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FGS








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




Division of Resource Assessment and Management
Edwin J. Conklin, Director




Florida Geological Survey
Walter Schmidt, State Geologist and Chief






SSpecial Publication No. 41


Shore-Breaking Wave Height Transformation

by

James H. Balsillie

and

Wavelength and Wave Celerity During Shore-Breaking

by

James H. Balsillie


Florida Geological Survey
Tallahassee, Florida
2000























































Printed for the
Florida Geological Survey

Tallahassee, Florida
2000

ISSN 0085-0640






ii








LETTER OF TRANSMnTTAL


Florida Geological Survey
Tallahassee

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

Dear Governor Bush:

The Florida Geological Survey, Division of Resource Assessment and Management,
Department of Environmental Protection is publishing two papers, in this, Special Publication No.
41: "Shore-breaking wave height transformation" and "Wavelength and wave celerity during
shore-breaking".

The first paper identifies a numerical methodology for transforming wave height during the
shore-breaking process. The second paper is a companion paper, which provides a numerical
methodology for transforming the wavelength and wave celerity during the shore-breaking
process.

Both papers provide new technology based on measured data (field and laboratory) and,
therefore, do not constitute theoretical approaches currently employed by coastal practitioners.
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, to name but a few.

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 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.

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 wave
characteristics. 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, does not only have a significantly long
shoreline, but it has a large wave energy variability and a significant level of coastal
development. 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.

James H. Balsillie
February 2000
























































































vi









CONTENTS



SHORE-BREAKING WAVE HEIGHT TRANSFORMATION


ABSTRACT ..................


INTRODUCTION...............


CLASSICAL WAVE THEORIES .....


WAVE HEIGHT GROWTH DURING SH


Terminal Boundary Condition .


ORE-BREA


Initial Boundary Condition ...........


Transformation of HIH, .............


WAVE CREST ELEVATION ABOVE THE DESIGN


Terminal Boundary Condition .........


Initial Boundary Condition ...........


Prediction of H'/H Transformation ....


CONCLUSIONS ......................


N O TE ..... ..... ..... ............ .


REFERENCES .........................


NOTATION .........................


KING










WATER


....


....


....


. . .


. . .


. . .


....


LEVEL


. . .


. . .


. . .


....


....


....


....


LIST OF TABLES


Table 1. Field and laboratory data used in analyses..


. . . . . . . . . . . . . 5


Table 2. Laboratory data of Putnam (1945)for evaluation of wave height above
the D W L. . . . . . . . . . . . . . . . . . . . . . . . .. 17


LIST OF FIGURES


Figure 1. Pertinent nearshore wave parameters; those at A illustrate conditions at
the initiation of alpha wave peaking, those at B represent conditions at
shore-breaking for plunging waves. ................................. 2


Page


. . 1


... 1


.. 2













S..11


S. .15


S. .16


. 19
. 11










..21


..23


..23


. 28







Figure 2. Wave transformation from deep water to shore-breaking, where the alpha
wave peaking process is denoted by ab. ............................. 4

Figure 3. Relationship between the water depth at shore-breaking, db, and the
shore-breaking wave height, Hb .................................... 7

Figure 4. Relationship for prediction of the shore-breaking wave height from the
initial equivalent wave steepness parameter. ........................... 8

Figure 5. Relationship between the equivalent wave steepness parameter evaluated
at initiation of alpha wave peaking and at shore-breaking. ................. 9

Figure 6. Evaluation of Munk's (1949) parameter for determining the initiation of
wave peaking in the shore-breaking process). ........................ 10

Figure 7. Relationship for the prediction of the relative depth of water at which
alpha wave peaking is initiated..................................... 10

Figure 8. Alpha wave peaking predictions using equation (7) plotted as the solid
curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured
data (solid circles) are for bed slope data of 0.072 and 0.054 from Putnam
(1945), and bed slope data of 0.0292 of Buhr Hansen and Svendsen (1979). ... 12

Figure 9. Dependence of design pier deck soffit elevation on wave crest elevation
above the design water level (DWL) for two commonly applied theoretical
approaches. ................................................ 14

Figure 10. Relationship between the wave height at shore-breaking, Hb, and the
trough depth just preceding the crest, db; field data are from Balsillie and
Carter (1980) and Balsillie (1980). laboratory data are from Iverson (1952). . 1 5

Figure 11. Relationship between the water depth at shore-breaking, db, and the total
water depth at shore-breaking, d.; data sources as for Figure 10. ......... 15

Figure 12. Relationship between the total depth at shore-breaking, d., and the
wave trough at shore-breaking, dt; data sources as for Figure 10. ........... 16

Figure 13. Relative depth equations for initiation of alpha wave peaking at (di/Hi),
and at initiation of wave height increase above the DWL at (di/H)'. ........... 17

Figure 14. Wave steepness parameter at (di/Hi)' relative to the wave steepness
parameter at (d H ). ............................................. 18

Figure 15. Relationship for prediction of the incident value of H'/H. ............. 18

Figure 16. Transformation of H'/H using equation (21) plotted as solid curves.
Vertical dashed lines represent dlH = 1.28 = db/Hb. Measured data (solid
circles) are from Putnam (1945). ........................... ...... 20








Figure 17. Predicted H' from equation (21) versus prototype Beach Erosion Board
wave tank data reported by Bretschneider (1960). ...................... 21

Figure 18. Predicted H' from equation (21) versus Lake Okeechobee hurricane wave
data reported by Bretschneider (1960). ............................. 21

Figure 19. Ratio of wave crest elevation above the design water level to wave
height from equation (21) ............. ........................ 22

Figure 20. Region of applicability for various wave theories and for the Alpha Wave
Peaking methodology of this work. ................................ 24



WAVELENGTH AND WAVE CELERITY DURING SHORE-BREAKING

Page

A BSTRA CT .......................................................31

INTRO DUCTIO N ...... ............................................ 31

DISCUSSION AND RESULTS .......................................... 31

Terminal Boundary Conditions .................................... 33

Initial Boundary Conditions ...................................... 34

Wave Celerity and Wavelength Transformation During Alpha Wave Peaking .... 36

CO NCLUSIO NS ...................................................37

REFERENCES .................................................... 37

NO TA TIO N .......................................................39

TABLE

Table of statistics relating measured and predicted wavelengths at the shore-
breaking position. ................... ................... ...... 34

FIGURES

Figure 1. Illustration of wave transformation from deep water to shore-breakinig,
where the alpha wave peaking process is given the notation ap. ........ .. .32

Figure 2. Evaluation of relationships for prediction of the wavelength at the
shore-breaking position. ........................................ 35







Figure 3. Comparison of predicted and measured wave speeds at the point of
initiation of alpha wave peaking; predicted data are from equation (4),
measured data from Hansen and Svendsen (1979). ................... .. 36

Figure 4. Illustration of wave speed attenuation during shore-breaking using
equation (15) for various initial wave steepness values.. ................. 37









SHORE-BREAKING WAVE HEIGHT

TRANSFORMATION

by

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


ABSTRACT
As waves begin to shore-break, the wave crest rapidly increases in height, reaching a maximum
at the shore-breaking position. This phenomenon, termed alpha wave peaking, is primarily dependent on
the wave steepness and may be predicted according to:

S1.0 0.4 In tanh 100 '
H, g 7 92}\\

where H, is the mean incident wave height, T is the wave period and Hb is the mean shore-breaking wave
height. For waves considered in this work, Hb ranged from 1.04 to 2.39 times as high as H,. The relative
depth of water, di /H defining the point at which alpha wave peaking begins (i.e., the initiation of the
shore-breaking process) is given by:

= {, In tanh65
H, H, 2 I g

Transformation of H/H, during alpha peaking, where H is the local mean wave height, is given by:

N Nb tanh f
H, H I H H,

in which d, is the water depth at shore-breaking, d is the local water depth, and solutions for 0, and tC2
are developed in the text.
Many coastal engineering design solutions requiring consideration of wave activity can be
accomplished only if the crest elevation of the design wave(s) is known relative to a design water level
(DWL). From analysis of field and laboratory data, it is determined that at the shore-breaker position
approximately 84% of the wave crest lies above the DWL. The amount of the wave that lies above the
DWL during shore-breaking may range from about 0.5 to 0.84. Transformation of H'IH, where H is the
local wave height and H' is the amount of H lying above the DWL during alpha wave peaking, may be
predicted by:

0H'4L- a) tanh 03 (d
HH H,

where H,' is the amount of Hb lying above the DWL and the solutions for 03 and c4 are developed in the
text.


INTRODUCTION wave height, beginning just prior to, and
reaching a maximum at shore-breaking can,
Generally, as waves approach the even over gentle bed slopes, be
shoreline, the height of the waves first tend "...remarkably sudden..." (Munk, 1949) and
to decrease and then to increase rapidly just accompanied by progressive distortion and
prior to shore-breaking. This increase in the asymmetry of the wave in profile view, has









been termed alpha wave peaking by Balsillie
(1980). Wave peaking has been observed
as characteristic activity in shore-breaking
wave mechanics (Scripps Institute of
Oceanography, 1944a, 1944b; Putnam,
1945; Munk, 1949; Iverson, 1952; Stoker,
1957; Kinsman, 1965; Byrne, 1969; Clifton
and others, 1971; Komar, 1976; Nakamura
and others, 1966; Van Dorn, 1978; Hansen
and Svendsen, 1979; Balsillie, 1980, 1983b,
1983c; etc.).

Two mechanisms occur during shore-
breaking: 1. the transformation of H/H,, and
2. the transformation of H'/H, where H is
the local wave height, H' is the amount of
the wave lying above the design water level
(DWL), and H, is the wave height at the
initiation of alpha wave peaking. Pertinent
wave height parameters are illustrated in
Figure 1 for plunging shore-breakers, where
at the shore-breaking position, the front face
of the wave crest becomes vertical (to
eventually curl, form an air pocket, and fall
into the wave trough fronting the crest).
Where spilling shore-breakers occur, the
shore-breaking position is reached when the
top of the wave crest becomes unstable and
aerated and turbulent water slips down and
across the front face of the crest. Other
shore-breaker types such as surging and
collapsing waves have also been identified


and defined (e.g., Galvin, 1968; Balsillie,
1985). The first of the above mechanisms
defines the first subject of this paper, the
latter is addressed in the second section.
Wavelength and wave celerity behavior
during shore-breaking are the subjects of
separate works (Balsillie 1984, 1999a).

CLASSICAL WAVE
THEORIES

It is to be noted that existing wave
theories have relevance in offshore waters
where wave distortion and boundary effects
are less constraining compared to conditions
in nearshore waters. Application of
quantifying methodology for the nearshore
zone, however, has been sorely lacking,
ostensibly because of the distorted nature of
waves and confining boundary conditions. It
would, therefore, appear appropriate to
recapitulate some of the more conspicuous
short-comings of "classical wave theories"
relevant to the problem at hand.

From an historical perspective, one
should have at least, some appreciation, of
early theoretical work of such researchers as
Gerstner (1802), Levi-Cevita (1924), Struik
(1926), Gaillard (1935), and Mason (1941).
For a more detailed account of early theorists
see the bibliography of Mason (1941).

B B


Figure 1. Pertinent nearshore wave height parameters; those at A illustrate
conditions at the initiation of alpha wave peaking, those at B represent conditions
at shore-breaking for plunging waves.









Subsequent to this early work, wave theories
have surfaced which currently are in general
use. Following are descriptions and
discussion of such theoritical approaches.

Linear (Airy) Wave Theory suffers
from the basic problem that the wave is
divided evenly above and below the still
water level (SWL). This condition clearly
does not hold for waves in shallow and
transitional water depths or, for that matter,
waves undergoing breaking in any depth of
water. Also, the calculating algorithum for
wavelength in any depth of water is, at best,
an envelope fit giving but approximating
results (Bretschneider, 1960; Balsillie,
1984a, 1999b).

Stokes' Wave Theory attempts to
more nearly depict the distortion of the wave
profile about the SWL, with the degree of
distortion increasing for higher order theory.
However, its applicability does not extend
into shallow water nor the shallower part of
transitional water depths. Dean (1974; fig.
7) demonstrates the same region of validity
for Stokes' 5th Order Theory which includes
Stokes' 3rd and 4th order region of Figure 1.

Cnoidal Wave Theory has been
popularly used for predicting shallow and
transitional water wave conditions.
However, depending upon the investigator,
there is variability in defining the region of
validity for the theory. Laitone (1963) notes
its validity for d/L < 0.125. Svendsen and
Hansen (1976) discuss the applicability of
Cnoidal theory using developments of
Skovgarrd and others (1974) for the
deformation of waves up to shore-
breaking (the theory is applicable where d/Lo
< 0.10 or d/L < 0.13 (see NOTATION
section at the end of the paper for
definitions), seaward of which they
recommend the use of Airy wave theory).
Svendsen and Hansen state "...even though
cnoidal theory seems to predict the wave
height variation reasonably well, no
information can be deduced from that theory


(or any other theory) about where breaking
occurs." Similarly, Skovgaard and others
(1974) note that where (d/g T2) > 0.014
(i.e., d/L > 0.13) "... Cnoidal theory is
meaningless". The Shore Protection Manual
(U. S. Army, 1984), on the other hand,
appears more restrictive suggesting the use
of the Ursell or Stokes parameter such that
the theory is valid where U = L2 H/d3 > 26.
In addition, as the wavelength becomes long
approaching infinity, Cnoidal Wave Theory
reduces to Solitary Wave Theory; and as H/d
becomes small, the wave profile approaches
the sinusoidal profile predicted by Linear
(Airy) Theory. Several other problems are
entrenched in the theory. First, it is not an
easy nor expedient theory in application.
Second, it requires that the wavelength is
known when, in fact, this wave parameter is
not quantitatively known for use in actual
design applications. Third, the wavelength
is first calculated from Linear (Airy) Theory in
deep water (i.e., Lo) then modified to the
local water depth of concern (i.e., L), which
if only because of the approach, suggests an
approximation at best.

Solitary Wave Theory is a special
case of Cnoidal Theory in which the
wavelength is infinitely long and the entire
wave crest lies above the SWL, is clearly not
applicable to natural cases of shore-breaking
where shoaling waves have periodicity.

More recently developed Stream
Function Theory (Chappelear, 1961; Dean,
1965a, 1965b, 1967; Monkmeyer, 1970)
constitutes an attempt to mitigate the
difficulty of expanding to higher orders the
Stokes'-type approach, rendering the "...
approach computationally simpler than
Chappelear's technique, ..., to have wave
theories that could be developed on the
computer to any order" (Dean and
Dalrymple, 1984, p. 305). Stream Function
Theory, which has "... some of the same
limitations ..." of higher order Stokes'
solutions (U. S. Army, 1984, p. 2-59) has
the capability to proceed to the breaking limit









in transitional water depths, but not in
shallow water.

WA VE HEIGHT GROWTH
DURING SHORE-BREAKING

It is, in part, the intent of this work to
provide a practicable solution to the
shortcomings of the previously discussed
wave theories.

Mathematical descriptions developed
in the following sections require that only the
initial wave height, H,, and wave period, T,
are known. In an earlier study, Balsillie
(1980) used H. or Hm as indicators of Hi,
where Ho is the deep water wave height and
Hm is the wave height measured in the
constant depth portion of a laboratory wave
channel. In many laboratory investigations
it has been found that initial wave
characteristics are in the range where Ho and
Hm and the resulting value of Hi are
approximately equivalent. Generally, this
occurs for waves with higher wave
steepness values. However, due to


refraction and frictional effects, etc., where
the generated wave steepness is small, Hi
can become significantly less than Ho or Hm.
The importance of the latter phenomenon is
illustrated by an example from the laboratory
data of Putnam (1945) in Figure 2.

In the earlier work (Balsillie, 1980), it
was reported that the alpha wave peaking
parameter, Hb/H,, is dependent on the
equivalent wave steepness parameter, Hi/(g
T2), and the bed slope, m The influence of
these parameters is included in ensuing
analyses. In addition, the continuous
transformation of H/Hi during shore-breaking
is investigated. First, however,
determination of where alpha wave peaking
is initiated and terminated require
identification. Where possible, both field
data and laboratory data are considered. It
is to be noted, however, that laboratory
information by far constitutes the bulk of
available data. However, since the studies
of Balsillie (1980, 1983b), new laboratory
data have become available (Table 1) to


I I I I I I I


**--Conlient Deth
-i- Deep Wales


D10
0.0110


cc


rcHtl .00 W
retllo* of Ways
aaf


Trovel I



0-- o
II


p d0
% 5: b


Data from Putanm (1451
T a0.965 i.


I -

H Hb


0.7'1 I I. . ..I I. i I I II
10 9 8 7 6 8 4 1 2 I
d/H

Figure 2. Wave transformation from deep water to shore-breaking,
alpha wave peaking process is denoted by ab.


where the


$I. V


09-


Corrected
H
,H;










Table 1. Field and laboratory data used in analyses.



T 4 4\-1 2. H3
Investigator m -
(al



FIELD DATA
Wood (1970, 1971)1 0.0556 3.49 1.58 4.23 2.00 451.2 286.2

LABORATORY DATA
Putnam (1945) 0.072 0.865 1.04 2.34 1.76 78.8 74.4
0.072 1.15 1.29 3.13 2.27 170.0 131.0
0.072 1.22 1.29 2.69 2.67 208.0 147.0
0.072 1.50 1.66 4.77 3.17 391.0 236.8
0.072 1.54 1.58 4.45 2.92 428.0 267.0
0.072 1.97 1.86 5.01 3.11 916.5 462.8
0.054 0.86 1.08 2.22 1.66 87.4 78.6
0.054 0.965 1.11 2.50 1.93 108.9 103.8
0.054 1.34 1.48 3.96 2.96 312.0 213.7
0.054 1.50 1.58 4.39 3.16 425.7 278.7
0.054 1.97 1.84 5.94 3.47 1039.1 557.9
Step" 1.05 1.16 --- 126.2 108.5
Step" 1.19 1.19 --- 142.6 119.4
Setp" 1.35 1.31 -- 237.4 181.5
Step" 1.50 1.60 --- .- 375.0 235.0
Step" 1.98 1.86 --- 997.8 441.0
Hansen and 0.0292 0.833 1.32 3.64 2.55 207.0 156.3
Svendsen (1979) 0.0292 1.00 1.13 2.57 1.66 104.2 91.8
0.0292 1.00 1.21 2.95 1.89 153.1 126.4
0.0292 1.00 1.35 3.92 2.66 258.4 191.8
0.0292 1.25 1.30 2.50 1.51 162.9 124.3
0.0292 1.25 1.37 3.15 2.04 229.2 167.6
0.0292 1.25 1.46 3.95 2.42 395.3 270.1
0.0292 1.67 1.46 3.57 2.29 283.2 194.8
0.0292 1.67 1.42 3.57 2.15 302.5 209.8
0.0292 1.67 1.47 3.84 2.27 340.3 233.5
0.0292 1.67 1.48 4.13 2.41 389.1 261.3
0.0292 1.67 1.66 5.05 3.09 675.7 408.2
0.0292 2.00 1.69 4.47 2.66 608.6 359.3
0.0292 2.00 1.95 5.50 3.01 1048.2 537.8
0.0292 2.50 1.84 5.22 2.63 874.9 479.7
0.0292 2.50 2.20 5.95 2.75 1531.4 704.1
0.0292 3.33 2.39 6.67 3.08 2544.5 1069.2
Singamsetti and 0.025 1.28 1.27 --- --- 170.8 134.9
Wind (1980) 0.025 1.55 1.25 173.8 138.8

0.050 1.038 1.21 --- --- 162.4 134.5
0.050 1.55 1.20 --- 162.9 135.5
0.050 1.55 1.28 2.43 1.56 216.0 168.2
0.050 1.55 1.27 2.68 1.22 210.2 165.8










Table 1. Field and laboratory data used in analyses (cont.).



T H, d, g (
Investigator m - 7
(s)
H, H, d,



Singamsetti and 0.100 1.035 1.11 --- --- 160.3 144.8
Wind (1980) 0.100 1.555 1.25 --- --- 173.8 139.3
0.200 1.038 1.35 --- --- 157.6 116.7
Wang and others 0.067 1.00 1.19 --- --- 122.5 103.2
(1982)'" 0.067 1.33 1.20 --- --- 148.2 123.8
0.067 1.34 1.28 --- --- 195.5 153.0
0.067 1.47 1.22 --- .- 178.0 146.0
0.067 1.56 1.49 --- .- 283.9 190.8
0.067 1.65 1.21 -- --- 196.2 161.7
0.067 1.89 1.32 --- -- 280.1 212.2
0.100 1.22 1.17 --- --- 145.9 124.7
0.100 1.50 1.34 -- 191.7 143.2
0.100 1.58 1.13 -- 222.4 197.3
0.100 1.61 1.33 --- 267.4 201.6
0.100 1.80 1.55 -- 396.9 256.1
0.100 1.89 1.52 --- --350.1 230.3
Nakagawa (1983)"' 0.364 1.40 1.27 --- --- 179.5 141.2
Mizuguchi (1986)"' 0.050 1.22 1.39 --- 331.5 239.1
Hattori (1986)"** 0.050 0.85 1.23 --- --- 160.9 131.1
0.050 1.00 1.38 -- --- 316.1 227.9
0.050 1.40 1.81 --- 738.8 408.7
0.050 1.00 1.30 --- --- 208.5 160.7
0.050 0.80 1.05 --- --- 96.5 92.2
0.050 0.85 1.20 --- --- 138.8 116.1
0.050 1.00 1.42 3.16 --- 257.9 181.5
0.050 1.40 1.96 --- --- 768.3 392.0
0.050 0.80 1.06 --- -- 96.3 90.9
0.050 0.84 1.02 --- --- 123.5 121.3
0.050 0.99 1.25 --- 240.1 192.1
0.050 1.40 1.70 5.04 --- 711.4 417.6

Watanabe and 0.050 1.19 1.40 2.91 1.6 252.3 180.2
Dibajnia (1988)'" 0.050 1.18 1.28 2.50 1.6 213.2 166.4
0.050 0.94 1.17 2.50 1.6 135.3 115.5

Takikawa and others 0.050 2.08 1.36 2.62 2.08 229.2 197.2
(1997)'"' _____

SBased on 400 consecutive wave measurements; step had a slope of 0.444, post-step slope was
0.009; '" New data.









represent a wide range in bed slope
conditions.

Terminal Boundary Condition

The terminal boundary of alpha wave
peaking is defined as the shore-breaking
point. Galvin (1968) provides a
comprehensive description of the various
types of shore-breaking waves. Of the
principal types, however, spilling and
plunging shore-breakers constitute those
more commonly applied in design
considerations. The shore-breaking point of
a plunging breaker is defined to occur when
the front face of the wave crest becomes
vertical (Figure 1); the shore-breaking point
of a spilling breaker occurs when the top of


o Laboral
X Field D0
10 T T oField 0O


0.01
0.01


the wave crest becomes unstable and water
and foam slides or spills down the front face
of the crest.

Two parameters identifying
termination of alpha wave peaking are db/Hb
and Hb/Hi. The first parameter may be
straightforwardly given by the McCowan
criterion (McCowan, 1894; Munk, 1949;
Balsillie, 1983a, Balsillie, 1999b, Balsillie and
Tanner, 1999), illustrated in Figure 3, and
given by:


H= 1.28
d6


where Hb is the shore-breaking wave height,
and db is the water depth at shore-breaking.


0.1 1
Hb (m)


Figure 3. Relationship between the water depth at shore-breaking,
d,, and the shore-breaking wave height, Hb (after Balsillie, 1999b).









Enhancement in precision of db/H, prediction
has been attempted by incorporating the bed
slope and wave steepness (Weggel, 1972a,
1972b; Mallard, 1978). Balsillie (1983a,
1999b) found, however, that equation (1) as
yet constitutes the most reliable predicting
equation, and that equation (1) applies
equally well to both spilling and plunging
shore-breakers. The second parameter,
Hb/Hi, describing the relative height attained
as a result of alpha wave peaking, is more
difficult to quantify. It is, however,
considered to be a terminal boundary
parameter since Hi is understood to be
specified as input.

In the previous work published by the
author (Balsillie, 1980), both wave steepness
and bed slope were indicated to affect alpha
wave peaking. However, based on new
data, and subsequent analyses and testing,
the following relationship can be
recommended:


/


illustrated in Figure 4.

Additional attention was given to the
bed slope and no refinement was found to
improve equation (2). In fact, equation (2) is
evaluated for a wide range of bed slope
conditions, wherein any scatter might easily
be attributed to the difficulty in identifying
where shore-breaking occurs (Balsillie,
1999b). Other work suggests that the bed
slope is probably more instrumental in
influencing the type of shore-breaker that
will be produced (Balsillie, 1984b, 1984c,
1985, 1999b).

Due to scale differences between
axes of Figure 4, wave steepness data from
Table 1 are plotted in Figure 5 where now
the axes are comparable. Dividing both
sides of equation (2) by g T2 yields:


gT. g 1.0 -0.4 ntanh 100H (3)
-t1 p7(2[ ( ff^~


1.0 0.4{In tanh o100
tn9 ^ I


2.5
Bed Slope
S0.025
a 0.0292
2 A 0.050
20.054 A
H b X 0.0556
H 00.067
S0.072 o
1.5 A 0.100 U Equation (2)
X 0.200
a 0.368
o step

1
10 100 1 1000 10000
gHi 2

Figure 4. Relationship for prediction of the shore-breaking wave height
from the initial equivalent wave steepness parameter.











1000


-1 A" "-% ta
(Hb\ 0.067
9T2 D 0.072
T 0 0.100
x 0.200 /
No.368
o step
A Equation (3)
100



100 Hi -1 1000
kgT2/

Figure 5. Relationship between the equivalent wave
steepness parameter evaluated at initiation of alpha wave
peaking and at shore-breaking.


which is superimposed upon the data of
Figure 5 to show excellent agreement.

Initial Boundary Condition

Various investigators (e.g., Stokes,
1880: Galvin, 1969; Dean, 1974) have
conducted studies to delineate constraints of
breaking. It was Munk (1949), however,
who considered in some detail wave peaking
in the shore-breaking process. He applied
the Rayleigh assumption (Eagleson and
Dean, 1966) given by:


d, ,
4 '1")


Equation (5) is plotted in Figure 6,
from which the non-representative nature of
the equation is apparent. Additional analysis
indicates that if we solve for d,/H, rather than
d,/d, of equation (5) and consider the
incident wave steepness (bed slope
produced no consistent results), the
following relationship, plotted in Figure 7,
provides good agreement:


d(4) = I in rtanh (65s- l (6)
4) / H, 27


C, E, = c, E,


where c is the phase speed (shallow water
condition only, where wave period is
conserved and no energy is lost), and E is
the total wave energy, and the subscripts i
and b refer to conditions at initiation of wave
peaking and at shore-breaking, respectively.
Using the Rayleigh assumption and Solitary
wave theory, Munk (1949) suggests that:


in which it is assumed that db/Hb = 1.28.
The equation represents a significant range
of bed slopes (i.e., 0.0292 to 0.072) for
data from a variety of sources.



























0 1 2 3 4
Hb
Hi
Figure 6. Evaluation of Munk's (1949)
parameter for determining the initiation of
wave peaking in the shore-breaking
process.









7 Bed Slope
S0.0292
6 *- -.


Transformation of H/H,


In addition to specification of the
boundary conditions, it is desirable to be
able to predict the continuous behavior of
alpha wave peaking. Such behavior, for
example, may be important in determining
horizontal and vertical impact loading
potential of shore-breaking waves, and in
sediment transport predictions.

Data tabulated by Putnam (1945)and
Hansen and Svendsen (1979) are used to
determine the nature of the transformation.
The general equation is given by:

H _H 2 tanh d (7)
H, H, H H i


where Q, is a coefficient which determines
where the transformation of H/Hj begins,
given by:

e (
01 (dlH,) -(d )(8)


-Ru 1Uu 1 uuu


Figure 7. Relationship for the prediction of the relative depth of
water at which alpha wave peaking is initiated.


JAAA


0nnnn









in which (dj/H,) is given by equation (6), e is
the Naperian constant, and Q2 determines
the local peaked height during the shore-
breaking process given by:

2= H- 1.0 (9)
H,

where Hb/Hi is given by equation (2).
Equation (7) is evaluated in Figure 8 for
various bed conditions. Data from
Singamsetti and Wind (1980) are not plotted
because the authors did not tabulate the
transformation information. Only four data
points are available for the field data of
Wood (1970, 1971) and are not plotted.
Data from Putnam (1945) for the step slope
could be plotted, but would require
considerable license in estimation to
determine the value of d, (since the waves
began to shore-break on the step slope over
which measurements were widely spaced).
Data of Wang and others (1982), Nakagawa
(1983), and Mizuguchi (1986) provide only
data for H,, H, and T. The more recent data
of Hattori (1986) are not plotted here; it is to
be noted, however, that equation (7)
resulted in essentially precise representation
of his data.

In many of the plots of Figure 8, the
laboratory data suggest that dlbHb is closer
to unity than to a value of 1.28. From
Figure 4, however, it is evident that
laboratory data "tend" toward a lower value.
This may be symptomatic of difficulties in
determining precisely when small laboratory
waves shore-break (i.e., since this must be
visually determined). The terminal boundary
condition of dl/Hb = 1.28 is, therefore,
maintained. Scatter of data relative to
equation (7) is noted in some of the plots.
Overall, however, the shape of the
transformation appears to be well
represented by equation (7).


WA VE CREST ELEVA TION ABOVE
THE DESIGN WATER LEVEL

While wave crest height is useful in
practical applications, there is also a need to
know the wave crest elevation above some
reference plane. For instance, where a
storm tide is used in assessing coastal
engineering design solutions, wave
crest/trough elevations relative to the known
storm tide still water level (i.e., DWL) are
needed.

For example, suppose that the task is
assigned to design the elevation of a fishing
pier deck where the shore-breaking wave
height at the structure from other
calculations is estimated to be 4.5 m
(approximately 15 feet). If, from theoretical
calculations, the sine wave assumption is
used (Figure 9) then one-half, or 2.25 m, of
the wave will lie above the design water
level (SWL). If, however, the Solitary wave
assumption is used, the entire 4.5 m wave
lies above the DWL. This results in a large
design uncertainty of 2.25 m (7.4 feet).
While the sine wave assumption may
actually be too low to insure a safe deck
elevation, the Solitary wave assumption may
be in excess, particularly in view of the high
costs associated with construction and
maintenance in the littoral zone.

The above example, though it states
the basic problem, is an over-simplification.
It is well known that in addition to the design
wave crest elevation above the DWL, other
considerations, in particular the expected
horizontal and vertical design wave impact
loads, should be applied. The latter is
possible only if the nature of wave
transformation during shore-breaking is
known.

An estimation of the amount of the
local wave crest that lies above the DWL,
H', can be attempted using various wave
theories. However, the applicability of
classical theories, although they have











I m = 0.0292 m = 0.0292
SHi = 0.064 m I Hi = 0.0388 m
I = 1.0 a T = 1.25 s


I I
I7


m = 0.0292 I m = 0.0292
H; = 0.0668 m H 0.0961 m-
: T 1.25 I T= 1.67 a

1 r
.... .-


m 0.0292
Hi = 0.0941 m
T = 1.25 s


m = 0.0292
HK = 0.09 m
T = 1.67 S


4k


IL


2.0 m = 0.0292 m = 0.0292
Hi = 0.0801 m Hi = 0.0403 m
ST = 1.67 a *1 T = 1.67 a







1.5 -
I
1.0 -




20m = 0.0292 m = 0.0292
SHi = 0.07 m H = 0.0644 m
T = 1.67 a T 2.0 s



1.5 0
1.0 .I 8 I




m = 0.0292 m = 0.0292





1.5
1. H 0.07 m 0 H 0.0428 m
T = 2.5 s T = 3.33 5 -

*,


0 1 2 3 4 5 6 7 8 9 0 1


2 3 4 5 6 7 8 9 10


d/H
Figure 8. Alpha wave peaking predictions using equation (7) plotted
as the solid curves. Vertical dashed lines represent d/H = 1.28 =
db/Hb. Measured data (solid circles) are for bed slope data of 0.072
and 0.054 from Putnam (1945), and bed slope data of 0.0292 from
Hansen and Svendsen (1979).




































H 1.5
H I
I.

1.0
20




'I







1.5 -

1 1
2.0


1.5 : ;



0 1 2 3 4 6
012346


Figure 8. Icont.)


d/H









Pier Deck


Solitary Wave Assumption

Sine Wave
Assumption


2 mn
Vertical 1 m
Scale
0m


Figure 9. Dependence of design pier deck soffit elevation on wave crest
elevation above the design water level (DWL) for two commonly applied
theoretical approaches.


demonstrated relevance for predicting
"deeper water" wave conditions, are not
specifically designed to predict wave
behavior during the shore-breaking process,
particularly since shore-breaking waves are
not symmetrical in profile view. Rather, the
crests become progressively asymmetrical
and distorted (e.g., Adeyemo, 1968, 1970;
Weggel, 1968).

Despite the underlying importance of
the issue considered, surprisingly little work
has been produced which addresses the
phenomenon. In fact, of the work done, the
paper of Bretschneider (1960), that by its
singularity, becomes a classical
accomplishment. Bretschneider's
nomograph (Bretschneider, 1960; U. S.
Army, 1984, p. 7-107) provides measures of
H' relative to the design water level (i.e.,
DWL which in this paper represents the
SWL). However, for the entire range of
conditions represented by the nomograph no
mathematical description has been
developed.

This author (Balsillie, 1983c)
attempted to formulate a mathematical


representation using a modification of
second order Stokes wave theory. The
results, however, were less than
satisfactory.

A problem associated with the data
used by Bretschneider is that his wave
measurements were single point source
samples. It was determined that, in addition,
it would be valuable to know how the value
of H'/H behaves as the wave progresses
across a shoaling bathymetry to shore-
breaking. Hence, the author (JHB)
conducted a re-analysis.

While the amount of continuous wave
height transformation data is not large, the
data of Putnam (1945) for two bed slopes in
conjunction with Bretschneider's (1960),
Iverson's (1952), Weishar's (1976),
Hansen's (1976) and Balsillie and Carter's
(1980, 1984a, 1984b) and Balsillie's (1980)
point source data provide sufficient
information on which to conduct an
investigation. As before, it becomes
necessary to identify initial and terminal
boundary conditions of the process.









Terminal Boundary Condition


The first boundary condition occurs
where the value of H'/H is evaluated at the
shore-breaking position, that is, the value of
Hb'/Hb.

Using the laboratory data of Iverson
(1952) and field data of Balsillie and Carter
(1980) and Balsillie (1980) the relationship
between the shore-breaking wave height and
the wave trough depth, d, (i.e., the vertical
distance from the wave trough located just
shoreward of the breaking wave crest to the
bottom, referenced to the DWL), may be
found. These data are plotted in Figure 10,
and indicate that dt/Hb = 1.092 for 88 wave
samples. Weishar (1976) reports that dt/Hb
= 1.124 for 116 field measurements. A
weighted average from the two groups of
data yields:


d,
1.11
H,
Iverson's (1952) laboratory
data and field data of Balsillie and
Carter (1980) and Balsillie (1980)
also suggest that the depth at
breaking can be related to the total
depth, de (i.e., the vertical distance
from the top of the wave crest at
shore-breaking to the bottom),
according to:


b- 0.590
do


(10)


(11)


as illustrated in Figure 11.

With the same data source
used to develop equation (11), dt
and d. can be related according to:


dt- 0.529
dc


(12)


which is illustrated in Figure 12.


Hb(m)
Figure 10. Relationship between the
wave height at shore-breaking, Hb, and
the trough depth just preceding the
crest, d4; field data are from Balsillie and
Carter (1980) and Balsillie (1980),
laboratory data are from Iverson (1952).


de-(m)
Figure 11. Relationship between the water depth
at shore-breaking, db, and the total water depth
at shore-breaking, d.; data sources as for Figure
10.

Combination of equations (1) and (10)
through (12), where Hb' = dc db, yields the
average result that:










= 0.84


which is referenced to the
Hansen (1976) reports that:

H (.85)


(13)


DWL.


(14)


where the values are referenced to
the mean water level (MWL) and still
water level (SWL) as indicated by the
subscripts (definitions of MWL and
SWL are those given by Galvin, Figur
1969). Regardless of the slight atsh
discrepancy between the DWL at sh
coefficients of equations (13) and 10.
(14), Hansen's result has been
presented to show that Hb'/Hb is in
the mid-eighty percent range and not some
significantly large or small value.

It is appropriate, also, to address the
effect of shore-breaker type. The field data
of Balsillie and Carter (1980) and Balsillie
(1980) represent plunging and spilling type
shore-breakers. However, no correlation
was found between Hb'/Hb and the shore-
breaker type. Weishar (1976) reports the
results of a field study using ground
photography including the shore-breaker
type. Using the established relationship of
db/Hb = 1.28, Weishar's values of Hb/db
may be transformed to yield:


H) P,


(0.88)n = (0.85)sn


and:


Z,


where the subscripts PI and Sp refer to
plunging and spilling shore-breaker types.
Weishar (1976) is careful to note, however,


do (m)
e 12. Relationship between the total depth
ore-breaking, dc, and the wave trough depth
ore-breaking, d, data sources as for Figure



that considerable variation occurred in the
data and that the difference between mean
values of the last two equations may not be
statistically significant. It is concluded,
therefore, that while there may be
dependence of the value of Hb'/Hb on the
shore-breaker type, sufficient data are not
yet available nor work on other methods
(e.g., Iwagaki and others, 1974)
accomplished to justify such a commitment.
Hence, equation (13) shall prevail as least
equivocal guidance.

Initial Boundary Condition

The other boundary condition occurs at
the beginning of the shore-breaking process.
It has been suggested (see Balsillie, 1983c,
p. 7) that the maximum value of H'/H in
deep water is about 0.64. This maximum
value, then, represents forced wave
conditions (i.e., the waves are subject to the
wind forces from which they were
generated, and maximum wave steepness is
maintained). A value of less than 0.64
represents free or coasting waves (i.e., the
waves are no longer subject to original
generating winds, but have left the
generation area and have or are undergoing


(0.86)MM = (0.82)SW,









dispersion). (See Mooers, 1976; Balsillie and
others, 1976)

Upon reaching transitional water
depths, the bottom slope begins to introduce
an additional effect on the value of H'/H.
Therefore, H'/H may have a value greater
than 0.5 when the wave reaches the point of
initiation of the shore-breaking process.
When a wave reaching the initiation point is
forced, one may expect the progressive
increase in the value H'/H to be minimal,
provided that bed slope conditions do not


change significantly during the
shore-breaking process. It was
found, however, that for free
or coasting waves H'IH does
not begin to significantly
increase in value until shore-
breaking begins, which has
been determined to occur
when the critical alpha wave
peaking depth is encountered.
Near the point of initiation of
shore-breaking, the value H'/H
shall be given the notation
(Hi'/H,)' which requires
quantification.


It was also found r"gure
(Balsillie, 1983b) that the alpha
increase in the wave height wave h
above the DWL begins
somewhat earlier (i.e., further offshore)
in the shore-propagating wave
transformation history than does the
initiation of the alpha wave peaking
process. In terms of design approach,
this complication should not be of undue
concern, since even though the height
of the wave above the DWL might be
slightly increasing, the total wave height
typically is still decreasing...until the
initiation of alpha wave peaking after
which H/Hi and H'/H both increase to
reach a maximum value at the shore-
breaking position. In addition, the initial
values of both H/Hi and H'/H are
significantly small. Even though vertical


changes might be slight in the design
application sense, it would be prudent to
account for this apparent discrepancy.

Using the data of Putnam (1945)
illustrated in Figure 13 and listed in Table 2,
the relative water depth (di/Hi)' indicating
where the increase in wave crest height
above the SWL appears to begin may be
approximated by:


( H,)


- In tanh 201 i
2 gQj


(17)


(AJ
13. Relative depth equations for initiation of
wave peaking at (di/Hi), and at initiation of
eight increase above the DWL at (di/Hi)'.
Table 2. Laboratory data of Putnam (1945)
for evaluation of wave height above the DWL.
m T (H,'lH,)' (dj/H, (H,/g T2)'
0.072 0.865 0.575 3.5 71.4
S 1.15 0.550 4.0 158.7
1.22 0.525 5.0 200.0
1.50 0.525 6.0 384.6
1.54 0.500 6.5 434.8
1.97 0.525 7.0 909.1
0.054 0.86 0.575 3.5 75.2
0.965 0.550 4.0 96.2
1.34 0.550 5.0 303.0
1.50 0.520 6.0 400.0
1.97 0.525 7.0 1000.0
NOTE: H{' is evaluated at the point where the height
above the DWL begins to rise, not at the initiation of
Alpha Wave Peaking ... use equation (20) for
transformation.


rl_..__









Also plotted in Figure 13 is the
relationship for d,/Hi from equation (6)
for comparative purposes.

The wave steepness parameter
at the two points are slightly different
in value. Using the data of Putnam
(1945), the transformation of wave
steepness may be approximated
(Figure 14) by:


( 1.462 ( 0ss


(18)


in which (Hi /g T2)' is the wave steep-
ness parameter where crest height
increase above the DWL is initiated,
and (Hi/g T2) is the wave steepness
parameter at the beginning of alpha
wave peaking.

The relative incident wave
height, (H,'/Hi)' conforming to relative
depth conditions, illustrated in Figure
15, may be given by:


0.5.+ 1.25 q' (19)
Hi 1 79(dfl


where (H2/(g di T2))' is evaluated at )
(di/Hi)'.

Analytically, it becomes of
value to be able to predict the
magnitude of H'/H at the point of
initiation of alpha wave peaking
(i.e., at di/Hi rather than at (d, /Hi)'). Figure
Using the notation (H'/Hi)a, incide
numerical analysis results in a
closely fitted value according to:


H- = 0.515 + 12 (20)
H, g T'

It is interesting to note that where
limiting conditions are imposed,
equation (19) provides consistent results.


10" / H, /H \1" -








r tan Erb
102- *t07
--








Figure 14. Wave steepness parameter at (d,/H,)'
relative to the wave steepness parameter at
(di/Hi).

0.7 -.. . ...... ......... . .....


Ji ., . .. ...I ,.,- J , .


\g di2/

S15. Relationship for prediction of
nt value of H'/H.


the


Suppose that a profile occurs where a wave,
initially in deep water, suddenly encounters
an abrupt slope change and it must shore-
break. Suppose, also, that the wave in deep
water is fully forced so that (Ho'/H) =
(Hi'/Hi)' = 0.64. For such conditions, from
equation (19), (Hi /g (di T2))'-' = 56.4. From
another viewpoint, where Hb/db = 1/1.28 =
0.78 and (H /g T2)' = 1/(14 r), then (Hi /(g


-Se text
.Equation (19)

00.72 "---s


m-
A 0.054
0 0.072


1 0.5








di T2))'-1 = 63.8, and (Hi'/Hi)' = 0.648.
Both solutions are close and represented by
the asterisk in Figure 15.

Also, where the maximum possible
value of (H'/H) in equation (20) is 0.84,
representing maximum forced wave
conditions in shallow water, the value of (Hi
/(g T2)) becomes 0.0271. Where Hb/Lb =
0.79 (Hb /(g T2))05 given by Balsillie (1984a),
then by substitution (H/L)max for shallow
water waves becomes 0.13 or 1/7.7. This
approximation is consistent with the Michell
(1893) criterion of 1/7.

Prediction of H'H Transformation

Incorporation of the preceding
boundary conditions leads to the following
development describing the H'/H
transformation during the shore-breaking
process. The general equation is given by:


H'
H


4{4 anh [ .- (
it 1 0, 9 b


where db/Hb = 1.28, and:
e
(d, H,)' (db/H,,)

in which e is the Naperian
is 1.28, and:


constant, (t


Hb H,
Hb H,)


steepness values tend to have larger initial
H'/H values). There is, however,
discrepancy between measured and
predicted values of H'/H as the shore-
breaking position is closely approached.
Putnam's (1945) laboratory data tend to
consistently underestimate the value of H'/H
very near and at the shore-breaking position.
One must recall, however, that the behavior
of the curve predicted by equation (21) is
determined by the terminal boundary
condition that Hb'/Hb = 0.84, which is
based on prototype wave data and results
from other investigations. Hence, laboratory
conditions or measurement techniques may
account for the apparently low values of
Putnam's data near shore-breaking. While
the terminal boundary condition must surely
be refined by future research efforts,
equation (21) would appear to provide a
satisfactory and, certainly, a useful method
for predicting H'/H during the shore-breaking
process.


(21)
(21) Equation (21) is also tested using the
prototype laboratory wave (Figure 17) and
field hurricane waves (Figure 18) reported
by Bretschneider (1960). Figure 17 indicates
very good agreement. Figure 18, however,
indicates that equation (21) underestimates
(22) H'/H. One must keep in mind that the
measured data represent the maximum wave
height occurring during one-minute recording
db/Hb) periods, not the average wave height.
Therefore, one would expect equation (21)
to underestimate the measured data, and
that the plotted line represents an expected
(23) upper limit as also found by Bretschneider
(1960).


where from equation (16) Hb'/Hb = 0.84,
(Hi'/Hi)' is given by equation (19), and (di/Hi)'
is given by equation (17).

Equation (21) is plotted in Figure 16
as the solid curves. It is to be noted from
these plots that equation (21) appears to
successfully represent initial values of H'/H
(i.e., incident waves with larger wave


Based on the success of equation
(21), the value of H'/H as a function of H/(g
T2) and d/(g T2) is given by Figure 19 for
transitional and shallow water depths.
Figure 19, then, provides an alternative to
the nomographic approach originally
proposed by Bretschneider (1960).
































H' *
0.9
H m = 0.072 m = 0.054
Hi = 0.041B4 m Hi = 0.0949 m
0.7 T = 1.97 s T = 0.965 s

I e** ( *
0.5 *" e -
0.0
0,9 -----
m = 0.054m 0054
"dHj = 0.05807 m Hi = 0.03803 m
0.7 T = 1.34 s T = 1.97


0.5 1 I I &
0.9 1
0 5 10 15 20 25
ee m m 0.054
Hi = 0.05513 m
0.7 T = 1.5 s


0 .5 1 F** '-* *-

0 5 10 15 20 25
d/H
Figure 16. Transformation of H'/H using equation (21) plotted as solid
curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured
data (solid circles) are from Putnam (1945).















H "
pred %

1 .. I





0 1 2
H'meas (M)
Figure 17. Predicted H' from equation
(21) versus prototype Beach Erosion
Board wave tank data reported by
Bretschneider (1960).


CONCLUSIONS

Classical wave theory, among certain
coastal practitioners, has been viewed as the
"last word" in the prognostication of water
wave behavior. However, when applied to
the progressive distortion of waves as they
approach and reach shore-breaking, classical
wave theory falls short of providing realistic
solutions. A major shortcoming of wave
theories is that they require specification of
variables not known or which can only be
approximated, in particular the wavelength.
Shortcomings of theories have been
presented in the second section of this work.

The results of this work, however,
provide a methodology for wave height
transformation during the wave breaking
process up to the point of shore-breaking,
requiring only the specification of initial wave
height and period; bed slope is apparently
not a factor affecting the transformation
(other research suggests that bed slope is
more nearly instrumental in influencing the
type of shore-breaker that will be produced
(e.g., Balsillie, 1999b)).


pred








1 1 2
""Bas (m)
Figure 18. Predicted H' from equation
(21) versus Lake Okeechobee hurricane
wave data reported by Bretschneider
(1960).


While wave generation equations
(e.g., for storms and hurricanes) and
classical theory can be used to generate and
propagate waves to the initial or incipient
shore-breaking depth (i.e., d/H,)', the
importance of the numerical methodology
developed here is that it requires no a priori
historical knowledge of wave height behavior
seaward of d/Hi or (d/H,)'. Equations can be
simply evaluated using a hand-held scientific
calculator; other equations may be as simply
evaluated, while wave height transformation
equations detailing the shore-breaking
process may be more easily evaluated using
a programmable calculator or computer.

For the data used in this study, wave
peaking transformation is initiated in water
depths of from about 1.5 to 7 times the
incident wave height. Two mechanisms of
wave breaking transformation have been
numerically quantified. The first deals with
change in total height of the wave, and the
second with the change in crest height
above the still water level. The first





















H
D.7



0.6



0.5 --L
Shallow and Tranrletlonal Water -- Deep
I I I Water
II l l I I I t ll I t I I I1 I I II I i
0.0001 0.001 0.01 0.1
d
gT2

Figure 19. Ratio of wave crest elevation above the design water level to wave height
from equation (21).


mechanism is based on a larger set of data
from a variety of sources and representing
many bed slope conditions. For the data
used in this study, the total wave height at
shore-breaking ranged from 1 to about 2.5
times the height of the incident wave. The
second mechanism is based on a much
smaller data set, but the reader should note
that it is essentially a "fine-tuned" version of
methodology presently existing. The amount
of the wave crest lying above the still water
level ranged from 0.5 to 0.84, with a
constant value of 0.84 at the shore-breaking
position.

Some of the major numerical
attributes of the introduced prediction
methodology which address shortcomings of
classical wave theories include:


1. The methodology does not
require that the wavelength is
known to calculate the basic
wave parameters through the
shore-breaking process.

2. The wavelength and wave
celerity, however, can be
calculated using empirically
derived expressions addressed
in a companion work (Balsillie,
1999a).

3. The point of initiation of
shore-breaking (i.e., incipient
shore-breaking given by di /Hi)
can be determined from Hi /g
T2. Therefore, one needs only
to know of the incident wave
height and period to









determined the water depth at
which shore-breaking begins.

4. The final shore-breaking
wave height can be
numerically determined from
the incident wave height and
wave period.

5. Continuous quantitative
peaking of the wave height
can be numerically determined
through the shore-breaking
process from incipient shore-
breaking to final shore-
breaking.

6. The amount of the wave
crest lying above the still
water level can be numerically
determined continuously
through the shore-breaking
process.

7. A shore-breaker
classification can be redefined
as a numerical continuum.

8. The methodology requires
no apriori knowledge of the
wave behavior seaward of di
/Hi rather based on incident
wave height and period, it is
dependent upon waterdepths.

Figure 20 is a modified version of
Figure 2-7 from the Shore Protection Manual
(U. S. Army, 1984), originally proposed by
LeMehaute (1969). It has become a
"standard" among coastal engineers for
identifying regions of validity for wave
theories. It is modified as follows. First, the
terminology validity is invalid, and is replaced
by "regions of applicability"; this is so
because theories by definition have no
validity since they are unproven. Second,
the "breaking" region in the upper left-hand
part of the figure is renamed "broken" since,
by definition, waves have broken when d/H


= 1.28 or H/d = 0.78 is reached. Third,
the "nonbreaking" region has been moved to
the right and, fourth, a "shore-breaking
region has been identified.

In addition to wave theories, the
region of applicability of results from this
work are plotted on Figure (20) as the stipled
region. The line labelled "incipient shore-
breaking" identifies the beginning of the
shore-breaking process given by equation
(6). It is to be noted that as waves become
longer (i.e., smaller values of H/g T2) wave
peaking becomes greater (i.e., Hb /Hi
becomes greater in value). Very steep
waves (i.e., large values of H/g T2) peak very
little or not at all.

The delineation of "shallow water"
and "transitional water" depths at a value of
d/L = 0.04 would appear to be
inappropriate. The results of this study
suggest that the delineation should be
dropped and that shallow water should begin
at the line segment BC of Figure 20 (which
is given by equation (6)).

NOTE

It is important to note that variables in
this paper represent average wave heights
and periods, etc. Moment variable statistics
relating to significant or maximum wave
heights, for instance, may be found in other
works (e.g., Balsillie and Carter, 1984a,
1984b; U. S. Army, 1974).

ACKNOWLEDGEMENTS

Florida Geological Survey staff
reviews were conducted by Paulette Bond,
Kenneth Campbell, Thomas M. Scott, Joel
Duncan, Edward Lane, and Steven Spencer,
whose editorial comments are gratefully
acknowledged.

The review comments of William F.
Tanner of Florida State University are
gratefully acknowledged. Nicholas C. Kraus










d
= 0.04
L
d
-g T2 0.00155
-Shallow Water


d
- = 0.13
L
d
-- = 0.014
alT


d
- 0.5
L
d
-- 0.0792
gT DWatr
- 4 Deep Watr -


0.001


0.005





0.0001


0.0005
01.0005


C 0.005 0.001 0.01 0.06 0.1 0.4
d
gT2


Figure 20. Regions of applicability for various wave theories and for the Alpha
Wave Peaking methodology of this work.


of the U. S. Army, Coastal Engineering
Research Center (now the Coastal
Engineering Laboratory) also reviewed the
manuscript.


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H
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NOTATION

The following symbols are used in
this paper:

c = local wave speed.

cb = wave speed at shore-breaking.

ci = wave speed at initiation of wave
peaking.

d = local water depth measured from the
DWL.

db = water depth at shore-breaking.

d, = water depth from the wave crest to
the bed at shore-breaking.

di = water depth at initiation of wave
peaking.

dt = water depth from the trough fronting
the wave crest to the bed at shore-
breaking.

dti = water depth from the trough fronting
the wave crest to the bed at initiation
of wave peaking.

DWL = design water level (in this paper it
is the SWL).


e = Naperian constant.

Eb = total wave energy at shore-breaking.

Ei = total wave energy at initiation of
wave peaking.

g = acceleration of gravity.

H = mean local wave height.

Hb = mean wave height at shore-breaking.

Hb' = mean amount of wave crest at
shore-breaking lying above the DWL.

Hi = mean wave height at the initiation of
wave peaking.

Hi' = mean amount of wave crest lying
above the DWL at the initiation of
wave peaking.

Ho = mean deep water wave height.

Hm = mean wave height measured in the
constant depth portion of a laboratory
wave channel.

Ht = water depth from wave crest to bed
at the initiation of wave peaking.

L = local wavelength.

Lo = deep water wavelength.

m = bed slope.

MWL = mean water level.

PL = subscript denoting the plunging-type
shore-breaking wave.

SP = subscript denoting the spilling-type
shore-breaking wave.

SWL = still water level.









T = wave period.

Q = relating coefficients developed in the
text.









WAVELENGTH AND WAVE CELERITY
DURING SHORE-BREAKING


by

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

ABSTRACT

Prediction of wave phase speed and, hence, wavelength at shore-breaking has remained
a controversial issue. Based on available field data (n = 47) and laboratory data (n = 40 to 71),
a family of relationships is derived for predicting wavelength at shore-breaking. Assuming
approximate linear wave speed attenuation, a method is derived for prediction of wave speed
during the shore-breaking process.


INTRODUCTION

Wave height, H, wavelength, L, wave
period, T, water depth, d, and bed slope tan
a, constitute the fundamental hydraulic
variables forming the basis for derivation of
composite parameters (e.g., wave
steepness, H/L; or equivalent wave
steepness parameter, H/(g T2); surf similarity
parameter, tana/(H/g T2)1/2; etc.) utilized in
most coastal engineering design
applications. It becomes not only desirable
to be able to provide for determination of
such parameters over a wide variety of
conditions in order to accurately describe a
natural process, but to be able to provide
the simplest and most straightforward
procedures possible.

As the number of basic variables
becomes large, the solution of any problem
invariably becomes proportionally more
complex. It becomes desirable, therefore, to
provide methods for predicting as many of
the variables as is feasible. One such
variable is the wavelength.

As will become evident,
determination of the local values of H and d
as waves shore-propagate over a shoaling
bed is complex, since following specification


of their values, which may exhibit a wide
range, H relative to d experiences additional
and significant progressive transformations
as shoaling continues. However, the wave
period, once initially specified, is considered
to be conserved (i.e., remains invariant)
across the shoaling bathymetry until
breaking occurs, and a simplifying condition
emerges. The wavelength, however,
experiences attenuation during shore-
breaking, thereby introducing additional
complexity. The wavelength not only
appears in many shoaling design wave
equations, but usually just when one has
little insight as to its local value short of
tedious theoretical calculations for obtaining
an estimation. Most importantly
wavelength is related to the wave speed and
wave energy.

It becomes important, therefore, to
provide methods) for prediction of the
wavelength and wave speed. In this paper
such prediction is investigated during the
shore-breaking process.

DISCUSSION AND RESULTS

As shore-propagating waves
approach the shoreline across shoaling










I 1 II1 I I F


= l.0110

{ --- -
~ `~ ~5 ~ ^ o


Hi

Direction of Wave
Travel


N


'0-0


S[ Data from Putnam (1945)
T = 0.065 s


I -








A I -

I t I


Adjusted
H
H.


Figure 1. Illustration of wave transformation from deep water to shore-breaking, where
the alpha wave peaking process (segment A B) is given the notation ap.


bathymetry, the wave height tends to
initially decrease due to a number of factors
such as friction, etc., and then begins to
increase rapidly in height just before shore-
breaking occurs (Scripps Institute of
Oceanography, 1944a, 1944b; Putnam,
1945; Munk, 1949; Iverson, 1952; Stoker,
1957; Kinsman, 1965; Balsillie, 1983a,
1983b, 1983c, 1999a). The transformation
is illustrated in Figure 1 (notation is defined
at the end of the paper). It is the increase in
wave height which accompanies the shore-
breaking process (note: waves may break in
relatively deeper water due only to critically
high wind stresses which cause waves to
become critically steep (i.e., forced waves);
shore-breaking occurs primarily because
water depths become critically shallow).
Shore-breaking wave mechanics are
described by the alpha wave peaking
concept (Balsillie, 1980, 1983b, 1983c, in
1999a) and denoted by the symbol a in
Figure 1. Alpha wave peaking, then,


describes the zone of interest
investigation of the wave celerity
wavelength.


for
and


The speed with which a group of
waves, comprising a wave train, travels is
not always equivalent to the speed of
individual waves within the group. The
individual wave speed, termed the phase
speed, is given by:


and group wave speed, cg, by:


C, = n


In deep water (i.e., d/L > 0.5) n = 0.5; in
intermediate water depths (i.e., 0.04 < d/L
< 0.5) n increases in value to become,
finally, n =1 in shallow water (i.e., d/L <


Constant Deptl

.-- Deep Water


tan cb = 0.054 I


1.0 --


0.7 6
10


9 8 7 6 6 4 3 2
d/H


I









0.04) and c = cg.


According to small amplitude (Airy)
wave theory, the phase speed and
wavelength in any depth of water may be
given by:


gT tanh 2wd
2w L


to have been proliferated in the literature.

Van Dorn (1978)found that at shore-
breaking, the wave speed was always
greater than the small amplitude speed of
equation (4), and smaller than the solitary
wave celerity of equation (5). He reports
that:


which is evaluated in the following section.

In this work the alpha wave peaking
process is assumed to occur in shallow
water where n = 1. In order to determine
the transformation of c and L during shore-
breaking, one first needs to have knowledge
of the governing boundary conditions.

Terminal Boundary Conditions

Equation (3) is has been applied to
predict conditions at shore-breaking. Shore-
breaking defines the terminal boundary
condition for the phenomena considered in
this work. A more appropriate application
from small amplitude (Airy) wave theory is
given by:


c7-
T


9d1


or where Solitary wave theory is applied, by:


Lb
cb-
T


g{db + H1)


where Hb' is that portion of the wave height
at shore-breaking lying above the design
water level (DWL). [Note: the entire wave
crest of a solitary wave lies above the still
water level and Hb' = Hb.] About equation
(5), Smith (1976) states: "Although this
equation is widely used in the literature on
wave theories and is generally accepted,
few discussions have been presented which
establishes its validity." The same appears
to be true of equation (4), while a general
misunderstanding about equation (3) seems


C& =2g1


which was found to agree roughly with that
predicted for limiting Stokes waves in deep
water.

Available field and laboratory data
(see the Table) are used to evaluate the
above equations. The wave celerity is
analyzed in terms of the wavelength rather
than the wave speed since the length yields
a much wider range of values (from 0.5 to
100 m, or 2 orders of magnitude). The data
are plotted in Figure 2. Figure 2 illustrates
that equation (3) does not predict Lb and,
hence cb, with the precision of the other
fitted relationships. It is to be noted,
however, that equation (3) was developed
from theoretical considerations to represent
an upper limit envelope curve (see equation
4 for a suggested correction factor and fig.
8, both from Bretschnieder, 1960). In
addition, because equation (3) is an
algorithm it is awkward to apply and not
generally recommended for use in design
work, at least not in the breaker zone.

Based on empirical evidence (field
and laboratory data of the Table), equations
that more successfully predict expected
values in shallow water are closely given by:


L, g d,



2


L
T-
T









Table of statistics relating
breaking position.


and:

Lb= T g (db+Hd


Where db 1.28 Hb (McCowan,
1894; Munk, 1949; Balsillie, 1983a, 1985,
1999a, 1999b)and Hb' = 0.84 Hb (Balsillie,
1983c, 1985), the previous three
relationships can be modified to yield two
additional equations, as:


Lb 3 gHr (10)


and


Lb, g (d + H,) (11)
4


measured and predicted wavelengths at the shore-


and we now have a family of design
relationships for the prediction of Lb and cb.


Initial Boundary Conditions


With the exception of the results of
Hansen and Svendsen (1979), there is little
if any, data available which will allow for
determination of the wave speed at the
point of initiation of alpha wave peaking
(i.e., at cb). Based on other alpha wave
peaking investigations (Balsillie, 1980,
1983b, 1983c, 1999a), it may be
reasonable to assume that ci can be related
to cb. However, the problem is encountered
that the difference between ci and cb is
slight, at least compared to natural
variability in the data and possible
measurement errors.


Investigator L = m TL~ Tg;H) Lb m Tg
and n
Category
m r m r m r
FIELD DATA
Gaillard (1904) 26 0.9462 0.9608 0.7514 0.9587 1.241 0.9404
Balsillie and Carter (1980) 21 1.226 0.9692 0.9101 0.9695 1.359 0.9689
Field Results 47 0.9832 0.9737 0.7737 0.9757 1.259 0.9720
LABORATORY DATA
Galvin and Eagleson (1965) 24 1.342* 0.3603* 0.8843* 0.3842* 1.178 0.3658
Eagleson (1965) 7 1.084* 0.8399* 0.8131* 0.8850* 1.233 0.7985
Van Dorn (1976, 1978) 12 1.264 0.9636 0.8895 0.9889 1.254 0.9933
Hansen and Svendsen (1979) 28 1.113 0.9947 0.8036 0.9962 1.162 0.9969
Laboratory Results 40-71 1,205 0.9858 0.8561 0.9947 1.211 0.9933
ALL DATA
Total Results 87-118 0.9972 0.9801 0.7794 0.9210 1.254 0.9836
Weighted Results 87-118 1.111 ----- 0.829 ----- 1.251 --
Adjusted Weighted Results** 87-118 1.1176 ----- 0.8374 ----- 1.2644
NOTES: Unless otherwise indicated all m and r are from regression analyses.
These results represent db referenced to MWL and are not used in determination of m, all others used in
the analysis are referenced to SWL.
** Adjusted values were determined so that resulting equations yield intra-consistent results.



























0.1 i I I i I. 'I I II I lI I i lI I I I II I 1
0.3 05 0.1 0.5 0. 0.5 0.1 0. 5 10 100
Predicted Lb (m)
Figure 2. Evaluation of relationships for prediction of the wavelength at the shore-
breaking position. From left to right figure equations are given by text equations (3),
(7), (9), and (8), respectively.


Another approach using theoretical
reasoning proves useful. The total energy of
a wave is the sum total of its kinetic and
potential energies. The kinetic energy is
that portion of the total energy due to water
particle velocities associated with wave
motion. Potential energy is that portion of
the total energy resulting from the wave
fluid mass lying above the still water level
(SWL). Based on solitary wave theory,
Balsillie (1984) determined expressions for
the kinetic, potential and total wave energies
at the initiation of alpha wave peaking and
at the shore-breaking position. Application
of the Rayleigh assumption (Eagleson and
Dean, 1966),.e., ... ci ETi = cb ETb (where
ci and ETi are the wave speed and total
wave energy, respectively, at the initiation
of alpha wave peaking, and cb and ETb are
their counterparts at shore-breaking),
resulted in a working relationship given by:


c, L Lr 1.84 1 +
c, L, L[, H


(12)


in which (H'/H)ia is the percent of the wave
crest height lying above the SWL at the
initiation of alpha wave peaking (Balsillie,
1983c, 1999a), given by:


( H'
H ),~


0.515 + 12


(13)


sgT


Comparison of the results from
equation (12) with the data of Hansen and
Svendsen (1979) suggests a slight
modification to the coefficient of equation
(12) by a factor of P, = 1.073, and:










C1 T L, H]
. 1.840 1 1 + (14)
Cb Lb Lb k1 ,J
T

Measured data and predicted results from
equation (14) using Hansen and Svendsen's
(1979) data are illustrated in Figure 3.

Wave Celerity and Wavelength
Transformation During Alpha
Wave Peaking

The data of Hansen and Svendsen
(1979) for a slope of 0.0292 and that of
Hedges and Kirkgdz (1981) for slopes of
0.2247,0.1404,0.102 and 0.0667 suggest
that the transformation of c, the local wave
celerity during shore-breaking, is non-linear
but only slightly so. For three bed slopes of
0.022, 0.040, and 0.083, Van Dorn (1978)
illustrates that the transformation of c is
only slightly non-linear and that it accelerated
at a rate close to -0.5 g tan ab.
Unfortunately, Van Dorn did not publish his
transformation data, Hedges and Kirg6z did
not publish their breaker data, and only the
data of Hansen and Svendsen, for a single
slope, are available.

A power curve is used to more nearly
represent the slightly non-linear attenuation
of the wave speed during shore-breaking.
Two points were used in the curve fitting
analysis: 1. ci /cb from equation (14) and,
2. c/cb = 1.0 at the shore-breaking position.
The following equations provide for
computation of wave speed (and, thereby,
wavelength) attenuation with a slightly non-
linear character:


c T L _3(d 2
Cb L, Lb H


(15)


1.5


1.0


0.5
0.5


Measured c, (m/s)


Figure 3. Comparison of predicted and
measured wave speeds at the point of
initiation of alpha wave peaking;
predicted data are from equation (14),
measured data from Hansen and
Svendsen (1979).

where c is the local wave speed,


02 = 0.1475tanh 0.00074 Hi-1 (16)
gT2


03 = 0.9987 0.2338,2


(17)


Examples from equation (15) are
illustrated in Figure 4. Equation (15) is valid
where (di /Hi) (d/H) 2 1.28 where di /Hi
is the relative water depth at which (H'/H),
occurs given (Balsillie, 1983b, 1999a) by:


d ,. td b W H, H )l
n tanh (65
H, H, 211 gjj


(18)


The shore-breaking wave height, Hb,
used in computation of the wave speed at
the shore-breaking position, cb, may be
evaluated from incident wave conditions
(Balsillie, 1983b, 1999a) according to:










-= 1.0 0.4 n tanh 100 H\\
His~l


CONCLUSIONS


Three issues concerning the
prediction of the wavelength and wave
speed during the shore-
breaking wave process 1.30
have been addressed.
1.25


First, a family of
relationships based on field
and laboratory data has
been defined for
determination of L and c at
the shore-breaking
position. These
relationships have been
used to refine theoretical
predictions from small
amplitude (Airy) and
solitary wave theories.
The family of derived
relationships provides for
alternate data sources to


1.20

1.15


c/cb


I.J tP .I UI I I I U A -


Third, using the above two equations
(19) as boundary conditions, the transformation
of c and L during alpha wave peaking may
be predicted using equation (15).


The results presented herein are
important because they provide a
methodology for predicting wavelength and
celerity requiring that only the incident wave

: ... . .


2 3 4 5 6 7
d/H


Figure 4. Illustration of wave speed attenuation during
shore-breaking using equation (15) for various initial wave
steepness values.


more closely facilitate the
needs of the coastal engineer whose
completeness in data may differ from project
to project. In addition, the commonly used
algorithm given by equation (3) is assessed.
Not only is the expression difficult to apply,
but in surf zone applications it has often
been incorrectly used since it represents an
upper limit envelope fit. In view of the
developments presented in this work, the
continued use of equation (3) in surf zone
applications is no longer necessary.

Second, based on consideration of
Solitary wave theory and the Rayleigh
assumption, the wavelength and wave
speed at the initiation of shore-breaking (i.e.,
beginning of alpha wave peaking where the
wave crest begins to increase in height) may
be predicted using equation (14).


height and period are known. As more data
become available, refinement or
corroboration of the relationship for
prediction of the initial or incipient
wavelength or celerity is desirable. The
same is true for prediction of wavelength
and celerity transformation throughout the
shore-breaking process, although as noted
the slight non-linearity in attenuation should
not pose significant problems.

ACKNOWLEDGEMENTS

Florida Geological Survey staff
reviews were conducted by Jon Arthur,
Paulette Bond, Kenneth Campbell, Joel
Duncan, Ed Lane, Jacqueline Lloyd, Deborah
MeKeel, Frank Rupert, and Thomas Scott.
Their editorial comments are greatefully
acknowledged.


Hi
T7


I ..









The review comments of William F.
Tanner of Florida State University are
gratefully acknowledged. Nicholas C. Kraus
of the U. S. Army, Coastal Engineering
Research Center (now the Coastal
Engineering Laboratory) also reviewed the
manuscript.

REFERENCES

Balsillie, J. H., 1980, The peaking of waves
accompanying shore-breaking: in
Tanner, W. F., ed., Shorelines Past
and Present, Department of Geology,
Tallahassee, FL, Florida State
University, v. 1, p. 183-247.

1983a, 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.

1983b, The transformation of
the wave height during shore-
breaking: Florida Department of
Natural Resources, Beaches and
Shores Technical and Design
Memorandum No. 83-4, 33 p.

1983c, Wave crest elevation
above the design water level during
shore-breaking: Florida Department
of Natural Resources, Beaches and
Shores Technical and Design
Memorandum No. 83-5, 41 p.

1984, Wave length and wave
celerity during shore-breaking:
Florida Department of Natural
Resources, Beaches and Shores
Technical and Design Memorandum
No. 84-1, 17 p.


1985, Redefinition of shore-
breaker classification as a numerical
continuum and a design shore-
breaker: Journal of Coastal
Research, v. 1, p. 247-254.

,_ 1999a, Wave height
transformation during shore-
breaking: Florida Geological Survey,
Special Publication No. 21, p. 1-30.


1999b, On the breaking of
nearshore waves: Florida Geological
Survey, Special Publication No. 45,
p. 1-155.

Balsillie, J. H., and Carter, R. W. G., 1980,
On the runup resulting from shore-
breaking wave activity: in W. F.
Tanner, ed., Shorelines Past and
Present, Tallahassee, FL,
Department of Geology, Florida State
University, v. 2, p. 269-341.

Balsillie, J. H., Campbell, K., Coleman, C.,
Entsminger, L., Glassen, R.,
Hajishafie, N., Huang, D., Tunsoy, A.
F., and Tanner, W. F., 1976, Wave
parameter gradients along the wave
ray: Marine Geology, v. 22. p. M17-
M21.

Bretschneider, C. L., 1960, Selection of
design wave for offshore structures:
Transactions, American Society of
Civil Engineers, v. 125, pt. 1, paper
no. 3026, p. 388-416.

Eagleson, P. S., 1965, Theoretical study of
longshore currents on a plane beach:
Massachusetts Institute of
Technology, School of Engineering,
Hyrdodynamics Laboratory Report
No. 82, 48 p.









Eagleson, P. S., and Dean, R. G., 1966,
Small amplitude wave theory. in A.
T. Ippen, ed., Esturay and Coastline
Hydrodynamics, New York,
McGraw-Hill Inc., chap. 1, p. 1-92.

Gaillard, D. D., 1904, Wave action in
relation to engineering structures: U.
S. Army Corps of Engineers
Professional Paper No. 31. p. 110-
123.

Galvin, C. J., Jr., and Eagleson, P. S.,
1965, Experimental study of
longshore currents on a plane beach:
Coastal Engineering Research Center
Technical Memorandum No. 10, 80
p.

Hansen, J., and Svendsen, I. A., 1979,
Regular waves in shoaling water:
Technical University of Denmark,
Institute of Hydrodynamics and
Hydraulic Engineering, Series Paper
No. 21, 233 p.

Hedges, T. S., and Kirkgdz, M. S., 1981,
Experimental study of the
transformation zone of plunging
breakers: Coastal Engineering, v. 4,
p. 319-333.

Iverson, H. W., 1952, Laboratory study of
breakers: in Gravity Waves, National
Bureau of Standards Circular 52, U.
S. Government Printing Office,
Washington, D. C., p. 9-32.

Kinsman, B., 1965, Wind Waves, Pentice-
Hall, Inc., Englewood Cliffs, New
Jersey, 676 p.

McGowan, J., 1894, On the highest wave
of permanent type: Philosophical
Magazine, Edinburgh, Series No. 5,
v. 32, p. 351-358.


Mooers, C. H. K., 1976, Wind-driven
currents of the continental margin: in
Stanley, D. J., and Swift, D. J. P.,
eds., Marine Sediment Transport and
Environmental Management, New
York, John Wiley and Sons, p. 29-
52.

Munk, W. H., 1949, The solitary wave and
its application to surf zone problems:
Annals of the New York Academy of
Sciences, v. 51, p. 376-424.

Putnam, J. A., 1945, Preliminary report of
model studies on the transition of
waves in shallow water: University
of California at Berkeley, College of
Engineering, Contract Nos. 16290,
HE-116-106 (declassified U. S. Army
document from the Coastal
Engineering Research Center), 35 p.

Scripps Institute of Oceanography, 1944a,
Waves in shoaling water: Scripps
Institute of Oceanography, La Jolla,
CA, S. I. O. Report No. 1, 28 p.

1944b, Effect of bottom slope
on breaker characteristics as
observed along the Scripps
Institution pier: Scripps Institute of
Oceanography, La Jolla, CA, S. I. O.
Report No. 24, 14 p.

Smith, R. M., 1976, Breaking wave criteria
on a sloping beach: M. S. Thesis, U.
S. Naval Postgraduate School,
Monterey, CA, 97 p.

Stoker, J. J., 1957, Water Waves, New
York, Interscience Publishers, Inc.,
567 p.

Van Dorn, W. G., 1976, Set-up and set-
down in shoaling breakers:
Proceedings of the 15th Conference
on Coastal Engineering, chap. 41, p.
738-751.









1978, Breaking invariants in
shoaling waves: Journal of
Geophysical Research, v. 83, p.
2981-2988.

NOTATION

Symbols

c local wave (phase) speed or celerity

cg group wave speed

d local water depth measured from the
DWL

DWL design water level

g acceleration of gravity

H mean local wave height

H' mean local wave height lying above
the DWL

L local wavelength

m coefficient

n number of data points comprising a
sample

r Pearson product-moment correlation
coefficient

SWL design water level represented by the
still water level

T wave period

tan a bed slope

ab alpha wave peaking

C(n coefficients

Subscripts

b parametervalue at the shore-breaking


position (i.e., at termination of alpha
wave peaking).

i parameter value at the beginning of
shore-breaking (i.e., at the initiation
of alpha wave peaking)

m parameter value before entering
transitional water depths




Shore-breaking wave height transformation
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 Material Information
Title: Shore-breaking wave height transformation
Added title page title: Wavelength and wave celerity during shore-breaking
Physical Description: x, 40 p. : ill. ; 28 cm.
Language: English
Creator: Balsillie, James H.
Publisher: Florida Geological Survey
Place of Publication: Tallahassee, Fla.
Publication Date: 2000
 Subjects
Subjects / Keywords: Water waves -- Mathematical models   ( lcsh )
Ocean waves -- Research   ( lcsh )
Marine sediments -- Analysis -- Florida   ( lcsh )
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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 Notes
Statement of Responsibility: by James H. Balsillie and, Wavelength and wave celerity during shore-breaking / by James H. Balsillie.
General Note: Florida Geological Survey special publication number 41
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Table of Contents
    Front Cover
        Front Cover 1
        Front Cover 2
    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
    Shore-breaking wave height transformation
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
    Wavelength and wave celerity during shore-breaking
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
    Back Matter
        Page 41
        Page 42
    Back Cover
        Page 43
        Page 44
Full Text




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




Division of Resource Assessment and Management
Edwin J. Conklin, Director




Florida Geological Survey
Walter Schmidt, State Geologist and Chief






Special Publication No. 41


Shore-Breaking Wave Height Transformation

by

James H. Balsillie

and

Wavelength and Wave Celerity During Shore-Breaking

by

James H. Balsillie


Florida Geological Survey
Tallahassee, Florida
2000


'V




























NO C


Printed for the
Florida Geological Survey

Tallahassee, Florida
2000

ISSN 0085-0640










LETTER OF TRANSMITTAL


Florida Geological Survey
Tallahassee

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

Dear Govemor Bush:

The Florida Geological Survey, Division of Resource Assessment and Management,
Department of Environmental Protection is publishing two papers, in this, Special Publication No.
41: "Shore-breaking wave height transformation" and "Wavelength and wave celerity during
shore-breaking".

The first paper identifies a numerical methodology for transforming wave height during the
shore-breaking process. The second paper is a companion paper, which provides a numerical
methodology for transforming the wavelength and wave celerity during the shore-breaking
process.

Both papers provide new technology based on measured data (field and laboratory) and,
therefore, do not constitute theoretical approaches currently employed by coastal practitioners.
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, to name but a few.

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 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.

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 wave
characteristics. 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, does not only have a significantly long
shoreline, but it has a large wave energy variability and a significant level of coastal
development. 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.

James H. Balsillie
February 2000










CONTENTS



SHORE-BREAKING WAVE HEIGHT TRANSFORMATION


Page


. 1


A BSTRA CT ..................................................


. . . . . . . . . . . . . . . . . . . . . . 1


CLASSICAL WAVE THEORIES ...........................


WAVE HEIGHT GROWTH DURING SHORE-BREAKING ..........


Terminal Boundary Condition .......................


Initial Boundary Condition .........................


Transformation of H/Hi ...........................


WAVE CREST ELEVATION ABOVE THE DESIGN WATER LEVEL . .


Terminal Boundary Condition .......................


Initial Boundary Condition .........................


Prediction of H'/H Transformation ...................


CONCLUSIONS .....................................


N O T E . . . . . . . . . . . . .. . .. .. . . . . . .


REFERENCES .......................................


NOTATION ........................................


. . . . . . 2


. . . . . . 4


. . . . . . 7


. . . . . . 9


. . . . .. . 10


. . . . . . 1 1


. . . . . . 15


. . . . . . 16


. . . . . . 19


. . . . . . 2 1


........... 23


........... 23


........... 28
.2o , ,































... .. .28


LIST OF TABLES


Table 1. Field and laboratory data used in analyses. .......................... 5


Table 2. Laboratory data of Putnam (1945) for evaluation of wave height above
the D W L. .. .. .. ............................. ......... ....... 17


LIST OF FIGURES


Figure 1. Pertinent nearshore wave parameters; those at A illustrate conditions at
the initiation of alpha wave peaking, those at B represent conditions at
shore-breaking for plunging waves. ................................ 2


INTRODUCTION










Figure 2. Wave transformation from deep water to shore-breaking, where the alpha
wave peaking process is denoted by b. .............................. 4

Figure 3. Relationship between the water depth at shore-breaking, db, and the
shore-breaking wave height, Hb .................. .................. 7

Figure 4. Relationship for prediction of the shore-breaking wave height from the
initial equivalent wave steepness parameter. ........................... 8

Figure 5. Relationship between the equivalent wave steepness parameter evaluated
at initiation of alpha wave peaking and at shore-breaking. ................. 9

Figure 6. Evaluation of Munk's (1949) parameter for determining the initiation of
wave peaking in the shore-breaking process). ........................ 10

Figure 7. Relationship for the prediction of the relative depth of water at which
alpha wave peaking is initiated. ................................... 10

Figure 8. Alpha wave peaking predictions using equation (7) plotted as the solid
curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured
data (solid circles) are for bed slope data of 0.072 and 0.054 from Putnam
(1945), and bed slope data of 0.0292 of Buhr Hansen and Svendsen (1979). .... 12

Figure 9. Dependence of design pier deck soffit elevation on wave crest elevation
above the design water level (DWL) for two commonly applied theoretical
approaches. ................................................ 14

Figure 10. Relationship between the wave height at shore-breaking, Hb, and the
trough depth just preceding the crest, db; field data are from Balsillie and
Carter (1980) and Balsillie (1980), laboratory data are from Iverson (1952). ... 15

Figure 11. Relationship between the water depth at shore-breaking, db, and the total
water depth at shore-breaking, de; data sources as for Figure 10. .......... 15

Figure 12. Relationship between the total depth at shore-breaking, dc, and the
wave trough at shore-breaking, dt; data sources as for Figure 10. ............ 16

Figure 13. Relative depth equations for initiation of alpha wave peaking at (di/Hi),
and at initiation of wave height increase above the DWL at (d/H,)'. ........... 17

Figure 14. Wave steepness parameter at (d1/Hi)' relative to the wave steepness
param eter at (d/H ). ........................................... 18

Figure 15. Relationship for prediction of the incident value of H'/H. ............. 18

Figure 16. Transformation of H'/H using equation (21) plotted as solid curves.
Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured data (solid
circles) are from Putnam (1945). .................................. 20










Figure 17. Predicted H' from equation (21) versus prototype Beach Erosion Board
wave tank data reported by Bretschneider (1960). ...................... 21

Figure 18. Predicted H' from equation (21) versus Lake Okeechobee hurricane wave
data reported by Bretschneider (1960). ............................. 21

Figure 19. Ratio of wave crest elevation above the design water level to wave
height from equation (21). ...................................... 22

Figure 20. Region of applicability for various wave theories and for the Alpha Wave
Peaking methodology of this work. ................................ 24



WAVELENGTH AND WAVE CELERITY DURING SHORE-BREAKING

Page

A BSTRA CT .......................................................31

INTRO DUCTION ................................................... 31

DISCUSSION AND RESULTS ........................................... 31

Terminal Boundary Conditions .................................... 33

Initial Boundary Conditions ...................................... 34

Wave Celerity and Wavelength Transformation During Alpha Wave Peaking .... 36

CONCLUSIONS ....................................................37

REFERENCES ................ ......................................37

NOTATION ..................................................... 39

TABLE

Table of statistics relating measured and predicted wavelengths at the shore-
breaking position. ............................................ 34

FIGURES

Figure 1. Illustration of wave transformation from deep water to shore-breakinig,
where the alpha wave peaking process is given the notation a ... ........ 32

Figure 2. Evaluation of relationships for prediction of the wavelength at the
shore-breaking position. ................................. ....... 35









Figure 3. Comparison of predicted and measured wave speeds at the point of
initiation of alpha wave peaking; predicted data are from equation (4),
measured data from Hansen and Svendsen (1979). ................... .. 36

Figure 4. Illustration of wave speed attenuation during shore-breaking using
equation (15) for various initial wave steepness values.. ................. 37











SHORE-BREAKING WAVE HEIGHT

TRANSFORMATION

by

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


ABSTRACT
As waves begin to shore-break, the wave crest rapidly increases in height, reaching a maximum
at the shore-breaking position. This phenomenon, termed alpha wave peaking, is primarily dependent on
the wave steepness and may be predicted according to:

S1.0 0.4 In nh (00 ,

where H, is the mean incident wave height, T is the wave period and Hb is the mean shore-breaking wave
height. For waves considered in this work, Hb ranged from 1.04 to 2.39 times as high as H,. The relative
depth of water, di /Hi defining the point at which alpha wave peaking begins (i.e., the initiation of the
shore-breaking process) is given by:

l = in ta nh 65 ,J
H, H, 2 g; 7-2

Transformation of H/H, during alpha peaking, where H is the local mean wave height, is given by:


HH, H ,

in which db is the water depth at shore-breaking, d is the local water depth, and solutions for 0, and P2
are developed in the text.
Many coastal engineering design solutions requiring consideration of wave activity can be
accomplished only if the crest elevation of the design wave(s) is known relative to a design water level
(DWL). From analysis of field and laboratory data, it is determined that at the shore-breaker position
approximately 84% of the wave crest lies above the DWL. The amount of the wave that lies above the
DWL during shore-breaking may range from about 0.5 to 0.84. Transformation of H'/H, where H is the
local wave height and H' is the amount of H lying above the DWL during alpha wave peaking, may be
predicted by:

HI Wb 40 tanh 43! j0
H H H Hb

where Hb' is the amount of Hb lying above the DWL and the solutions for P3 and 04 are developed in the
text.


INTRODUCTION wave height, beginning just prior to, and
reaching a maximum at shore-breaking can,
Generally, as waves approach the even over gentle bed slopes, be
shoreline, the height of the waves first tend "...remarkably sudden..." (Munk, 1949) and
to decrease and then to increase rapidly just accompanied by progressive distortion and
prior to shore-breaking. This increase in the asymmetry of the wave in profile view, has










been termed alpha wave peaking by Balsillie
(1980). Wave peaking has been observed
as characteristic activity in shore-breaking
wave mechanics (Scripps Institute of
Oceanography, 1944a, 1944b; Putnam,
1945; Munk, 1949; Iverson, 1952; Stoker,
1957; Kinsman, 1965; Byrne, 1969; Clifton
and others, 1971; Komar, 1976; Nakamura
and others, 1966; Van Dorn, 1978; Hansen
and Svendsen, 1979; Balsillie, 1980, 1983b,
1983c; etc.).

Two mechanisms occur during shore-
breaking: 1. the transformation of H/H,, and
2. the transformation of H'/H, where H is
the local wave height, H' is the amount of
the wave lying above the design water level
(DWL), and Hi is the wave height at the
initiation of alpha wave peaking. Pertinent
wave height parameters are illustrated in
Figure 1 for plunging shore-breakers, where
at the shore-breaking position, the front face
of the wave crest becomes vertical (to
eventually curl, form an air pocket, and fall
into the wave trough fronting the crest).
Where spilling shore-breakers occur, the
shore-breaking position is reached when the
top of the wave crest becomes unstable and
aerated and turbulent water slips down and
across the front face of the crest. Other
shore-breaker types such as surging and
collapsing waves have also been identified


and defined (e.g., Galvin, 1968; Balsillie,
1985). The first of the above mechanisms
defines the first subject of this paper, the
latter is addressed in the second section.
Wavelength and wave celerity behavior
during shore-breaking are the subjects of
separate works (Balsillie 1984, 1999a).

CLASSICAL WA VE
THEORIES

It is to be noted that existing wave
theories have relevance in offshore waters
where wave distortion and boundary effects
are less constraining compared to conditions
in nearshore waters. Application of
quantifying methodology for the nearshore
zone, however, has been sorely lacking,
ostensibly because of the distorted nature of
waves and confining boundary conditions. It
would, therefore, appear appropriate to
recapitulate some of the more conspicuous
short-comings of "classical wave theories"
relevant to the problem at hand.

From an historical perspective, one
should have at least, some appreciation, of
early theoretical work of such researchers as
Gerstner (1802), Levi-Cevita (1924), Struik
(1926), Gaillard (1935), and Mason (1941).
For a more detailed account of early theorists
see the bibliography of Mason (1941).


Figure 1. Pertinent nearshore wave height parameters; those at A illustrate
conditions at the initiation of alpha wave peaking, those at B represent conditions
at shore-breaking for plunging waves.










Subsequent to this early work, wave theories
have surfaced which currently are in general
use. Following are descriptions and
discussion of such theoritical approaches.

Linear (Airy) Wave Theory suffers
from the basic problem that the wave is
divided evenly above and below the still
water level (SWL). This condition clearly
does not hold for waves in shallow and
transitional water depths or, for that matter,
waves undergoing breaking in any depth of
water. Also, the calculating algorithum for
wavelength in any depth of water is, at best,
an envelope fit giving but approximating
results (Bretschneider, 1960; Balsillie,
1984a, 1999b).

Stokes' Wave Theory attempts to
more nearly depict the distortion of the wave
profile about the SWL, with the degree of
distortion increasing for higher order theory.
However, its applicability does not extend
into shallow water nor the shallower part of
transitional water depths. Dean (1974; fig.
7) demonstrates the same region of validity
for Stokes' 5th Order Theory which includes
Stokes' 3rd and 4th order region of Figure 1.

Cnoidal Wave Theory has been
popularly used for predicting shallow and
transitional water wave conditions.
However, depending upon the investigator,
there is variability in defining the region of
validity for the theory. Laitone (1963) notes
its validity for d/L < 0.125. Svendsen and
Hansen (1976) discuss the applicability of
Cnoidal theory using developments of
Skovgarrd and others (1974) for the
deformation of waves up to shore-
breaking (the theory is applicable where d/Lo
< 0.10 or d/L < 0.13 (see NOTATION
section at the end of the paper for
definitions), seaward of which they
recommend the use of Airy wave theory).
Svendsen and Hansen state "...even though
cnoidal theory seems to predict the wave
height variation reasonably well, no
information can be deduced from that theory


(or any other theory) about where breaking
occurs." Similarly, Skovgaard and others
(1974) note that where (d/g T2) > 0.014
(i.e., d/L > 0.13) "... Cnoidal theory is
meaningless". The Shore Protection Manual
(U. S. Army, 1984), on the other hand,
appears more restrictive suggesting the use
of the Ursell or Stokes parameter such that
the theory is valid where U = L2 H/d3 > 26.
In addition, as the wavelength becomes long
approaching infinity, Cnoidal Wave Theory
reduces to Solitary Wave Theory; and as H/d
becomes small, the wave profile approaches
the sinusoidal profile predicted by Linear
(Airy) Theory. Several other problems are
entrenched in the theory. First, it is not an
easy nor expedient theory in application.
Second, it requires that the wavelength is
known when, in fact, this wave parameter is
not quantitatively known for use in actual
design applications. Third, the wavelength
is first calculated from Linear (Airy) Theory in
deep water (i.e., Lo) then modified to the
local water depth of concern (i.e., L), which
if only because of the approach, suggests an
approximation at best.

Softary Wave Theory is a special
case of Cnoidal Theory in which the
wavelength is infinitely long and the entire
wave crest lies above the SWL, is clearly not
applicable to natural cases of shore-breaking
where shoaling waves have periodicity.

More recently developed Stream
Function Theory (Chappelear, 1961; Dean,
1965a, 1965b, 1967; Monkmeyer, 1970)
constitutes an attempt to mitigate the
difficulty of expanding to higher orders the
Stokes'-type approach, rendering the "...
approach computationally simpler than
Chappelear's technique, ..., to have wave
theories that could be developed on the
computer to any order" (Dean and
Dalrymple, 1984, p. 305). Stream Function
Theory, which has "... some of the same
limitations ..." of higher order Stokes'
solutions (U. S. Army, 1984, p. 2-59) has
the capability to proceed to the breaking limit











in transitional water depths, but not in
shallow water.

WA VE HEIGHT GROWTH
DURING SHORE-BREAKING

It is, in part, the intent of this work to
provide a practicable solution to the
shortcomings of the previously discussed
wave theories.

Mathematical descriptions developed
in the following sections require that only the
initial wave height, Hi, and wave period, T,
are known. In an earlier study, Balsillie
(1980) used Ho or Hm as indicators of H,,
where Ho is the deep water wave height and
Hm is the wave height measured in the
constant depth portion of a laboratory wave
channel. In many laboratory investigations
it has been found that initial wave
characteristics are in the range where Ho and
Hm and the resulting value of H, are
approximately equivalent. Generally, this
occurs for waves with higher wave
steepness values. However, due to




1.2 C-- Conltont Depth -


1.1
1.0----pWot---------
Mo 0t 10
9 011Dirlo
OT DIfeelloa
Tr4

H _
-H- --i- -


refraction and frictional effects, etc., where
the generated wave steepness is small, H,
can become significantly less than Ho or Hm.
The importance of the latter phenomenon is
illustrated by an example from the laboratory
data of Putnam (1945) in Figure 2.

In the earlier work (Balsillie, 1980), it
was reported that the alpha wave peaking
parameter, Hb/H,, is dependent on the
equivalent wave steepness parameter, H,/(g
T2), and the bed slope, m The influence of
these parameters is included in ensuing
analyses. In addition, the continuous
transformation of H/H, during shore-breaking
is investigated. First, however,
determination of where alpha wave peaking
is initiated and terminated require
identification. Where possible, both field
data and laboratory data are considered. It
is to be noted, however, that laboratory
information by far constitutes the bulk of
available data. However, since the studies
of Balsillie (1980, 1983b), new laboratory
data have become available (Table 1) to



I I I

* 0.054
Mb

T2 0.0086
Wove


Figure 2. Wave transformation from deep water to shore-breaking,
alpha wave peaking process is denoted by ab.


Corrected
H
NH


where the












Table 1. Field and laboratory data used in analyses.



T H( d. d,H
Investigator m -T
(s) E




FIELD DATA
Wood (1970, 1971)' 0.0556 3.49 1.58 4.23 2.00 451.2 286.2

LABORATORY DATA
Putnam (1945) 0.072 0.865 1.04 2.34 1.76 78.8 74.4
0.072 1.15 1.29 3.13 2.27 170.0 131.0
0.072 1.22 1.29 2.69 2.67 208.0 147.0
0.072 1.50 1.66 4.77 3.17 391.0 236.8
0.072 1.54 1.58 4.45 2.92 428.0 267.0
0.072 1.97 1.86 5.01 3.11 916.5 462.8
0.054 0.86 1.08 2.22 1.66 87.4 78.6
0.054 0.965 1.11 2.50 1.93 108.9 103.8
0.054 1.34 1.48 3.96 2.96 312.0 213.7
0.054 1.50 1.58 4.39 3.16 425.7 278.7
0.054 1.97 1.84 5.94 3.47 1039.1 557.9
Step" 1.05 1.16 --- --- 126.2 108.5
Step" 1.19 1.19 --- 142.6 119.4
Setp" 1.35 1.31 -- 237.4 181.5
Step" 1.50 1.60 --- --- 375.0 235.0
Step" 1.98 1.86 --- --- 997.8 441.0
Hansen and 0.0292 0.833 1.32 3.64 2.55 207.0 156.3
Svendsen (1979) 0.0292 1.00 1.13 2.57 1.66 104.2 91.8
0.0292 1.00 1.21 2.95 1.89 153.1 126.4
0.0292 1.00 1.35 3.92 2.66 258.4 191.8
0.0292 1.25 1.30 2.50 1.51 162.9 124.3
0.0292 1.25 1.37 3.15 2.04 229.2 167.6
0.0292 1.25 1.46 3.95 2.42 395.3 270.1
0.0292 1.67 1.46 3.57 2.29 283.2 194.8
0.0292 1.67 1.42 3.57 2.15 302.5 209.8
0.0292 1.67 1.47 3.84 2.27 340.3 233.5
0.0292 1.67 1.48 4.13 2.41 389.1 261.3
0.0292 1.67 1.66 5.05 3.09 675.7 408.2
0.0292 2.00 1.69 4.47 2.66 608.6 359.3
0.0292 2.00 1.95 5.50 3.01 1048.2 537.8
0.0292 2.50 1.84 5.22 2.63 874.9 479.7
0.0292 2.50 2.20 5.95 2.75 1531.4 704.1
0.0292 3.33 2.39 6.67 3.08 2544.5 1069.2
Singamsetti and 0.025 1.28 1.27 --- --- 170.8 134.9
Wind (1980) 0.025 1.55 1.25 --- 173.8 138.8

0.050 1.038 1.21 --- 162.4 134.5
0.050 1.55 1.20 --- --- 162.9 135.5
0.050 1.55 1.28 2.43 1.56 216.0 168.2
0.050 1.55 1.27 2.68 1.22 210.2 165.8












Table 1. Field and laboratory data used in analyses (cont.).



T Hb d, d, ( )
Investigator m -d
H, H, db



Singamsetti and 0.100 1.035 1.11 --- --- 160.3 144.8
Wind (1980) 0.100 1.555 1.25 --- --- 173.8 139.3
0.200 1.038 1.35 --- --- 157.6 116.7
Wang and others 0.067 1.00 1.19 --- --- 122.5 103.2
(1982)"' 0.067 1.33 1.20 --- --- 148.2 123.8
0.067 1.34 1.28 --- --- 195.5 153.0
0.067 1.47 1.22 --- --- 178.0 146.0
0.067 1.56 1.49 --- --- 283.9 190.8
0.067 1.65 1.21 --- --- 196.2 161.7
0.067 1.89 1.32 --- -- 280.1 212.2
0.100 1.22 1.17 --- --- 145.9 124.7
0.100 1.50 1.34 --- --- 191.7 143.2
0.100 1.58 1.13 --- --- 222.4 197.3
0.100 1.61 1.33 -- 267.4 201.6
0.100 1.80 1.55 --- --- 396.9 256.1
0.100 1.89 1.52 --- --- 350.1 230.3
Nakagawa (1983)"" 0.364 1.40 1.27 -- 179.5 141.2
Mizuguchi (1986)"' 0.050 1.22 1.39 --- -- 331.5 239.1
Hattori (1986)"' 0.050 0.85 1.23 --- --- 160.9 131.1
0.050 1.00 1.38 --- --- 316.1 227.9
0.050 1.40 1.81 --- --- 738.8 408.7
0.050 1.00 1.30 --- --- 208.5 160.7
0.050 0.80 1.05 --- --- 96.5 92.2
0.050 0.85 1.20 --- --- 138.8 116.1
0.050 1.00 1.42 3.16 --- 257.9 181.5
0.050 1.40 1.96 --- --- 768.3 392.0
0.050 0.80 1.06 --- 96.3 90.9
0.050 0.84 1.02 --- 123.5 121.3
0.050 0.99 1.25 --- --- 240.1 192.1
0.050 1.40 1.70 5.04 --- 711.4 417.6

Watanabe and 0.050 1.19 1.40 2.91 1.6 252.3 180.2
Dibajnia (1988)'" 0.050 1.18 1.28 2.50 1.6 213.2 166.4
0.050 0.94 1.17 2.50 1.6 135.3 115.5

Takikawa and others 0.050 2.08 1.36 2.62 2.08 229.2 197.2
(1997)"'____ __

' Based on 400 consecutive wave measurements; step had a slope of 0.444, post-step slope was
0.009; New data.









represent a wide range in bed slope
conditions.

Terminal Boundary Condition

The terminal boundary of alpha wave
peaking is defined as the shore-breaking
point. Galvin (1968) provides a
comprehensive description of the various
types of shore-breaking waves. Of the
principal types, however, spilling and
plunging shore-breakers constitute those
more commonly applied in design
considerations. The shore-breaking point of
a plunging breaker is defined to occur when
the front face of the wave crest becomes
vertical (Figure 1); the shore-breaking point
of a spilling breaker occurs when the top of


Laborat
x Field Di
10 o Field Di
SAField Di
Field Di
*Field Da
Field Dg


WAVES CAN BREAK IN THIS
REGION DUE TO CRITICALLY
HIGH WIND STRESSES
1


0.01 L
0.01


the wave crest becomes unstable and water
and foam slides or spills down the front face
of the crest.

Two parameters identifying
termination of alpha wave peaking are db/Hb
and Hb/Hi. The first parameter may be
straightforwardly given by the McCowan
criterion (McCowan, 1894; Munk, 1949;
Balsillie, 1983a, Balsillie, 1999b, Balsillie and
Tanner, 1999), illustrated in Figure 3, and
given by:


H 1.28
d,


where Hb is the shore-breaking wave height,
and db is the water depth at shore-breaking.


0.1 1 10
Hb (m)


Figure 3. Relationship between the water depth at shore-breaking,
db, and the shore-breaking wave height, Hb (after Balsillie, 1999b).









Enhancement in precision of db/Hb prediction
has been attempted by incorporating the bed
slope and wave steepness (Weggel, 1972a,
1972b; Mallard, 1978). Balsillie (1983a,
1999b) found, however, that equation (1) as
yet constitutes the most reliable predicting
equation, and that equation (1) applies
equally well to both spilling and plunging
shore-breakers. The second parameter,
Hb/Hj, describing the relative height attained
as a result of alpha wave peaking, is more
difficult to quantify. It is, however,
considered to be a terminal boundary
parameter since Hi is understood to be
specified as input.

In the previous work published by the
author (Balsillie, 1980), both wave steepness
and bed slope were indicated to affect alpha
wave peaking. However, based on new
data, and subsequent analyses and testing,
the following relationship can be
recommended:


illustrated in Figure 4.

Additional attention was given to the
bed slope and no refinement was found to
improve equation (2). In fact, equation (2) is
evaluated for a wide range of bed slope
conditions, wherein any scatter might easily
be attributed to the difficulty in identifying
where shore-breaking occurs (Balsillie,
1999b). Other work suggests that the bed
slope is probably more instrumental in
influencing the type of shore-breaker that
will be produced (Balsillie, 1984b, 1984c,
1985, 1999b).

Due to scale differences between
axes of Figure 4, wave steepness data from
Table 1 are plotted in Figure 5 where now
the axes are comparable. Dividing both
sides of equation (2) by g T2 yields:


S2_ T2 1.0
gr gT2


- 0.4 tanh(l00


H
- 0.4 n tanh 100


1rann


4annn


1U IUU H IUUU Iv UU


Figure 4. Relationship for prediction of the shore-breaking wave height
from the initial equivalent wave steepness parameter.


Hb
= 1.0
H,















/Hb o0.067
- _T2) a 0.072
gT A0.100
X 0.200
O.6388
step
A Equation (3)
100



100 -1 1000
/-H

g T2
Figure 5. Relationship between the equivalent wave
steepness parameter evaluated at initiation of alpha wave
peaking and at shore-breaking.


which is superimposed upon the data of
Figure 5 to show excellent agreement.

Initial Boundary Condition

Various investigators (e.g., Stokes,
1880: Galvin, 1969; Dean, 1974) have
conducted studies to delineate constraints of
breaking. It was Munk (1949), however,
who considered in some detail wave peaking
in the shore-breaking process. He applied
the Rayleigh assumption (Eagleson and
Dean, 1966) given by:


d-


Equation (5) is plotted in Figure 6,
from which the non-representative nature of
the equation is apparent. Additional analysis
indicates that if we solve for d,/H, rather than
d,/db of equation (5) and consider the
incident wave steepness (bed slope
produced no consistent results), the
following relationship, plotted in Figure 7,
provides good agreement:


c, E, = c, E,


d, d,
(4) H H
HI Hb,


{ In tanh (65-gH)
\ [ g^}\


where c is the phase speed (shallow water
condition only, where wave period is
conserved and no energy is lost), and E is
the total wave energy, and the subscripts i
and b refer to conditions at initiation of wave
peaking and at shore-breaking, respectively.
Using the Rayleigh assumption and Solitary
wave theory, Munk (1949) suggests that:


in which it is assumed that db/Hb = 1.28.
The equation represents a significant range
of bed slopes (i.e., 0.0292 to 0.072) for
data from a variety of sources.














3 0 N
Of


1-2 x
db l
^4 .^Bed Slope
n 0.0292
A 0.050
x 0.0556
Equation (5) o 0.072


0 1 2 3 4
Hb
Hi
Figure 6. Evaluation of Munk's (1949)
parameter for determining the initiation of
wave peaking in the shore-breaking
process.


Transformation of H/H,


In addition to specification of the
boundary conditions, it is desirable to be
able to predict the continuous behavior of
alpha wave peaking. Such behavior, for
example, may be important in determining
horizontal and vertical impact loading
potential of shore-breaking waves, and in
sediment transport predictions.

Data tabulated by Putnam (1945)and
Hansen and Svendsen (1979) are used to
determine the nature of the transformation.
The general equation is given by:


HHb 02 tanh [) (7)
H Jd d )
H, H, H HJ


where 0, is a coefficient which determines
where the transformation of H/Hi begins,
given by:


e


4nnn


innnn


IU IUU I ,W1


Figure 7. Relationship for the prediction of the relative depth of
water at which alpha wave peaking is initiated.











in which (d,/H,) is given by equation (6), e is
the Naperian constant, and (2 determines
the local peaked height during the shore-
breaking process given by:
Hb
0I2 1.0 (9)
H1

where Hb/H, is given by equation (2).
Equation (7) is evaluated in Figure 8 for
various bed conditions. Data from
Singamsetti and Wind (1980) are not plotted
because the authors did not tabulate the
transformation information. Only four data
points are available for the field data of
Wood (1970, 1971) and are not plotted.
Data from Putnam (1945) for the step slope
could be plotted, but would require
considerable license in estimation to
determine the value of di (since the waves
began to shore-break on the step slope over
which measurements were widely spaced).
Data of Wang and others (1982), Nakagawa
(1983), and Mizuguchi (1986) provide only
data for H,, H. and T. The more recent data
of Hattori (1986) are not plotted here; it is to
be noted, however, that equation (7)
resulted in essentially precise representation
of his data.

In many of the plots of Figure 8, the
laboratory data suggest that db/Hb is closer
to unity than to a value of 1.28. From
Figure 4, however, it is evident that
laboratory data "tend" toward a lower value.
This may be symptomatic of difficulties in
determining precisely when small laboratory
waves shore-break (i.e., since this must be
visually determined). The terminal boundary
condition of db/Hb = 1.28 is, therefore,
maintained. Scatter of data relative to
equation (7) is noted in some of the plots.
Overall, however, the shape of the
transformation appears to be well
represented by equation (7).


WAVE CREST ELEVATION ABOVE
THE DESIGN WA TER LEVEL

While wave crest height is useful in
practical applications, there is also a need to
know the wave crest elevation above some
reference plane. For instance, where a
storm tide is used in assessing coastal
engineering design solutions, wave
crest/trough elevations relative to the known
storm tide still water level (i.e., DWL) are
needed.

For example, suppose that the task is
assigned to design the elevation of a fishing
pier deck where the shore-breaking wave
height at the structure from other
calculations is estimated to be 4.5 m
(approximately 15 feet). If, from theoretical
calculations, the sine wave assumption is
used (Figure 9) then one-half, or 2.25 m, of
the wave will lie above the design water
level (SWL). If, however, the Solitary wave
assumption is used, the entire 4.5 m wave
lies above the DWL. This results in a large
design uncertainty of 2.25 m (7.4 feet).
While the sine wave assumption may
actually be too low to insure a safe deck
elevation, the Solitary wave assumption may
be in excess, particularly in view of the high
costs associated with construction and
maintenance in the littoral zone.

The above example, though it states
the basic problem, is an over-simplification.
It is well known that in addition to the design
wave crest elevation above the DWL, other
considerations, in particular the expected
horizontal and vertical design wave impact
loads, should be applied. The latter is
possible only if the nature of wave
transformation during shore-breaking is
known.

An estimation of the amount of the
local wave crest that lies above the DWL,
H', can be attempted using various wave
theories. However, the applicability of
classical theories, although they have













2.0 '- I I I I I 1 I 1 I I I
2.0 Ism 0.29 !
= 0.0292 m = 0.0292
: Hi = 0.064 m Hi = 0.0388 m
T = 1.0 s T = 1.25 s
1.5 I


1.0 I _
2.0 m = 0.0292 m = 0.0292
Hi = 0.0668 m I Hi = 0.0961 m
T = 1.25 I T = 1.67 s
1.5


1.0-
20 m 0.0292 m = 0.0292
Hi = 0.0941 m Hi = 0.09 m
I T 1.25 s T = 1.67 s


1.0 8 -

2.0 m = 0.0292 m =0.0292
Hi = 0.0801 m Hi = 0.0403 m
ST = 1.67 a I\ T = 1.67 s
1.5


1.0
2.0
2 I m = 0.0292 m = 0.0292
SHi 0.07 m Hi = 0.0644 m
1. T 1.67 s T = 2.0 s



1.0- I
2.0 m = 0.0292 \ m = 0.0292
-,Hi = 0.0374 m Hi = 0.04 m
T = 2.0 s T = 2.5 s
1.5 I

.0 I -
1.0 -
2.0 -
m = 0.0292 I m = 0.0292
: *Hi = 0.07 m Hi= 0.0428 m
.\& T = 2.5 s \ T = 3.33 s
1.5 -


l n_ I i i i i I I I P I I


0 1 2 3 4 5 6 7 8 9 0 1 2


3 4 5 6 7 8 9 10


d/H
Figure 8. Alpha wave peaking predictions using equation (7) plotted
as the solid curves. Vertical dashed lines represent d/H = 1.28 =
db/Hb. Measured data (solid circles) are for bed slope data of 0.072
and 0.054 from Putnam (1945), and bed slope data of 0.0292 from
Hansen and Svendsen (1979).










-1 I I I I I I I I I I I
Sm = 0.072 I m = 0.072
Hi = 0.093 m Hi = 0.0701 m
T = 0.865 I T= 1.22 a


II I

m = 0.072 m = 0.072
Hi = 0.0762 m Hi = 0.0564 m
T= 1.15 T = 1.5 .
I

____-,,_______-*_*___ __ I 0*" e

m = 0.072 m = 0.054
I Hi = 0.0543 m Hi = 0.0829 m
T = 1.54 a T = 0.86 s
'-\


I I* *,*o *o* _-

*-'! m = 0.072 1 m = 0.054
Hi = 0.0415 m Hi = 0.0838 m
T = 1.97 T = 0.965




Sm = 0.054 m = 0.054
SH = 0.0564 m Hi = 0.0366 m
T = 1.34 a T = 1.97 s




) I m = 0.054 I m = 0.0292
H = 0.0518 m Hi 0.0329 m
T = 1.5 T = 0.833 a




Sm m = 0.0292
H = 0.0942 m Hi = 0.0379 m
ST' 1.05 T=1.0a
5 -


; .________k,,1


0


1 2 38 4 3 6 7


Figure 8. (cont.)


1 2 3 4 5 6 7 8 9 0
12345d/H
d/H


n


a I-


1.(









Pier Deck


Solitary Wave Assumption

Sine Wave


2 m
Vertical 1 m
Scale O
0m-1


Figure 9. Dependence of design pier deck soffit elevation on wave crest
elevation above the design water level (DWL) for two commonly applied
theoretical approaches.


demonstrated relevance for predicting
"deeper water" wave conditions, are not
specifically designed to predict wave
behavior during the shore-breaking process,
particularly since shore-breaking waves are
not symmetrical in profile view. Rather, the
crests become progressively asymmetrical
and distorted (e.g., Adeyemo, 1968, 1970;
Weggel, 1968).

Despite the underlying importance of
the issue considered, surprisingly little work
has been produced which addresses the
phenomenon. In fact, of the work done, the
paper of Bretschneider (1960), that by its
singularity, becomes a classical
accomplishment. Bretschneider's
nomograph (Bretschneider, 1960; U. S.
Army, 1984, p. 7-107) provides measures of
H' relative to the design water level (i.e.,
DWL which in this paper represents the
SWL). However, for the entire range of
conditions represented by the nomograph no
mathematical description has been
developed.

This author (Balsillie, 1983c)
attempted to formulate a mathematical


representation using a modification of
second order Stokes wave theory. The
results, however, were less than
satisfactory.

A problem associated with the data
used by Bretschneider is that his wave
measurements were single point source
samples. It was determined that, in addition,
it would be valuable to know how the value
of H'/H behaves as the wave progresses
across a shoaling bathymetry to shore-
breaking. Hence, the author (JHB)
conducted a re-analysis.

While the amount of continuous wave
height transformation data is not large, the
data of Putnam (1945) for two bed slopes in
conjunction with Bretschneider's (1960),
Iverson's (1952), Weishar's (1976),
Hansen's (1976) and Balsillie and Carter's
(1980, 1984a, 1984b) and Balsillie's (1980)
point source data provide sufficient
information on which to conduct an
investigation. As before, it becomes
necessary to identify initial and terminal
boundary conditions of the process.









Terminal Boundary Condition


The first boundary condition occurs
where the value of H'/H is evaluated at the
shore-breaking position, that is, the value of
Hb'/Hb.

Using the laboratory data of Iverson
(1952) and field data of Balsillie and Carter
(1980) and Balsillie (1980) the relationship
between the shore-breaking wave height and
the wave trough depth, d, (i.e., the vertical
distance from the wave trough located just
shoreward of the breaking wave crest to the
bottom, referenced to the DWL), may be
found. These data are plotted in Figure 10,
and indicate that dt/Hb = 1.092 for 88 wave
samples. Weishar (1976) reports that dt/Hb
= 1.124 for 116 field measurements. A
weighted average from the two groups of
data yields:


d,
1.11
Hb
Iverson's (1952) laboratory
data and field data of Balsillie and
Carter (1980) and Balsillie (1980)
also suggest that the depth at
breaking can be related to the total
depth, dc (i.e., the vertical distance
from the top of the wave crest at
shore-breaking to the bottom),
according to:


d- 0.590
d.


(10)


(11)


as illustrated in Figure 11.


Hb(m)
Figure 10. Relationship between the
wave height at shore-breaking, Hb, and
the trough depth just preceding the
crest, d,; field data are from Balsillie and
Carter (1980) and Balsillie (1980),
laboratory data are from Iverson (1952).


With the same data source
used to develop equation (11), dt
and dc can be related according to:


dt 0.529
d,


(12)


which is illustrated in Figure 12.


de- (m)
Figure 11. Relationship between the water depth
at shore-breaking, db, and the total water depth
at shore-breaking, d.; data sources as for Figure
10.

Combination of equations (1) and (10)
through (12), where Hb' = dc db, yields the
average result that:










0.84
H,

which is referenced to the
Hansen (1976) reports that:

(0.85),, = (0.82)swL
Hb


(13)


DWL.


(14)


where the values are referenced to
the mean water level (MWL) and still
water level (SWL) as indicated by the
subscripts (definitions of MWL and
SWL are those given by Galvin,
1969). Regardless of the slight
discrepancy between the DWL
coefficients of equations (13) and
(14), Hansen's result has been
presented to show that Hb'/Hb is in


the mid-eighty percent range and not some
significantly large or small value.

It is appropriate, also, to address the
effect of shore-breaker type. The field data
of Balsillie and Carter (1980) and Balsillie
(1980) represent plunging and spilling type
shore-breakers. However, no correlation
was found between Hb'/Hb and the shore-
breaker type. Weishar (1976) reports the
results of a field study using ground
photography including the shore-breaker
type. Using the established relationship of
db/Hb = 1.28, Weishar's values of Hb/db
may be transformed to yield:


Hib
C bP


and:


H,,P
r,


(0.88),M = (0.85)s,


= (0.86)m = (0.82)s,


where the subscripts PI and Sp refer to
plunging and spilling shore-breaker types.
Weishar (1976) is careful to note, however,


(ml* Field Data
.. '" D Lab Data (n =48)

So0.5 1.0 1.
dc (m)
Figure 12. Relationship between the total depth
at shore-breaking, dc, and the wave trough depth
at shore-breaking, dt data sources as for Figure
10.


that considerable variation occurred in the
data and that the difference between mean
values of the last two equations may not be
statistically significant. It is concluded,
therefore, that while there may be
dependence of the value of Hb'/Hb on the
shore-breaker type, sufficient data are not
yet available nor work on other methods
(e.g., Iwagaki and others, 1974)
accomplished to justify such a commitment.
Hence, equation (13) shall prevail as least
equivocal guidance.

Initial Boundary Condition

The other boundary condition occurs at
the beginning of the shore-breaking process.
It has been suggested (see Balsillie, 1983c,
p. 7) that the maximum value of H'/H in
deep water is about 0.64. This maximum
value, then, represents forced wave
conditions (i.e., the waves are subject to the
wind forces from which they were
generated, and maximum wave steepness is
maintained). A value of less than 0.64
represents free or coasting waves (i.e., the
waves are no longer subject to original
generating winds, but have left the
generation area and have or are undergoing










dispersion). (See Mooers, 1976; Balsillie and
others, 1976)

Upon reaching transitional water
depths, the bottom slope begins to introduce
an additional effect on the value of H'/H.
Therefore, H'/H may have a value greater
than 0.5 when the wave reaches the point of
initiation of the shore-breaking process.
When a wave reaching the initiation point is
forced, one may expect the progressive
increase in the value H'/H to be minimal,
provided that bed slope conditions do not
change significantly during the
shore-breaking process. It was


found, however, that for free
or coasting waves H'/H does
not begin to significantly
increase in value until shore-
breaking begins, which has
been determined to occur
when the critical alpha wave
peaking depth is encountered.
Near the point of initiation of
shore-breaking, the value H'/H
shall be given the notation
(Hi'/Hi)' which requires
quantification.


It
(Balsillie,
increase


was also found
1983b) that the
in the wave height


Figure
alpha
wave I


above the DWL begins
somewhat earlier (i.e., further offshore)
in the shore-propagating wave
transformation history than does the
initiation of the alpha wave peaking
process. In terms of design approach,
this complication should not be of undue
concern, since even though the height
of the wave above the DWL might be
slightly increasing, the total wave height
typically is still decreasing...until the
initiation of alpha wave peaking after
which H/Hi and H'/H both increase to
reach a maximum value at the shore-
breaking position. In addition, the initial
values of both H/Hi and H'/H are
significantly small. Even though vertical


changes might be slight in the design
application sense, it would be prudent to
account for this apparent discrepancy.

Using the data of Putnam (1945)
illustrated in Figure 13 and listed in Table 2,
the relative water depth (di/Hi)' indicating
where the increase in wave crest height
above the SWL appears to begin may be
approximated by:


(Y =
IH/


SIn tanh _201--
2 Lj


(17)


T
13. Relative depth equations for initiation of
wave peaking at (di/Hi), and at initiation of
height increase above the DWL at (di/H,)'.
Table 2. Laboratory data of Putnam (1945)
for evaluation of wave height above the DWL.
m T (H,'H)' (d,/H,)' (H/g T)'
0.072 0.865 0.575 3.5 71.4
1.15 0.550 4.0 158.7
1.22 0.525 5.0 200.0
1.50 0.525 6.0 384.6
1.54 0.500 6.5 434.8
1.97 0.525 7.0 909.1
0.054 0.86 0.575 3.5 75.2
0.965 0.550 4.0 96.2
1.34 0.550 5.0 303.0
1.50 0.520 6.0 400.0
1.97 0.525 7.0 1000.0
NOTE: H,' is evaluated at the point where the height
above the DWL begins to rise, not at the initiation of
Alpha Wave Peaking ... use equation (20) for
transformation.











Also plotted in Figure 13 is the
relationship for di/Hi from equation (6)
for comparative purposes.

The wave steepness parameter
at the two points are slightly different
in value. Using the data of Putnam
(1945), the transformation of wave
steepness may be approximated
(Figure 14) by:


f 1462 \ 1.055
( H, = 1.462 1.
[g gr1Tg


(18)


in which (Hi /g T2)' is the wave steep-
ness parameter where crest height
increase above the DWL is initiated,
and (Hi/g T2) is the wave steepness
parameter at the beginning of alpha
wave peaking.

The relative incident wave
height, (Hi'/Hi)' conforming to relative
depth conditions, illustrated in Figure
15, may be given by:


(- 0.5 + 1.25 g (19)


where (Hi2/(g di T2))' is evaluated at
(di/H,)'.

Analytically, it becomes of
value to be able to predict the
magnitude of H'/H at the point of
initiation of alpha wave peaking
(i.e., at d,/H, rather than at (d, /Hi)').
Using the notation (H'/Hi)o,
numerical analysis results in a
closely fitted value according to:


S= 0.515 + 12
Ha s g


-1
T'
Figure 14. Wave steepness parameter at (di/Hi)'
relative to the wave steepness parameter at
(d/Hi-).


'"---- -See text
0.-5 / Equation (19)
(/-) 8 0.054 -^ f-


H .72 o HA \h
00.0<72 -o--~ ~ _B .
..5 ---------
, .. . , .j I , , , I
10 IoX 103 104

g d1T2/
Figure 15. Relationship for prediction of the
incident value of H'/H.


(20)


It is interesting to note that where
limiting conditions are imposed,
equation (19) provides consistent results.


Suppose that a profile occurs where a wave,
initially in deep water, suddenly encounters
an abrupt slope change and it must shore-
break. Suppose, also, that the wave in deep
water is fully forced so that (Ho'/H) =
(Hi'/Hi)' = 0.64. For such conditions, from
equation (19), (Hi /g (di T2))'1 = 56.4. From
another viewpoint, where Hb /db = 1/1.28 =
0.78 and (Hi /g T2)' = 1/(14 n), then (Hi /(g










di T2))'- = 63.8, and (Hi'/Hi)' = 0.648.
Both solutions are close and represented by
the asterisk in Figure 15.

Also, where the maximum possible
value of (H'/H) in equation (20) is 0.84,
representing maximum forced wave
conditions in shallow water, the value of (Hi
/(g T2)) becomes 0.0271. Where Hb/Lb =
0.79 (Hb /(g T2))0.5 given by Balsillie (1984a),
then by substitution (H/L)max for shallow
water waves becomes 0.13 or 1/7.7. This
approximation is consistent with the Michell
(1893) criterion of 1/7.

Prediction of H'H Transformation

Incorporation of the preceding
boundary conditions leads to the following
development describing the H'/H
transformation during the shore-breaking
process. The general equation is given by:


H H 4 tanh '( -


where db/Hb = 1.28, and:

3 = ee
(dIHi)' (dblH,)


(21)


(22)


in which e is the Naperian constant, (db/Hb)
is 1.28, and:


H H/
H,,r


(23)


where from equation (16) Hb'/Hb = 0.84,
(Hi'/Hi)' is given by equation (19), and (d,/H,)'
is given by equation (17).

Equation (21) is plotted in Figure 16
as the solid curves. It is to be noted from
these plots that equation (21) appears to
successfully represent initial values of H'/H
(i.e., incident waves with larger wave


steepness values tend to have larger initial
H'/H values). There is, however,
discrepancy between measured and
predicted values of H'/H as the shore-
breaking position is closely approached.
Putnam's (1945) laboratory data tend to
consistently underestimate the value of H'/H
very near and at the shore-breaking position.
One must recall, however, that the behavior
of the curve predicted by equation (21) is
determined by the terminal boundary
condition that Hb'/Hb = 0.84, which is
based on prototype wave data and results
from other investigations. Hence, laboratory
conditions or measurement techniques may
account for the apparently low values of
Putnam's data near shore-breaking. While
the terminal boundary condition must surely
be refined by future research efforts,
equation (21) would appear to provide a
satisfactory and, certainly, a useful method
for predicting H'/H during the shore-breaking
process.

Equation (21) is also tested using the
prototype laboratory wave (Figure 17) and
field hurricane waves (Figure 18) reported
by Bretschneider (1960). Figure 17 indicates
very good agreement. Figure 18, however,
indicates that equation (21) underestimates
H'/H. One must keep in mind that the
measured data represent the maximum wave
height occurring during one-minute recording
periods, not the average wave height.
Therefore, one would expect equation (21)
to underestimate the measured data, and
that the plotted line represents an expected
upper limit as also found by Bretschneider
(1960).

Based on the success of equation
(21), the value of H'/H as a function of H/(g
T2) and d/(g T2) is given by Figure 19 for
transitional and shallow water depths.
Figure 19, then, provides an alternative to
the nomographic approach originally
proposed by Bretschneider (1960).


































0.9
H m =0.072 m = 0.054
Hi = 0.04184 m Hi = 0.0949 m
0.7 T= 1.97 s T = 0.965 a


0.5 I
0.9 I
m = 0.054 m = 0.054
Hi = 0.05807 m H = 0.03803 m
0.7 T = 1.34 a T = 1.97 s


0.5 1
0.9 .I I
m I 0.054 5 10 15 20 25
Hj = 0.05513 m
0.7 T = 1.5 s
I .. ,
0.5 I * *

0 5 10 15 20 25
d/H
Figure 16. Transformation of H'/H using equation (21) plotted as solid
curves. Vertical dashed lines represent d/H = 1.28 = db/Hb. Measured
data (solid circles) are from Putnam (1945).
















Hprod
Hpred S*

(m) *





0 1 2
H"mMe (m)
Figure 17. Predicted H' from equation
(21) versus prototype Beach Erosion
Board wave tank data reported by
Bretschneider (1960).


CONCLUSIONS

Classical wave theory, among certain
coastal practitioners, has been viewed as the
"last word" in the prognostication of water
wave behavior. However, when applied to
the progressive distortion of waves as they
approach and reach shore-breaking, classical
wave theory falls short of providing realistic
solutions. A major shortcoming of wave
theories is that they require specification of
variables not known or which can only be
approximated, in particular the wavelength.
Shortcomings of theories have been
presented in the second section of this work.

The results of this work, however,
provide a methodology for wave height
transformation during the wave breaking
process up to the point of shore-breaking,
requiring only the specification of initial wave
height and period; bed slope is apparently
not a factor affecting the transformation
(other research suggests that bed slope is
more nearly instrumental in influencing the
type of shore-breaker that will be produced
(e.g., Balsillie, 1999b)).


H .a

(m)

1 -



0

Figure 18.
(21) versus
wave data
(1960).


H;m.a (m)
Predicted H' from equation
Lake Okeechobee hurricane
reported by Bretschneider


While wave generation equations
(e.g., for storms and hurricanes) and
classical theory can be used to generate and
propagate waves to the initial or incipient
shore-breaking depth (i.e., d/H,)', the
importance of the numerical methodology
developed here is that it requires no a priori
historical knowledge of wave height behavior
seaward of d/H, or (d/H,)'. Equations can be
simply evaluated using a hand-held scientific
calculator; other equations may be as simply
evaluated, while wave height transformation
equations detailing the shore-breaking
process may be more easily evaluated using
a programmable calculator or computer.

For the data used in this study, wave
peaking transformation is initiated in water
depths of from about 1.5 to 7 times the
incident wave height. Two mechanisms of
wave breaking transformation have been
numerically quantified. The first deals with
change in total height of the wave, and the
second with the change in crest height
above the still water level. The first





















H'

0.7



0.6-



0o
Shallow and Tranmltlonal Water 4 Deep
I I I 11111 I I I I1iII I I I gpI l Water
0.0001 0.001 0.01 0.1
d
gT2

Figure 19. Ratio of wave crest elevation above the design water level to wave height
from equation (21).


mechanism is based on a larger set of data
from a variety of sources and representing
many bed slope conditions. For the data
used in this study, the total wave height at
shore-breaking ranged from 1 to about 2.5
times the height of the incident wave. The
second mechanism is based on a much
smaller data set, but the reader should note
that it is essentially a "fine-tuned" version of
methodology presently existing. The amount
of the wave crest lying above the still water
level ranged from 0.5 to 0.84, with a
constant value of 0.84 at the shore-breaking
position.

Some of the major numerical
attributes of the introduced prediction
methodology which address shortcomings of
classical wave theories include:


1. The methodology does not
require that the wavelength is
known to calculate the basic
wave parameters through the
shore-breaking process.

2. The wavelength and wave
celerity, however, can be
calculated using empirically
derived expressions addressed
in a companion work (Balsillie,
1999a).

3. The point of initiation of
shore-breaking (i.e., incipient
shore-breaking given by di /Hi)
can be determined from Hi /g
T2. Therefore, one needs only
to know of the incident wave
height and period to









determined the water depth at
which shore-breaking begins.

4. The final shore-breaking
wave height can be
numerically determined from
the incident wave height and
wave period.

5. Continuous quantitative
peaking of the wave height
can be numerically determined
through the shore-breaking
process from incipient shore-
breaking to final shore-
breaking.

6. The amount of the wave
crest lying above the still
water level can be numerically
determined continuously
through the shore-breaking
process.

7. A shore-breaker
classification can be redefined
as a numerical continuum.

8. The methodology requires
no apriori knowledge of the
wave behavior seaward of di
/Hi rather based on incident
wave height and period, it is
dependent upon water depths.

Figure 20 is a modified version of
Figure 2-7 from the Shore Protection Manual
(U. S. Army, 1984), originally proposed by
LeMehaute (1969). It has become a
"standard" among coastal engineers for
identifying regions of validity for wave
theories. It is modified as follows. First, the
terminology validity is invalid, and is replaced
by "regions of applicability"; this is so
because theories by definition have no
validity since they are unproven. Second,
the "breaking" region in the upper left-hand
part of the figure is renamed "broken" since,
by definition, waves have broken when d/H


= 1.28 or H/d = 0.78 is reached. Third,
the "nonbreaking" region has been moved to
the right and, fourth, a "shore-breaking
region has been identified.

In addition to wave theories, the
region of applicability of results from this
work are plotted on Figure (20) as the stipled
region. The line labelled "incipient shore-
breaking" identifies the beginning of the
shore-breaking process given by equation
(6). It is to be noted that as waves become
longer (i.e., smaller values of H/g T2) wave
peaking becomes greater (i.e., Hb /Hi
becomes greater in value). Very steep
waves (i.e., large values of H/g T2) peak very
little or not at all.

The delineation of "shallow water"
and "transitional water" depths at a value of
d/L = 0.04 would appear to be
inappropriate. The results of this study
suggest that the delineation should be
dropped and that shallow water should begin
at the line segment BC of Figure 20 (which
is given by equation (6)).

NOTE

It is important to note that variables in
this paper represent average wave heights
and periods, etc. Moment variable statistics
relating to significant or maximum wave
heights, for instance, may be found in other
works (e.g., Balsillie and Carter, 1984a,
1984b; U. S. Army, 1974).

ACKNOWLEDGEMENTS

Florida Geological Survey staff
reviews were conducted by Paulette Bond,
Kenneth Campbell, Thomas M. Scott, Joel
Duncan, Edward Lane, and Steven Spencer,
whose editorial comments are gratefully
acknowledged.

The review comments of William F.
Tanner of Florida State University are
gratefully acknowledged. Nicholas C. Kraus












d
=0.04
L
d
-'2 0.00155
gT


d
- = 0.13
L
d
- = 0.014
9T2


d
- 0.5
L


d
g 0.0792
gT2


.-Shallow Water Intermediate Water Deep Water -
I


0.01
d
g T2


0.05 0.1 0.4


Figure 20. Regions of applicability for various wave theories and for the Alpha
Wave Peaking methodology of this work.


of the U. S. Army, Coastal Engineering
Research Center (now the Coastal
Engineering Laboratory) also reviewed the
manuscript.


REFERENCES

Adeyemo, M. D., 1968, Effect of beach
slope and shoaling on wave
asymmetry: Proceedings of the 11 th
Conference on Coastal Engineering,
p. 145-172.


0.04





0.01


0.05


H
g T2


0.001


0.005




0.001


0.0005

0.0005


0.005 0.001










1970, Velocity fields in the
wave breaker zone: Proceedings of
the 12th Coastal Engineering
Conference, p. 435-460.

Balsillie, J. H., 1980, The peaking of waves
accompanying shore-breaking: in
Tanner, W. F., ed., Proceedings of a
Symposium on Shorelines Past and
Present, Tallahassee, FL, Department
of Geology, Florida State University,
p. 183-247.

1983a, 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.

1983b, The transformation of
the wave height during shore-
breaking: the alpha wave peaking
process: Florida Department of
Natural Resources, Beaches and
Shores Technical and Design
Memorandum No. 83-4, 33 p.

1983c, Wave crest elevation
above the design water level during
shore-breaking: Florida Department
of Natural Resources, Beaches and
Shores Technical and Design
Memorandum No. 83-5, 41 p.

1984a, Wave length and wave
celerity during shore-breaking:
Florida Department of Natural
Resources, Beaches and Shores
Technical and Design Memorandum
No. 84-1, 17 p.

1984b, Attenuation of wave
characteristics following shore-
breaking on longshore sand bars:
Florida Department of Natural
Resources, Beaches and Shores
Technical and Design Memorandum
No. 84-3, 62 p.


1984c, A multiple shore-
breaking wave transformation
computer model: Florida Department
of Natural Resources, Beaches and
Shores Technical and Design
Memorandum No. 84-4, 81 p.

1985, Redefinition of shore-
breaker classification as a numerical
continuum and a design shore-
breaker: Journal of Coastal
Research, v. 1, no. 3, p. 247-254.

1999a, Wavelength and wave
celerity during shore-breaking:
Florida Geological Survey, Special
Publication No. 41, p. 31-40.

1999b, On the breaking of
nearshore waves: Florida Geological
Survey, Special Publication No. 45, p.
1-155.

Balsillie, J. H., and Carter, R. W. G., 1980,
On the runup resulting from shore-
breaking wave activity: in Tanner,
W. F., ed., Proceedings of a
Symposium on Shorelines Past and
Present, Tallahassee, FL, Department
of Geology, Florida State University,
p. 269-341.

1984a, Observed wave data:
the shore-breaker height: Florida
Department of Natural Resources,
Beaches and Shores Technical and
Design Memorandum No. 84-2, 70 p.

1984b, The visual estimation of
shore-breaking wave heights:
Coastal Engineering, v. 8, p. 367-
385.











Balsillie, J. H., Campbell, K., Coleman, C.,
Entsminger, L., Glassen, R.,
Hajishafie, N., Huang, D., Tunsoy, A.
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NOTATION

The following symbols are used in
this paper:

c = local wave speed.

cb = wave speed at shore-breaking.

ci = wave speed at initiation of wave
peaking.

d = local water depth measured from the
DWL.

db = water depth at shore-breaking.

dc = water depth from the wave crest to
the bed at shore-breaking.

di = water depth at initiation of wave
peaking.

dt = water depth from the trough fronting
the wave crest to the bed at shore-
breaking.

dti = water depth from the trough fronting
the wave crest to the bed at initiation
of wave peaking.

DWL = design water level (in this paper it
is the SWL).


e = Naperian constant.

Eb = total wave energy at shore-breaking.

E, = total wave energy at initiation of
wave peaking.

g = acceleration of gravity.

H = mean local wave height.

Hb = mean wave height at shore-breaking.

Hb' = mean amount of wave crest at
shore-breaking lying above the DWL.

Hi = mean wave height at the initiation of
wave peaking.

Hi' = mean amount of wave crest lying
above the DWL at the initiation of
wave peaking.

Ho = mean deep water wave height.

Hm = mean wave height measured in the
constant depth portion of a laboratory
wave channel.

Hti = water depth from wave crest to bed
at the initiation of wave peaking.

L = local wavelength.

Lo = deep water wavelength.

m = bed slope.

MWL = mean water level.

PL = subscript denoting the plunging-type
shore-breaking wave.

SP = subscript denoting the spilling-type
shore-breaking wave.

SWL = still water level.












T = wave period.

( = relating coefficients developed in the
text.









WAVELENGTH AND WAVE CELERITY
DURING SHORE-BREAKING


by

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

ABSTRACT

Prediction of wave phase speed and, hence, wavelength at shore-breaking has remained
a controversial issue. Based on available field data (n = 47) and laboratory data (n = 40 to 71),
a family of relationships is derived for predicting wavelength at shore-breaking. Assuming
approximate linear wave speed attenuation, a method is derived for prediction of wave speed
during the shore-breaking process.


INTRODUCTION

Wave height, H, wavelength, L, wave
period, T, water depth, d, and bed slope tan
a, constitute the fundamental hydraulic
variables forming the basis for derivation of
composite parameters (e.g., wave
steepness, H/L; or equivalent wave
steepness parameter, H/(g T2); surf similarity
parameter, tana/(H/g T2)1/2; etc.) utilized in
most coastal engineering design
applications. It becomes not only desirable
to be able to provide for determination of
such parameters over a wide variety of
conditions in order to accurately describe a
natural process, but to be able to provide
the simplest and most straightforward
procedures possible.

As the number of basic variables
becomes large, the solution of any problem
invariably becomes proportionally more
complex. It becomes desirable, therefore, to
provide methods for predicting as many of
the variables as is feasible. One such
variable is the wavelength.

As will become evident,
determination of the local values of H and d
as waves shore-propagate over a shoaling
bed is complex, since following specification


of their values, which may exhibit a wide
range, H relative to d experiences additional
and significant progressive transformations
as shoaling continues. However, the wave
period, once initially specified, is considered
to be conserved (i.e., remains invariant)
across the shoaling bathymetry until
breaking occurs, and a simplifying condition
emerges. The wavelength, however,
experiences attenuation during shore-
breaking, thereby introducing additional
complexity. The wavelength not only
appears in many shoaling design wave
equations, but usually just when one has
little insight as to its local value short of
tedious theoretical calculations for obtaining
an estimation. Most importantly
wavelength is related to the wave speed and
wave energy.

It becomes important, therefore, to
provide methods) for prediction of the
wavelength and wave speed. In this paper
such prediction is investigated during the
shore-breaking process.

DISCUSSION AND RESULTS

As shore-propagating waves
approach the shoreline across shoaling









I I I I I I I I


a-- Constant Depth-

s-- Deep Water -


S0.0110



0----~ 0


tan ab = 0.054


Hi
0.0086-
OTZ
Direction of Wave
Travel


Data from Putnam (1945)
T = 0.065 s


Hb
gT2 *


d-= = 1.28
H Hb


Figure 1. Illustration of wave transformation from deep water to shore-breaking, where
the alpha wave peaking process (segment A B) is given the notation ap.


bathymetry, the wave height tends to
initially decrease due to a number of factors
such as friction, etc., and then begins to
increase rapidly in height just before shore-
breaking occurs (Scripps Institute of
Oceanography, 1944a, 1944b; Putnam,
1945; Munk, 1949; Iverson, 1952; Stoker,
1957; Kinsman, 1965; Balsillie, 1983a,
1983b, 1983c, 1999a). The transformation
is illustrated in Figure 1 (notation is defined
at the end of the paper). It is the increase in
wave height which accompanies the shore-
breaking process (note: waves may break in
relatively deeper water due only to critically
high wind stresses which cause waves to
become critically steep (i.e., forced waves);
shore-breaking occurs primarily because
water depths become critically shallow).
Shore-breaking wave mechanics are
described by the alpha wave peaking
concept (Balsillie, 1980, 1983b, 1983c, in
1999a) and denoted by the symbol a in
Figure 1. Alpha wave peaking, then,


describes the zone of interest
investigation of the wave celerity
wavelength.


The speed with which a group of
waves, comprising a wave train, travels is
not always equivalent to the speed of
individual waves within the group. The
individual wave speed, termed the phase
speed, is given by:


L
C-
7-

and group wave speed, c9, by:

L
C9 = n-
r


In deep water (i.e., d/L > 0.5) n = 0.5; in
intermediate water depths (i.e., 0.04 < d/L
< 0.5) n increases in value to become,
finally, n =1 in shallow water (i.e., d/L <


1.0


0.9 I-


0.8 -


0.7 I
10


Adjusted
H
H1


9 8 7 6 5 4 3 2
d/H


for
and










0.04) and c = Cg.


According to small amplitude (Airy)
wave theory, the phase speed and
wavelength in any depth of water may be
given by:


=L = g tanh 2d
T 2w L


which is evaluated in the following section.

In this work the alpha wave peaking
process is assumed to occur in shallow
water where n = 1. In order to determine
the transformation of c and L during shore-
breaking, one first needs to have knowledge
of the governing boundary conditions.

Terminal Boundary Conditions

Equation (3) is has been applied to
predict conditions at shore-breaking. Shore-
breaking defines the terminal boundary
condition for the phenomena considered in
this work. A more appropriate application
from small amplitude (Airy) wave theory is
given by:

Lb
Cb = i (4)

or where Solitary wave theory is applied, by:


Lb- g(d + H;) (5)

where Hb' is that portion of the wave height
at shore-breaking lying above the design
water level (DWL). [Note: the entire wave
crest of a solitary wave lies above the still
water level and Hb' = Hb.] About equation
(5), Smith (1976) states: "Although this
equation is widely used in the literature on
wave theories and is generally accepted,
few discussions have been presented which
establishes its validity." The same appears
to be true of equation (4), while a general
misunderstanding about equation (3) seems


to have been proliferated in the literature.

Van Dorn (1978) found that at shore-
breaking, the wave speed was always
greater than the small amplitude speed of
equation (4), and smaller than the solitary
wave celerity of equation (5). He reports
that:


C, = 2H


which was found to agree roughly with that
predicted for limiting Stokes waves in deep
water.

Available field and laboratory data
(see the Table) are used to evaluate the
above equations. The wave celerity is
analyzed in terms of the wavelength rather
than the wave speed since the length yields
a much wider range of values (from 0.5 to
100 m, or 2 orders of magnitude). The data
are plotted in Figure 2. Figure 2 illustrates
that equation (3) does not predict Lb and,
hence cb, with the precision of the other
fitted relationships. It is to be noted,
however, that equation (3) was developed
from theoretical considerations to represent
an upper limit envelope curve (see equation
4 for a suggested correction factor and fig.
8, both from Bretschnieder, 1960). In
addition, because equation (3) is an
algorithm it is awkward to apply and not
generally recommended for use in design
work, at least not in the breaker zone.

Based on empirical evidence (field
and laboratory data of the Table), equations
that more successfully predict expected
values in shallow water are closely given by:


L, b=T d,
5


L, = T 2 g Hi
2^









Table of statistics relating measured and predicted wavelengths at the shore-
breaking position.

Investigator L" = m TV L, = m Tr L, m TgH
and n
Category
m r m r m r
FIELD DATA
Gaillard (1904) 26 0.9462 0.9608 0.7514 0.9587 1.241 0.9404
Balsillie and Carter (1980) 21 1.226 0.9692 0.9101 0.9695 1.359 0.9689
Field Results 47 0.9832 0.9737 0.7737 0.9757 1.259 0.9720
LABORATORY DATA
Galvin and Eagleson (1965) 24 1.342* 0.3603* 0.8843* 0.3842* 1.178 0.3658
Eagleson (1965) 7 1.084* 0.8399* 0.8131* 0.8850* 1.233 0.7985
Van Dom (1976, 1978) 12 1.264 0.9636 0.8895 0.9889 1.254 0.9933
Hansen and Svendsen (1979) 28 1.113 0.9947 0.8036 0.9962 1.162 0.9969
Laboratory Results 40-71 1.205 0.9858 0.8561 0.9947 1.211 0.9933
ALL DATA
Total Results 87-118 0.9972 0.9801 0.7794 0.9210 1.254 0.9836
Weighted Results 87-118 1.111 ----- 0.829 ----- 1.251 --
Adjusted Weighted Results** 87-118 1.1176 .---- 0.8374 ----- 1.2644 ---
NOTES: Unless otherwise indicated all m and r are from regression analyses.
These results represent db referenced to MWL and are not used in determination of m, all others used in
the analysis are referenced to SWL.
** Adjusted values were determined so that resulting equations yield intra-consistent results.


and we now have a family of design
relationships for the prediction of Lb and cb.


and:

L = T 2 ( + H
9


Initial Boundary Condtions


Where db = 1.28 Hb (McCowan,
1894; Munk, 1949; Balsillie, 1983a, 1985,
1999a, 1999b) and Hb' = 0.84 Hb (Balsillie,
1983c, 1985), the previous three
relationships can be modified to yield two
additional equations, as:


L, v H (10)
5

and


L, -T g (d, + H) (11)
S4


With the exception of the results of
Hansen and Svendsen (1979), there is little
if any, data available which will allow for
determination of the wave speed at the
point of initiation of alpha wave peaking
(i.e., at cb). Based on other alpha wave
peaking investigations (Balsillie, 1980,
1983b, 1983c, 1999a), it may be
reasonable to assume that ci can be related
to cb. However, the problem is encountered
that the difference between ci and cb is
slight, at least compared to natural
variability in the data and possible
measurement errors.






























0.3 0.5 0.1 0.5 0.1 0.5 0.1 0.5 1.0 5 10 100
Predicted Lb (m)
Figure 2. Evaluation of relationships for prediction of the wavelength at the shore-
breaking position. From left to right figure equations are given by text equations (3),
(7), (9), and (8), respectively.


Another approach using theoretical
reasoning proves useful. The total energy of
a wave is the sum total of its kinetic and
potential energies. The kinetic energy is
that portion of the total energy due to water
particle velocities associated with wave
motion. Potential energy is that portion of
the total energy resulting from the wave
fluid mass lying above the still water level
(SWL). Based on solitary wave theory,
Balsillie (1984) determined expressions for
the kinetic, potential and total wave energies
at the initiation of alpha wave peaking and
at the shore-breaking position. Application
of the Rayleigh assumption (Eagleson and
Dean, 1966),i.e., ... ci ETi = cb ETb (where
ci and ETi are the wave speed and total
wave energy, respectively, at the initiation
of alpha wave peaking, and cb and ETb are
their counterparts at shore-breaking),
resulted in a working relationship given by:


C/
Cb


L/
L = 1.841 .8
L,- L -H
T


(12)


in which (H'/H)ia is the percent of the wave
crest height lying above the SWL at the
initiation of alpha wave peaking (Balsillie,
1983c, 1999a), given by:


SH'
H),


(13)


H
0.515 + 12 H-
g


Comparison of the results from
equation (12) with the data of Hansen and
Svendsen (1979) suggests a slight
modification to the coefficient of equation
(12) by a factor of 0, = 1.073, and:










LI
S 1.84 1, 1 + 14)
Cb Lb Lb
T

Measured data and predicted results from
equation (14) using Hansen and Svendsen's
(1979) data are illustrated in Figure 3.

Wave Celerity and Wavelength
Transformation During Alpha
Wave Peaking

The data of Hansen and Svendsen
(1979) for a slope of 0.0292 and that of
Hedges and Kirkg6z (1981) for slopes of
0.2247,0.1404,0.102 and 0.0667 suggest
that the transformation of c, the local wave
celerity during shore-breaking, is non-linear
but only slightly so. For three bed slopes of
0.022, 0.040, and 0.083, Van Dorn (1978)
illustrates that the transformation of c is
only slightly non-linear and that it accelerated
at a rate close to -0.5 g tan ab.
Unfortunately, Van Dorn did not publish his
transformation data, Hedges and Kirg6z did
not publish their breaker data, and only the
data of Hansen and Svendsen, for a single
slope, are available.

A power curve is used to more nearly
represent the slightly non-linear attenuation
of the wave speed during shore-breaking.
Two points were used in the curve fitting
analysis: 1. ci /Cb from equation (14) and,
2. c/cb = 1.0 at the shore-breaking position.
The following equations provide for
computation of wave speed (and, thereby,
wavelength) attenuation with a slightly non-
linear character:


Measured ci (m/s)


Figure 3.
measured
initiation
predicted
measured
Svendsen


Comparison of predicted and
wave speeds at the point of
of alpha wave peaking;
data are from equation (14),
data from Hansen and
(1979).


where c is the local wave speed,


I -0.3
02 = 0.1475 tanh .00074 (H'- (16)
0074)


and


03 = 0.9987 0.233802


(17)


Examples from equation (15) are
illustrated in Figure 4. Equation (15) is valid
where (di /Hi) (d/H) 2 1.28 where di /Hi
is the relative water depth at which (H'/H),
occurs given (Balsillie, 1983b, 1999a) by:


d, d,
H, Ho


W {n[tanh(65 --)
2 I


(18)


The shore-breaking wave height, Hb,
used in computation of the wave speed at
the shore-breaking position, cb, may be
evaluated from incident wave conditions
(Balsillie, 1983b, 1999a) according to:


L
c
C, Lb
T


L
Lb,


03 2


(15)











Hb = 1.0 0.4n tanh 100 HlO
H, [[ g2 )


CONCLUSIONS

Three issues concerning the
prediction of the wavelength and wave
speed during the shore-
breaking wave process 1.30 I
have been addressed.


First, a family of
relationships based on field
and laboratory data has
been defined for
determination of L and c at
the shore-breaking
position. These
relationships have been
used to refine theoretical
predictions from small
amplitude (Airy) and
solitary wave theories.
The family of derived
relationships provides for
alternate data sources to


l/cb


Third, using the above two equations
(19) as boundary conditions, the transformation
of c and L during alpha wave peaking may
be predicted using equation (15).


The results presented herein are
important because they provide a
methodology for predicting wavelength and
celerity requiring that only the incident wave


1 2 3 4 5 6 7
d/H
Figure 4. Illustration of wave speed attenuation during
shore-breaking using equation (15) for various initial wave
steepness values.


more closely facilitate the
needs of the coastal engineer whose
completeness in data may differ from project
to project. In addition, the commonly used
algorithm given by equation (3) is assessed.
Not only is the expression difficult to apply,
but in surf zone applications it has often
been incorrectly used since it represents an
upper limit envelope fit. In view of the
developments presented in this work, the
continued use of equation (3) in surf zone
applications is no longer necessary.

Second, based on consideration of
Solitary wave theory and the Rayleigh
assumption, the wavelength and wave
speed at the initiation of shore-breaking (i.e.,
beginning of alpha wave peaking where the
wave crest begins to increase in height) may
be predicted using equation (14).


height and period are known. As more data
become available, refinement or
corroboration of the relationship for
prediction of the initial or incipient
wavelength or celerity is desirable. The
same is true for prediction of wavelength
and celerity transformation throughout the
shore-breaking process, although as noted
the slight non-linearity in attenuation should
not pose significant problems.

ACKNOWLEDGEMENTS

Florida Geological Survey staff
reviews were conducted by Jon Arthur,
Paulette Bond, Kenneth Campbell, Joel
Duncan, Ed Lane, Jacqueline Lloyd, Deborah
MeKeel, Frank Rupert, and Thomas Scott.
Their editorial comments are greatefully
acknowledged.









The review comments of William F.
Tanner of Florida State University are
gratefully acknowledged. Nicholas C. Kraus
of the U. S. Army, Coastal Engineering
Research Center (now the Coastal
Engineering Laboratory) also reviewed the
manuscript.

REFERENCES

Balsillie, J. H., 1980, The peaking of waves
accompanying shore-breaking: in
Tanner, W. F., ed., Shorelines Past
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NOTATION

Symbols

c local wave (phase) speed or celerity

cg group wave speed

d local water depth measured from the
DWL

DWL design water level

g acceleration of gravity

H mean local wave height

H' mean local wave height lying above
the DWL

L local wavelength

m coefficient

n number of data points comprising a
sample

r Pearson product-moment correlation
coefficient

SWL design water level represented by the
still water level

T wave period

tan a bed slope

ab alpha wave peaking

0q1 coefficients

Subscripts

b parametervalue at the shore-breaking


position (i.e., at termination of alpha
wave peaking).

i parameter value at the beginning of
shore-breaking (i.e., at the initiation
of alpha wave peaking)

m parameter value before entering
transitional water depths


828 8 "5234
84/29/1 3476 -B F