were plotted in support
of equation (4).

Through ensuing years investigators








Table 1. Some early general observations reported by Gailard (1904)* and Scripps
Institution of Oceanography (1945) .
INVESTIGATION RESULTS
J. Scott Russell, wave tank db/Hb = 1.0, tan ab = 0.02
J. Scott Russell, field data db/Hb = 1.0, tan ob = 0.02
Thomas Stevenson, Firth of Forth field data* (db/Hb)avg = 0.715
Thomas Stevenson, Scarborough field data, 1870* db/Hb = 2.3
Henri Bazin, wave tank, 1859 db/Hb = 3/2, tan ab = 0.015
William Shield, Peterhead field data, 1888 db/Hb = 1.27
British Admiralty, Swell Forecast Section, Comdr. (dbHb)avg = 1.5
Suthons, R. N., 1944*
British Superintendent of Mine Design** d/Hb = 4/3
Canadian Report on "Beach intelligence in the 1.2 < d/Hb b 1.45
Mediterranean area"*
University of California wave tank study, 1945** Decrease of db/Hb with decreasing wave steepness
and increasing bed slope.

Estero Bay, California landing craft field study, Lt. (db/Hb)avg = 1.30, although data showed a
Munch** decrease in db/Hb with decreasing wave steepness.
Scripps Institution field study on July 1, 1944** (db/Hb)avg = 1.30, but show an increase in db/Hb
with decreasing wave steepness.

Monterey Harbor, California photo study, 1945" db/Hb = 1.14
Oceanside, California field study, 1945* db/Hb = 1.55


variously conducted experiments to
numerically quantify where shore-breaking
occurs. Largely, these studies were
conducted by engineers apparently
convinced that waves were simply not depth
limited .. that the correct answer had to be
more complicated. Often only small data
sets were considered in their quests. A list
of studies and wave parameters considered
is given in Table 2.

Galvin (1969) suggested that where
at the shore-breaking position, the bed slope,
tan ab, is tan ab > 0.07, then:


0.92
H,


(5a)


and where tan ab < 0.07:

= 1.4 6.85 tan a.
Hb


(5b)


which are both referenced to the mean water
level (MWL) rather than SWL. Galvin (1969)
suggested that for tan ab on the order of
from 0.05 to 0.1, SWL is higher than MWL
by a factor of 0.04 Hb, and where tan ab is
about 0.2 by a factor of 0.08 Hb.

Collins and Weir (1979) found:


S= (0.72 + 5.6 tan oa,)-
H,
and Mallard (1978) concluded:










Table 2. Table of independent variables related to the water depth
at shore-breaking.
Wave Parameter types Wave parameters used
and Investigators in investigations
Wave parameters at Hb T tan b Hb T2
shore-breaking. b
McCowan (1894) X
Scripps (1945), Munk (1949) X
Galvin (1969) X X
Collins and Weir (1969) X X
Weggel (1972a, 1972b) X X X
Mallard (1978) X X
Balsillie (1983) X X X X
Deep water wave parameters. Ho HolLo Fo
Smith and Kraus (1990, 1991) X X X X
Nelson (1994), and Gourlay (1994) X X X
Notes: Ho = deep water wave height, Lo = deep water wave length,
fo = deep water surf similarity parameter.


db = [0.73 + 2.87 (tan ab0997]-1 (7)
Hb
Equations (5a) through (7) are plotted
in Figure 4. For these equations, values of
db/Hb are close for tan ab less than about
0.01. However, for tan ab > 0.01 there is
disparity.

An additional parameter which had
been earlier considered as a possibly
influential factor was the equivalent wave
steepness, Hb/(g T2), whose derivation is
given by Battjes (1974, p. 469). Weggel
(1972a, 1972b) introduced this parameter
into his investigations, and suggested:


db
Hbn
4m


H+-1
+ c, g
g T2


where:


c = c g (1.0 e-1an oA)


(8a)


in which c2 = 4.462 m2/s = 1.36 ft2/sec
are constants for use in assuring unit
consistency, and:


C3 = 1.56 (1.0 + e-1.5 an h)-


(8b)


Balsillie (1983) undertook an
investigation to compare the relative validity
of the preceding predictive methods. He
excluded Galvin's (1969) equation because:
1) it was referenced to MSL, 2) had a
discontinuous behavior for tan ab > 0.07,
and 3) was represented by other predictive
methods that were more comprehensive in
their data coverage. The Collins and Weir
(1979) method was not considered because
Mallard's (1978) result was so similar, and
included more comprehensive data coverage.
The result was that McCowan's equation
was slightly more successful in quantifying
the relationship for shore-breaker water

















1





0
0.0


01


tan ab
Figure 4. Comparison, of results of predicted
dblHb using the relationships of Galvin (1969
Collins and Weir (1969), and Mallard (1971
which consider bed slope, tan ab, onll
described in text.


depth. Since that work, considerably more
data has been collected, primarily in the
laboratory range of possibilities, although
some prototypical laboratory results have
surfaced (Maruyama and others, 1983;
Stive, 1985; Nadaoka, 1986; Takikama and
others ,1997). For this and other reasons,
the subject of where shore-breaking occurs
has been revisited.

Smith and Kraus (1990, 1991)
conducted laboratory studies of shore-
breaking over longshore bars from which
they (Smith and Kraus, 1991) suggest:

(0.41 + 0.98 to)-1 (9a)
H,
where 0.3 fo < 0.85, and:

d,
(1.45 0.22 o)-' (9b)
H1
where 1.6 fo hybrid form of the surf similarity parameter
(comprised partly of deep water
characteristics and partly of breaker zone
characteristics) evaluated as:


- I I I I I 1|111 1 -- I | 7 1 I I 1 1 11i1

Galvin (1969)




Collins and Weir (1969) "L" -
Mallard ( 8)
Mallard (1978) -.
. ...i . . ....I . .....I


tan a,
So (9c)

in which Ho and Lo are the deep water
wave height and wave length,
respectively. They note a correlation
coefficient of 0.85 between measured
and predicted values of db/Hb for the
above regions of validity for equation (9a)
and (9b). For the region 0.85 < fo <
1.6, which they term the transition
Region, they seem to suggest the use of
equation (9b) although they do note the
ed highly scattered nature of their data.
I),
8) Based on laboratory data,
Kaminsky and Kraus (1993) confirmed
the use of equation (4) as an average
representation for typical field beach
slopes.

Nelson (1982, 1994) and Gourlay
(1994) have published rather convoluted
accounts using mostly laboratory data (one
"field" study by Nelson, 1994) in which they
posit that . for waves propagating over
a horizontal bottom . the maximum value
of Hb/db never exceeds 0.55." Recasting
this yields:

d, k 1.82 H,, (10)
Moreover, Gourlay (1994) notes that the use
of equation (4) rather than (10) "
overestimates the wave energy reaching a
structure by a factor of 2 . .", which it
does. He further suggests that equation (4)
is, therefore, ". . unnecessarily
conservative". Both assertions are
significantly surprising, The depth of water
at shore-breaking significantly affects the
energy content of the wave form, particularly
in light of new findings concerning how
wave energy is partitioned between the
breaking wave crest and trough (Balsillie,
1997, 1999). Hence, this has prompted the
author to revisit the issue of where shore-
breaking occurs. We shall term this last
result the Nelson-Gourlay horizontal bed
slope breaking anomaly which will be


' """' ~ """'









readdressed later in this work.

THE DATA

General characteristics of the data
considered in this work are listed in Table 3.
They constitute the largest data compilation
amassed to date for assessment of where
shore-breaking occurs. All the major shore-
breaking wave types (i.e., spilling, plunging,
and surging breakers) are represented by
these data.

Laboratory Data

This work includes data from 27
laboratory investigations, to include 640 data
sets, 16 of which have prototype dimensions
and 624 of which are much smaller waves
of typical laboratory size. Generally,
laboratory investigators are quite specific
about reporting experimental physical
attributes and measurement techniques.
Occasionally, a researcher may fail to report
a crucial factor because it seemed obvious at
the time of final document preparation. By-
and-large, however, final documentation is
usefully complete. Rather than reiterating
such information here, which would require
significant space, the reader is referred to
each referenced study.

Field Data

It is the author's impression, because
of the lack of inclusion of field data in
existing treatments on this subject, that field
data are not accepted with the veracity
accorded data collected in the laboratory
wave tank. Field data significantly increase
the domain of data coverage. As it turns
out, the laboratory data occupy but only
3.36% of the dimensional domain of the
data available. Scientific pursuits require
consideration of all viable data. Given some
of the characteristics of all the data which
requires anthropic observation (to be
discussed later), the author does not quite
understand why the exclusion of field data in


most works published on this subject should
be so. And so, it appears necessary to
provide some detailed discussion about field
data considered in this account.

Data from six field investigators are
used in this work, totalling 131 data sets for
db, Hb, T, tan ab, and 172 data sets for db
and Hb.

Gailard Data

The first comprehensive compilation
of the data on the subject was authored by
Gaillard (1904). There is no reported record
of how he measured his data. Even so, his
results are sincerely and sensitively complete
with regard to documentation and global
extent. They are, at the very least, viable
observations which, in the ensuing 94 years
have not been discredited. His data in the
scientific sense, therefore, cannot be
dismissed.

Scrpps Leica Data

These data, collected as part of the
war effort during World War II were
photographed at Scripps Institution of
Oceanography (1944a, 1944b, 1945) at La
Jolla, California. The Leica Type I and II data
were collected on a daily basis from January
9 to April 15, 1944, and depended upon
profile conditions where the breakers were
measured. Type II measurements were
made for breakers occurring over the stoss
slope of longshore bars. Type I
measurements represent waves shore-
breaking on the linear nearshore slope just
offshore from the shoreline. Photographs
were taken at a point about 7.62 m above
the pier deck at a distance of about 75 m
south of the shore end of the pier. A large
clock suspended from a boom was in the
field of view and clearly visible in the photos.
Wide planks clearly marked at 0.61 m
vertical intervals were affixed to the Scripps
Pier piles. Daily profiles were surveyed along
the pier and the SWL was determined using








Table 3. List of general characteristics of field and laboratory data used in this work.

db Hb T
Investigator n ) () ( tan ab
(W) (i) (s)

FIELD DATA

Gaillard (1904) 25 0.792- 5.349 0.61 -3.353 3.85- 10.98 0.0154- 0.029
12 1.189- 5.456 0.61 -3.962 -------

Scripps (1944a,
1944b,1945) Leica 56 1.554- 4.450 1.219- 3.475 6.5- 13.7 0.0159
Type I

Scripps (1944a,
1944b, 1945) 18 1.68 3.72 1.28 2.74 7.0- 13.0 0.049
Leica Type II

Scripps (1945) 29 0.49- 1.26 0.27- 1.13 ----- ---
Spec. Meas.

Basillie and Carter 30 0.11 -0.758 0.057- 0.541 1.33-8.57 0.017- 0.462
(1980)

Weishar (1978) 2 0.94, 1.01 0.60, 0.77 7.7 0.05

Total 172

LABORATORY DATA PROTOTYPE DIMENSIONS

Maruyama and 1 2.0 1.29 3.1 0.0340
others (1983)

Stive (1985) 2 0.2, 1.90 0.18, 1.50 1.8, 5.0 0.0250

Nadaoka (1986) 12 0.14-0.605 0.103 0.509 0.92- 2.99 0.05

Takikama and
Takikamaand 1 0.232 0.215 2.08 0.05
others (1997)

Total 16

LABORATORY DATA

Munk (1949) BEB 0.03, 0.049,
Munk(1949)BEB 37 0.043- 0.187 0.031-0.13 0.73- 1.093,03 049
data 0.159

Munk (1949) 0.0541,
ekel d 16 0.0631 0.1451 0.068- 0.1 0.86 1.98 0.0
Berkeley data 0.0719, 0.09

Putnam and others 0.066, 0.098,
(1949) 37 0.058- 0.229 0.037- 0.143 0.72- 2.32 0.10, 0.139,
0.143, 0.241

Iverson (1952) 63 0.043- 0.165 0.043- 0.128 0.74- 2.67 0.02, 0.033,
0.05, 0.10

Morison and
Morson and 6 0.07- 0.129 0.056- 0.113 0.78-2.62 0.02, 0.10
Crooke (1953)











Table 3. List of general characteristics of field and laboratory data used in this work
(cont).

db Hb T
Investigator n (m) ( tan b

LABORATORY DATA (CONT.)

Galvin and
Galvin and24 0.021 0.081 0.03- 0.091 1.0- 1.50 0.10
Eagleson (1965)

Eagleson (1965) 7 0.058- 0.123 0.044- 0.095 0.79- 1.57 0.10

Horikawa and Kuo 0.0125,
Horikawa and Kuo 97 0.06- 0.263 0.06- 0.182 1.2 -2.3 0.0125,
(1967)* 0.0333, 0.050

Bowen and others 0.042 0.097 0.04- 0.13 0.82 2.37 0.082
(1968)

Komar and 44 0.034- 0.213 0.03-0.17 0.81 -2.37 0.036, 0.07,
Simmons (1968)** 0.086, 0.105

Galvin (1969) 17 0.039 0.114 0.038- 0.115 1.0 6.0 0.05,010,
0.20

Weggel and 9 0.087- 0.169 0.089- 0.162 1.27 2.05 0.051
Maxwell (1970)

Saekiand Sasaki 2 0.097- 0.164 0.099- 0.106 1.3-2.5 0.020
(1973)

Iwagaki and others 23 0.06- 0.158 0.044- 0.128 1.0- 2.0 0.03,0.05,
(1974) 0.10

Walker (1974) 15 0.031 0.125 0.024- 0.116 1.17-2.33 0.033

Van Dorn (1978) 12 0.093 0.217 0.108 0.166 1.65-4.8 0.022, 0.04
0.083

Hansen and16 0.047- 0.149 0.043- 0.14 0.83- 3.33 0.0282
Svendsen (1979)

Singamsetti and 95 0.078- 0.22 0.073 0.193 1.03- 1.73 0.025, 0.05,
Wind (1980) 0.10, 0.20

Mizugauchi (1981) 1 0.083 0.10 1.2 0.10

Visser (1982) 7 0.088 0.122 0.58- 0.108 0.7 -2.01 0.05, 0.10

Watanabe and
Watanabe and 3 0.1 -0.11 0.075 0.082 0.94- 1.19 0.05
Dibajnia (1988)

Smith and Kraus 77 0.091 0.271 0.088 0.216 1.02-2.49 0.08- 0.412
(1990) bars

Smith and Kraus
Smith and Kraus 5 0.131 0.216 0.082- 0.165 1.02-2.49 0.033
(1990) plane beach

Total 624

*Data listed in Smith and Kraus (1990); *data listed in Gaughan and others (1973).










a tide gauge. General bed slope conditions
for the field experiments and other details of
the field effort are documented by the
Scripps Institution of Oceanography (1944a,
1944b).

Scripps Special Measurements

A number of precise simultaneous
measurements of db and Hb were made near
the Scripps Pier and included in the Scripps
Institution of Oceanography (1945, p. 22-
23) report. Water depth at breaking was
measured using a hollow tube (internal to
which was a float) in the surf to measure the
still water level. Breaker height was
simultaneously measured using a scale
affixed to the outside of the tube. Three
observers would make simultaneous readings
of water depth, elevation of trough, and of
breaker crest. Unless the wave broke right
at the tube, measurements were eliminated.

Balsillie and Carter Data

Beach and nearshore shore-normal
profile surveying was the first task in the
quantitative measurement portion of these
field experiments (Balsillie and Carter, 1980;
Balsillie and Carter, 1984a, 1984b). Next, a
site was selected and surveyed at a location
considerably seaward of the surf zone,
where waves were as undistorted as
possible. Thirty elevations of wave crest
and trough measured from the bed were
recorded. These data, when averaged,
became as closely as was possible an
estimate of the SWL for the experiment.
The investigators were careful to note the
presence of separate wave trains, and
breaker zone widths for each wave train
were identified on the surveyed shore-normal
profile. For each wave train, 30 shore-
breaker crest height and trough elevations
were measured using a special rod fitted
with a foot that would not sink into the
sandy bed. Wave period for each wave train
was determined by measuring the time it
took for 11 wave crests to pass a stationary


point in the surf zone. When divided by 10,
an estimate of the wave period resulted.
Several such sets of these measurements
were taken by each participant to ensure a
more representative quantification. Bed
slopes were determined from the surveyed
profile data.

Weishar Data

Weishar (1976), Weishar and Byrne
(1979) analyzed the contents of a one-hour
16 mm film of wave activity from the
Virginia Beach, Virginia, pier, filmed by
Robert Byrne eight years earlier. The filming
was focused upon a rectangular grid
comprised of 0.61 m by 0.61 m segments
mounted on steel pipes jetted into the
nearshore (from the foreshore to 30.5 m
offshore) along a shore-normal azimuth at 3
m centers. Every 20 minutes the nearshore
profile was surveyed. During the period of
filming a relatively well defined swell was
present, with the wave crest traces being
almost parallel with the shoreline; very little
local wind wave activity was present.
Weisher reported the characteristics of 120
"consecutive" individual waves for the one-
hour film. Certainly, his procedure was
highly selective, since it can be estimated
that some 400 to 500 waves were probably
present during the filming. He did not,
however, identify his data by wave train.
The author, therefore, conducted a modal
analysis by decomposing the cumulative
probability distribution using the Method of
Differences (Tanner, 1959; see also Balsillie,
1995, p. 59-61) which, in the author's
opinion, is far superior to spectral analysis.
From this analysis, it was determined that
Weishar primarily measured waves for two
wave trains for which quantitatively
representative statistics were determined.

ANAL YTICAL PROCEDURE

Prior to engaging in analytical pursuits
there are two scientific analytical tools that
the author should like to introduce to the










analysis of several selected methods used to
predict where shore-breaking occurs.


Occam's Razor


Most of us dealing with science are
familiar with the terminology "Occam's
Razor". How many have more detailed
knowledge about the terminology is
unknown. The concept, however, is of
such importance to the present treatment
that some discussion is here presented.

William (of) Ockham or Occam
(Ockham is near Guildford, southwest of
London, England) was a medieval scholastic
monk (1285-1349)who is generally credited
with the law of parsimony which states
pluaralitas non est ponenda sine necessitas
(plurality should not be posited without
necessity). Others before Occam (e.g.,
French Dominican theologian Durand de
Saint-Porcain and Occam's teacher Duns
Scotus) and after Occam (e.g., Galileo
Galilei) also posited the principle but Occam
is credited as its originator because he used
it so frequently and employed it so sharply.
(e.g., Thorburn, 1915, 1918; Burns, 1915;
Encyclopaedia Britannica, 1981.) So it is to
be sharply employed in this work. Simply
stated . if multiple mechanisms explain
the facts equally well then the simplest
mechanism is that, in the scientific sense,
which requires acceptance.

Doyles' Principle

Sir Arthur Conan Doyle (1859-1930)
coined the phrase through his super sleuth
Sherlock Holmes, which states: When you
have eliminated the impossible, whatever
remains, however improbable, must be the
truth. This, the reader shall come to
understand, has merit in this work.

Procedure

Two analytical procedures are utilized
in this work. The first is stepwise
regression. The second is a comparative


STEPWISE REGRESSION


Stepwise regression is a powerful
statistical tool for ranking variables by their
relative importance (Harrison and Krumbein,
1964; Krumbein and Graybill, 1965) when
considered synergistically. In addition, a
basic contribution of this statistical tool is
that it readily provides for the evaluation of
data redundancy. Redundant variables are
those that, in large part, restate what some
other variable has already measured.
Redundancy is common in the early stages
of scientific quantification, when the physical
meaning of multiple interrelated variables
may not be clearly discernible (Harrison and
Krumbein, 1964; Krumbein and Graybill,
1965). Matrix operations performed to
derive relating equation coefficients are less
affected by roundoff errors; moreover,
derived coefficients are less sensitive to
small errors in the data (Miesch and Conner,
1968).

Three sets of data were selected for
stepwise regression analysis. In the first,
the water depth at shore-breaking, db, is
selected as the dependent variable and
based on the preceding accounts of
attempted identification of input variables,
Hb, T, tan ab, Hb/(g T2), and fb are selected
as independent variables. Note that
predictive methods which use deep water
wave parameters are not considered in this
analysis. It is the author's opinion that they
introduce additional complications that are
simply not necessary. Also, please
understand that, at this stage in the
investigation, one is interested in using the
stepwise regression tool as the means for a
substantive search for identifying the
contributory importance of variables, rather
than for finding a formal predicting equation
(e.g., Krumbein and Graybill, 1965, p. 395
and 398). Hence, in the first data set the
dependent variable (db) is in units of length


study.







as is one independent variable (Hb), another
independent variable is in units of time (T),
and the remainder (tan ab, Hb/(g T2), and fb)
are dimensionless. Determination of a formal
equation (i.e., unit consistency across the
equation) is a later step.

The stepwise regression application
used in this work (all possible regressions)
follows the development of Krumbein and
Graybill (1965, p. 391-399). The computer
application (written by the author) is
described by Balsillie and Tanner (1999).
The first stepwise regression equation for
consideration has the form:


de = Po + (P1 Hb) + (PA T)
H
+ (p3 tan a) +(4 P,
+ (P5s )


(11)


In the second data selection, the ratio
db/Hb was selected as the dependent
variable (actually a parameter), and the
stepwise regression equation assumes the
form:


=Po (P1 T) + (p2 tan a)

+ (P4 + (Ps )


+ (tanh 0.4 t,)
In the third data selection, the bed
slope, tan ab, was selected as the dependent
variable, and the general equation for
stepwise regression assessment is:
tan, = Po + (PI d,) + (2 H,)

+ (p3 T) (p ) (13)

Stepwise regression, through
algebraic matrix inversion, allows for the
determination of values of ,f so that the best
fit predictive outcomes as closely as possible
approach the measured independent variable
or parameter. All possible nonrecurring
combinations of independent variables are


processed, and relative assessment statistics
are then determined. The order in which
input independent variables are introduced to
the statistical procedure makes no
difference, nor do the units involved unless,
perhaps, they are transformed in some
unusual manner. Assessment statistics can
be the Pearson product-moment correlation
coefficient r, or r2, the sum of squares of db
accounted for. The later expressed as a
percentage (i.e., 100 r2) is preferred for this
method (e.g., Krumbein and Graybill, 1965;
Draper and Smith, 1980). Both, however,
indicate the degree of success of prediction,
i.e., the degree of agreement between
predicted and measured values of db (simply
put: 0 or 0% for no agreement, 1.0 or
100% for perfect agreement).


Assessment of Equation (11)

There are 771 sets of data (the
largest data base amassed to date for this
subject) available for analytical treatment
(Table 3). Only the Scripps (1945) Special
Measurements and part of Gaillard's (1904)
data set are excluded from stepwise
regression, because only db and Hb were
measured. Stepwise regression results for
equation (11) are listed in Table 4 in which
each line in Table 4 represents results for a
predictive equation and where assessment
statistics 100 r2 and r are given in columns
(7) and (8). In addition, the relative percent
contributory importance of independent input
variables is given in columns (2) through (6).
Note that in this later relative assessment,
some variables can have a contributory value
exceeding 100%. This occurs because
others have negative values (since ,f
coefficients can have negative assignments).
Each line, however, totals to 100%. The
later allows the reader the advantage to
readily determine the magnitude of
contribution, or to see how secondary
contributions are "cancelled out".

Now, we need only to decide upon an
upper limit value relative to which we select










Table 4. Stepwise regression results for equation (11).

Mean percent contributions of variables
in predicting db (7)
Percent of (8)
(2) (3) (4) (5) (6) Sum of Correlation
(1) Squares of db Coefficient,
Eq. Hb T tan b Hb/(g T2 b Accounted for r
No. (m) (s) (100 r2)


Independent Variables Taken One at a Time

1 100 94.26* 0.9709*
2 100 74.25 0.8617
3 100 6.79 0.2606
4 100 3.66 0.1914
5 100 4.87 0.2208

Independent Variables Taken Two at a Time

6 104.72 -4.72 94.27* 0.9709*
7 101.15 -1.15 94.26* 0.9709*
8 97.99 2.01 94.26* 0.9709*
9 100.83 -0.83 94.26* 0.9709*
10 103.25 -3.25 74.31 0.8620
11 71.72 28.28 78.58 0.8865
12 119.72 -19.72 76.01 0.8719
13 43.06 46.94 8.43 0.2904
14 92.21 7.79 6.81 0.2029
15 58.43 41.57 9.90 0.3146

Independent Variables Taken Three at a Time

16 106.50 -5.07 -1.43 94.27* 0.9710*
17 104.65 -4.69 0.04 94.27* 0.9709*
18 104.93 -4.57 -0.36 94.27* 0.9709*
19 98.98 -1.50 2.52 94.27* 0.9709*
20 101.13 -1.29 -0.16 94.26* 0.9709*
21 98.73 1.85 -0.58 94.26* 0.9709*
22 74.20 -5.17 30.97 78.92 0.8884
23 108.19 37.22 -45.41 73.38 0.8853
24 79.34 28.96 -8.30 79.43 0.8912
25 -4.53 59.93 44.60 9.91 0.3149

Independent Variables Taken Four at a Time

26 105.49 -4.59 -1.48 0.58 94.27* 0.9710*
27 108.06 -7.00 -4.15 3.08 94.28* 0.9710*
28 104.96 -4.59 -0.02 -0.36 94.27* 0.9709*
29 96.89 -4.44 4.38 3.17 94.27* 0.9709*
30 83.07 11.79 23.66 -18.53 79.71 0.8928

Independent Variables Taken Five at a Time

31 103.99 -5.5 -5.53 2.74 4.36 94.28* 0.9710*

NOTES: Each line represents results for a predicting equation. The correlation coefficient
measures the degree of agreement between predicted db outcomes from step-wise regression
equations and the measured db values.









Table 5. Correlation matrix for results forthcoming from equation (11). Numerical
values above and to the right of the zig-zag diagonal line are Pearson product-moment
correlation coefficients (r), those below and to the left are 100 r2. See text for
application.
db Hb T tan Hb(gT2) b
(m) (m) ()a T
db (m) 00 0.9709 0.8617 -0.2606 -0.1914 -0.2208
Hb (m) 94.26% 100% 1.00 0.8926 -0.2631 -0.2037 -0.2233
T (s) 74.25% 79.68% 100%1.00 -0.2759 -0.4391 -0.1031
tan ab 6.79% 6.92% 7.61% 100% 100 0.2593 0.8204
Hb/(g T2) 3.66% 4.15% 19.28% 6.72% 100%00 -0.1387

Fb 4.87% 4.99% 1.06% 67.30% 1.92% 100% 00


appropriate contributing combinations. For
this investigation, the selection of r2 > 94%
( or r > 0.97) is straightforward (values are
tagged with asterisks). All others can be
dismissed as being predicting combinations
with subservient import. Normally, the next
task is to determine the net contributions of
ranked variables. However, for the data
assessed in this work, not even that is
necessary (although we shall do so later).

For equation (11) there are 31
mechanisms, or equations in Table 4, from
which to choose. The equations become, in
groups of independent variables taken n at a
time (1 n n < 5), progressively more complex
from equations (1) to (31). More complex
successful equations (i.e., where 100 r2 >
0.94, or where r > 0.97) deviate from a
successful solution given by the simplest a
maximum of two-hundredths of a percent.
A deviation of 0.02% is simply not grounds
for selecting a more convoluted predicting
solution. We can, then, without the
slightest hesitation, employ Occam's Razor
and select equation (1) given by:

db = o + (p1 H,) (14)
as the best solution (see Appendix II for the
/8 value table). We can, in addition, because
of the small magnitude of fo (8o = -0.017),


dispense with the term and suggest:

db = 1 Hb = 1.372Hb


(15)


although we shall use another statistical
fitting procedure to determine a formal form
for the relationship between measured data
and predicted results.

Data Redundancy

We can inspect for data redundancy
among the independent variables and
parameters (IVs and IPs) using the
correlation matrix of Table 5. As an
example, the relationship between Hb and
tan ab is 6.92%. For this case, we state
that Hb accounts for 6.92% of the sum of
squares of tan ab (we can also state it the
other way around). There is, therefore, little
redundancy between the two variables.
However, a rather large 100 r2 value of
67.03% exists between tan ab and fb. This
might be expected since fb is an IP
containing tan ab. The effect is evident, in
terms of the percent contribution of IVs, for
equations 15, 25, 28, and 30 of Table 4.
All, however, are not in the acceptance
region (i.e., 100 r2 > 94%), except for
equation 28 of Table 4 in which the
contributory percentages are both very small.
The only other "noticeable" indication of










data redundancy is between T and Hb/(g T2),
which may be applicable to equation 13 of
Table 4; equation 13 of Table 4 is not,
however, within the acceptance region.

Net Contributions

Let us now, to be complete in our
analysis, determine the net contribution of
ranked independent variables for equation
(11) following the method of Krumbein and
Graybill (1965, p. 397-398).

Determination of net contributions of
the independent variables or parameters
starts with the selection of the strongest of
the independent variables taken one at a
time which is equation 1 of Table 4 for Hb
where 100 r2 = 94.26%.

Next, from the independent variables
taken two at a time series, we select the
highest 100 r2 value of 94.27% which
represents equation 6 of Table 4 in which T
is in the presence of Hb. Hence, the net
contribution of T becomes: 94.27% minus
the 100 r2'value for equation 1 of Table 4, or
94.27%- 94.26% = 0.01%.

Now, from the independent variables
taken three at a time series, we select in the
presence of Hb and T, the next highest 100
r2 value for a IV or IP not yet selected.
(Note that as this procedure progresses we
can be faced with the situation that the
choice becomes less clear, so one may have
to make a tentative selection.) We select
equation 16 of Table 4 for tan ab which has
a 100 r2 value of 94.27%, and the net
contribution of tan ab becomes: 94.27%
minus the 100 r2 value for equation 6 of
Table 4 or 94.27% 94.27% = 0.00%.

Applying the same technique for
independent variables taken four at a time,
we find that equation 27 is the strongest
selection in which fb is in the presence of
Hb, T, and tan ab where 100 r2 =94.28%.
The net contribution of fb becomes 94.28%


Table 6. Net contributions
of independent variables.
Variable Net
Contribution
Hb 94.26%
T 0.01%
tan ab 0.00%
fb 0.01%
Hb/(g T2) 0.00%


minus the 100 r2 value for equation 16 of
Table 4, or 94.28% 94.27% = 0.01%.
Finally, the last remaining variable
needing net contributory assessment is Hb/(g
T2) in the presence of Hb, T, tan ab, and fb.
Only one equation remains in Table 4 since
it represents independent variables taken five
at a time. The net contribution of Hb/(g T2)
becomes 94.28% minus the 100 r2 value for
equation 16 of Table 4 or 94.28% 94.28%
= 0.00%.

The net contributions are listed in
Table 6. Two of the values appear with a
net contribution of 0.00% because we have
not carried the results enough places to the
right of the decimal point. The exercise
makes little difference, however, because
except for Hb, contributions of the other
variables or parameters are negligible. These
results tell us, for the example at hand, no
more than the conclusion previously reached
that the best predictor, by far, are the results
given by equations (14) or (15). On the
other hand, the exercise is highly useful
because one would not submit to "high level
agency or corporate management", Table 4
and accompanying discussion as the final
assessment. Managers are concerned with
making, with all dispatch possible, decisions
based on the best simply defendable facts
available, and the Table 4 assessment would
probably still be too involved. Certainly, it
would require some time to explain and,
probably, one they would not want to
reiterate. Hence, while the preceding needs










to be formalized in a support document, the
final results presented to management is to
be reported as for Table 6, because they are
straightforwardly the simplest.

Some Observations

For those skeptics who doubt the
accuracy of field measurements in favor of
"more accurate" laboratory wave tank data,
here are some results that may be surprising.
If Gaillard's data (n = 25) are excluded
because we do not know how they were
measured and, therefore, may be unsure of
their accuracy, please be advised that
stepwise regression results become less
improved by an "across the board" average
value of 2.82% relative to Table 4 values
(see Appendix II for data and results).
Further, if Scripps Leica Type I and II data (n
= 74) are excluded because they are
represented by general bed slope conditions,
and Weishar's data (n = 2) are excluded
because of the manner in which the author
included them in this account, stepwise
regression results become less improved
"across the board" by an average value of
1.15% relative to Table 4 results. If all field
data are excluded (n = 131), stepwise
regression results "across the board"
become less improved by an average of
0.99% than those of Table 4. Finally, if
laboratory data only are considered and data
of prototypical dimensions (Maruyama and
others (1983), Stive (1985), Nadaoka
(1986), and Takikama and others (1997)) are
excluded (n = 16), stepwise regression
results "across the board" become less
improved by an average value of 20.22%
than those of Table 4! This results in an
100 r2 value of 74.04% which certainly is
not a convincing magnitude. These results
(particularly the last one) are significant
because it is the larger waves (field and
prototypical laboratory waves) which
significantly increases the strength of the
fitted relationship, even though they
comprise but 19.07% of the data. In terms
of the ranges in the domains of the data


(3.45 m for Hb, and 5.33 m for db),
however, the field plus prototype laboratory
data represent close to 96.64% of the
domain, the small laboratory waves but the
remaining 3.36%.

Data Set Listings and Stepwise
Regression Results

Appendix II provides additional
stepwise regression results for various data
groupings used in this work (see Table of
Contents). Data listings by investigator and
stepwise regression results are given in
Appendix Ill. All of Weishar's (1976) data
are listed and analyzed rather than the two
data groupings used in this work. Several
sets of laboratory investigations were so
small, or had no inverse matrix algebraic
solutions (see A Note on Matrix Algebra to
follow), and were grouped together.

Stepwise Assessment of Individual
Data Sets

The preceding has dealt with
stepwise regression analysis for all available
data. In addition, it would appear to be of
value to assess each individual data set
using stepwise regression. The results
(using Appendix III data listings) are given in
Table 7 as net contributions, as are simple
and weighted averages (the latter based on
sample size, n) for various data groupings.

From the results listed in Table 7, it is
apparent, for the most part, that there is no
guarantee that any single randomly selected
data set will result in a predictive outcome in
which one can have confidence. A strong
outcome is given by the prototype laboratory
data, and it can be used as a clue. This data
set, however, has but 16 data entries.

The clue alluded to is that the
strength of the previous result using all the
data is because of the large range in the data
domain.












Table 7. Net contributions of independent variables in predicting db from. stepwise
regression analysis for equation (11). Results are listed for each investigation and/or
data groupings and averages associated with data groupings.


Net Contribution in Predicting db (%)


) tanab Hbg T2) b Total

FIELD DATA


Gaillard (1904) 25 81.80 0.62 2.81 0.01 5.47 90.71
Scripps (1944a) Leica Type I 56 71.00 0.00 1.20 0.07 0.51 73.32
Scripps (1944a) Leica Type II 18 37.15 10.99 5.86 7.17 7.08 68.25
Balsillie and Carter (1980) 30 83.84 0.05 0.92 0.04 0.22 85.07

Total 129
Simple Averages 68.45 2.92 2.70 1.82 3.32 79.34
Weighted Averages 71.36 1.67 2.10 1.04 2.32 78.72

LABORATORY DATA PROTOTYPE DIMENSIONS

Various investigators' 16 98.00 0.86 0.64 0.17 0.13 99.80

Field and Prototype Laboratory Data

Total 145
Simple Averages 74.36 2.50 2.29 1.49 2.68 83.43
Weighted Averages 74.30 1.58 1.94 0.95 2.08 81.04

LABORATORY DATA

Munk (1949) BEB data 37 91.27 0.17 0.04 0.80 0.12 92.40
Munk (1949) Berkeley data 16 52.69 21.46 5.38 0.01 6.03 85.57
Putnam and others (1949) 37 90.48 3.85 0.02 3.86 0.03 98.26
Iverson (1952) 63 77.23 1.26 8.37 2.03 0.27 89.16
Morison and Crooke (1953) 6 53.96 4.92 7.70 9.41 24.01 100.00
Galvin and Eagleson (1965) 24 2.43 2.73 8.44 67.66 0.37 81.54
Eagleson (1965) 7 2.83 2.44 0.83 93.43 0.00 99.53
Horikawa and Kuo (1967) 97 70.88 0.00 13.19 0.29 0.08 84.44
Bowen and others (1968) 11 89.74 0.96 1.21 5.53 0.27 97.44
Komar and Simmons (1968) 44 91.15 0.02 0.65 2.74 2.83 97.39
Galvin (1969) 17 70.18 0.18 0.44 2.34 6.68 79.82
Weggel and Maxwell (1970) 9 25.64 0.01 0.20 0.40 72.85 99.10
Iwagaki and others (1974) 23 83.59 0.06 5.26 4.31 0.38 93.60
Walker (1974) 15 91.82 0.05 0.59 0.00 1.57 94.03
Van Dorn (1978) 12 64.62 0.38 16.79 10.87 0.50 93.16
Hansen and Svendsen (1979) 16 93.02 0.00 0.34 5.19 0.16 98.71
Singamsetti and Wind (1980) 95 70.53 0.71 9.64 2.73 5.81 89.42
Smith and Kraus (1990) 82 35.39 0.79 3.77 0.42 0.05 40.42
Small laboratory data sets*" 13 18.76 6.83 2.26 1.17 31.15 80.51

Total 624
Simple Averages 61.91 2.47 4.48 11.22 8.06 89.18
Weighted Averages 66.57 1.48 6.08 5.62 3.49 83.67

Field, Prototype Laboratory, and Laboratory Data

Total 769
Simple Averages 64.50 2.47 4.02 9.19 6.94 87.99
Weighted Averages 68.03 1.50 5.30 4.74 3.23 83.17

'Maruyama and others (1983), Stive (1985), Nadaoka (1986), Takikama and others (1997).
**Saeki and Sasaki (1973), Mizugauchi (1981), Visser (1982), Watanabe and Dibajnia (1988).











Evaluation of a Special Data Set

It has been identified that where or
when waves shore-break is a surrogate
qualitative determination, not a quantitative
measurement. Let us assume, for the
moment, that some investigators are more
precise in their surrogate assessment than
are others, and that this is indicated by the
stepwise regression results (it must be
understood, however, that there is no way
to prove this to be so). The result might be
that db is, for such data sets, predicted by a
more complex equation.

The author screened individual data
sets of Appendix III, selecting those which
exhibit significantly large values and ranges
in values for 100 r2. Twelve data sets (all


laboratory data) were found. Values of 100
r2 are listed in Table 8 for the simplest
predictive equation (i.e., equation 1 of the
stepwise regression tables for db versus Hb)
and the most complex predicting equation
(i.e., equation 31 of the stepwise regression
tables for db versus all independent
variables). All acceptable values for other
combinations of independent variables will
have 100 r2 values lying between the two.
Also listed in Table 8 are the differences for
the above, and sample size for each data
set.

Three data sets (data set I.D. Nos. 1,
2, and 3 of Table 8; n = 69) would, without
any question, result in db versus Hb being
the best outcome. These data sets would
not, therefore, be appropriate for


Table 8. Characteristics of data and assessments for a special data set.
100l 2 100 r2 for
for Eq. 1 of Eq. 31 of Differences
ID. Investor Stepwise Stepwise of preceing
No. Regression Regression two columns
Table Table (%)
(%) (%)
Range of minimum and maximum values of 100 r2 is minimal.

1 Munk (1949) BEB 91.27 92.40 1.13 37
2 Walker (1974) 91.82 94.03 2.21 15
3 Small Lab Data Sets* 98.53 99.48 0.95 17

Average 1.43
Total 69
Range of minimum and maximum values of 100 r2 is large.

4 Putnam and others (1949) 90.46 98.26 7.80 37
5 Eagleson (1965) 55.39 99.63 44.14 7
6 Bowen and others (1968) 89.74 97.44 7.70 11
7 Komar and Simmons (1968) 91.15 97.89 6.74 44
8 Weggel and Maxwell (1970) 89.90 99.10 9.20 9
9 Iwagaki and others (1974) 83.59 93.60 10.01 23
10 Van Dorn (1978) 64.62 93.16 28.54 12
11 Hansen and Svendsen (1979) 93.02 98.71 5.69 16
12 Nadaoka (1986) 91.61 99.69 8.08 12

Average 14.21
Total 171

*Includes the small data sets of Saeki and Sasaki (1973), Mizugauchi (1981), Visser (1982),
Maruyama and others (1983), Watanabe and Dibajnia (1988) and Takikama and others (1997).










consideration.

Nine data sets (data set I.D. Nos. 4 -
12 of Table 8; n = 171) were found to have
differences between the simplest and most
complex prediction outcomes ranging from
5.69% to 44.14%, with an average
difference of 14.21%. Stepwise regression
results for these nine combined data sets are
detailed in Appendix IV; net contributions are
listed in Table 9. Results of Table 9 indicate
that the independent variable Hb remains
most strongly related to db with a small
contribution attributable to Hb/(g T2) and but
a minor contribution due to tan ab. It would
seem reasonable that secondary contributing
variables or parameters should have
contributions of from 10% to 40%. The
results do not, therefore, appear to be
convincing that a more complex predicting
equation for determining where breaking
occurs can be identified with certainty.
Moreover, this combined data set comprises
but 26.73% of all laboratory data or but
22.18% of all data considered elsewhere in
this work.


Table 9. Net contribution
of independent variables for
special data set.
Variable Net
Contribution

Hb 88.23%
Hb/(g T2) 4.82%
fb 0.03%
tan ob 0.16%
T 0.00%



Assessment of Equations (12) and (13)

The percent sum of squares of the
independent variable (or parameter)
accounted for are, in every instance so low,
that it is inappropriate to even consider any
viably causative relationships) forthcoming
from equations (12) or (13). Stepwise


regression analysis results for equation (12)
where the dependent parameter is given by
db/Hb are given in Appendix V. Results for
equation (13) in which tan ab is the
dependent variable are given in Appendix VI.

A Note on Matrix Algebra

There is one important occurrence of
which the reader should be aware. Not all
matrices have an inverse (see Krumbein and
Graybill, 1965, p. 269-275). The reason
may or may not be obvious. If, for instance,
for a data set. all the values for a column of
data representing a variable or parameter are
identical (e.g., tan ab), there may be no
inverse. A small introduced change to one
of the values can correct the problem. For
other data sets, the answer may not be
obvious, and another solution will be
required (e.g., combine it with some other
data set or data sets).

COMPARATIVE ANAL YSIS
OF
PREDICTIVE METHODS

In addition to the stepwise regression
analysis we shall conduct comparative
analyses for some of the predictive
methodologies identified earlier. This was
the analytical procedure used in an earlier
work (Balsillie, 1983) except that now more
data are available.

As in the earlier work (Balsillie,
1983) we shall employ a highly useful,
although not commonly applied, statistical
tool. The commonly utilized statistical tool
for applications is predictive regression in
which one parameter or variable is defined to
be independent upon which the other is
dependent. Under these circumstances
regression is determined using the sum of
squares of the distances of data points from
the regressed line assessed only in the
direction parallel to the axis of the
independent- variable. In actual practice,
however, we utilize db and Hb










interchangeably, or more precisely, we
consider them both to be independent (i.e.,
a bivariate distribution). For such
circumstances, we need to use the
regression techniques as detailed by Ricker
(1973) which he calls functional regression
but which can also be termed cubic least
squares regression. It is based on the
consideration (e.g., Tessier, 1948) that the
central tendency of the data is more
precisely determined by minimizing the sum
of the products of both the vertical and
horizontal distances of each data point from
the regressed line. Further, if it is
determined that the data pass through the
origin (i.e., 0, 0) of the plot, as they should
for our data, the slope of the regressed cubic
least squares (CLS) fit precisely becomes:


M db
E Hb


(16)


which for all the data of this study becomes:


db = 1.3533 H.


(17)


It is, however, to be recognized,
when dealing with regression techniques,
that larger data values introduce greater
"weight" (i.e., more influence) over the
resulting regression outcome. For many data
sets, the effect is inconsequential. For
others, it is sensitively influential. The
influence is evident for the data sample of
this work because it encompasses over 2.5
orders of dimensional magnitude. Hence,
the much larger field data magnitudes exert
more numerical influence upon the resulting
regression outcome than do the small
laboratory wave tank data. How, then, does
one provide for a fair assessment?

A sensible solution is to subdivide the
sample into logical subsets with smaller
domains and determine representative
regression coefficients for each subset. For
the data of this work it would seem
appropriately fair to consider three data
subsets: 1) small scale laboratory wave tank
data, 2) prototype scale laboratory wave


tank data, and 3) field data. (Note:
Laboratory waves were considered to have
prototypical dimensions at about Hb > 0.20
m . Data sets were not split up. Only
those data sets for which the majority of
entries met the criterion are considered to
have prototypical magnitudes.) Results are
listed in Table 10, which include simple and
weighted average regressions and the
regression result given by equation (17)
which is a viable outcome. By taking
combinations, it is seen that averages
converge, from which a grand average
regression emerges. Hence, the final
regression equation becomes:


db = 1.2767 Hb


(18)


which is 0.26% less than McCowan's
coefficient of 1.28 (r = 0.9709).

Table 10. Determination of final reg-
ression coefficient relating db and Hb.
Cubic
Least
Data Grouping aes n
Squares
Regression
Prototype Laboratory Data 1.2423 16
Small Laboratory Data 1.1548 624
Field Data 1.4045 172

Simple Average 1.2672 812
Weighted Average 1.2094 812
Equation (17) 1.3533 812
Grand Average 1.2767 812


For reasons identified earlier, we shall
be selective in choosing the methods to be
compared. They are: 1) the McCowen form
of the equation, 2) Weggel's equation, and
3) Mallard's equation.

Functional regression results are listed
in Table 11 for all data, field data, and
laboratory data. As in the earlier
comparative analysis (Balsillie, 1983), the
form of the equation of McCowan does best
in predicting db than do either of Mallard's












Table 11. Comparative analysis results for three prediction methods of db.

e Jo R, Functional Regression Natural Variabty
No Variabity Analysis Results Analysis Results

n m r Withdi Above Below Withh Abov Below
Envelope Envelope Envelope Envelope Envelope Envelope
McCowan (1894) form of Relating Equation: db = 1.277 Hb
All Data
1 239 200 383 452 115 245
812 1.353 0.9697 0.0764 2 430 124 258 713 34 65
S3 594 66 152 791 14 7
Field Data

1 93 63 16 102 56 14
172 1.405 0.9186 0.1451 2 132 38 2 142 30 0
3 151 21 0 158 14 0
Laboratory Data

1 277 84 279 350 59 231
640 1.162 0.9665 0.1234 2 487 16 137 571 4 65
3 599 2 39 633 0 7

Mallard's (1978) Formula: db 1.094 (fn Hb, tan abl)
All Data
1 230 242 299 500 146 125
771 1.103 0.9651 0.0665 2 434 163 174 683 78 10
3 590 121 60 728 43 0

Field Data
1 60 44 27 71 39 21
131 1.107 0.8922 0.1306 2 106 23 2 111 20 0
3 119 12 0 122 9 0

Laboratory Data
1 359 122 159 429 107 104
640 1.089 0.9644 0.1248 2 531 73 34 572 58 10
3 592 46 2 606 34 0

Weggers (1972a, 1972b) Formula: db 1.198 (fnn RH. tan b. Hb(g T2)

All Data

1 289 223 249 527 148 96
771 1.251 0.9600 0.0802 2 531 147 93 689 75 7
3 639 106 26 729 41 1
Field Data
1 57 56 18 63 54 14
131 1.251 0.8812 0.1546 2 96 35 0 97 34 0
3 111 20 0 112 19 0

Laboratory Data
1 319 144 177 464 92 82
640 1.136 0.9298 0.1008 2 525 74 41 592 41 7
3 587 45 8 617 22 1










which considers Hb and
tan ab, or Weggel's
equation which
considers Hb, tan ab,
and Hb(g T2). Mallard's
and Weggel's equations
do not differ greatly in
terms of the correlation
coefficient from the best
solution. The stepwise
regression analysis
shows, however, that
this occurs because
variables other than Hb
do not introduce
important contributions.
(Note, however, that
Mallard's equation


Table 12. Determination of corrective regression
coefficients for Mallard's and Weggel's equations.
Mallard's Equation Weggel's Equation

Dat GroupingCubic Least Cubic Least
Data Grouping Squares n Squares n
Regression Regression
Prototype Laboratory Data 1.0475 16 0.9403 16
Small Laboratory Data 1.0937 624 1.1602 624
Field Data 1.1068 131 1.2513 131

A. Simple Average 1.0827 771 1.1173 771
B. Weighted Average 1.0950 771 1.1711 771
C. CLS Regression Coeff. 1.1030 771 1.2247 771
Grand Average of A, B, & C 1.0936 771 1.1979 771
,, L=


underestimates measured db by 8.56%, and
Weggel's equation underestimates measured
db by 16.52%. See Table 12.) Hence,
again, we are responsibly bound to apply
Occam's Razor, and accept that the simplest
approach be selected as the best predicting
equation. Equation (18) is only 0.26% less
than McCowan's solution given by equation
(4). It would, in the author's opinion, be
improper to suggest equation (20) in lieu of
equation (4), the later of which it is posited,
stands as the standard instrument for
prediction.

The plotted results (Figure 5)
encompass 812 data points, 172 of which
represent field data, 16 of which are
laboratory data with prototypical dimensions,
and 624 that are of the small laboratory
variety. The domain of values for each of db
and Hb, covers 2.5 orders of magnitude.

A Logarithmic Relationship

Visual inspection of Figure 5 suggests
that the trend of data might be better
represented (relative to equation (4)) by a
logarithmic relationship. The relationship for
the data of Figure 5 (dash-dot-dash line) is
given by:


d' = o0.275 + 1.058 In Hb
cff = e


(19)


whose goodness of fit is represented by a
significantly large product-moment
correlation coefficient, r, of 0.9689, which is
comparable to the r attained for equations
(17) and (18). It is to be noted, however,
that the departure of equation (19) from
equation (4) is minimal for the domain of
plotted data. Given the data variability,
small departure, and comparable
correlations, the use of equation (19) instead
of equation (4) does not appear (again, in
terms of Occam's Razor) to be justified.

Variability Analysis

Functional Regression Variabbity
Analysis

Following the functional regression
analysis, the plus and minus standard
deviation, s, about the regression line is
given (Ricker, 1973) by:


s = [dp n 2)1 m


(20)


where ta/2 is the Student's t value for a two-
tailed test, n is the sample size, r is the























1



db
(m)




0.1









0.01
0.01


0.1
Hb (m)


Figure 5. The McCowen equation as it represents the data of Table 3 for db and Hb.
Data include 812 total points, 172 field data points, 16 laboratory prototype data
pairs, and 624 small wave laboratory data pairs; 3 sb indicate the maximum limits
of natural shore-breaker wave height variability within a shore-breaking wave train
(applies to McCowan equation).


Pearson product-moment correlation
coefficient, and m is the regression slope.
Equation (20) cannot be effectively evaluated
if the sample size becomes too large, at least
where large values of r are concerned (i.e., r
approaching unity). Following Ricker (1973),


therefore, s was determined by calculating
the grand average of 60 random samples
taken from a data grouping with n number of
data points, where the number of random
data selections was 2 n. For example, for
the fifth entry of Table 11 where n = 131,










60 random samples of were taken 2n or 262
times from which a grand average for s was
determined.

If we define envelopes 1, 2, and
3 three standard deviations about the
regressed line then, assuming the data are
Gaussian, 68.27% of the data should lie
within 1 s of the mean, 95.45% should lie
within 2 s of the mean, and 99.73% of
the data should lie within 3 s of the mean.
These are statistical expectations. We apply
it in a relative manner. That is, the larger the
amount of data included within the
envelopes, the more confidence we have in
the regression (i.e., in addition to the
correlation coefficient). While variously large
values for data inclusion within the envelope
do occur (Table 11), the statistical variability
analysis is limited because it is self-
containedwithin its own statistical system
and because it is not comparable across data
groupings (i.e., all data, field data, and
laboratory data). That is, even though we
have kept the functional regression
coefficient constant for each predictive
method (see Table 11), the standard
deviations and correlation coefficients
associated with the regression coefficient
differs for each data grouping; hence, s will
be different. Fortunately, there is an
alternative, more realistic approach.

Natural Variability Analysis

This method is more applicable
related to the issue at hand because it
assesses the natural variability associated
with ocean wave heights. Let us, first,
inspect the case where a wave gauge is
sited in a water depth of, say, 10 m. The
gauge is not capable of selectively measuring
each of the several to many wave trains
(generally non-breaking) which encounter it.
It, therefore, measures the spectral record.
The analysis of many records indicate that
the standard deviation, s, associated with
the mean wave height, H, becomes (e.g., U.
S. Army, 1984):


s, = 0.4 H


(21)


In the surf zone, however, conditions
are distinctly different. We have learned that
at shore-breaking waves are depth limited.
Hence, a wave train characterized by a
shore-breaker height of 1 m will break in a
water depth of close to 1.28 m. A wave
train with 0.5 m shore-breakers will break in
a water depth of 0.64 m. Wave
superposition interference between shore
propagating or reflected waves can occur
periodically in which the wave heights
become precisely additive. If the
interference is not precisely in phase then
partial precise additive superposition occurs.
The point is that there can be variability in
results due to wave interference, but if
observations are carefully conducted it can
be kept to a minimum. Balsillie and Carter
(1984a, 1984b) investigated the natural
variability of shore-breaker heights (single
wave trains) in the field by measuring at the
shore-breaker position 30 elevations each of
the distance from the bed to breaker trough
and breaker crest and found:


sb = 0.21 Hb


(22)


in which Hb is the mean shore-breaker height
for a wave train, and sb is its associated
standard deviation. Field data for 47 wave
trains (2,820 individual measurements)
leading to the quantification of equation (22)
are plotted in Figure 6.

Quantities such as ss/H and sb/Hb are
termed the relative dispersion (or coefficient
of dispersion). For natural distributions
relative dispersions of 0.5 or less are
considered as having excellent homogeneity,
i.e., a "tight" distribution (see Balsillie, 1995,
p. 17 and 38-39), which lends much greater
credence to our assessment.

If we now define envelopes of 1,
2, and 3 standard deviations about the
regressed line, again assuming the data are
Gaussian, then 68.27% of the data for 1
sb, 95.45% of the data for 2 s, and














sb 0.15

(m) 0.1 --
0.05-e S r = 0.9255
o"0.05 J- "" n = 47 wave trains

S 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Hb (m)
Figure 6. Relationship between the mean shore-breaking wave height, Hb, for
single wave trains and their associated standard deviations, sb, (after Balsillie
and Carter, 1984a, 1984b).


99.73% of the data for 3 sb should lie
within each of the envelopes. The data,
listed in Table 8 (Natural Variability Analysis)
show that this is a much better variability
analytical treatment.

In fact, for the most part the amount
of data lying within the envelopes (Table 8)
are all sufficiently large that, in addition to
the significantly large correlation
coefficients, allow us to have increased
confidence in the regressed results (Ricker,
1973).

DISCUSSION

The regression line conforming to
McCowan's equation is plotted through the
.data of Figure 4, as are lines (dashed) for
plus and minus 3 sb (the 1 sb and 2 sb
envelopes are not plotted to keep the plot
simplified). It is evident that the bulk of data
lies between these indicators of expected
natural breaker height variability, which we
have noted lends greater credibility to our
results. It is also recognized that the field
data (and prototype laboratory data) show
less variability than do the small wave
laboratory data. This consideration is not
trivial in terms of the location of the
regression line and its effect on wave


energy, E. Examples of how the wave
energy (note: E oc Hb2) is affected are listed
in Table 13. If one takes the position that a
more conservative approach (i.e., responsibly
higher versus arbitrarily lower energy) is
more desirable, we can posit some
comments about the resulting outcome. We
are not greatly concerned with data lying
above (and to the left) of the regression line
because 1) they yield lower breaking wave
energies not conducive to a responsibly
conservative design solution, and 2) waves
can break in this region due to critically high
wind stresses (and other esoteric causes).
We become, in fact, more concerned with
the data lying below (and to the right) of the
regression line. Note that the field and
laboratory prototype data are less "ragged"
in their distribution below than above the
regression line, and plot closer to the line
than do the small wave laboratory data. The
latter are about evenly distributed about the
regression line. Based on his years of
experience in the field, the auther has found
that larger waves (up to about 1 m in height)
are more precisely measurable than very
small waves. In fact, only 14 of the 172
field and prototype waves (8.13%) lie below
the -1 sb envelope line, while 231 of the 640
small laboratory waves (36.09%) lie below
the line. None of the field or prototype










Table 13. Examples of the degree on which shore-breaking wave energy levels are dependent
on the depth in which the waves break. Assessment is made in terms of sb given by equation
(20), where the wave energy is directly proportional to the wave height squared.
Wave Energy Additional Ramifications for
Position Relative to
Regreosion Uae to Regression Lne of Relative to Final Shore-Breaking Wave
Regression Une Figure 4 n b McCowan Swash Energy Dissipation
gure 4. Equation Potential

db = 1.28 Hb + 3 sb = 1.91 Hb 0.45 Waves break further offshore,
Above regression swash potentially wider
line. db = 1.28 Hb + 2 sb = 1.70 Hb 0.57 allowing greater energy
dissipation; damage potential is
db = 1.28 Hb + 1 sb = 1.49 Hb 0.74 lessened.

On regression line. db = 1.28 Hb 1.00 Status quo.
db = 1.28 Hb 1 sb = 1.07 Hb 1.43 Waves break closer to shore,
Below regression swash narrower and energy
line. db = 1.28 Hb 2 sb = 0.86 Hb 2.22 dissipation less; damage
Sdb = 1.28 Hb 3b = 0.65 Hb 3.87 potential increases.
db = 1.28 H, 3 s, = 0.65 H, 3.87


waves lie below the -2 sb line, while 65
small laboratory waves (10.16) lie below 2
sb. The author posits, given the large
sample size, that this result strengthens the
position of the plotted regression line,
particularly since the field and prototype
waves comprise 96.64% of the measured
domain of the data.

With the proper measurement
equipment and techniques, whether in the
field or the laboratory, we can, within
acceptable limits, measure each shore-
breaking characteristic such as Hb, db, tan
ab, cb, and uc with precision. It is the
kinematic stability criterion (equation (1))
that provides us with the better answer as to
when breaking occurs. With rare exception,
however, we do not use this approach.
Rather, we have relied upon a set of
surrogate definitions concerning the
appearance of various shore-breaking wave
types which aid us more nearly to determine
where shore-breaking occurs. It is an
observed determination, not a measured
quantity; no gauge can measure where
waves break in terms of the surrogate
definitions. Employing Sir A. C. Doyle's
principle, then, this must be the principle


cause of the bulk of variability or error of the
data.

Low slope conditions greatly increase
the cognitive difficulty in determining
precisely when shore-breaking happens.
This is particularly true for small laboratory
waves, and is why laboratory nearshore
slopes are usually exaggerated relative to
natural nearshore bed slopes. That is, the
steeper bed slopes induce a response in
wave behavior that is easier to perceive and,
therefore, to "measure". This was, in fact,
recognized in early field studies (e.g., Scripps
Institution of Oceanography, 1945).

It has been demonstrated that where
or when nearshore waves shore-break,
remarkably affects wave energy constraints.
Let us, also, illustrate an analog to energy
constraints by looking at wave induced
impact pressures. Pressures produced by
waves are described to consist of a first
extremely high pressure of very short
duration termed shock or impact pressure or
gifle by Larras (1937), followed by a second
pressure that is significantly less in
magnitude and longer in duration termed
dynamic pressure or bourrage by Larras










(1937). As an example, short of
constructing a Galveston, Texas, seawall,
manmade structures simply cannot be
designed to withstand impact pressures
within reasonable funding constraints.
Results from field studies (Miller and others,
1974a, 1974b; Miller, 1976) indicate that
impact forces from shore-breaking and
broken waves significantly exceed those
from non-breaking waves. Highest impact
pressures occur in post-breaking waves (i.e.,
bores), with greater pressures occurring from
plunger-generated than spilling-generated
bores, Shore-breaking waves produced next
highest impact pressures, with greater
pressures occurring for plunging than spilling
shore-breakers. The difference between
breaking and post-breaking impact pressures
is the elevation at which the maximum
horizontal impact occurs. For post-breaking
bores, the maximum pressure is nearer the
still water level. For waves at the shore-
breaking position, it occurs in the upper
portion of the wave crest. (Balsillie, 1985).
The author (Balsillie, 1985) found that at the
shore-breaking position the shore-breaker
with the greatest impact pressure had a
value of fb = 1.0 which occurred at an
elevation of 0.45 Hb above the still water
level or 1.7 Hb above the bed.

It should, therefore, be evident that
while nearshore waves are depth limited, the
breaker type is dependent upon both bed
slope and equivalent wave steepness.

The Nelson-Gourlay Horizontal
Bed Slope Breaking Anomaly

It is acknowledged that the subject of
shore-breaking on a horizontal bed is one
deserving of study. It is very difficult to
measure such an occurrence, even in the
field, let alone in the laboratory. In fact, as
previously noted, laboratory bed slopes over
which breaking is induced are exaggerated
so that the researcher is better able to
recognize a sudden change in crest
deformation aiding in identifying when shore-


breaking occurs.

Bearing this in mind, the results of
Nelson (1982, 1994) and Gourlay (1994)
pose some difficulties. While the list is
longer, following are some of the major
concerns.

1) Their laboratory wave tank
designss, while termed a reef, are what are
more commonly described as a step profile.
As nearly as can be determined, the first
encountered "seaward" or stoss slope of the
step ranged from 0.0467 5 tan ab <
0.2222. The step top, or "test slope" over
which breaking was assessed, ranged from
tan ab = 0.0358 to horizontal. It is the
slope immediately preceding breaking which
clearly causes shore-breaking. It is also clear
that it was the stoss slope which caused
their waves to break. These are not, then,
the proper conditions for assessing shore-
breaking on a horizontal bed.

2) Their wave tank was some 30 m
in length. However, the distance from the
wave generator to the front of the step stoss
slope was but 5 m. This does not seem to
be an adequate distance for waves to
undergo wave dispersion mechanics. They
may, then, represent forced waves, which
explains breaking in greater water depths.

3) They did not define their criteria
for identifying when shore-breaking occurs.

4) Their relating coefficient (i.e.,
db/Hb = 1.82) lies to the left and above the
regression line of Figure 4. It, therefore,
results in a non-conservative assessment of
wave energy. Results of this study represent
a much larger and dimensionally broader data
set than they used. Even so, their
coefficient lies within + 3 sb (precisely at
2.48 sb), which may not be an anomalous
value when assessed in terms of natural
shore-breaker height variability except forthe
previous concerns.










5) Gourlay (1994) modeled a field
site in a small-scale laboratory wave tank,
then scaled his waves to prototype
dimensions. Such scaling is subject to
question. It would have been much more
useful had he reported the original wave tank
data. It is not, for example, clear if the
error/variability associated with his
determination of gauge determined wave
heights is on the order of variability
associated with d/H.

6) Nelson's (1994) reported field
study results are not clearly reported in
support of his conclusionss.

CONCLUSIONS

This work has been conducted to find
a least equivocal, best representative
predicting numerical relationship identifying
where shore-breaking occurs. It may well be
that variables or parameters other than Hb
will provide a refining and strengthening role
in determining where shore-breaking occurs
(although it is the authors' suspicion they
will be of secondary, tertiary, etc.,
importance). The results of this study,
however, indicate that as long as we persist
to use the surrogate method (i.e., visually
determined shore-breaking wave types to
determine where shore-breaking occurs),
please be advised that the issue of
identifying a more complex predicting
relationship for determining when or where
waves shore-break will not be forthcoming (a
suggestion as to why not is given in
Appendix VII). To ultimately resolve the
issue will require measurement of the
kinematic stability parameter (equation (1)),
perhaps even the dynamic stability
parameter. The former has, to some extent,
been measured under controlled conditions
by various researchers (e.g., Adyemo, 1970;
Divoky and others, 1970; Iwagaki and
others, 1974; Van Dorn, 1978; Sakai and
Iwagaki, 1978; Easson and others, 1988;
etc.). Hence, while we know that it is
possible, such a program of research will be


resource intensive. Given, however, the
importance of where shore-breaking occurs
on wave energy constraints, such a research
effort would seem justifiable.

There is the possibility, of course,
that measurement of the kinematic stability
parameter will do no better than the
surrogate methodology. The reality is, of
course, that such work has not been
accomplished, and we must posit our
conclusions based on available data. Many
investigators have dealt with the subject of
the cause of shore-breaking, quite often
using only their own data sets. To ignore
existing data or exclude existing data
without sound reasoning and justification is
simply not part of the scientific process. It
is fortunate that a significantly large amount
of data are available (i.e., all data on the
subject known to the author) upon which it
is possible to conclude,, in terms of reliability,
statistically significant results. Two lines of
statistical assessment were employed in this
work: 1) stepwise regression, and 2)
comparative analysis of selected existing
prediction methodologies using functional
regression. It was determined that the best
predicting equation is given by db = 1.277
Hb. This result is but 0.26% larger than
McCowan's (1894) result given by equation
(4). Hence, it is concluded that McCowan's
relationship remains as the standard
instrument for prediction.

The establishment of a standard,
particularly one of import, is more often than
not, a tricky proposition at best. This one,
however, significantly affects wave energy
and impact pressures, both of which are
fundamental shore-breaking outcomes of
significant and pragmatically important
proportions. It is anologous to filing our
annual U. S. income tax forms in which we
must first know of the number of deductions
to be claimed before we can proceed.
Hence, before we can do anything else
concerning a littoral design solution or
coastal processes result, we must know










where the waves shore-break; in one way or
another it is a requirement of almost all
coastal geology and engineering applications.
We must ask, therefore, the following
question. Do we become tolerant of
resulting multiple prediction methods based
on small data sets which will pose future
accounting problems, or do we establish a
standard based on all available data that in
the future, should it be refined, can easily be
transformed?

It must be concluded, therefore,
based on the existing evidence, that
nearshore waves are water-depth limited.
The consideration of other variables or
parameters are highly useful in determining
the type of shore-breaking wave (i.e.,
spilling, plunging, surging, etc.) that occurs
(see equation (3)). It is, in fact, here
suggested that McCowan's equation (i.e.,
equation (4)) be formally termed McCowan's
Lemma.

ACKNOWLEDGEMENTS

This work may not have been pursued
had it not been for the encouragement of
William F. Tanner, Regents Professor,
Department of Geology, Florida State
University and Visiting Scientist, Florida
Geological Survey. His guidance, both
personal and through published work on the
application and merits of stepwise
regression, was invaluable. His concerted
interest and countless hours of discussion
are more than acknowledged with thanks.
Special thanks are also to be extended to
Paulette Bond for her attention to the subject
and advice, and to Paulette Bond, Don L.
Hargrove, Jacqueline M. Lloyd, and William
C. Parker for their necessary presence in
collecting field data for extending the data
coverage of Figure 3 of this work.

Florida Geological Survey staff
reviews were conducted by Paulette Bond,
Kenneth Campbell, L. James Ladner, Ed
Lane, Jacqueline Lloyd, Deborah Mekeel,


Walter Schmidt, and Thomas Scott. The
outside review and encouragement of
Nicholas C. Kraus with the Coastal
Engineering Laboratory, is also
acknowledged.

REFERENCES

Adeymo, M. D., 1970, Velocity fields in the
wave breaker zone: Proceedings of
the 12th Coastal Engineering
Conference, v. 1, chap. 27, p. 435-
460.

Balsillie, J. H., 1983, On the determination
of when waves break in shallow
water: Florida Department of Natural
Resources, Beaches and Shores
Technical and Design Memorandum
No. 83-3, 25 p.

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APPENDIX I


Example calculations of the effect of shore-breaking water depth
on wave energy constraints.





























































38











Example Calculations of the
Water Depth on Wave



First, let us evaluate all shore-
breaking cases at a water depth, db, of 1.28
m. Second, we wish to evaluate final results
in terms of the energy density, Eb, which for
Airy waves is given by:


,- pg H2
Eb --8-Hb
8
in which pf is the fluid mass density, ar
is the acceleration gravity assigned the v
of 9.8 m/s2 for Earth-bound applicati'
Hence, for our purposes Eb oc Hb2. Tt
we wish to assess the examples relative
some standard which has been arbiter
selected to be given by:

db = c, H1, = 1.28 H,,


Effect of Shore-Breaking
Energy Constraints


"b2 2
H 2
b1


Eb2 6.5536
S- 6.55
E= 1.00


That is, the wave energy occurring at c2 =
0.50 is 6.55 times that where c, = 1.28.


(I-1) EXAMPLE 2.


id g Let us next evaluate the case where:
alue
3ns. d, = c H3 = 3.00 H, (1-4)
iird,
e to Height and energy conditions for Hbl apply
arily precisely here as they appear for Example 1.
For equation (1-4):
db 1.28
(1-2) H 3.00 3.00 0.427m
3.00 30


EXAMPLE 1.


Let us evaluate the case where:


db = c2 H, = 0.50 Hb


(1-3)


Now, for equation (1-2):

db 1.28
H, = 8 1.0m
1.28 1.28
and for equation (1-3)

d, 1.28
Ht -1.2 =2.56m
0.50 0.05


H 32 = E, = 0.4272 = 0.1823


Comparatively, then:


H 2
b3
H 2
t'I


E 0.1823-= 0.182
Eb 1.00


and the wave energy at c3 is 0.182 times
that where c, = 1.28.


Moreover, the following can be stated:

Hb,2 = E,, = 1.002 = 1.00

and

H,2 = E, = 2.562 = 6.5536


Comparatively, then:
































































40












APPENDIX II

Stepwise regression results for data sets and subsets



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (11) of the text.

100 r2 = 100 r2.

The dependent variable, db, is the water depth at the shore-breaking position.

Independent variables:

(1) Shore-breaking wave height, Hb
(2) Wave period, T
(3) Bed slope leading to shore-breaking, tan ab
(4) Equivalent wave steepness, Hb/g T2
(5) Surf similarity parameter, tan ab/[Hb/(g T2)]0.5

Correlation Matrix.

Correlation matrix treatment is given by Table 5 of the text.

Variables:

(1) db (2) Hb (3) T (4) tan ab

(5) Hb/(g T2) (6) tan ab/[Hb/(g T2)]05































































42













ALL DATA
n = 771

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 94.26 0.9709
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 74.25 0.8617
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 6.79 0.2606
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 3.66 0.1914
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 4.87 0.2208
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 104.72 -4.72 0.00 0.00 0.00 0.00 0.00 0.00 94.27 0.9709
7 101.15 0.00 -1.15 0.00 0.00 0.00 0.00 0.00 94.26 0.9709
8 97.99 0.00 0.00 2.01 0.00 0.00 0.00 0.00 94.26 0.9709
9 100.83 0.00 0.00 0.00 -0.83 0.00 0.00 0.00 94.26 0.9709
10 0.00 103.25 -3.25 0.00 0.00 0.00 0.00 0.00 74.31 0.8620
11 0.00 71.72 0.00 28.28 0.00 0.00 0.00 0.00 78.58 0.8865
12 0.00 119.72 0.00 0.00 -19.72 0.00 0.00 0.00 76.01 0.8719
13 0.00 0.00 53.06 46.94 0.00 0.00 0.00 0.00 8.43 0.2904
14 0.00 0.00 92.21 0.00 7.79 0.00 0.00 0.00 6.81 0.2609
15 0.00 0.00 0.00 58.43 41.57 0.00 0.00 0.00 9.90 0.3146
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 106.50 -5.07 -1.43 0.00 0.00 0.00 0.00 0.00 94.27 0.9710
17 104.65 -4.69 0.00 0.04 0.00 0.00 0.00 0.00 94.27 0.9709
18 104.93 -4.57 0.00 0.00 -0.36 0.00 0.00 0.00 94.27 0.9709
19 98.98 0.00 -1.50 2.52 0.00 0.00 0.00 0.00 94.27 0.9709
20 101.13 0.00 -1.29 0.00 0.16 0.00 0.00 0.00 94.26 0.9709
21 98.73 0.00 0.00 1.85 -0.58 0.00 0.00 0.00 94.26 0.9709
22 0.00 74.20 -5.17 30.97 0.00 0.00 0.00 0.00 78.92 0.8884
23 0.00 108.19 37.22 0.00 -45.41 0.00 0.00 0.00 78.38 0.8853
24 0.00 79.34 0.00 28.96 -8.30 0.00 0.00 0.00 79.43 0.8912
25 0.00 0.00 -4.53 59.93 44.60 0.00 0.00 0.00 9.91 0.3149
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 105.49 -4.59 -1.48 0.58 0.00 0.00 0.00 0.00 94.27 0.9710
27 108.06 -7.00 -4.15 0.00 3.08 0.00 0.00 0.00 94.28 0.9710
28 104.96 -4.59 0.00 -0.02 -0.36 0.00 0.00 0.00 94.27 0.9709
29 96.89 0.00 -4.44 4.38 3.17 0.00 0.00 0.00 94.27 0.9709
30 0.00 83.07 11.79 23.66 -18.53 0.00 0.00 0.00 79.71 0.8928
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 103.99 -5.56 -5.53 2.74 4.36 0.00 0.00 0.00 94.28 0.9710


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9709 0.8617 -0.2606 -0.1914 -0.2208

(2) 94.2587 1.0000 0.8926 -0.2631 -0.2037 -0.2233

(3) 74.2542 79.6819 1.0000 -0.2759 -0.4391 -0.1031

(4) 6.7908 6.9235 7.6131 1.0000 0.2593 0.8204

(5) 3.6632 4.1475 19.2788 6.7219 1.0000 -0.1387

(6) 4.8738 4.9881 1.0629 67.3045 1.9247 1.0000

Numbers in Parentheses are the Variable or Parameter Number













ALL DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (88 X8)


BO B1 B2 B3 B4 B5 B6 B7 B8


1.372 0.000 0.000 0.000 0.000
0.000 0.322 0.000 0.000 0.000
0.000 0.000 -3.543 0.000 0.000
0.000 0.000 0.000 -57.112 0.000
0.000 0.000 0.000 0.000 -0.188
1.402 -0.009 0.000 0.000 0.000
1.370 0.000 -0.075 0.000 0.000
1.374 0.000 0.000 1.970 0.000
1.371 0.000 0.000 0.000 -0.004
0.000 0.319 -0.336 0.000 0.000
0.000 0.360 0.000 69.113 0.000
0.000 0.317 0.000 0.000 -0.114
0.000 0.000 -3.075 -39.614 0.000
0.000 0.000 -3.305 0.000 -0.018
0.000 0.000 0.000 -67.551 -0.215
1.402 -0.010 -0.090 0.000 0.000
1.402 -0.009 0.000 0.037 0.000
1.401 -0.009 0.000 0.000 -0.002
1.371 0.000 -0.100 2.439 0.000
1.370 0.000 -0.084 0.000 0.001
1.373 0.000 0.000 1.800 -0.003
0.000 0.355 -0.828 72.222 0.000
0.000 0.338 3.881 0.000 -0.308
0.000 0.353 0.000 62.701 -0.080
0.000 0.000 0.366 -70.608 -0.235
1.399 -0.009 -0.094 0.541 0.000
1.414 -0.013 -0.260 0.000 0.013
1.401 -0.009 0.000 -0.015 -0.002
1.373 0.000 -0.302 4.339 0.014
0.000 0.354 1.680 49.090 -0.172
1.408 -0.011 -0.359 2.588 0.018


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


1 -0.017
2 -0.340
3 0.765
4 0.794
5 0.715
6 -0.005
7 -0.010
8 -0.028
9 -0.012
10 -0.308
11 -0.805
12 -0.192
13 0.939
14 0.768
15 1.105
16 0.004
17 -0.005
18 -0.003
19 -0.022
20 -0.011
21 -0.024
22 -0.746
23 -0.314
24 -0.657
25 1.117
26 0.000
27 0.007
28 -0.003
29 -0.034
30 -0.609
31 -0.010













ALL DATA MINUS GAILLARDS (1904) FIELD DATA
n = 746
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 97.00 0.9849
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 81.40 0.9022
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 5.75 0.2397
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 5.13 0.2264
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 3.73 0.1930
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 86.59 13.41 0.00 0.00 0.00 0.00 0.00 0.00 97.10 0.9854
7 101.11 0.00 -1.11 0.00 0.00 0.00 0.00 0.00 97.00 0.9849
8 101.32 0.00 0.00 -1.32 0.00 0.00 0.00 0.00 97.00 0.9849
9 100.14 0.00 0.00 0.00 -0.14 0.00 0.00 0.00 97.00 0.9849
10 0.00 101.40 -1.40 0.00 0.00 0.00 0.00 0.00 81.41 0.9023
11 0.00 73.00 0.00 27.00 0.00 0.00 0.00 0.00 85.32 0.9237
12 0.00 118.23 0.00 0.00 -18.23 0.00 0.00 0.00 82.98 0.9109
13 0.00 0.00 42.78 57.22 0.00 0.00 0.00 0.00 8.64 0.2939
14 0.00 0.00 103.51 0.00 -3.51 0.00 0.00 0.00 5.75 0.2398
15 0.00 0.00 0.00 63.34 36.66 0.00 0.00 0.00 10.32 0.3212
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 86.99 13.40 -0.39 0.00 0.00 0.00 0.00 0.00 97.10 0.9854
17 76.77 17.03 0.00 6.21 0.00 0.00 0.00 0.00 97.13 0.9856
18 87.29 14.51 0.00 0.00 -1.80 0.00 0.00 0.00 97.11 0.9854
19 101.99 0.00 -0.97 -1.02 0.00 0.00 0.00 0.00 97.00 0.9849
20 100.80 0.00 -2.85 0.00 2.06 0.00 0.00 0.00 97.00 0.9849
21 101.75 0.00 0.00 -1.42 -0.33 0.00 0.00 0.00 97.00 0.9849
22 0.00 74.97 -4.01 29.04 0.00 0.00 0.00 0.00 85.52 0.9248
23 0.00 106.51 38.96 0.00 -45.47 0.00 0.00 0.00 85.76 0.9261
24 0.00 80.52 0.00 27.54 -8.05 0.00 0.00 0.00 86.10 0.9279
25 0.00 0.00 -13.21 67.43 45.78 0.00 0.00 0.00 10.45 0.3232
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 77.19 17.26 -1.10 6.65 0.00 0.00 0.00 0.00 97.14 0.9856
27 84.17 16.87 4.46 0.00 -5.51 0.00 0.00 0.00 97.12 0.9855
28 77.46 17.82 0.00 6.11 -1.39 0.00 0.00 0.00 97.14 0.9856
29 100.41 0.00 -3.14 0.40 2.33 0.00 0.00 0.00 97.00 0.9849
30 0.00 85.93 17.11 19.82 -22.87 0.00 0.00 0.00 86.67 0.9310
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 77.51 18.08 0.95 5.67 -2.21 0.00 0.00 0.00 97.14 0.9856


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9849 0.9022 -0.2397 -0.2264 -0.1930

(2) 96.9971 1.0000 0.9019 -0.2386 -0.2260 -0.1953

(3) 81.3960 81.3346 1.0000 -0.2543 -0.4474 -0.0749

(4) 5.7465 5.6927 6.4689 1.0000 0.2598 0.8173

(5) 5.1263 5.1073 20.0174 6.7481 1.0000 -0.1432

(6) 3.7260 3.8155 0.5604 66.7958 2.0497 1.0000

Numbers in Parentheses are the Variable or Parameter Number













ALL DATA MINUS GAILLARDS (1904) FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

B1 B2 B3 B4 B5 B6 B7 B8


EQ BO
NO
1 -0.008
2 -0.312
3 0.620
4 0.705
5 0.571
6 -0.039
7 -0.003
8 -0.002
9 -0.007
10 -0.300
11 -0.685
12 -0.195
13 0.812
14 0.619
15 0.948
16 -0.037
17 -0.084
18 -0.033
19 0.001
20 -0.004
21 0.000
22 -0.647
23 -0.305
24 -0.567
25 0.978
26 -0.080
27 -0.043
28 -0.077
29 -0.006
30 -0.510
31 -0.076


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


0.000 0.000 0.000 0.000
0.291 0.000 0.000 0.000
0.000 -2.729 0.000 0.000
0.000 0.000 -56.452 0.000
0.000 0.000 0.000 -0.138
0.024 0.000 0.000 0.000
0.000 -0.057 0.000 0.000
0.000 0.000 -1.008 0.000
0.000 0.000 0.000 0.000
0.290 -0.125 0.000 0.000
0.323 0.000 55.250 0.000
0.288 0.000 0.000 -0.090
0.000 -2.209 -43.887 0.000
0.000 -2.811 0.000 0.006
0.000 0.000 -64.667 -0.164
0.024 -0.022 0.000 0.000
0.034 0.000 5.695 0.000
0.026 0.000 0.000 -0.006
0.000 -0.049 -0.773 0.000
0.000 -0.147 0.000 0.007
0.000 0.000 -1.083 -0.001
0.321 -0.534 57.384 0.000
0.307 3.498 0.000 -0.266
0.318 0.000 50.273 -0.064
0.000 0.959 -72.747 -0.217
0.034 -0.068 6.058 0.000
0.031 0.252 0.000 -0.020
0.035 0.000 5.526 -0.006
0.000 -0.163 0.307 0.008
0.319 1.979 34.045 -0.172
0.035 0.057 5.119 -0.009


1.274
0.000
0.000
0.000
0.000
1.187
1.273
1.273
1.274
0.000
0.000
0.000
0.000
0.000
0.000
1.187
1.159
1.179
1.272
1.273
1.273
0.000
0.000
0.000
0.000
1.157
1.163
1.153
1.273
0.000
1.151













ALL DATA MINUS GAILLARDS (1904), SCRIPPS LEICA TYPE I AND II (1944a, 1944b,
1945), AND WEISHARS (1976) FIELD DATA
n = 670

STEPUISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 93.18 0.9653
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 16.22 0.4027
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.18 0.0421
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 2.32 0.1525
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.17 0.0409
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 105.38 -5.38 0.00 0.00 0.00 0.00 0.00 0.00 93.28 0.9658
7 106.13 0.00 -6.13 0.00 0.00 0.00 0.00 0.00 93.56 0.9673
8 96.78 0.00 0.00 3.22 0.00 0.00 0.00 0.00 93.24 0.9656
9 105.64 0.00 0.00 0.00 -5.64 0.00 0.00 0.00 93.55 0.9672
10 0.00 101.82 -1.82 0.00 0.00 0.00 0.00 0.00 16.23 0.4029
11 0.00 58.70 0.00 41.30 0.00 0.00 0.00 0.00 38.40 0.6197
12 0.00 131.38 0.00 0.00 -31.38 0.00 0.00 0.00 19.54 0.4420
13 0.00 0.00 -45.59 145.59 0.00 0.00 0.00 0.00 2.89 0.1701
14 0.00 0.00 58.25 0.00 41.75 0.00 0.00 0.00 0.19 0.0436
15 0.00 0.00 0.00 104.35 -4.35 0.00 0.00 0.00 2.33 0.1528
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 113.76 -6.95 -6.81 0.00 0.00 0.00 0.00 0.00 93.69 0.9679
17 104.09 -4.75 0.00 0.65 0.00 0.00 0.00 0.00 93.28 0.9658
18 107.21 -1.85 0.00 0.00 -5.35 0.00 0.00 0.00 93.56 0.9672
19 101.23 0.00 -6.64 5.41 0.00 0.00 0.00 0.00 93.70 0.9680
20 106.56 0.00 -3.65 0.00 -2.91 0.00 0.00 0.00 93.59 0.9674
21 103.63 0.00 0.00 1.68 -5.31 0.00 0.00 0.00 93.56 0.9673
22 0.00 61.07 -5.64 44.57 0.00 0.00 0.00 0.00 39.59 0.6292
23 0.00 108.92 63.13 0.00 -72.05 0.00 0.00 0.00 29.75 0.5454
24 0.00 65.95 0.00 42.74 -8.69 0.00 0.00 0.00 41.32 0.6428
25 0.00 0.00 -95.26 118.19 77.08 0.00 0.00 0.00 4.50 0.2120
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 106.75 -3.51 -6.81 3.56 0.00 0.00 0.00 0.00 93.72 0.9681
27 116.29 -9.68 -9.74 0.00 3.13 0.00 0.00 0.00 93.71 0.9680
28 104.46 -0.56 0.00 1.39 -5.28 0.00 0.00 0.00 93.56 0.9673
29 100.04 0.00 -8.22 6.44 1.75 0.00 0.00 0.00 93.71 0.9680
30 0.00 70.14 13.51 36.86 -20.52 0.00 0.00 0.00 42.51 0.6520
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 108.05 -6.23 -10.92 4.71 4.39 0.00 0.00 0.00 93.75 0.9683


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9653 0.4027 -0.0421 0.1525 -0.0409

(2) 93.1835 1.0000 0.4453 0.0201 0.1334 0.0202

(3) 16.2205 19.8285 1.0000 -0.0755 -0.5767 0.3260

(4) 0.1768 0.0403 0.5707 1.0000 0.2084 0.8087

(5) 2.3242 1.7788 33.2640 4.3432 1.0000 -0.2069

(6) 0.1670 0.0408 10.6303 65.3993 4.2787 1.0000

Numbers in Parentheses are the Variable or Parameter Number













ALL DATA MINUS GAILLARDS (1904), SCRIPPS LEICA TYPE I AND II (1944a, 1944b,
1945), AND WEISHARS (1976) FIELD DATA (CONT)

TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (81 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


1.322 0.000 0.000 0.000 0.000 0.000
0.000 0.057 0.000 0.000 0.000 0.000
0.000 0.000 -0.071 0.000 0.000 0.000
0.000 0.000 0.000 5.664 0.000 0.000
0.000 0.000 0.000 0.000 -0.004 0.000
1.343 -0.005 0.000 0.000 0.000 0.000
1.324 0.000 -0.103 0.000 0.000 0.000
1.318 0.000 0.000 0.897 0.000 0.000
1.324 0.000 0.000 0.000 -0.006 0.000
0.000 0.057 -0.020 0.000 0.000 0.000
0.000 0.104 0.000 21.417 0.000 0.000
0.000 0.066 0.000 0.000 -0.020 0.000
0.000 0.000 -0.130 6.261 0.000 0.000
0.000 0.000 -0.044 0.000 -0.002 0.000
0.000 0.000 0.000 5.589 -0.001 0.000
1.349 -0.006 -0.109 0.000 0.000 0.000
1.339 -0.004 0.000 0.171 0.000 0.000
1.330 -0.002 0.000 0.000 -0.006 0.000
1.317 0.000 -0.117 1.437 0.000 0.000
1.324 0.000 -0.061 0.000 -0.003 0.000
1.321 0.000 0.000 0.438 -0.006 0.000
0.000 0.105 -0.188 22.444 0.000 0.000
0.000 0.102 1.151 0.000 -0.087 0.000
0.000 0.112 0.000 21.228 -0.019 0.000
0.000 0.000 -0.554 10.400 0.030 0.000
1.333 -0.003 -0.115 0.909 0.000 0.000
1.358 -0.008 -0.153 0.000 0.003 0.000
1.324 0.000 0.000 0.359 -0.006 0.000
1.315 0.000 -0.146 1.729 0.002 0.000
0.000 0.120 0.448 18.515 -0.045 0.000
1.341 -0.005 -0.183 1.195 0.005 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


1 -0.016
2 0.042
3 0.142
4 0.104
5 0.141
6 -0.011
7 -0.008
8 -0.021
9 -0.008
10 0.044
11 -0.156
12 0.053
13 0.112
14 0.142
15 0.106
16 -0.001
17 -0.012
18 -0.007
19 -0.014
20 -0.007
21 -0.011
22 -0.148
23 -0.020
24 -0.144
25 0.086
26 -0.008
27 0.001
28 -0.010
29 -0.015
30 -0.147
31 -0.007













ALL LABORATORY DATA
n = 640
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 93.42 0.9665
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 5.15 0.2270
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.23 0.0480
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 7.29 0.2700
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 1.70 0.1305
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 115.82 -15.82 0.00 0.00 0.00 0.00 0.00 0.00 93.87 0.9689
7 106.15 0.00 -6.15 0.00 0.00 0.00 0.00 0.00 93.81 0.9686
8 94.71 0.00 0.00 5.29 0.00 0.00 0.00 0.00 93.57 0.9673
9 106.93 0.00 0.00 0.00 -6.93 0.00 0.00 0.00 93.89 0.9690
10 0.00 104.05 -4.05 0.00 0.00 0.00 0.00 0.00 5.19 0.2278
11 0.00 57.67 0.00 42.33 0.00 0.00 0.00 0.00 33.14 0.5757
12 0.00 151.59 0.00 0.00 -51.59 0.00 0.00 0.00 8.85 0.2974
13 0.00 0.00 -34.24 134.24 0.00 0.00 0.00 0.00 8.51 0.2918
14 0.00 0.00 -203.04 0.00 303.04 0.00 0.00 0.00 2.96 0.1720
15 0.00 0.00 0.00 128.22 -28.22 0.00 0.00 0.00 7.96 0.2822
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 128.80 -20.24 -8.56 0.00 0.00 0.00 0.00 0.00 94.41 0.9717
17 127.94 -22.82 0.00 -5.12 0.00 0.00 0.00 0.00 93.91 0.9691
18 118.63 -12.68 0.00 0.00 -5.95 0.00 0.00 0.00 94.15 0.9703
19 98.99 0.00 -6.98 7.99 0.00 0.00 0.00 0.00 94.11 0.9701
20 107.18 0.00 -1.69 0.00 -5.50 0.00 0.00 0.00 93.90 0.9690
21 102.15 0.00 0.00 4.05 -6.20 0.00 0.00 0.00 93.97 0.9694
22 0.00 59.64 -5.10 45.45 0.00 0.00 0.00 0.00 34.50 0.5873
23 0.00 113.23 78.16 0.00 -91.39 0.00 0.00 0.00 22.14 0.4705
24 0.00 63.94 0.00 43.79 -7.74 0.00 0.00 0.00 36.06 0.6005
25 0.00 0.00 -54.90 124.69 30.21 0.00 0.00 0.00 8.75 0.2958
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 132.87 -22.55 -8.66 -1.65 0.00 0.00 0.00 0.00 94.42 0.9717
27 142.23 -34.64 -21.86 0.00 14.26 0.00 0.00 0.00 94.58 0.9725
28 129.27 -18.55 0.00 -4.43 -6.29 0.00 0.00 0.00 94.17 0.9704
29 97.34 0.00 -9.12 9.31 2.47 0.00 0.00 0.00 94.12 0.9702
30 0.00 69.02 13.45 37.96 -20.43 0.00 0.00 0.00 37.34 0.6110
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 137.38 -32.02 -21.99 2.08 14.55 0.00 0.00 0.00 94.59 0.9726


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9665 0.2270 -0.0480 0.2700 -0.1305

(2) 93.4155 1.0000 0.3017 0.0156 0.2404 -0.0636

(3) 5.1543 9.1028 1.0000 -0.1307 -0.6267 0.2456

(4) 0.2308 0.0243 1.7080 1.0000 0.2222 0.8374

(5) 7.2874 5.7807 39.2754 4.9361 1.0000 -0.1846

(6) 1.7027 0.4045 6.0342 70.1186 3.4068 1.0000

Numbers in Parentheses are the Variable or Parameter Number













ALL LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 Xl) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


1.319 0.000 0.000 0.000 0.000
0.000 0.042 0.000 0.000 0.000
0.000 0.000 -0.074 0.000 0.000
0.000 0.000 0.000 9.269 0.000
0.000 0.000 0.000 0.000 -0.014
1.349 -0.013 0.000 0.000 0.000
1.321 0.000 -0.098 0.000 0.000
1.306 0.000 0.000 1.369 0.000
1.313 0.000 0.000 0.000 -0.007
0.000 0.041 -0.029 0.000 0.000
0.000 0.119 0.000 23.308 0.000
0.000 0.050 0.000 0.000 -0.021
0.000 0.000 -0.176 10.135 0.000
0.000 0.000 0.317 0.000 -0.032
0.000 0.000 0.000 8.739 -0.009
1.355 -0.015 -0.115 0.000 0.000
1.368 -0.017 0.000 -1.028 0.000
1.337 -0.010 0.000 0.000 -0.006
1.302 0.000 -0.117 1.971 0.000
1.315 0.000 -0.026 0.000 -0.006
1.305 0.000 0.000 0.971 -0.007
0.000 0.120 -0.185 24.256 0.000
0.000 0.107 1.337 0.000 -0.107
0.000 0.126 0.000 23.003 -0.019
0.000 0.000 -0.354 11.796 0.013
1.361 -0.016 -0.113 -0.318 0.000
1.387 -0.024 -0.272 0.000 0.012
1.354 -0.014 0.000 -0.871 -0.006
1.302 0.000 -0.156 2.335 0.003
0.000 0.137 0.483 20.024 -0.050
1.381 -0.023 -0.283 0.393 0.013


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


1 -0.017
2 0.062
3 0.133
4 0.073
5 0.144
6 0.000
7 -0.009
8 -0.024
9 -0.007
10 0.066
11 -0.193
12 0.075
13 0.083
14 0.140
15 0.087
16 0.012
17 0.010
18 0.004
19 -0.017
20 -0.007
21 -0.012
22 -0.183
23 -0.019
24 -0.178
25 0.072
26 0.015
27 0.020
28 0.012
29 -0.019
30 -0.179
31 0.017













LABORATORY DATA MINUS PROTOTYPE LABORATORY DATA
n = 624

STEPUISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 74.04 0.8605
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.40 0.0631
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.0023
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 15.17 0.3895
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 2.93 0.1712
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 115.32 -15.32 0.00 0.00 0.00 0.00 0.00 0.00 75.57 0.8693
7 106.78 0.00 -6.78 0.00 0.00 0.00 0.00 0.00 75.70 0.8700
8 89.42 0.00 0.00 10.58 0.00 0.00 0.00 0.00 75.95 0.8715
9 109.37 0.00 0.00 0.00 -9.37 0.00 0.00 0.00 76.89 0.8768
10 0.00 93.20 6.80 0.00 0.00 0.00 0.00 0.00 0.41 0.0640
11 0.00 51.63 0.00 48.37 0.00 0.00 0.00 0.00 35.93 0.5994
12 0.00 515.34 0.00 0.00 -415.34 0.00 0.00 0.00 4.23 0.2056
13 0.00 0.00 -18.78 118.78 0.00 0.00 0.00 0.00 16.15 0.4018
14 0.00 0.00 -474.15 0.00 574.15 0.00 0.00 0.00 9.97 0.3158
15 0.00 0.00 0.00 123.26 -23.26 0.00 0.00 0.00 16.28 0.4035
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 129.26 -19.92 -9.34 0.00 0.00 0.00 0.00 0.00 77.83 0.8822
17 95.02 -3.80 0.00 8.77 0.00 0.00 0.00 0.00 75.99 0.8717
18 119.44 -10.78 0.00 0.00 -8.65 0.00 0.00 0.00 77.52 0.8805
19 94.36 0.00 -8.23 13.87 0.00 0.00 0.00 0.00 78.60 0.8866
20 109.02 0.00 2.76 0.00 -11.78 0.00 0.00 0.00 76.96 0.8772
21 98.46 0.00 0.00 9.29 -7.75 0.00 0.00 0.00 78.08 0.8836
22 0.00 53.53 -5.26 51.72 0.00 0.00 0.00 0.00 37.67 0.6137
23 0.00 115.66 110.17 0.00 -125.83 0.00 0.00 0.00 24.81 0.4981
24 0.00 58.40 0.00 50.13 -8.53 0.00 0.00 0.00 40.20 0.6340
25 0.00 0.00 -2.27 123.14 -20.87 0.00 0.00 0.00 16.28 0.4035
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 98.98 -3.03 -8.48 12.53 0.00 0.00 0.00 0.00 78.62 0.8867
27 129.97 -20.68 -10.06 0.00 0.77 0.00 0.00 0.00 77.83 0.8822
28 94.74 2.49 0.00 10.48 -7.71 0.00 0.00 0.00 78.10 0.8837
29 93.45 0.00 -9.44 14.59 1.40 0.00 0.00 0.00 78.61 0.8866
30 0.00 65.74 18.60 41.92 -26.26 0.00 0.00 0.00 42.98 0.6556
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 100.66 -5.66 -11.79 13.20 3.58 0.00 0.00 0.00 78.67 0.8870


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8605 0.0631 0.0023 0.3895 -0.1712

(2) 74.0435 1.0000 0.2136 0.1505 0.2996 -0.0030

(3) 0.3988 4.5618 1.0000 -0.1223 -0.6874 0.2708

(4) 0.0005 2.2639 1.4968 1.0000 0.2514 0.8363

6(5) 15.1705 8.9761 47.2546 6.3188 1.0000 -0.1732

(6) 2.9311 0.0009 7.3352 69.9318 3.0005 1.0000

Numbers in Parentheses are the Variable or Parameter Number













LABORATORY DATA MINUS PROTOTYPE LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 Xl) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


1.087 0.000 0.000 0.000 0.000 0.000
0.000 0.005 0.000 0.000 0.000 0.000
0.000 0.000 0.001 0.000 0.000 0.000
0.000 0.000 0.000 5.874 0.000 0.000
0.000 0.000 0.000 0.000 -0.008 0.000
1.121 -0.010 0.000 0.000 0.000 0.000
1.111 0.000 -0.084 0.000 0.000 0.000
1.032 0.000 0.000 2.182 0.000 0.000
1.086 0.000 0.000 0.000 -0.008 0.000
0.000 0.005 0.007 0.000 0.000 0.000
0.000 0.049 0.000 12.377 0.000 0.000
0.000 0.009 0.000 0.000 -0.009 0.000
0.000 0.000 -0.066 6.261 0.000 0.000
0.000 0.000 0.311 0.000 -0.026 0.000
0.000 0.000 0.000 5.594 -0.005 0.000
1.157 -0.012 -0.099 0.000 0.000 0.000
1.053 -0.003 0.000 1.737 0.000 0.000
1.109 -0.007 0.000 0.000 -0.006 0.000
1.049 0.000 -0.108 2.758 0.000 0.000
1.076 0.000 0.032 0.000 -0.009 0.000
1.042 0.000 0.000 1.758 -0.007 0.000
0.000 0.050 -0.088 13.030 0.000 0.000
0.000 0.042 0.709 0.000 -0.055 0.000
0.000 0.054 0.000 12.457 -0.010 0.000
0.000 0.000 -0.007 5.661 -0.004 0.000
1.065 -0.002 -0.108 2.412 0.000 0.000
1.160 -0.012 -0.106 0.000 0.001 0.000
1.029 0.002 0.000 2.035 -0.007 0.000
1.050 0.000 -0.126 2.931 0.001 0.000
0.000 0.061 0.304 10.356 -0.029 0.000
1.081 -0.004 -0.150 2.536 0.003 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


1 0.007
2 0.110
3 0.118
4 0.084
5 0.127
6 0.019
7 0.012
8 0.000
9 0.017
10 0.109
11 -0.028
12 0.115
13 0.088
14 0.123
15 0.092
16 0.027
17 0.005
18 0.023
19 0.004
20 0.017
21 0.010
22 -0.026
23 0.062
24 -0.024
25 0.092
26 0.008
27 0.027
28 0.007
29 0.003
30 -0.023
31 0.008













PROTOTYPE LABORATORY DATA
n = 16
STEPUISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1) (2)

100.00 0.00
0.00 100.00
0.00 0.00
0.00 0.00
0.00 0.00

147.96 -47.96
97.30 0.00
94.50 0.00
126.16 0.00
0.00 -210.52
0.00 72.22
0.00-5719.66
0.00 0.00
0.00 0.00
0.00 0.00

171.04 -57.03
467.90 -289.62
136.80 -61.52
99.62 0.00
103.05 0.00
125.71 0.00
0.00 148.93
0.00 116.36
0.00 191.25
0.00 0.00


11







31


11



-1



2

-11
11

13


26 370.85 -230.04 2
27********24321.612164
28 339.27 -242.06
29 104.11 0.00 5
30 0.00 115.95 10

31-4544.24 4134.88 249


(3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME


0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
00.00 0.00 0.00 0.00 0.00
0.00 100.00 0.00 0.00 0.00
0.00 0.00 100.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN TWO AT A
0.00 0.00 0.00 0.00 0.00
2.70 0.00 0.00 0.00 0.00
0.00 5.50 0.00 0.00 0.00
0.00 0.00 -26.16 0.00 0.00
10.52 0.00 0.00 0.00 0.00
0.00 27.78 0.00 0.00 0.00
0.00 0.00 5819.66 0.00 0.00
11.61 -11.61 0.00 0.00 0.00
80.36 0.00 19.64 0.00 0.00
0.00 16.21 83.79 0.00 0.00
INDEPENDENT VARIABLES TAKEN THREE AT
14.01 0.00 0.00 0.00 0.00
0.00 -78.28 0.00 0.00 0.00
0.00 0.00 24.72 0.00 0.00
5.81 6.19 0.00 0.00 0.00
'1.83 0.00 -24.88 0.00 0.00
0.00 0.14 -25.85 0.00 0.00
4.03 65.10 0.00 0.00 0.00
0.17 0.00 -126.54 0.00 0.00
0.00 36.60 -127.85 0.00 0.00
3.06 -17.48 -15.59 0.00 0.00
INDEPENDENT VARIABLES TAKEN FOUR AT A
3.00 -63.81 0.00 0.00 0.00
40.99 0.00******** 0.00 0.00
0.00 -52.97 55.75 0.00 0.00
2.77 -10.46 -46.42 0.00 0.00
11.04 2.99 -119.98 0.00 0.00
INDEPENDENT VARIABLES TAKEN FIVE AT A
p5.82 406.15-2392.60 0.00 0.00


CORRELATION

(1) (2) (3)


1.0000

98.0014

57.1710

45.2711

3.1360

29.1949


0.9900

1.0000

66.8677

46.5568

2.0523

24.9090


0.7561

0.8177

1.0000

39.2992

12.7513

0.0060


N MATRIX

(4)


-0.6728

-0.6823

-0.6269

1.0000

1.2939

33.0902


0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
A TIME
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
TIME
0.00
0.00
0.00
0.00
0.00
TIME
0.00


98.00
57.17
45.27
3.14
29.19

98.86
98.00
98.13
98.29
63.68
80.08
85.74
51.79
48.78
33.94

98.88
99.43
98.99
98.13
98.35
98.29
83.50
89.73
88.10
52.24

99.44
99.63
99.56
98.44
89.76

99.80


Numbers in Parentheses are the Variable or Parameter Number


0.9900
0.7561
0.6728
0.1771
0.5403

0.9943
0.9900
0.9906
0.9914
0.7980
0.8949
0.9260
0.7196
0.6984
0.5826

0.9944
0.9971
0.9949
0.9906
0.9917
0.9914
0.9138
0.9473
0.9386
0.7228

0.9972
0.9982
0.9978
0.9922
0.9474

0.9990


(5) (6)


0.1771 -0.5403

0.1433 -0.4991

-0.3571 -0.0077

0.1137 0.5752

1.0000 -0.6382

40.7358 1.0000













PROTOTYPE LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 Xl) + (B2 X2) + (83 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


1.423 0.000 0.000 0.000 0.000 0.000
0.000 0.447 0.000 0.000 0.000 0.000
0.000 0.000 -43.809 0.000 0.000 0.000
0.000 0.000 0.000 16.164 0.000 0.000
0.000 0.000 0.000 0.000 -1.744 0.000
1.613 -0.095 0.000 0.000 0.000 0.000
1.428 0.000 0.321 0.000 0.000 0.000
1.416 0.000 0.000 3.287 0.000 0.000
1.379 0.000 0.000 0.000 -0.199 0.000
0.000 0.326 -21.328 0.000 0.000 0.000
0.000 0.555 0.000 46.774 0.000 0.000
0.000 0.445 0.000 0.000 -1.725 0.000
0.000 0.000 -45.712 23.453 0.000 0.000
0.000 0.000 -35.229 0.000 -0.739 0.000
0.000 0.000 0.000 -25.840 -2.327 0.000
1.601 -0.097 -1.061 0.000 0.000 0.000
1.871 -0.211 0.000 -12.495 0.000 0.000
1.759 -0.144 0.000 0.000 0.221 0.000
1.405 0.000 -0.664 3.488 0.000 0.000
1.407 0.000 2.413 0.000 -0.236 0.000
1.379 0.000 0.000 0.060 -0.197 0.000
0.000 0.460 -15.634 44.008 0.000 0.000
0.000 0.583 24.518 0.000 -2.419 0.000
0.000 0.493 0.000 20.657 -1.257 0.000
0.000 0.000 -52.737 34.151 0.531 0.000
1.892 -0.214 0.950 -12.995 0.000 0.000
2.146 -0.353 -13.930 0.000 1.043 0.000
2.027 -0.264 0.000 -12.634 0.232 0.000
1.426 0.000 5.851 -5.717 -0.442 0.000
0.000 0.579 22.401 3.271 -2.285 0.000
2.197 -0.364 -9.768 -7.838 0.804 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


1 -0.067
2 -0.449
3 2.471
4 0.311
5 1.393
6 0.057
7 -0.084
8 -0.095
9 0.055
10 0.776
11 -1.105
12 0.477
13 2.340
14 2.472
15 1.945
16 0.114
17 0.313
18 -0.016
19 -0.062
20 -0.046
21 0.054
22 -0.168
23 -0.560
24 -0.064
25 2.280
26 0.272
27 0.464
28 0.239
29 -0.047
30 -0.556
31 0.479













ALL FIELD DATA
n = 131
STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 82.76 0.9097
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 34.48 0.5872
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 15.51 0.3938
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 16.71 0.4088
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 15.42 0.3926
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 111.40 -11.40 0.00 0.00 0.00 0.00 0.00 0.00 82.98 0.9109
7 99.24 0.00 0.76 0.00 0.00 0.00 0.00 0.00 82.77 0.9098
8 90.80 0.00 0.00 9.20 0.00 0.00 0.00 0.00 83.45 0.9135
9 100.10 0.00 0.00 0.00 -0.10 0.00 0.00 0.00 82.76 0.9097
10 0.00 114.32 -14.32 0.00 0.00 0.00 0.00 0.00 38.54 0.6208
11 0.00 64.70 0.00 35.30 0.00 0.00 0.00 0.00 74.99 0.8659
12 0.00 113.69 0.00 0.00 -13.69 0.00 0.00 0.00 41.91 0.6474
13 0.00 0.00 -76.34 176.34 0.00 0.00 0.00 0.00 27.78 0.5270
14 0.00 0.00 58.89 0.00 41.11 0.00 0.00 0.00 15.98 0.3997
15 0.00 0.00 0.00 147.65 -47.65 0.00 0.00 0.00 24.85 0.4985
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 110.45 -11.07 0.62 0.00 0.00 0.00 0.00 0.00 82.99 0.9110
17 69.36 15.77 0.00 14.87 0.00 0.00 0.00 0.00 83.65 0.9146
18 111.32 -11.46 0.00 0.00 0.14 0.00 0.00 0.00 82.98 0.9109
19 90.23 0.00 0.70 9.07 0.00 0.00 0.00 0.00 83.47 0.9136
20 98.36 0.00 5.71 0.00 -4.07 0.00 0.00 0.00 82.92 0.9106
21 90.25 0.00 0.00 9.30 0.44 0.00 0.00 0.00 83.46 0.9136
22 0.00 64.76 -0.12 35.35 0.00 0.00 0.00 0.00 74.99 0.8660
23 0.00 102.84 25.43 0.00 -28.28 0.00 0.00 0.00 44.48 0.6669
24 0.00 65.12 0.00 35.37 -0.49 0.00 0.00 0.00 75.04 0.8663
25 0.00 0.00 -117.31 177.26 40.05 0.00 0.00 0.00 28.45 0.5334
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 68.07 16.21 0.86 14.86 0.00 0.00 0.00 0.00 83.68 0.9148
27 107.26 -8.45 3.65 0.00 -2.46 0.00 0.00 0.00 83.03 0.9112
28 68.86 15.74 0.00 14.95 0.45 0.00 0.00 0.00 83.66 0.9147
29 90.33 0.00 1.23 8.87 -0.44 0.00 0.00 0.00 83.47 0.9136
30 0.00 65.34 3.82 34.02 -3.18 0.00 0.00 0.00 75.30 0.8677
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 67.08 17.16 2.54 14.58 -1.37 0.00 0.00 0.00 83.70 0.9149


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9097 0.5872 -0.3938 0.4088 -0.3926

(2) 82.7559 1.0000 0.6831 -0.4464 0.3642 -0.4292

(3) 34.4802 46.6613 1.0000 -0.3490 -0.3278 -0.2153

(4) 15.5070 19.9298 12.1827 1.0000 -0.1601 0.9353

(5) 16.7102 13.2635 10.7464 2.5616 1.0000 -0.2931

(6) 15.4164 18.4171 4.6364 87.4857 8.5882 1.0000

Numbers in Parentheses are the Variable or Parameter Number













ALL FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


1.325 0.000 0.000 0.000 0.000
0.000 0.280 0.000 0.000 0.000
0.000 0.000 -11.030 0.000 0.000
0.000 0.000 0.000 336.641 0.000
0.000 0.000 0.000 0.000 -0.364
1.389 -0.031 0.000 0.000 0.000
1.335 0.000 0.431 0.000 0.000
1.278 0.000 0.000 73.560 0.000
1.323 0.000 0.000 0.000 -0.003
0.000 0.244 -6.023 0.000 0.000
0.000 0.385 0.000 554.783 0.000
0.000 0.251 0.000 0.000 -0.258
0.000 0.000 -9.439 292.223 0.000
0.000 0.000 -5.940 0.000 -0.180
0.000 0.000 0.000 264.607 -0.276
1.396 -0.030 0.334 0.000 0.000
1.122 0.055 0.000 136.634 0.000
1.391 -0.031 0.000 0.000 0.003
1.287 0.000 0.423 73.524 0.000
1.333 0.000 3.281 0.000 -0.101
1.284 0.000 0.000 75.192 0.012
0.000 0.384 -0.139 553.529 0.000
0.000 0.288 14.042 0.000 -0.677
0.000 0.381 0.000 545.942 -0.024
0.000 0.000 -15.957 323.146 0.236
1.128 0.058 0.603 139.913 0.000
1.381 -0.023 1.991 0.000 -0.058
1.128 0.056 0.000 139.206 0.013
1.288 0.000 0.745 71.893 -0.011
0.000 0.390 4.490 536.568 -0.162
1.120 0.062 1.800 138.336 -0.042


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 .0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


NO
1 0.089
2 0.128
3 2.705
4 1.297
5 2.602
6 0.218
7 0.055
8 -0.047
9 0.094
10 0.637
11 -2.291
12 0.580
13 1.794
14 2.668
15 1.755
16 0.190
17 -0.395
18 0.212
19 -0.080
20 0.039
21 -0.073
22 -2.274
23 0.126
24 -2.210
25 1.746
26 -0.460
27 0.151
28 -0.430
29 -0.078
30 -2.307
31 -0.481












APPENDIX III


Data listings and stepwise regression results for each investigation



Notes:

n = number of data sets.

Stepwise Regression Results.

Stepwise regression results are those for equation (11) of the text.

100 r*2 = 100 r2.

The dependent variable, db, is the water depth at the shore-breaking position.

Independent variables:

(1) Shore-breaking wave height, Hb
(2) Wave period, T
(3) Bed slope leading to shore-breaking, tan ab
(4) Equivalent wave steepness, Hb/g T2
(5) Surf similarity parameter, tan cb[Hb/(g T2)]05

Correlation Matrix.

Correlation matrix treatment is given by Table 5 of the text.

Variables:

(1) db (2) Hb (3) T (4) tan ab

(5) Hb/(g T2) (6) tan ab/[Hb/(g T2)]o05






























































58











GAILLARD (1904) FIELD DATA
n= 25

Water Breaker Wave Bed Equivalent Surf
Depth Height Period SLope Uave Sim
(m) (m) (s) Steepness Parm

1.5390 0.9140 4.4200 0.0200 0.00477 0.28946
1.8140 1.0670 5.0700 0.0200 0.00424 0.30730
1.9200 1.1280 6.5300 0.0200 0.00270 0.38495
1.9350 1.2190 4.6600 0.0200 0.00573 0.26426
2.4380 1.3720 5.6300 0.0160 0.00442 0.24075
4.5420 1.6760 5.9800 0.0250 0.00478 0.36151
3.8400 1.8290 6.8000 0.0250 0.00404 0.39351
4.5570 1.9810 5.1700 0.0250 0.00756 0.28748
4.3890 2.0730 8.3300 0.0250 0.00305 0.45279
4.2210 2.1340 6.6100 0.0250 0.00498 0.35413
4.2670 2.3770 10.9800 0.0250 0.00201 0.55737
5.1970 2.4380 8.3900 0.0290 0.00353 0.48782
4.1450 2.5910 9.4400 0.0250 0.00297 0.45898
5.1050 2.7430 7.1900 0.0290 0.00541 0.39412
5.3490 2.8960 6.8400 0.0290 0.00632 0.36489
4.4200 3.0480 6.4700 0.0250 0.00743 0.29003
4.9990 3.3530 7.2300 0.0290 0.00655 0.35845
0.7920 0.6100 4.9600 0.0250 0.00253 0.49702
0.9150 0.7620 5.2500 0.0250 0.00282 0.47069
0.8230 0.7920 5.8000 0.0250 0.00240 0.51006
1.2500 0.8530 5.6000 0.0250 0.00278 0.47453
1.4790 0.9140 4.8400 0.0250 0.00398 0.39621
1.3410 1.0060 3.8500 0.0250 0.00693 0.30041
3.0480 2.2860 6.0000 0.0154 0.00648 0.19131
3.0480 2.4380 4.7500 0.0154 0.01103 0.14666











GAILLARD (1904) FIELD DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 81.80 0.9044
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 40.76 0.6384
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 18.38 0.4287
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 8.31 0.2883
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.05 0.0214
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 75.18 24.82 0.00 0.00 0.00 0.00 0.00 0.00 83.31 0.9127
7 63.93 0.00 36.07 0.00 0.00 0.00 0.00 0.00 84.61 0.9198
8 118.45 0.00 0.00 -18.45 0.00 0.00 0.00 0.00 83.47 0.9136
9 82.66 0.00 0.00 0.00 17.34 0.00 0.00 0.0X) 83.15 0.9119
10 0.00 62.89 37.11 0.00 0.00 0.00 0.00 0.00 44.47 0.6669
11 0.00 70.18 0.00 29.82 0.00 0.00 0.00 0.00 74.04 0.8604
12 0.00 214.59 0.00 0.00 -114.59 0.00 0.00 0.00 60.21 0.7759
13 0.00 0.00 76.72 23.28 0.00 0.00 0.00 0.00 35.67 0.5973
14 0.00 0.00 170.72 0.00 -70.72 0.00 0.00 0.00 35.05 0.5920
15 0.00 0.00 0.00 44.25 55.75 0.00 0.00 0.00 23.09 0.4805
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 57.37 12.87 29.76 0.00 0.00 0.00 0.00 0.00 85.23 0.9232
17 104.70 8.30 0.00 -13.00 0.00 0.00 0.00 0.00 83.49 0.9137
18 76.29 16.30 0.00 0.00 7.41 0.00 0.00 0.00 83.40 0.9132
19 73.82 0.00 32.50 -6.32 0.00 0.00 0.b0 0.00 84.95 0.9217
20 63.20 0.00 39.47 0.00 -2.67 0.00 0.00 0.00 84.63 0.9199
21 110.98 0.00 0.00 -14.98 4.00 0.00 0.00 0.00 83.48 0.9137
22 0.00 47.34 28.46 24.20 0.00 0.00 0.00 0.00 81.40 0.9022
23 0.00 80.26 104.09 0.00 -84.35 0.00 0.00 0.00 90.69 0.9523
24 0.00 65.33 0.00 30.30 4.37 0.00 0.00 0.00 74.13 0.8610
25 0.00 0.00 106.83 16.43 -23.26 0.00 0.00 0.00 36.26 0.6022
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 42.93 21.30 29.30 6.48 0.00 0.00 0.00 0.00 85.34 0.9238
27 1.80 78.23 102.05 0.00 -82.07 0.00 0.00 0.00 90.70 0.9524
28 100.21 7.38 0.00 -10.81 3.22 0.00 0.00 0.00 83.50 0.9138
29 118.29 0.00 118.97 -49.87 -87.39 0.00 0.00 0.00 86.67 0.9310
30 0.00 80.04 103.61 0.17 -83.82 0.00 0.00 0.00 90.69 0.9523
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 2.73 78.51 103.95 -1.05 -84.15 0.00 0.00 0.00 90.71 0.9524


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9044 0.6384 0.4287 0.2883 -0.0214

(2) 81.7990 1.0000 0.5968 0.2970 0.4465 -0.1508

(3) 40.7607 35.6187 1.0000 0.3942 -0.3831 0.5454

(4) 18.3769 8.8234 15.5382 1.0000 -0.2634 0.6630

(5) 8.3114 19.9390 14.6732 6.9360 1.0000 -0.8260

(6) 0.0459 2.2741 29.7449 43.9627 68.2242 1.0000

Numbers in Parentheses are the Variable or Parameter Number











GAILLARD (1904) FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 Xl) + (B2 X2) + (B3 X3) + ... + (B8 X8)

B1 B2 B3 B4 B5 B6 B7 B8


EQ BO
NO
1 0.000
2 -0.731
3 -0.925
4 2.063
5 3.216
6 -0.606
7 -1.468
8 0.296
9 -0.723
10 -2.201
11 -4.399
12 0.511
13 -3.532
14 -1.246
15 -3.791
16 -1.668
17 0.060
18 -0.718
19 -1.051
20 -1.475
21 0.125
22 -6.719
23 -3.559
24 -4.805
25 -2.665
26 -2.419
27 -3.484
28 -0.069
29 0.470
30 -3.575
31 -3.350


1.739
0.000
0.000
0.000
0.000
1.563
1.639
1.863
1.773
0.000
0.000
0.000
0.000
0.000
0.000
1.535
1.785
1.634
1.720
1.623
1.852
0.000
0.000
0.000
0.000
1.330
0.066
1.781
1.745
0.000
0.099


0.000 0.000 0.000 0.000
0.610 0.000 0.000 0.000
0.000 169.533 0.000 0.000
0.000 0.000 215.952 0.000
0.000 0.000 0.000 -0.327
0.146 0.000 0.000 0.000
0.000 69.412 0.000 0.000
0.000 0.000 -108.119 0.000
0.000 0.000 0.000 1.792
0.531 82.885 0.000 0.000
0.839 0.000 467.780 0.000
0.884 0.000 0.000 -8.017
0.000 214.433 322.918 0.000
0.000 312.564 0.000 -8.312
0.000 0.000 637.882 10.391
0.098 59.786 0.000 0.000
0.040 0.000 -82.577 0.000
0.099 0.000 0.000 0.765
0.000 56.828 -54.879 0.000
0.000 76.073 0.000 -0.331
0.000 0.000 -93.132 0.322
0.741 117.806 497.165 0.000
0.852 292.082 0.000 -15.194
0.823 0.000 501.005 0.935
0.000 259.494 198.133 -3.627
0.187 68.127 74.792 0.000
0.821 283.145 0.000 -14.618
0.037 0.000 -71.592 0.276
0.000 131.721 -274.040 -6.211
0.851 291.450 2.378 -15.136
0.807 282.550 -14.135 -14.682


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000































































62








SCRIPPS (1944a, 1944b, 1945) LEICA TYPE I FIELD DATA
n = 56


Water
Depth
(m)

2.2555
1.6154
2.3165
2.7737
2.1946
1.5545
2.6518
2.2860
2.7737
2.0117
2.0117
1.9812
2.3165
1.7374
2.2250
2.9870
3.1394
2.1946
2.1641
2.2250
2.0117
2.4994
2.9870
2.8651
2.9870
3.8710
3.2918
3.7186
2.6822
1.8288
3.0480
2.9566
2.6822
1.8288
2.4079
3.3528
2.4079
2.4689
2.5603
3.6881
1.9507
2.5603
2.7127
2.1336
3.7186
2.8042
2.7737
2.6213
3.2004
3.6881
3.4442
3.4138
3.0175
3.0175
2.4994
4.4501


Breaker Wave
Height Period
(m) (s)

2.2555 13.7000
1.4630 12.0000
1.6459 13.3000
2.2555 12.7000
1.9507 12.2000
1.2192 10.2000
2.1336 11.6000
1.7678 12.0000
2.3165 11.5000
1.7069 10.0000
2.0117 10.0000
1.7069 10.0000
2.0117 11.2000
1.4021 9.2000
1.5850 9.0000
2.0117 10.2000
2.6213 10.5000
2.0117 10.0000
1.4630 9.5000
2.1336 9.6000
1.7678 9.5000
1.9507 9.4000
2.4384 9.5000
2.1946 9.6000
2.4384 10.3000
2.7432 10.5000
2.1946 10.5000
2.6822 9.6000
2.1336 9.8000
1.4021 8.1000
2.8651 10.3000
2.0726 9.0000
2.5603 9.4000
2.0726 9.0000
1.5240 7.7000
2.9870 9.0000
1.8288 8.5000
2.3774 8.8000
2.1946 8.8000
2.4384 10.0000
1.7678 8.0000
2.0117 7.2000
2.4384 9.0000
1.8898 8.0000
2.6213 9.2000
2.3774 8.8000
2.9870 8.5000
2.1946 8.0000
2.3774 7.5000
3.4747 9.0000
3.0480 8.2000
2.7432 7.5000
2.4384 7.2000
2.8651 8.0000
2.1946 6.5000
3.3528 7.8000


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm

0.0159 0.00123 0.45374
0.0159 0.00104 0.49381
0.0159 0.00095 0.51601
0.0159 0.00143 0.42091
0.0159 0.00134 0.43478
0.0159 0.00120 0.45980
0.0159 0.00162 0.39529
0.0159 0.00125 0.44923
0.0159 0.00179 0.37609
0.0159 0.00174 0.38099
0.0159 0.00205 0.35094
0.0159 0.00174 0.38099
0.0159 0.00164 0.39305
0.0159 0.00169 0.38673
0.0159 0.00200 0.35583
0.0159 0.00197 0.35796
0.0159 0.00243 0.32281
0.0159 0.00205 0.35094
0.0159 0.00165 0.39094
0.0159 0.00236 0.32713
0.0159 0.00200 0.35564
0.0159 0.00225 0.33500
0.0159 0.00276 0.30282
0.0159 0.00243 0.32256
0.0159 0.00235 0.32832
0.0159 0.00254 0.31555
0.0159 0.00203 0.35280
0.0159 0.00297 0.29176
0.0159 0.00227 0.33395
0.0159 0.00218 0.34049
0.0159 0.00276 0.30288
0.0159 0.00261 0.31116
0.0159 0.00296 0.29241
0.0159 0.00261 0.31116
0.0159 0.00262 0.31046
0.0159 0.00376 0.25920
0.0159 0.00258 0.31286
0.0159 0.00313 0.28408
0.0159 0.00289 0.29568
0.0159 0.00249 0.31876
0.0159 0.00282 0.29949
0.0159 0.00396 0.25268
0.0159 0.00307 0.28688
0.0159 0.00301 0.28967
0.0159 0.00316 0.28284
0.0159 0.00313 0.28408
0.0159 0.00422 0.24480
0.0159 0.00350 0.26880
0.0159 0.00431 0.24211
0.0159 0.00438 0.24032
0.0159 0.00463 0.23378
0.0159 0.00498 0.22539
0.0159 0.00480 0.22950
0.0159 0.00457 0.23525
0.0159 0.00530 0.21840
0.0159 0.00562 0.21203











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE I FIELD DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 71.00 0.8426
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 3.60 0.1897
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.82 0.0906
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 34.25 0.5853
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 33.29 0.5770
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 103.21 -3.21 0.00 0.00 0.00 0.00 0.00 0.00 71.03 0.8428
7 0.34 0.00 99.66 0.00 0.00 0.00 0.00 0.00 72.05 0.8488
8 99.36 0.00 0.00 0.64 0.00 0.00 0.00 0.00 71.00 0.8426
9 104.10 0.00 0.00 0.00 -4.10 0.00 0.00 0.00 71.05 0.8429
10 0.00 -0.43 100.43 0.00 0.00 0.00 0.00 0.00 3.65 0.1911
11 0.00 62.18 0.00 37.82 0.00 0.00 0.00 0.00 54.58 0.7388
12 0.00-2930.74 0.00 0.00 3030.74 0.00 0.00 0.00 66.41 0.8149
13 0.00 0.00 101.04 -1.04 0.00 0.00 0.00 0.00 34.27 0.5854
14 0.00 0.00 99.52 0.00 0.48 0.00 0.00 0.00 33.56 0.5793
15 0.00 0.00 0.00 -327.88 427.88 0.00 0.00 0.00 35.03 0.5919
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 0.31 0.01 99.67 0.00 0.00 0.00 0.00 0.00 72.10 0.8491
17 120.82 -14.60 0.00 -6.22 0.00 0.00 0.00 0.00 71.09 0.8431
18 103.70 19.80 0.00 0.00 -23.50 0.00 0.00 0.00 71.09 0.8432
19 0.33 0.00 99.68 -0.01 0.00 0.00 0.00 0.00 72.09 0.8491
20 0.33 0.00 99.66 0.00 0.01 0.00 0.00 0.00 72.07 0.8489
21 120.05 0.00 0.00 -5.79 -14.26 0.00 0.00 0.00 71.12 0.8433
22 0.00 0.26 99.59 0.14 0.00 0.00 0.00 0.00 57.51 0.7583
23 0.00 0.47 99.99 0.00 -0.46 0.00 0.00 0.00 68.23 0.8260
24 0.00 306.91 0.00 39.15 -246.06 0.00 0.00 0.00 68.19 0.8258
25 0.00 0.00 99.89 -0.22 0.33 0.00 0.00 0.00 35.14 0.5928
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 0.31 0.01 99.67 0.00 0.00 0.00 0.00 0.00 72.10 0.8491
27 0.24 0.11 99.74 0.00 -0.10 0.00 0.00 0.00 72.28 0.8502
28 117.52 6.40 0.00 -4.92 -19.01 0.00 0.00 0.00 71.12 0.8433
29 0.34 0.00 99.69 -0.01 -0.01 0.00 0.00 0.00. 72.10 0.8491
30 0.00 0.40 99.83 0.06 -0.29 0.00 0.00 0.00 70.65 0.8406
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 0.21 0.15 99.74 0.01 -0.11 0.00 0.00 0.00 72.32 0.8504


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.8426 -0.1897 0.0906 0.5853 -0.5770

(2) 70.9951 1.0000 -0.2038 -0.0146 0.6883 -0.6643

(3) 3.5990 4.1544 1.0000 -0.3663 -0.7932 0.8516

(4) 0.8203 0.0213 13.4164 1.0000 0.1756 -0.2438

(5) 34.2544 47.3792 62.9198 3.0821 1.0000 -0.9309

(6) 33.2898 44.1235 72.5268 5.9443 86.6506 1.0000

Numbers in Parentheses are the Variable or Parameter Number











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE I FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 0.342 1.057 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 3.401 0.000 -0.077 0.000 0.000 0.000 0.000 0.000 0.000
3 -603.013 0.000 0.00038093.775 0.000 0.000 0.000 0.000 0.000
4 1.806 0.000 0.000 0.000 321.226 0.000 0.000 0.000 0.000
5 4.338 0.000 0.000 0.000 0.000 -5.096 0.000 0.000 0.000
6 0.425 1.053 -0.008 0.000 0.000 0.000 0.000 0.000 0.000
7 -687.745 1.059 0.00043276.234 0.000 0.000 0.000 0.000 0.000
8 0.347 1.049 0.000 0.000 5.527 0.000 0.000 0.000 0.000
9 0.489 1.032 0.000 0.000 0.000 -0.273 0.000 0.000 0.000
10 -159.478 0.000 -0.07310241.944 0.000 0.000 0.000 0.000 0.000
11 -1.911 0.000 0.299 0.000 643.562 0.000 0.000 0.000 0.000
12 2.815 0.000 0.443 0.000 0.000 -13.355 0.000 0.000 0.000
13 85.829 0.000 0.000-5284.744 322.437 0.000 0.000 0.000 0.000
14 360.569 0.000 0.000********* 0.000 -5.211 0.000 0.000 0.000
15 2.834 0.000 0.000 0.000 198.128 -2.128 0.000 0.000 0.000
16 -748.012 1.066 0.01047059.845 0.000 0.000 0.000 0.000 0.000
17 0.645 1.111 -0.031 0.000 -46.805 0.000 0.000 0.000 0.000
18 0.659 0.947 0.042 0.000 0.000 -1.445 0.000 0.000 0.000
19 -722.595 1.084 0.00045467.279 -15.401 0.000 0.000 0.000 0.000
20 -718.018 1.074 0.00045174.926 0.000 0.157 0.000 0.000 0.000
21 0.757 1.043 0.000 0.000 -41.139 -0.834 0.000 0.000 0.000
22-1258.331 0.000 0.34578987.835 675.525 0.000 0.000 0.000 0.000
23 -974.955 0.000 0.47761481.921 0.000 -13.675 0.000 0.000 0.000
24 1.293 0.000 0.443 0.000 200.261 -10.360 0.000 0.000 0.000
25 232.023 0.000 0.000******** 190.888 -2.310 0.000 0.000 0.000
26 -750.954 1.062 0.01147243.891 3.123 0.000 0.000 0.000 0.000
27 -796.122 0.888 0.09550109.815 0.000 -2.448 0.000 0.000 0.000
28 0.767 1.016 0.013 0.000 -34.757 -1.106 0.000 0.000 0.000
29 -706.742 1.081 0.00044477.232 -27.515 -0.225 0.000 0.000 0.000
30-1152.452 0.000 0.48372529.651 236.892 -10.190 0.000 0.000 0.000
31 -843.777 0.804 0.13253098.787 40.998 -2.908 0.000 0.000 0.000
































































66











SCRIPPS (1944a, 1944b, 1945) LEICA TYPE II FIELD DATA
n = 18

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

3.1400 1.7700 13.0000 0.0489 0.00107 1.49551
3.0200 1.9500 12.5000 0.0490 0.00127 1.37310
3.2300 2.3200 12.0000 0.0490 0.00164 1.20850
3.7200 2.2000 10.5000 0.0490 0.00204 1.08589
3.4700 2.5000 11.2000 0.0490 0.00203 1.08657
3.5400 1.8300 10.0000 0.0490 0.00187 1.13392
3.3200 1.8300 10.0000 0.0490 0.00187 1.13392
1.6800 1.2800 8.8000 0.0490 0.00169 1.19313
3.6300 2.3800 9.3000 0.0490 0.00281 0.92471
3.4400 2.5600 9.6000 0.0490 0.00283 0.92037
3.5000 2.3200 10.5000 0.0490 0.00215 1.05744
3.7200 2.1000 10.0000 0.0490 0.00214 1.05852
2.8300 2.3800 9.5000 0.0490 0.00269 0.94459
3.6000 2.5000 8.9000 0.0490 0.00322 0.86343
3.3500 2.3200 9.0000 0.0490 0.00292 0.90637
3.1700 2.2600 9.0000 0.0490 0.00285 0.91833
3.4400 2.3200 8.0000 0.0490 0.00370 0.80567
3.3200 2.7400 7.0000 0.0490 0.00571 0.64868


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 37.15 0.6095
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.16 0.0395
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.60 0.0774
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 4.41 0.2099
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 8.07 0.2841
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 72.04 27.96 0.00 0.00 0.00 0.00 0.00 0.00 43.12 0.6566
7 -1.83 0.00 101.83 0.00 0.00 0.00 0.00 0.00 38.42 0.6199
8 118.75 0.00 0.00 -18.75 0.00 0.00 0.00 0.00 44.32 0.6657
9 77.73 0.00 0.00 0.00 22.27 0.00 0.00 0.00 41.18 0.6417
10 0.00 0.28 99.72 0.00 0.00 0.00 0.00 0.00 1.42 0.1190
11 0.00 71.25 0.00 28.75 0.00 0.00 0.00 0.00 19.98 0.4470
12 0.00 710.20 0.00 0.00 -610.20 0.00 0.00 0.00 50.63 0.7116
13 0.00 0.00 97.31 2.69 0.00 0.00 0.00 0.00 4.41 0.2101
14 0.00 0.00 99.20 0.00 0.80 0.00 0.00 0.00 9.02 0.3004
15 0.00 0.00 0.00 19.81 80.19 0.00 0.00 0.00 9.62 0.3102
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 -191.45 -73.66 365.12 0.00 0.00 0.00 0.00 0.00 43.12 0.6567
17 111.95 4.28 0.00 -16.22 0.00 0.00 0.00 0.00 44.34 0.6659
18 106.52 -295.00 0.00 0.00 288.48 0.00 0.00 0.00 55.85 0.7474
19 -4.18 0.00 103.56 0.62 0.00 0.00 0.00 0.00 44.76 0.6690
20 -21.56 0.00 127.53 0.00 -5.97 0.00 0.00 0.00 41.19 0.6418
21 147.23 0.00 0.00 -30.06 -17.17 0.00 0.00 0.00 44.53 0.6673
22 0.00 1.29 98.26 0.45 0.00 0.00 0.00 0.00 24.02 0.4901
23 0.00 -4.56 100.49 0.00 4.07 0.00 0.00 0.00 51.47 0.7174
24 0.00 2292.32 0.00 -72.75-2119.57 0.00 0.00 0.00 50.90 0.7134
25 0.00 0.00 98.76 0.27 0.97 0.00 0.00 0.00 13.43 0.3665
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 -2.95 0.39 102.01 0.56 0.00 0.00 0.00 0.00 44.90 0.6701
27 1.94 -4.39 97.89 0.00 4.56 0.00 0.00 0.00 62.77 0.7923
28 153.37 -312.01 0.00 -19.55 278.18 0.00 0.00 0.00 63.31 0.7957
29 -1.44 0.00 100.39 0.44 0.62 0.00 0.00 0.00 46.23 0.6799
30 0.00 -2.38 99.53 0.21 2.64 0.00 0.00 0.00 52.91 0.7274
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 4.23 -8.35 96.67 -0.41 7.87 0.00 0.00 0.00 68.25 0.8262












SCRIPPS (1944a, 1944b, 1945) LEICA TYPE II FIELD DATA (CONT)

CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000

(2) 37.1489

(3) 0.1558

(4) 0.5991

(5) 4.4058


0.6095 0.0395 0.0774 0.2099 -0.2841

1.0000 -0.3156 0.3034 0.6704 -0.7010

9.9620 1.0000 -0.5011 -0.8380 0.8933

9.2059 25.1075 1.0000 0.3296 -0.5570

44.9444 70.2181 10.8623 1.0000 -0.9152


(6) 8.0738 49.1456 79.7898 31.0300 83.7589 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (83 X3) + ... + (B8 X8)


EQ BO B1
NO
1 1.511 0.807
2 3.165 0.000
3 -64.835 0.000
4 3.057 0.000
5 3.963 0.000
6 0.495 0.914
7 105.636 0.854
8 1.199 1.127
9 0.265 1.068
10 -111.231 0.000
11 0.210 0.000
12 2.666 0.000
13 -5.059 0.000
14 107.326 0.000
15 4.973 0.000
16 4.335 0.915
17 1.108 1.109
18 6.813 -1.710
19 63.215 1.141
20 14.133 1.064
21 1.639 1.102
22 -208.801 0.000
23 99.561 0.000
24 3.094 0.000
25 251.729 0.000
26 93.815 1.216
27 345.981 -3.029
28 10.482 -5.023
29 167.566 1.075
30 183.054 0.000
31 300.532 -5.718


B2 B3 B4 B5 B6 B7 B8


0.000 0.000 0.000 0.000
0.012 0.000 0.000 0.000
0.000 1390.374 0.000 0.000
0.000 0.000 91.997 0.000
0.000 0.000 0.000 -0.652
0.079 0.000 0.000 0.000
0.000-2127.387 0.000 0.000
0.000 0.000 -158.193 0.000
0.000 0.000 0.000 0.645
0.032 2330.876 0.000 0.000
0.220 0.000 357.579 0.000
0.442 0.000 0.000 -3.624
0.000 165.712 90.665 0.000
0.000-2106.509 0.000 -0.801
0.000 0.000 -135.337 -1.300
0.078 -78.299 0.000 0.000
0.009 0.000 -142.866 0.000
1.048 0.000 0.000 -9.768
0.000-1266.776 -151.109 0.000
0.000 -282.383 0.000 0.621
0.000 0.000 -200.088 -0.271
0.275 4253.431 389.047 0.000
0.442-1974.688 0.000 -3.760
0.439 0.000 -55.926 -3.865
0.000-5008.225 -270.941 -2.305
-0.035-1884.851 -205.268 0.000
1.513-6846.899 0.000 -14.980
2.261 0.000 569.129 -19.212
0.000-3366.040 -289.633 -0.972
0.430-3651.808 -156.277 -4.551
2.500-5865.058 494.481 -22.438


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000











BALSILLIE AND CARTER (1980) FIELD DATA
n= 30

Water Breaker Wave Bed Equivalent Surf
Depth Height Period SLope Wave Sim
(m) (m) (s) Steepness Parm

0.3750 0.2560 3.5000 0.0820 0.00213 1.77572
0.1600 0.1760 4.8300 0.1013 0.00077 3.65101
0.7580 0.4580 4.1900 0.0824 0.00266 1.59706
0.3700 0.1740 3.5300 0.0393 0.00142 1.04113
0.3500 0.2720 5.1400 0.0172 0.00105 0.53066
0.2000 0.1540 3.5300 0.1080 0.00126 3.04124
0.4000 0.3510 3.2700 0.0630 0.00335 1.08855
0.3000 0.2490 4.2900 0.0534 0.00138 1.43718
0.4500 0.3760 6.2100 0.4615 0.00099 14.63128
0.2600 0.2050 2.5700 0.1689 0.00317 3.00123
0.5400 0.4460 5.2900 0.0210 0.00163 0.52074
0.5400 0.3280 5.2900 0.0210 0.00120 0.60723
0.2000 0.1040 1.5100 0.0246 0.00465 0.36059
0.4300 0.2080 5.6300 0.0480 0.00067 1.85494
0.1600 0.1120 2.6500 0.1016 0.00163 2.51851
0.1100 0.0570 1.5700 0.0874 0.00236 1.79923
0.6300 0.5410 7.3000 0.1125 0.00104 3.49534
0.4800 0.3940 6.4300 0.0557 0.00097 1.78620
0.1800 0.1750 8.5700 0.0951 0.00024 6.09895
0.1700 0.0990 2.6700 0.0951 0.00142 2.52631
0.2100 0.1590 2.3200 0.1118 0.00301 2.03631
0.4400 0.2730 7.3200 0.0525 0.00052 2.30251
0.1800 0.1150 2.1100 0.0526 0.00264 1.02455
0.1300 0.0790 1.5800 0.0364 0.00323 0.64056
0.5600 0.4770 4.1900 0.0507 0.00277 0.96289
0.4300 0.3370 3.7500 0.0213 0.00245 0.43073
0.3300 0.3030 3.0000 0.1842 0.00344 3.14270
0.2600 0.1740 2.1700 0.0492 0.00377 0.80124
0.1600 0.0970 1.3300 0.1818 0.00560 2.43037
0.3200 0.3240 3.9100 0.0570 0.00216 1.22572









BALSILLIE AND CARTER (1980) FIELD DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 83.8' 0.9156
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 28.89 0.5375
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.19 0.0434
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 6.20 0.2489
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.0016
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 99.41 0.59 0.00 0.00 0.00 0.00 0.00 0.00 83.84 0.9156
7 106.14 0.00 -6.14 0.00 0.00 0.00 0.00 0.00 84.76 0.9207
8 102.18 0.00 0.00 -2.18 0.00 0.00 0.00 0.00 83.88 0.9159
9 104.89 0.00 0.00 0.00 -4.89 0.00 0.00 0.00 84.73 0.9205
10 0.00 108.55 -8.55 0.00 0.00 0.00 0.00 0.00 29.62 0.5443
11 0.00 71.53 0.00 28.47 0.00 0.00 0.00 0.00 36.60 0.6050
12 0.00 118.33 0.00 0.00 -18.33 0.00 0.00 0.00 33.70 0.5805
13 0.00 0.00 9.06 90.94 0.00 0.00 0.00 0.00 6.35 0.2520
14 0.00 0.00 411.01 0.00 -311.01 0.00 0.00 0.00 1.21 0.1101
15 0.00 0.00 0.00 86.79 13.21 0.00 0.00 0.00 6.78 0.2604
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 104.66 1.46 -6.13 0.00 0.00 0.00 0.00 0.00 84.77 0.9207
17 112.34 -6.50 0.00 -5.84 0.00 0.00 0.00 0.00 83.94 0.9162
18 98.35 7.38 0.00 0.00 -5.73 0.00 0.00 0.00 84.93 0.9216
19 108.21 0.00 -6.25 -1.97 0.00 0.00 0.00 0.00 84.80 0.9208
20 105.89 0.00 -3.85 0.00 -2.04 0.00 0.00 0.00 84.79 0.9208
21 111.93 0.00 0.00 -5.82 -6.12 0.00 0.00 0.00 84.98 0.9218
22 0.00 74.63 -4.99 30.36 0.00 0.00 0.00 0.00 38.09 0.6172
23 0.00 98.53 67.58 0.00 -66.11 0.00 0.00 0.00 52.61 0.7253
24 0.00 79.25 0.00 28.61 -7.86 0.00 0.00 0.00 41.39 0.6434
25 0.00 0.00 -71.15 102.03 69.12 0.00 0.00 0.00 8.38 0.2895
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 112.71 -2.81 -6.36 -3.54 0.00 0.00 0.00 0.00 84.80 0.9209
27 92.83 12.01 5.46 0.00 -10.30 0.00 0.00 0.00 84.97 0.9218
28 108.11 2.43 0.00 -4.44 -6.11 0.00 0.00 0.00 84.98 0.9219
29 113.92 0.00 5.01 -8.53 -10.40 0.00 0.00 0.00 85.02 0.9220
30 0.00 89.71 51.29 9.96 -50.95 0.00 0.00 0.00 53.47 0.7312
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 102.65 8.03 8.44 -5.84 -13.28 0.00 0.00 0.00 85.07 0.9223


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9156 0.5375 -0.0434 -0.2489 0.0016

(2) 83.8381 1.0000 0.5835 0.0574 -0.2492 0.1041

(3) 28.8871 34.0415 1.0000 0.0784 -0.7840 0.3804

(4) 0.1881 0.3295 0.6153 1.0000 0.0167 0.9133

'(5) 6.1974 6.2107 61.4635 0.0281 1.0000 -0.2992

(6) 0.0003 1.0838 14.4731 83.4090 8.9512 1.0000

Numbers in Parentheses are the Variable or Parameter Number






0








BALSILLIE AND CARTER (1980) FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8


1 0.049
2 0.149
3 0.344
4 0.405
5 0.336
6 0.048
7 0.064
8 0.057
9 0.060
10 0.162
11 -0.096
12 0.151
13 0.411
14 0.348
15 0.423
16 0.062
17 0.077
18 0.052
19 0.071
20 0.063
21 0.080
22 -0.092
23 -0.055
24 -0.093
25 0.450
26 0.079
27 0.041
28 0.073
29 0.085
30 -0.124
31 0.066


1.152 0.000 0.000 0.000 0.000
0.000 0.047 0.000 0.000 0.000
0.000 0.000 -0.086 0.000 0.000
0.000 0.000 0.000 -32.304 0.000
0.000 0.000 0.000 0.000 0.000
1.148 0.000 0.000 0.000 0.000
1.159 0.000 -0.190 0.000 0.000
1.145 0.000 0.000 -2.872 0.000
1.164 0.000 0.000 0.000 -0.006
0.000 0.047 -0.170 0.000 0.000
0.000 0.078 0.000 58.060 0.000
0.000 0.055 0.000 0.000 -0.015
0.000 0.000 -0.077 -32.219 0.000
0.000 0.000 -0.533 0.000 0.016
0.000 0.000 0.000 -35.411 -0.005
1.150 0.001 -0.191 0.000 0.000
1.170 -0.004 0.000 -7.154 0.000
1.123 0.005 0.000 0.000 -0.007
1.153 0.000 -0.189 -2.461 0.000
1.161 0.000 -0.120 0.000 -0.002
1.149 0.000 0.000 -7.018 -0.007
0.000 0.080 -0.243 61.308 0.000
0.000 0.097 3.008 0.000 -0.114
0.000 0.085 0.000 57.986 -0.015
0.000 0.000 0.919 -54.676 -0.034
1.163 -0.002 -0.186 -4.299 0.000
1.098 0.009 0.183 0.000 -0.013
1.140 0.002 0.000 -5.499 -0.007
1.146 0.000 0.143 -10.089 -0.011
0.000 0.104 2.688 21.640 -0.103
1.112 0.005 0.259 -7.431 -0.016


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000. 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000































































72









WEISHAR (1976) ORIGINAL FIELD DATA
n = 111

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

4.9000 2.6000 5.0300 0.0000 0.01049 0.00000
2.3000 2.0000 11.0300 0.0700 0.00168 1.70912
3.8000 2.6000 9.1800 0.0500 0.00315 0.89113
2.6000 2.5000 6.6200 0.0500 0.00582 0.65535
3.3000 2.9000 11.1700 0.0600 0.00237 1.23202
4.3000 4.2000 7.3400 0.0000 0.00795 0.00000
1.4000 1.6000 8.1400 0.0800 0.00246 1.61164
2.6000 2.7000 5.7500 0.0500 0.00833 0.54773
4.1000 5.1000 8.3400 0.0100 0.00748 0.11561
2.3000 1.4000 8.3700 0.0300 0.00204 0.66435
2.8000 2.7000 5.9100 0.0600 0.00789 0.67557
4.4000 4.0000 3.7500 0.0100 0.02902 0.05870
4.4000 1.6000 8.5600 0.0400 0.00223 0.84740
2.7000 2.5000 5.9000 0.0500 0.00733 0.58407
4.4000 4.0000 3.9100 0.0100 0.02670 0.06120
4.4000 3.4000 7.4300 0.0100 0.00628 0.12614
4.2000 4.1000 5.7400 0.0000 0.01270 0.00000
4.6000 2.4000 6.7800 0.0500 0.00533 0.68503
4.3000 2.0000 7.6600 0.0500 0.00348 0.84781
3.3000 2.4000 11.9700 0.0800 0.00171 1.93505
2.6000 3.1000 7.8300 0.0500 0.00516 0.69609
4.1000 4.0000 3.6700 0.0100 0.03030 0.05744
4.8000 2.0000 4.7800 0.0300 0.00893 0.31743
3.8000 2.8000 9.1800 0.0500 0.00339 0.85871
3.3000 3.1000 7.8200 0.0800 0.00517 1.11232
4.2000 3.2000 5.4300 0.0200 0.01107 0.19005
4.4000 3.3000 6.0600 0.0400 0.00917 0.41772
4.4000 4.2000 9.0200 0.0300 0.00527 0.41335
2.4000 1.5000 1.2000 0.0400 0.10629 0.12269
4.3000 3.4000 5.1800 0.0000 0.01293 0.00000
3.1000 2.7000 9.0200 0.0500 0.00339 0.85923
2.8000 3.7000 7.9800 0.0600 0.00593 0.77923
3.2000 3.9000 6.3100 0.0500 0.00999 0.50013
2.3000 1.0000 7.0200 0.0700 0.00207 1.53833
1.3000 0.4000 3.2700 0.0800 0.00382 1.29485
2.5000 3.2000 2.1500 0.0500 0.07064 0.18813
2.1000 1.0000 9.3800 0.0700 0.00116 2.05548
1.6000 1.0000 4.9500 0.0700 0.00416 1.08472
3.9000 3.9000 3.4300 0.0700 0.03383 0.38060
2.5000 0.9000 12.9500 0.0500 0.00055 2.13664
4.1000 3.5000 6.3000 0.0100 0.00900 0.10542
2.8000 2.0000 8.3800 0.0330 0.00291 0.61215
2.3000 2.1000 6.1400 0.0300 0.00568 0.39792
2.3000 2.0000 5.5100 0.0300 0.00672 0.36591
3.3000 2.0000 10.3000 0.0800 0.00192 1.82400
2.3000 1.3000 8.6100 0.0800 0.00179 1.89119
3.3000 2.0000 3.6000 0.0800 0.01575 0.63752
4.1000 3.2000 4.4000 0.0000 0.01687 0.00000
2.3000 1.1000 8.0600 0.0200 0.00173 0.48115
2.7000 2.4000 6.2200 0.0500 0.00633 0.62845
1.3000 0.7000 8.2200 0.0800 0.00106 2.46051
4.3000 2.5000 2.7200 0.0000 0.03448 0.00000
4.8000 3.0000 9.6500 0.0500 0.00329 0.87207
3.2000 1.1000 8.3800 0.0500 0.00160 1.25064
4.4000 2.2000 5.9100 0.0300 0.00643 0.37421
2.4000 1.2000 7.3800 0.0300 0.00225 0.63270
2.6000 1.5000 8.8600 0.0500 0.00195 1.13233
4.4000 1.0000 3.2700 0.0100 0.00954 0.10237
4.3000 3.2000 9.5000 0.0000 0.00362 0.00000
2.8000 1.0000 13.6500 0.0600 0.00055 2.56388
2.8000 2.1000 10.9300 0.0600 0.00179 1.41669
2.5000 2.0000 4.9500 0.0500 0.00833 0.54786
2.3000 0.9000 6.5400 0.0300 0.00215 0.64743
4.3000 3.8000 5.9800 0.0000 .0.01084 0.00000
4.4000 2.7000 8.3000 0.0200 0.00400 0.31626
4.2000 3.3000 9.3400 0.0200 0.00386 0.32191









WEISHAR (1976) ORIGINAL FIELD DATA (CONT)


Water
Depth
(m)

4.4000
6.1000
3.3000
3.2000
3.3000
4.4000
0.8000
2.8000
4.5000
2.3000
2.4000
2.3000
1.0000
2.5000
3.9000
4.6000
3.3000
3.2000
2.4000
3.2000
2.5000
2.7000
2.8000
4.4000
4.3000
2.9000
2.5000
2.8000
2.3000
2.6000
4.0000
4.4000
3.8000
3.8000
3.8000
4.2000
2.1000
4.1000
4.7000
2.3000
2.4000
2.8000
1.4000
3.1000
2.7000


Breaker Wave
Height Period
(m) (s)

3.6000 4.3900
2.0000 6.3000
1.1000 7.1800
3.1000 7.1900
2.3000 5.1000
3.0000 7.1100
0.8000 9.8900
2.5000 6.1500
2.5000 7.1800
0.6000 7.9800
2.0000 5.3400
2.8000 7.5000
2.5000 11.0900
2.9000 7.9000
3.6000 12.5300
2.4000 7.1000
1.4000 12.2900
1.8000 11.9700
1.9000 15.0000
2.8000 11.0200
1.8000 9.7300
1.0000 7.5000
2.2000 7.6600
3.9000 5.8300
3.0000 5.5100
2.8000 11.4900
2.4000 6.7000
2.6000 11.9100
1.2000 7.6600
2.6000 3.6700
3.5000 6.6300
3.9000 7.2600
3.1000 8.8600
2.9000 7.0200
3.4000 7.0200
2.9000 7.0200
2.0000 13.0000
3.5000 4.9500
3.1000 8.1400
2.0000 4.7900
1.4000 6.5400
1.6000 2.4000
1.3000 7.3400
3.2000 8.3800
2.5000 7.7400


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm

0.0100 0.01906 0.07243
0.0200 0.00514 0.27891
0.0800 0.00218 1.71447
0.0500 0.00612 0.63919
0.0600 0.00902 0.63164
0.0100 0.00606 0.12851
0.1100 0.00083 3.80765
0.0600 0.00674 0.73058
0.0100 0.00495 0.14216
0.0700 0.00096 2.25756
0.0300 0.00716 0.35462
0.0700 0.00508 0.98219
0.0900 0.00207 1.97614
0.0400 0.00474 0.58090
0.0700 0.00234 1.44714
0.0500 0.00486 0.71736
0.0800 0.00095 2.60130
0.0500 0.00128 1.39650
0.0300 0.00086 1.02199
0.0500 0.00235 1.03083
0.0400 0.00194 0.90813
0.0300 0.00181 0.70436
0.0600 0.00383 0.97002
0.0200 0.01171 0.18483
0.0500 0.01008 0.49794
0.0600 0.00216 1.28975
0.0400 0.00546 0.54155
0.0300 0.00187 0.69368
0.0700 0.00209 1.53232
0.0500 0.01970 0.35626
0.0600 0.00812 0.66565
0.0100 0.00755 0.11508
0.0500 0.00403 0.78765
0.0500 0.00600 0.64524
0.0600 0.00704 0.71509
0.0100 0.00600 0.12905
0.0700 0.00121 2.01437
0.0100 0.01458 0.08283
0.0500 0.00477 0.72365
0.0700 0.00889 0.74222
0.0300 0.00334 0.51910
0.0600 0.02834 0.35638
0.0800 0.00246 1.61223
0.0500 0.00465 0.73325
0.0500 0.00426 0.76622








WEISHAR (1976) ORIGINAL FIELD DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS


100 r*2 r


(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 124.11 -24.11 0.00 0.00 0.00 0.00 0.00 0.00
7 473.98 0.00 -373.98 0.00 0.00 0.00 0.00 0.00
8 101.02 0.00 0.00 -1.02 0.00 0.00 0.00 0.00
9 183.17 0.00 0.00 0.00 -83.17 0.00 0.00 0.00
10 0.00 -0.64 100.64 0.00 0.00 0.00 0.00 0.00
11 0.00 97.42 0.00 2.58 0.00 0.00 0.00 0.00
12 0.00-1128.88 0.00 0.00 1228.88 0.00 0.00 0.00
13 0.00 0.00 98.66 1.34 0.00 0.00 0.00 0.00
14 0.00 0.00 70.13 0.00 29.87 0.00 0.00 0.00
15 0.00 0.00 0.00 11.38 88.62 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 433.17 10.47 -343.64 0.00 0.00 0.00 0.00 0.00
17 163.46 -54.50 0.00 -8.96 0.00 0.00 0.00 0.00
18 112.13 72.32 0.00 0.00 -84.45 0.00 0.00 0.00
19 587.31 0.00 -460.81 -26.50 0.00 0.00 0.00 0.00
20 504.32 0.00 -389.98 0.00 -14.34 0.00 0.00 0.00
21 270.42 0.00 0.00 -26.31 -144.10 0.00 0.00 0.00
22 0.00 4.33 93.84 1.83 0.00 0.00 0.00 0.00
23 0.00 -107.16 102.08 0.00 105.08 0.00 0.00 0.00
24 0.00 -464.02 0.00 15.00 549.03 0.00 0.00 0.00
25 0.00 0.00 59.54 5.86 34.60 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 3117.43 -460.41-2358.53 -198.50 0.00 0.00 0.00 0.00
27 377.67 42.75 -287.18 0.00 -33.24 0.00 0.00 0.00
28 156.70 63.94 0.00 -10.96 -109.67 0.00 0.00 0.00
29 892.93 0.00 -648.92 -52.90 -91.12 0.00 0.00 0.00
30 0.00 -81.35 87.31 3.92 90.13 0.00 0.00 0.00
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 1883.11 -162.99-1378.89 -121.95 -119.29 0.00 0.00 0.00


CORRELATION

(1) (2) (3)


MATRIX

(4)


35.28 0.5940


3.66
38.43
1.01
35.10

36.26
50.47
35.34
44.20
38.43
3.69
40.24
38.48
40.90
37.08

50.48
37.30
45.99
50.96
50.49
45.94
38.50
42.88
40.34
41.75

51.13
50.55
46.51
51.14
42.97

51.17


0.1912
0.6199
0.1003
0.5924

0.6021
0.7104
0.5945
0.6648
0.6199
0.1922
0.6344
0.6204
0.6395
0.6090

0.7105
0.6107
0.6781
0.7139
0.7105
0.6778
0.6205
0.6548
0.6351
0.6462

0.7150
0.7110
0.6820
0.7151
0.6555

0.7153


(5) (6)


(1) 1.0000 0.5940

(2) 35.2841 1.0000

(3) 3.6553 2.4905

(4) 38.4305 21.3193

(5) 1.0070 4.3098

(6) 35.0969 35.0774

Numbers in Parentheses


-0.1912 -0.6199

-0.1578 -0.4617

1.0000 0.3120

9.7328 1.0000

36.6888 3.9424

38.7351 64.8794

are the Variable or


0.1003

0.2076

-0.6057

-0.1986

1.0000

15.0995

Parameter


-0.5924

-0.5923

0.6224

0.8055

-0.3886

1.0000

Number











WEISHAR (1976) ORIGINAL FIELD DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 Xl) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 1.767 0.611 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 3.808 0.000 -0.073 0.000 0.000 0.000 0.000 0.000 0.000
3 4.365 0.000 0.000 -24.983 0.000 0.000 0.000 0.000 0.000
4 3.202 0.000 0.000 0.000 7.703 0.000 0.000 0.000 0.000
5 3.952 0.000 0.000 0.000 0.000 -0.859 0.000 0.000 0.000
6 2.090 0.595 -0.038 0.000 0.000 0.000 0.000 0.000 0.000
7 3.058 0.403 0.000 -17.704 0.000 0.000 0.000 0.000 0.000
8 1.769 0.616 0.000 0.000 -1.842 0.000 0.000 0.000 0.000
9 2.750 0.385 0.000 0.000 0.000 -0.537 0.000 0.000 0.000
10 4.359 0.000 0.001 -25.014 0.000 0.000 0.000 0.000 0.000
11 3.865 0.000 -0.079 0.000 -1.874 0.000 0.000 0.000 0.000
12 3.339 0.000 0.111 0.000 0.000 -1.120 0.000 0.000 0.000
13 4.388 0.000 0.000 -25.172 -1.817 0.000 0.000 0.000 0.000
14 4.293 0.000 0.000 -16.379 0.000 -0.384 0.000 0.000 0.000
15 4.117 0.000 0.000 0.000 -11.740 -0.945 0.000 0.000 0.000
16 3.038 0.403 0.003 -17.807 0.000 0.000 0.000 0.000 0.000
17 2.350 0.610 -0.067 0.000 -9.928 0.000 0.000 0.000 0.000
18 2.555 0.325 0.069 0.000 0.000 -0.751 0.000 0.000 0.000
19 3.093 0.413 0.000 -18.084 -5.539 0.000 0.000 0.000 0.000
20 3.073 0.396 0.000 -17.055 0.000 -0.035 0.000 0.000 0.000
21 2.921 0.380 0.000 0.000 -10.987 -0.622 0.000 0.000 0.000
22 4.439 0.000 -0.007 -25.028 -2.596 0.000 0.000 0.000 0.000
23 3.787 0.000 0.076 -12.099 0.000 -0.686 0.000 0.000 0.000
24 3.429 0.000 0.102 0.000 -2.962 -1.122 0.000 0.000 0.000
25 4.376 0.000 0.000 -15.030 -7.881 -0.481 0.000 0.000 0.000
26 3.233 0.417 -0.020 -17.592 -7.886 0.000 0.000 0.000 0.000
27 3.017 0.383 0.014 -16.219 0.000 -0.103 0.000 0.000 0.000
28 2.731 0.342 0.046 0.000 -7.097 -0.735 0.000 0.000 0.000
29 3.158 0.392 0.000 -15.869 -6.890 -0.123 0.000 0.000 0.000
30 3.875 0.000 0.067 -12.089 -2.889 -0.688 0.000 0.000 0.000
31 3.213 0.402 -0.012 -16.397 -7.724 -0.078 0.000 0.000 0.000











MUNK (1949) BEACH EROSION BOARD LABORATORY DATA
n = 37


Water Breaker Wave
Depth Height Period
(m) (m) (s)


0.0610
0.0790
0.0460
0.0470
0.0810
0.0520
0.0510
0.0630
0.0430
0.0550
0.0430
0.0450
0.0810
0.0670
0.1050
0.0880
0.0540
0.1390
0.0980
0.0550
0.0630
0.0640
0.1870
0.1130
0.0480
0.0580
0.0440
0.1020
0.0820
0.0820
0.0440
0.0530
0.1020
0.0580
0.1110
0.1260
0.1700


0.0430
0.0540
0.0330
0.0340
0.0510
0.0380
0.0330
0.0410
0.0310
0.0650
0.0510
0.0430
0.0660
0.0620
0.0840
0.0690
0.0440
0.1000
0.0790
0.0440
0.0470
0.0490
0.1300
0.0900
0.0350
0.0520
0.0430
0.0640
0.0510
0.0550
0.0340
0.0390
0.0800
0.0490
0.0940
0.0950
0.1210


Bed Equivalent Surf
Slope Wave Sim
Steepness Parm


1.0300 0.0300
1.0300 0.0300
0.8500 0.0300
1.0300 0.0300
1.0300 0.0300
0.8500 0.0300
0.7500 0.0300
0.8500 0.0300
0.7500 0.0300
1.0800 0.0490
1.0800 0.0490
0.9600 0.0490
1.0800 0.0490
0.9700 0.0490
1.0800 0.0490
0.9500 0.0490
0.7300 0.0490
1.0800 0.0490
0.9700 0.0490
0.7500 0.0490
0.7400 0.0490
0.7500 0.0490
1.0800 0.0490
0.9700 0.0490
0.9700 0.1590
1.0800 0.1590
1.0800 0.1590
1.0800 0.1590
1.0800 0.1590
0.9700 0.1590
0.7500 0.1590
0.7400 0.1590
0.9600 0.1590
0.7400 0.1590
1.0900 0.1590
0.9700 0.1590
1.0900 0.1590


0.00414 0.46648
0.00519 0.41627
0.00466 0.43944
0.00327 0.52460
0.00491 0.42834
0.00537 0.40951
0.00599 0.38774
0.00579 0.39424
0.00562 0.40005
0.00569 0.64979
0.00446 0.73358
0.00476 0.71014
0.00577 0.64485
0.00672 0.59756
0.00735 0.57160
0.00780 0.55476
0.00843 0.53383
0.00875 0.52388
0.00857 0.52938
0.00798 0.54846
0.00876 0.52359
0.00889 0.51972
0.01137 0.45947
0.00976 0.49597
0.00380 2.58076
0.00455 2.35739
0.00376 2.59239
0.00560 2.12493
0.00446 2.38039
0.00596 2.05874
0.00617 2.02457
0.00727 1.86513
0.00886 1.68942
0.00913 1.66396
0.00807 1.76959
0.01030 1.56646
0.01039 1.55971











MUNK (1949) BEACH EROSION BOARD LABORATORY DATA (CONT)


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 91.27 0.9554
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 22.92 0.4787
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 2.50 0.1580
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 48.59 0.6971
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.0039
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 117.29 -17.29 0.00 0.00 0.00 0.00 0.00 0.00 91.44 0.9562
7 99.21 0.00 0.79 0.00 0.00 0.00 0.00 0.00 91.29 0.9554
8 96.98 0.00 0.00 3.02 0.00 0.00 0.00 0.00 91.30 0.9555
9 99.36 0.00 0.00 0.00 0.64 0.00 0.00 0.QO 91.28 0.9554
10 0.00 96.05 3.95 0.00 0.00 0.00 0.00 0.00 23.77 0.4875
11 0.00 63.49 0.00 36.51 0.00 0.00 0.00 0.00 86.28 0.9289
12 0.00 104.22 0.00 0.00 -4.22 0.00 0.00 0.00 23.96 0.4894
13 0.00 0.00 5.93 94.07 0.00 0.00 0.00 0.00 49.46 0.7033
14 0.00 0.00 1084.88 0.00 -984.88 0.00 0.00 0.00 30.29 0.5503
15 0.00 0.00 0.00 93.95 6.05 0.00 0.00 0.00 49.66 0,7047
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 116.53 -17.64 1.12 0.00 0.00 0.00 0.00 0.00 91.46 0.9564
17 -382.66 322.71 0.00 159.94 0.00 0.00 0.00 0.00 92.24 0.9604
18 118.12 -19.66 0.00 0.00 1.54 0.00 0.00 0.00 91.48 0.9565
19 96.13 0.00 0.80 3.06 0.00 0.00 0.00 0.00 91.31 0.9556
20 99.16 0.00 2.10 0.00 -1.26 0.00 0.00 0.00 91.29 0.9555
21 95.59 0.00 0.00 3.55 0.85 0.00 0.00 0.00 91.32 0.9556
22 0.00 63.57 -0.10 36.53 0.00 0.00 0.00 0.00 86.28 0.9289
23 0.00 97.11 72.18 0.00 -69.29 0.00 0.00 0.00 66.15 0.8133
24 0.00 63.63 0.00 36.49 -0.12 0.00 0.00 0.00 86.28 0.9289
25 0.00 0.00 -17.18 95.45 21.73 0.00 0.00 0.00 49.92 0.7065
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 -339.53 296.59 -4.10 147.04 0.00 0.00 0.00 0.00 92.33 0.9609
27 138.77 -39.63 -17.20 0.00 18.05 0.00 0.00 0.00 91.61 0.9571
28 -334.13 294.85 0.00 143.66 -4.38 0.00 0.00 0.00 92.36 0.9610
29 94.31 0.00 -2.71 5.01 3.39 0.00 0.00 0.00 91.32 0.9556
30 0.00 63.93 0.91 36.14 -0.99 0.00 0.00 0.00 86.29 0.9289
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 -335.01 300.86 16.56 137.62 -20.03 0.00 0.00 0.00 92.40 0.9613


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9554 0.4787 0.1580 0.6971 -0.0039

(2) 91.2725 1.0000 0.5371 0.1528 0.7185 -0.0151

(3) 22.9185 28.8462 1.0000 0.1394 -0.1796 0.2003

(4) 2.4970 2.3336 1.9432 1.0000 0.0935 0.9559

(5) 48.5907 51.6187 3.2250 0.8745 1.0000 -0.1520

(6) 0.0015 0.0229 4.0124 91.3657 2.3106 1.0000

Numbers in Parentheses are the Variable or Parameter Number











HUNK (1949) BEACH EROSION BOARD LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)


BO B1 B2 B3 B4 B5 B6 B7 B8


1.350 0.000 0.000 0.000 0.000 0.000
0.000 0.127 0.000 0.000 0.000 0.000
0.000 0.000 0.098 0.000 0.000 0.000
0.000 0.000 0.000 11.550 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000
1.387 -0.013 0.000 0.000 0.000 0.000
1.348 0.000 0.008 0.000 0.000 0.000
1.328 0.000 0.000 0.366 0.000 0.000
1.351 0.000 0.000 0.000 0.000 0.000
0.000 0.123 0.058 0.000 0.000 0.000
0.000 0.165 0.000 13.407 0.000 0.000
0.000 0.132 0.000 0.000 -0.005 0.000
0.000 0.000 0.058 11.405 0.000 0.000
0.000 0.000 1.163 0.000 -0.084 0.000
0.000 0.000 0.000 11.813 0.005 0.000
1.385 -0.013 0.009 0.000 0.000 0.000
2.169 -0.115 0.000 -8.008 0.000 0.000
1.392 -0.015 0.000 0.000 0.001 0.000
1.325 0.000 0.008 0.373 0.000 0.000
1.343 0.000 0.020 0.000 -0.001 0.000
1.324 0.000 0.000 0.435 0.001 0.000
0.000 0.165 -0.003 13.417 0.000 0.000
0.000 0.165 1.395 0.000 -0.106 0.000
0.000 0.166 0.000 13.394 0.000 0.000
0.000 0.000 -0.190 13.052 0.019 0.000
2.207 -0.121 0.019 -8.441 0.000 0.000
1.476 -0.026 -0.131 0.000 0.011 0.000
2.219 -0.123 0.000 -8.425 0.002 0.000
1.323 0.000 -0.027 0.621 0.003 0.000
0.000 0.166 0.027 13.220 -0.002 0.000
2.241 -0.126 -0.079 -8.129 0.008 0.000


0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000
0.000 0.000


1 -0.003
2 -0.043
3 0.069
4 0.000
5 0.077
6 0.007
7 -0.003
8 -0.004
9 -0.003
10 -0.044
11 -0.169
12 -0.043
13 -0.004
14 0.068
15 -0.007
16 0.007
17 0.111
18 0.007
19 -0.004
20 -0.003
21 -0.005
22 -0.169
23 -0.083
24 -0.169
25 -0.014
26 0.116
27 0.014
28 0.117
29 -0.006
30 -0.168
31 0.117
































































80











MUNK (1949) BERKELEY LABORATORY DATA
n = 16


Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.1256 0.0988 0.8700 0.0719 0.01331 0.62313
0.1070 0.0985 1.1500 0.0719 0.00760 0.82496
0.0951 0.0994 1.2200 0.0719 0.00681 0.87114
0.0823 0.0945 1.5000 0.0719 0.00429 1.09836
0.0832 0.0872 1.5400 0.0719 0.00375 1.17401
0.0713 0.0823 1.9700 0.0791 0.00216 1.70046
0.1384 0.0917 0.8600 0.0541 0.01266 0.48086
0.1106 0.0911 0.9650 0.0541 0.00999 0.54137
0.0747 0.0826 1.3400 0.0541 0.00469 0.78963
0.0747 0.0792 1.5000 0.0541 0.00359 0.90242
0.0631 0.0683 1.9700 0.0541 0.00180 1.27687
0.1433 0.0997 1.0500 0.0900 0.00922 0.93705
0.1430 0.0975 1.0900 0.0900 0.00838 0.98333
0.1451 0.0985 1.3500 0.0900 0.00551 1.21222
0.1445 0.0939 1.5000 0.0900 0.00426 1.37932
0.1180 0.0872 1.9800 0.0900 0.00227 1.88943


STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 52.69 0.7259
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 31.16 0.5582
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 34.75 0.5895
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 34.78 0.5897
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 2.60 0.1613
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 107.19 -7.19 0.00 0.00 0.00 0.00 0.00 0.00 53.45 0.7311
7 81.07 0.00 18.93 0.00 0.00 0.00 0.00 0.00 58.07 0.7621
8 93.77 0.00 0.00 6.23 0.00 0.00 0.00 0.00 55.86 0.7474
9 97.35 0.00 0.00 0.00 2.65 0.00 0.00 0.00 53.22 0.7295
10 0.00 -248.80 348.80 0.00 0.00 0.00 0.00 0.00 75.32 0.8679
11 0.00 -25.73 0.00 125.73 0.00 0.00 0.00 0.00 34.81 0.5900
12 0.00 240.33 0.00 0.00 -140.33 0.00 0.00 0.00 81.28 0.9015
13 0.00 0.00 73.93 26.07 0.00 0.00 0.00 0.00 77.77 0.8819
14 0.00 0.00 164.41 0.00 -64.41 0.00 0.00 0.00 69.27 0.8323
15 0.00 0.00 0.00 49.36 50.64 0.00 0.00 0.00 60.21 0.7760
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 125.35 86.50 -111.86 0.00 0.00 0.00 0.00 0.00 79.53 0.8918
17 70.15 18.45 0.00 11.40 0.00 0.00 0.00 0.00 58.39 0.7641
18 52.47 108.80 0.00 0.00 -61.28 0.00 0.00 0.00 85.27 0.9234
19 -52.40 0.00 112.21 40.19 0.00 0.00 0.00 0.00 78.50 0.8860
20 -166.48 0.00 448.73 0.00 -182.25 0.00 0.00 0.00 69.97 0.8365
21 56.30 0.00 0.00 20.18 23.52 0.00 0.00 0.00 68.93 0.8302
22 0.00 -20.43 95.89 24.54 0.00 0.00 0.00 0.00 78.22 0.8844
23 0.00 210.79 19.81 0.00 -130.60 0.00 0.00 0.00 81.47 0.9026
24 0.00 303.50 0.00 -12.61 -190.90 0.00 0.00 0.00 81.55 0.9031
25 0.00 0.00 76.11 25.54 -1.65 0.00 0.00 0.00 77.78 0.8819
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 266.89 128.05 -258.72 -36.21 0.00 0.00 0.00 0.00 81.35 0.9020
27 61.62 102.93 -11.44 0.00 -53.11 0.00 0.00 0.00 85.56 0.9250
28 50.81 104.31 0.00 1.73 -56.85 0.00 0.00 0.00 85.37 0.9240
29 -142.98 0.00 222.69 53.77 -33.48 0.00 0.00 0.00 79.14 0.8896
30 0.00 259.44 13.23 -7.75 -164.93 0.00 0.00 0.00 81.58 0.9032
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 59.93 102.04 -10.06 0.61 -52.52 0.00 0.00 0.00 85.57 0.9250







p











MUNK (1949) BERKELEY LABORATORY DATA (CONT)


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.7259 -0.5582 0.5895 0.5897 -0.1613

(2) 52.6872 1.0000 -0.6807 0.5439 0.6200 -0.3177

(3) 31.1605 46.3335 1.0000 0.1250 -0.9355 0.8833

(4) 34.7533 29.5804 1.5626 1.0000 -0.1060 0.5553

(5) 34.7763 38.4397 87.5134 1.1226 1.0000 -0.7937

(6) 2.6004 10.0908 78.0288 30.8311 62.9970 1.0000

Numbers in Parentheses are the Variable or Parameter Number



TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (88 X8)

EQ BO 81 B2 B3 B4 B5 B6 B7 88
NO
1 -0.113 2.429 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.169 0.000 -0.045 0.000 0.000 0.000 0.000 0.000 0.000
3 0.021 0.000 0.000 1.199 0.000 0.000 0.000 0.000 0.000
4 0.077 0.000 0.000 0.000 4.888 0.000 0.000 0.000 0.000
5 0.120 0.000 0.000 0.000 0.000 -0.012 0.000 0.000 0.000
6 -0.075 2.157 -0.010 0.000 0.000 0.000 0.000 0.000 0.000
7 -0.108 1.926 0.000 0.563 0.000 0.000 0.000 0.000 0.000
8 -0.082 1.958 0.000 0.000 1.881 0.000 0.000 0.000 0.000
9 -0.126 2.511 0.000 0.000 0.000 0.006 0.000 0.000 0.000
10 0.079 0.000 -0.052 1.362 0.000 0.000 0.000 0.000 0.000
11 0.085 0.000 -0.004 0.000 4.482 0.000 0.000 0.000 0.000
12 0.194 0.000 -0.152 0.000 0.000 0.116 0.000 0.000 0.000
13 -0.024 0.000 0.000 1.341 5.467 0.000 0.000 0.000 0.000
14 0.019 0.000 0.000 1.997 0.000 -0.054 0.000 0.000 0.000
15 -0.024 0.000 0.000 0.000 10.343 0.064 0.000 0.000 0.000
16 0.243 -1.871 -0.086 2.089 0.000 0.000 0.000 0.000 0.000
17 -0.180 2.223 0.039 0.000 5.222 0.000 0.000 0.000 0.000
18 0.387 -1.618 -0.223 0.000 0.000 0.164 0.000 0.000 0.000
19 0.007 -0.582 0.000 1.560 6.456 0.000 0.000 0.000 0.000
20 0.068 -0.729 0.000 2.459 0.000 -0.069 0.000 0.000 0.000
21 -0.110 1.353 0.000 0.000 7.012 0.049 0.000 0.000 0.000
22 0.005 0.000 -0.015 1.352 3.997 0.000 0.000 0.000 0.000
23 0.219 0.000 -0.172 -0.305 0.000 0.140 0.000 0.000 0.000
24 0.170 0.000 -0.139 0.000 1.258 0.114 0.000 0.000 0.000
25 -0.023 0.000 0.000 1.367 5.303 -0.002 0.000 0.000 0.000
26 0.163 -1.642 -0.052 1.991 3.221 0.000 0.000 0.000 0.000
27 0.379 -1.844 -0.204 0.428 0.000 0.138 0.000 0.000 0.000
28 0.417 -1.737 -0.237 0.000 -0.855 0.169 0.000 0.000 0.000
29 0.043 -1.018 0.000 1.984 5.537 -0.021 0.000 0.000 0.000
30 0.187 0.000 -0.151 -0.146 0.985 0.126 0.000 0.000 0.000
31 0.389 -1.863 -0.211 0.391 -0.276 0.142 0.000 0.000 0.000











PUTNAM AND OTHERS (1949) LABORATORY DATA
n = 37

Water Breaker Wave Bed Equivalent Surf
Depth Height Period Slope Wave Sim
(m) (m) (s) Steepness Parm

0.2290 0.1430 1.0000 0.0660 0.01459 0.54637
0.1340 0.0980 1.0600 0.0660 0.00890 0.69960
0.1710 0.1220 1.1400 0.0660 0.00958 0.67434
0.1250 0.0940 1.1500 0.0660 0.00725 0.77498
0.1190 0.0910 1.2500 0.0660 0.00594 0.85614
0.1220 0.0980 1.3200 0.0660 0.00574 0.87120
0.1130 0.0880 1.4000 0.0660 0.00458 0.97509
0.0730 0.0490 1.9000 0.1440 0.00139 3.86929
0.0700 0.0460 2.1300 0.1440 0.00103 4.47689
0.0730 0.0460 2.2200 0.1440 0.00095 4.66605
0.1460 0.0850 0.7200 0.2410 0.01673 1.86317
0.1580 0.1070 0.9200 0.2410 0.01290 '2.12191
0.0850 0.0670 1.1400 0.2410 0.00526 3.32275
0.0820 0.0670 1.2200 0.2410 0.00459 3.55592
0.0980 0.0730 0.9900 0.1000 0.00760 1.14706
0.0820 0.0670 1.3200 0.1000 0.00392 1.59643
0.0700 0.0490 1.6300 0.1000 0.00188 2.30517
0.0670 0.0490 1.9800 0.1000 0.00128 2.80014
0.1310 0.0850 0.8300 0.1390 0.01259 1.23879
0.1010 0.0700 0.9100 0.1390 0.00863 1.49665
0.0880 0.0670 1.0000 0.1390 0.00684 1.68109
0.0730 0.0610 1.1200 0.1390 0.00496 1.97324
0.0760 0.0610 1.3500 0.1390 0.00342 2.37846
0.1890 0.1040 0.8000 0.2600 0.01658 2.01911
0.1310 0.0880 0.9000 0.2600 0.01109 2.46938
0.1250 0.0850 0.9800 0.2600 0.00903 2.73592
0.0790 0.0610 1.2300 0.2600 0.00411 4.05347
0.0700 0.0670 1.2700 0.2600 0.00424 3.99349
0.1100 0.0790 0.9500 0.0980 0.00893 1.03693
0.0820 0.0640 1.3300 0.0980 0.00369 1.61288
0.0610 0.0490 1.6700 0.0980 0.00179 2.31450
0.0580 0.0370 1.9900 0.0980 0.00095 3.17389
0.1430 0.1010 1.0800 0.1430 0.00884 1.52129
0.1160 0.0880 1.3600 0.1430 0.00485 2.05233
0.0820 0.0610 1.5800 0.1430 0.00249 2.86379
0.0790 0.0610 1.9100 0.1430 0.00171 3.46192
0.0910 0.0670 2.3200 0.1430 0.00127 4.01236











PUTNAM AND OTHERS (1949) LABORATORY DATA (CONT)

STEPWISE REGRESSION ANALYSIS RESULTS

Eq 100 r*2 r
No

(1) (2) (3) (4) (5) (6) (7) (8)
INDEPENDENT VARIABLES TAKEN ONE AT A TIME
1 100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 90.48 0.9512
2 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.00 37.15 0.6095
3 0.00 0.00 100.00 0.00 0.00 0.00 0.00 0.00 0.01 0.0116
4 0.00 0.00 0.00 100.00 0.00 0.00 0.00 0.00 75.23 0.8674
5 0.00 0.00 0.00 0.00 100.00 0.00 0.00 0.00 35.24 0.5936
INDEPENDENT VARIABLES TAKEN TWO AT A TIME
6 99.98 0.02 0.00 0.00 0.00 0.00 0.00 0.00 90.48 0.9512
7 94.63 0.00 5.37 0.00 0.00 0.00 0.00 0.00 91.16 0.9548
8 83.88 0.00 0.00 16.12 0.00 0.00 0.00 0.00 94.34 0.9713
9 94.79 0.00 0.00 0.00 5.21 0.00 0.00 0.00 90.98 0.9539
10 0.00 85.61 14.39 0.00 0.00 0.00 0.00 0.00 39.48 0.6283
11 0.00 42.53 0.00 57.47 0.00 0.00 0.00 0.00 79.99 0.8943
12 0.00 64.90 0.00 0.00 35.10 0.00 0.00 0.00 44.20 0.6648
13 0.00 0.00 -64.36 164.36 0.00 0.00 0.00 0.00 80.47 0.8970
14 0.00 0.00 -168.05 0.00 268.05 0.00 0.00 0.00 50.76 0.7125
15 0.00 0.00 0.00 138.28 -38.28 0.00 0.00 0.00 76.98 0.8774
INDEPENDENT VARIABLES TAKEN THREE AT A TIME
16 89.65 4.48 5.87 0.00 0.00 0.00 0.00 0.00 91.30 0.9555
17 52.81 26.42 0.00 20.77 0.00 0.00 0.00 0.00 98.21 0.9910
18 97.28 -3.52 0.00 0.00 6.24 0.00 0.00 0.00 91.06 0.9542
19 84.34 0.00 -2.33 18.00 0.00 0.00 0.00 0.00 94.40 0.9716
20 93.86 0.00 4.00 0.00 2.14 0.00 0.00 0.00 91.21 0.9550
21 79.17 0.00 0.00 14.17 6.66 0.00 0.00 0.00 95.02 0.9748
22 0.00 48.01 -19.79 71.77 0.00 0.00 0.00 0.00 84.57 0.9196
23 0.00 153.90 160.70 0.00 -214.60 0.00 0.00 0.00 54.55 0.7386
24 0.00 62.28 0.00 57.63 -19.91 0.00 0.00 0.00 85.35 0.9238
25 0.00 0.00 -82.45 134.70 47.74 0.00 0.00 0.00 82.11 0.9061
INDEPENDENT VARIABLES TAKEN FOUR AT A TIME
26 52.64 26.71 -0.85 21.50 0.00 0.00 0.00 0.00 98.23 0.9911
27 82.24 13.75 12.07 0.00 -8.06 0.00 0.00 0.00 91.41 0.9561
28 52.43 27.07 0.00 21.04 -0.54 0.00 0.00 0.00 98.21 0.9910
29 70.92 0.00 -21.41 26.18 24.31 0.00 0.00 0.00 97.63 0.9881
30 0.00 66.94 7.95 51.94 -26.82 0.00 0.00 0.00 85.44 0.9243
INDEPENDENT VARIABLES TAKEN FIVE AT A TIME
31 54.86 22.96 -4.22 22.37 4.04 0.00 0.00 0.00 98.26 0.9912


CORRELATION MATRIX

(1) (2) (3) (4) (5) (6)


(1) 1.0000 0.9512 -0.6095 0.0116 0.8674 -0.5936

(2) 90.4806 1.0000 -0.6409 -0.0744 0.7835 -0.6788

(3) 37.1460 41.0738 1.0000 -0.2611 -0.8393 0.6388

(4) 0.0135 0.5531 6.8166 1.0000 0.2675 0.5394

(5) 75.2337 61.3902 70.4429 7.1574 1.0000 -0.5581

(6) 35.2366 46.0736 40.8038 29.0972 31.1425 1.0000

Numbers in Parentheses are the Variable or Parameter Number











PUTNAM AND OTHERS (1949) LABORATORY DATA (CONT)


TABLE OF RELATING BETA COEFFICIENTS FOR EACH EQUATION
Y = BO + (B1 X1) + (B2 X2) + (B3 X3) + ... + (B8 X8)

EQ BO B1 B2 B3 B4 B5 B6 B7 B8
NO
1 -0.015 1.595 0.000 0.000 0.000 0.000 0.000 0.000 0.000
2 0.179 0.000 -0.055 0.000 0.000 0.000 0.000 0.000 0.000
3 0.104 0.000 0.000 0.007 0.000 0.000 0.000 0.000 0.000
4 0.058 0.000 0.000 0.000 7.555 0.000 0.000 0.000 0.000
5 0.149 0.000 0.000 0.000 0.000 -0.019 0.000 0.000 0.000
6 -0.015 1.595 0.000 0.000 0.000 0.000 0.000 0.000 0.000
7 -0.023 1.605 0.000 0.048 0.000 0.000 0.000 0.000 0.000
8 -0.001 1.180 0.000 0.000 2.754 0.000 0.000 0.000 0.000
9 -0.030 1.705 0.000 0.000 0.000 0.003 0.000 0.000 0.000
10 0.197 0.000 -0.059 -0.091 0.000 0.000 0.000 0.000 0.000
11 -0.008 0.000 0.036 0.000 10.486 0.000 0.000 0.000 0.000
12 0.178 0.000 -0.035 0.000 0.000 -0.011 0.000 0.000 0.000
13 0.075 0.000 0.000 -0.136 8.108 0.000 0.000 0.000 0.000
14 0.129 0.000 0.000 0.269 0.000 -0.028 0.000 0.000 0.000
15 0.075 0.000 0.000 0.000 6.781 -0.005 0.000 0.000 0.000
16 -0.035 1.663 0.005 0.057 0.000 0.000 0.000 0.000 0.000
17 -0.059 1.153 0.033 0.000 5.508 0.000 0.000 0.000 .0.000
18 -0.025 1.680 -0.003 0.000 0.000 0.004 0.000 0.000 0.000
19 0.003 1.144 0.000 -0.016 2.965 0.000 0.000 0.000 0.000
20 -0.027 1.647 0.000 0.037 0.000 0.001 0.000 0.000 0.000
21 -0.018 1.297 0.000 0.000 2.820 0.004 0.000 0.000 0.000
22 0.012 0.000 0.034 -0.128 10.803 0.000 0.000 0.000 0.000
23 0.050 0.000 0.065 0.618 0.000 -0.053 0.000 0.000 0.000
24 -0.006 0.000 0.052 0.000 10.291 -0.010 0.000 0.000 0.000
25 0.059 0.000 0.000 -0.266 10.109 0.010 0.000 0.000 0.000
26 -0.057 1.133 0.033 -0.010 5.619 0.000 0.000 0.000 0.000
27 -0.043 1.618 0.015 0.124 0.000 -0.005 0.000 0.000 0.000
28 -0.059 1.140 0.034 0.000 5.557 0.000 0.000 0.000 0.000
29 -0.024 1.219 0.000 -0.192 5.464 0.014 0.000 0.000 0.000
30 -0.015 0.000 0.061 0.066 10.038 -0.014 0.000 0.000 0.000
31 -0.053 1.150 0.027 -0.046 5.693 0.003 0.000 0.000 0.000
































































86