 Front Cover 
 Title Page 
 Preface 
 The numerical prediction of shoreline... 
 Acknowledgement 
 Abstract 
 Table of Contents 
 Introduction 
 The theory of longshore currents... 
 Discussion of how the volume transport... 
 Linear stability analysis of governing... 
 Finite difference form of governing... 
 Structure and listing of computer... 
 Application of model to St. George... 
 Application of model to Jupiter... 
 Comments and suggestions 
 Reference 

Full Citation 
Material Information 

Title: 
Littoral drift and the prediction of shoreline changes 

Series Title: 
Technical paper  Florida Sea Grant 

Physical Description: 
x, 112 p. : ill., maps ; 28 cm. 

Language: 
English 

Creator: 
Pfeffer, Richard L 

Publisher: 
State University System of Florida, Sea Grant College Program 

Place of Publication: 
Gainesville 

Publication Date: 
1976 
Subjects 

Subject: 
Coast changes  Florida ( lcsh ) Beach erosion  Florida ( lcsh ) 

Genre: 
government publication (state, provincial, terriorial, dependent) ( marcgt ) bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Bibliography: 
Bibliography: p. 108112. 

Statement of Responsibility: 
Richard L. Pfeffer. 

Funding: 
Florida sea grant technical paper (unnumb.) 
Record Information 

Bibliographic ID: 
UF00072267 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Holding Location: 
University of Florida 

Rights Management: 
All rights reserved, Board of Trustees of the University of Florida 

Resource Identifier: 
aleph  000501499 oclc  04655518 notis  ACS1160 

Table of Contents 
Front Cover
Front Cover
Title Page
Page i
Preface
Page ii
Page iii
Page iv
Page v
Page vi
The numerical prediction of shoreline changes due to waveinduced longshore sediment transport
Page vii
Acknowledgement
Page viii
Abstract
Page ix
Table of Contents
Page x
Introduction
Page 1
Page 2
Page 3
Page 4
The theory of longshore currents and the equations describing strandline movement
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
Discussion of how the volume transport rate of sediment is related empirically to the volume transport rate of water
Page 19
Page 20
Page 21
Page 22
Page 23
Linear stability analysis of governing equations
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Finite difference form of governing equations and numerical scheme for their integration
Page 30
Page 31
Page 32
Page 33
Page 34
Structure and listing of computer program
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Page 52
Page 53
Page 54
Page 55
Page 56
Application of model to St. George Island
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Page 63
Page 64
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Page 83
Page 84
Page 85
Application of model to Jupiter Island
Page 86
Page 87
Page 88
Page 89
Page 90
Page 91
Page 92
Page 93
Page 94
Page 95
Page 96
Page 97
Page 98
Page 99
Page 100
Page 101
Page 102
Page 103
Page 104
Page 105
Comments and suggestions
Page 106
Page 107
Reference
Page 108
Page 109
Page 110
Page 111
Page 112

Full Text 
I r.,
i,~ p tifljpn
Florida Sea Grant
Technical Paper
LITTORAL DRIFT AND THE PREDICTION OF SHORELINE CHANGES
Richard L. Pfeffer
Principal Investigator
The Geophysical Fluid Dynamics Institute
Florida State University
TPllahassee, Florida
The information contained in this paper was developed Lijir the
auspices of the Florida Sea Grant College Program, with support from
the liOAA'Office of Sea Grant, U. S. Department of Commerce, :grant
number n4615844 This document is a Technical Paper of the'State
University System .of Florida Sea Grant College Program, 2001 McCarty
Hall, University of Florida, Gainesville, FL 32611. Technical Papers"
are duplicated in limited quantities for specialized audiences requiring
rapidaccess to information, which may be unedited.
Deceriber 1976
PREFACE
This report consists of two parts. Part I contains
5 technical papersk.by Barcilon, Lau, Miller, Tam and Travis,
all of which are published in scientific journals with high
standards of review. The first four of these cover results
of our early work under theSea Grant program. These papers
provide insight into the mechanisms of formation of transverse
sand bars, submarine longshore bars and rip currents. The
fifth paper is a thorough review paper in which theoretical and
experimental modelling of physical processes pertaining to the
near shore are discussed and compared with field observations.
Part II, by.Christopher Miller, presents the final computer model
for predicting changes in the plan shape of shorelines due to
the littoral drift component. This part of the report includes
a discussion of how the sediment transport rate is related empiri
cally to the water flow; the range of incident wave angles for
which the governing equations are stable, the finite difference
scheme, as well as a listing of the computer program. It also
includes a test of the model on specific coastal sites in Florida.
PART I: The study of longshore bars has led to the con
viction that the dredging of certain submarine longshore :bar
systems may actually lead to severe erosion of the shoreline.
A further suggcoion has, been made concerning the possibility
of reversing erosion trends in some beach locations by properly
*Editorialnote: The five papers are included by reference only, ds they
appear in the literature already.
References
Barcilon, A., aid J. P. Lau. 1973. A Model for Formation of Transverse
Bars. J. Geophys. Res. 78(15):26562664.
Lau, J. P., and B. Travis. 1973. Slowlyvarying Stokes Waves and
Submarine Longshore Bars. J. Geophys. Res. 78(21):44894497.
Lau, J. P., and A. Barcilon. 1973. "Harmonic Generation of Shallow.
Water Waves over Topography. J. Phys. Ocean. 2(4):405410.
Miller, C., and A. Barcilon. 1976. The Dynamics of the Littoral
Zone. Rev. Geophys. Space Phys. 14(1):8191.
Tam, C. K. W. 1973. The Dynamics of Rip Currents. J. Geophys. Res.
78(12):19371943.
contouringl the bottom topography to conform with wavereso
nant equilibrium patterns determined by the theory. It remains,
however, to determine the logistics and economics of such a
venture. Specific abstracts of each of the 5 individual papers
in this part of the final report are as follows:
The paper by Lau and Barcilon (1972) investigates the
reflection and nonlinear interaction between the first and
second harmonics of a twodimensional BousSinesq wave train.
Effects of topography are included, with the depth departing
from a constant in a finite region. It is'found that topogra
phy can speed up or reward energy transfer between the first
and second harmonics. The reflection coefficient is signifi
cantly different from the one obtained by using linear theory.
In the paper by Barcilon and Lau (1973).an extension
of Kennedy's potential model is used to investigate the forma
tion of sand bars normal to a gently sloping beach. The results
show that the spacing between the transverse bars depends upon
the inverse of the beach slope and upon th& square of the drift
velocities across the bars. In spite of certain drawbacks the
theoretical predictions compare well with several observational
studies.
The paper by Lau and Travis (1973) investigates the mass
transport velocity in the Stokes boundary layer due to slowly
var :. Sto 'av.s 1ipingrl ingr on and reflecting from a plane
tion is interpreted to indicate the possible locations of sub
marine longshore sandbar formation. It is found that the num
ber of bars is likely to increase when the;bottom gradient is
slight and that the spacing between the crests of the bars in
creases seaward. for some distance offshore. These results are in
qualitative agreement with field observations.
The.paper by Tam (1973) investigates the dynamics of 
rip currents using shallow water equations with a horizontal
eddy viscosity term. In this paper similarity solutions of the
model equations are found which appear to..give reasonable repre
sentations of the velocity profile and other characteristics of
rip currents.
The comprehensive reviauwpaper by. Miller and Barcilon
(1976) covers the present knowledge concerning the dynamics of
rip currents, longshorecurrents and computer modelZling of
beach deformation due. to waveinduced erosion and accretion.
PART II: As mentioned above, this part of the Final
Report (by Miller) contains'the final predictive computer model
including a listing of the computer.program and.a.test of the
model on specific coastal sites in Florida. .The numerical model
is.based upon recent developments in the theory of longshore
currents and Lagrangian description of shoreline deformation.
The computer program requires as input data the breaker character
istics (i.e., height, angle and duration for each wave consider,
ed) computed from raw data. The choice of sites for testing.the
predictive characteristics of this program:was dependent on the
availability of wave and bathymetric data. During 1975 deep
water ship wave data for the Gulf of Mexico were analyzed for
longterm (100 years) study of St. George Island. In addition,
a beach nourishment project at Jupiter Island'provided the.
opportunity for a shortterm. (8 months to 5 years) study. The
field observations on .wve climate .and transverse profiles
gathered by our collaborators at the University of 11crida were
provided to our group for analysis during the course of the year.
Observed changes in the St. George and Jupiter Island plan
profiles were compared with test predictions of the computer
model and are presented in this part of the Final Report. More
over, the computer prediction was .extended.beyond the present time
for these beaches by 20 years and 5.years, respectively, and will
require further monitoring of the beach.morphology to verify the
future predictions.
One result of the study was the definition of a
reasonable range for the empirical coefficient linking the
sediment and water motions. It was found for St. George
Island that, over the time period of interest, the longshore
mode of sand transport dominated and therefore, good predic
tions could be made if the nearshore wave field were known well.
For Jupiter Island, on the short term, the onshoreoffshore
component of sand movement predominated, thus making it possi
ble to model only general.trends due to the reworking of the
strandline by the longshore drift. Special attention is affor
ded the endpoint condition in each case.
The major conclusion of this study is that the present
numericalmodel is a viable predictor of shoreline movement
if (1) the predominant direction of sand transport is longshore.
(2) the nearshore wave climate can be adequately resolved (3)
SSpo in boundaries are teated in a physically ealistic t:;r
The Numerical Prediction of Shoreline Changes Due
to WaveInduced Longshore Sediment Transport
Christopher D. Miller
Geophysical Fluid Dynamics Institute
Florida State University
Tallahassee, Florida
1975
vii
Acknowledgements
The author would like to extend his, appreciation to the
following people:
Dr. Charles Quon of the Bedford Institute of Oceanography,
Dr. Paul Schwarztrauber of NCAR for helpful discussions; Todd L.
Walton of the University of Florida for the use of his wave
modification program; Ziya Ceylanli of the University of Florida
for valuable advice and assistance on the Jupiter Island project;
Dr. Richard Pfe.ffer for making available the facilities and
resources of the Geophysical Fluid Dynamics Institute, Florida
State University.
viii
Abstract
We have attempted to quantify numerically the changes
which occur in ..the plan shape of beaches due to.waveinduced
longshore sand transport. The approach of this study has been
to draw upon recent developments in the theory of longshore
currents, beach .deformation, and sediment transport to synthe
size a numerical model which can be calibrated in accordance
wifh field observation and laboratory studies and, subsequently,
used to make predictions of.shifts in a shoreline, given cer
tain bathymetric and wave data as input. The work of Longuet
Higgins (1970a,b) on longshore currents and that of LeBlond
(1972) on shoreline evolution constitute the fram:ewor: within
which we build our numerical model. The model is applied to
two Florida coastal regions, the Apalachicola Bay region in the
Panhandle and Jupiter Island on the southeast coast.
Table of Contents
ckno ledgeents . . .
Abstract. . . . . . .
SectLion
1. introduction . . . .. .
II. The Theory of Longshore Currents and the
Equations Describing Strandline Movement ...
III. Discussion of How the Volume Transport Rate
of Sediment is Related Empirically to the
Volume Transport Rate of Water . . .
IV. Linear Stability Analysis of Governing Equations
V. Finite Difference Form of Governing Equations and
and Numerical Scheme for their Integration ..
VI. Structure and Listing of Computer Program. .
VII. Application of Model to St. George Island .
... r..t,. ation of Modei to Jupiter TIsland '
'ccr ; ; 3 S es o .oa .....
Page
S. xiii
. ix
* 1.
.* 5.
* 19.
. .. 24.
* 30.
* 35.
* 57.
* 6,6.
. T .
108.
I. Introduction
A beachface can only achieve a state of quasi
equilibrium. Acted on by varying.wave climate, waveinduced
.circulations, tidal currents, windgenerated currents, etc.,
it undergoes modification on both short and .long time scales.
Movements of sediment are induced in both the onshoreoffshore
direction and the longshore direction. There is much evidence
that the sediment motion normal to shore is cyclical in nature
(e.g., the classic wintersummer variations in the transverse
profile) and that over the course of a year the net loss or
gain of sand to the beach system in this direction approximates
zero. Exceptions to this tenuous rule occur when the sand moved
offshore is made unavailable for eventual transport shoreward,
for instance, when a nearshore canyon acts as a sink for the
sand flow (as.in Southern California) or when storm waves remove
the sand to such a depththat the 'summer' accretivee) waves
c~..n... ef.c * shorward iration of the resul.t.ng semi
perman.ent offshore sand bars (e.g., off the west coast of
Florida), etc. In this study we.are concerned with time scales
of, at least, 1 year and longer. We assume implicitly that over
the period of a year there is no net displacement of the shore
line due to the onshoreoffshore shifting of sand. If this were
not the case then we would have to establish criteria, both
theoretical and empirical, governing the movement of sand normal
to the beach; examples of such approaches are provided in
section III.
There are several agents which can be responsible for
the introduction of sind into or the removal of sand from a
beach system. A river can discharge enormous quantities of
sediment into the coastal zone replenishing the beaches con
tinuously. A tidal inlet with its delicate balance between
currents and sand transport can act as an impasse to the long
shore 'river of sand' flowing by its mouth, trapping a substantial
amount in shoals both outside and inside of the inlet channel.
Violent storms (e.g., hurricanes) with their associated surge
and high waves can carry beach sand landward ('washover') or
far seaward making it inaccessible to the normal accretive pro
cesses. The refraction and diffraction of waves around barriers,
manmade (jetties, groins) and natural (tips of islands and
spits), conveys sand into quiescent 'shadow' regions.where it is
sheltered from wave attack. In other words, in order to model
c.rre.tl. the ch.g in C coastal ar'C.., one. must. be famr iliar
;,iL.;" the h:yr.a: yT i .L: .d marine gomorphology peculiar to rhac.
locale.
Confinin ou attention! to t Ihe longhore drift of sand
which we view in this study to be the principal means by which the
particulate matter at the coast is redistributed we seek to
apply recent developments in the theory of longshore currents
(LonguetHiggins, 1970a,b) and shoreline deformation (LeBlond,
1972) to the practical problem of predicting the change in the
shape of a shoreline over a period of time. The procedure.entails
choosing a coastal site for which adequate data on wave climate
and shoreline development exist. With wave and bathymetric data
3
serving as inputs the distribution of breaker wave characteristics
along the shore is computed; using this information a forcing,
function for the longshore flux of.water and sand can be derived.
The longshore divergence of these flows leads to local accumu
lations or deficits of sediment and the subsequent movement in
time of the strandline can be monitored. A predictive computer
model is developed based on these simple concepts and is applied
to St George Island, a barrier island fronting the Apalachicola
Bay, Florida, and Jupiter Island on the southeast coast of
Florida. St. George Island is presently undergoing developmental
pressures. Jupiter Island, plagued by erosion problems, is the
site of a recent beach fill project; the question naturally arises
as to whether or not the local beach system can retain this
artificially deposited sand. Our approach to each site differs
because of the nature of the inputs (source and analysis of wave
data, bathymetry), the scale of the motions, and the time period
o intcr.ast. We elaborate on these points insections VII and
VIII.
Previous numerical studies on the molding of a coastline
composed of loose material by waveinduced.forces include Price, et al.
(1972), Komar (1973), and LeBlond (1972). Price, et al. and Komar
formulated onedimensional Lagrangian descriptions for beach
change, i.e., the translation of the coordinate points which
define the shoreline was restricted totboandfro motion along
a line parallel to one of the fixed coordinate axes. Figure 1
from Komar indicates how the shoreline is represented .and its
movement normal to itself. In LeBlond's model the beach points
.t I
*( e.c di ie of 71r
;i
.
{I
' Y
 L L ~ W ~
L 9,
Discretizat ion of shoreline according to KoImar (1973)
Figre 1.
5
are free to move in the entire horizontalplane (a twodimensional
Lagrangian formulation); this allows.for a more accurate tCra;cking
of the evolving beach shape as well as a truer modeling of end
point boundary conditions (see Figure 4). Among the disad
vantages are the possibility .of very irregular spacing between
points and the merging of adjacent points (note: our computer
model allows for 'resetting' the beach if this:is warranted).
The general format for this study involves a discussion
of the equations governing the fluid and sediment motions, the.
finitedifference form of these equations and a scheme for their
.integration, a treatment of the empiricism which links the magni
tude of the sand flow to the longshore current, an explanation
and listing of the computer program, application of this numerical
model to specific beaches and conclusions.
II. Longshore Currents and Beach Deformation
Over the years various approaches have been adopted.in
attempts to describe how the orbital motion of water waves is
converted into the circulation velocities found in and near.the
surf zone. Longshore currents, which are prominent when wave
crests break skew to the bathymetric contours, have been treated
theoretically by considering the balances of mass, momentum, and/or.
energy in the wavebreaking region. Galvin (1967) has provided
a summary of longshore current theory and supporting lab and
ffeld data up to 1967. Galvin'.s conclusion that both theory
6
and data are wholly inadequate led to more sophisticated theo
retical models by Thornton (1969), Bowen (1969) and Longuet
Higgins (1970a,b). Based on the conservation of momentum and
the "radiation stress" concepts introduced by LonguetiIiggins
and Stewart (1960,1961,1962,1964) these three researchers
independently developed models to explain how the longshore
current is generated and what accounts for its crossstream
(shorenormal) profile. Each postulated that the main forcing
for this current is the oblique approach of a longcrested
breaking wave front. The steady state balance was taken to be
between the shorenormal gradient of the component of excess
momentum flux (radiation stress) parallel to shore and retarding
bottom and lateral friction. Bowen used a bottom friction term
proportional to the longshore current, v, and a constant hori
zontal eddy viscosity coefficient.. Thornton and LonguetHiggins
e1riv.d e. :oressions .fr the bot.toi_ stress and laeral couple n
:::.. .ifer froi. o n' a d .. icn ae ore plausible
physically. They..showed that the bottom friction is proportional
to the product uv where u is the ampiitude of the local orbital
velocity perpendicular to the shoreline. The eddy .coefficient
was assigned an offshore dependence, tending to zero at the shore
and increasing monotonically toward the breaker line, where
maximum mixing is to be expected. In Thornton's model the mixing
coefficient was allowed to realistically decay seaward of the
breaker line, whereas in LonguetHiggins' model it increased
continuously from shore seaward. Thornton solved his.equations
7
numerically; Bowen and LonguetHiggins obtained analytical
solutions. The longshore current profiles in each study were
supported by the available data, such as the laboratory results
of Galvin and Eagleson (1965).
For our purposes the equations of LonguetHiggins seem
the most appropriate because of the .physical bases for his
derivation and the ease of applying.his results. To be specific,
his expression for the longshore current as a function of the
nondimensional offshore coordinate, x* = x/xb (modified slightly
by inverting the xaxis so :as to have a positive depth gradient)
is
V +0
C:)
.,here x. : width of surf zone
5 a 1/2 1/2 3/2
K g s
K 8. C g
a = ratio between the wave amplitude and the local mean
depth in the surf zone, 2C.4
C = bottom stress coefficient 0.01
g = gravitational constant
s = beach slope tan. where 8 .is :the beach angle
= angle b ween the gradient of the local depth con
tours, and the wave propogation vector at the' breaker
line
P = TNSS/C(. aC) i a measure of the strength of lateral
mixing relative to bottom friction where N is a :Iumner
indicating the magnitude of the eddy coefficient.
P equal to 0.4 is a special case in which (1) is rodif.ied t
include a. logarithmic term.
 1 3,, :
Equation (1) is strictly valid only when. w b is small so that
cos b, ~1 It is worthwhile to take note of the assumptions and
simplifications which lead to (1), To enumerate:
1. Linear shallow water theory is employed in the surf
zone and immediately seaward.
2. The wave amplitude in the surf zone is taken to'be a
... The e.ch is p.ane and constant' sloping and _s act,<
on by a monochromatic wave train.,
4.. The angle of wave incidence is assumed to vary little
across the surf zone (due to .uch influence as re
fraction, wavecurrent interaction, etc.)
5. The horizontal eddy viscosity coefficient is set pro
portional to puL where p is the density., u a. charc
teristic velocity taken to be the wave velocity and. L a.
9
characteristic length .taken to be the horizontal
coordinate, x, i.e.,
where :h is the local depth.. .N is dependent on the level
.of turbulence in the water, a reasonable r.n.e based on
field measurements (Inman, et al,. 1971) being 0
If the .condition of small b is relaxed (as it. must be for any
practical study) then the expressions in (1) are multiplied by
cos Ob. We will return to some of these points later. as they
affect our model.
The merit of LonguetHiggins' model is that it removes
muchof the previous dependence on empiricism. Battjes (1972.)
and Earle (1974) have extended LonguetHiggins' analysis to
include a wave field characterized by a Rayleigh wave amplitude
distribution.
Under special circumstances nonunikforit ies of the wave
el.{ .n t:e ., ..re dir.ecti (,n c . rQ',uc nornn ligi e
gradients in the radiation stress atthe breaker line and force
a longshore current. O'Rourke and LeBlond (1972) have studied
the nature .of these .additional functions in .the. setting .of an
idealized semicircular bay.and concluded .that, whereas the stress
due to.the obliqueness of longcrested waves is dominant, the
contribution made by a 'lngshcre modulation in the.wave height
can be significant, with a longshore variation in the angle of.
wave incidence playing a minor role. LeSlond (1972) has expanded
10
on LonguetHiggins' (1970b) analytical expressions for a long
shore current to take into account all three types of drive.
The quantities of interest to us are the volume transport
rates 'of the longshore current and its sediment load. We assume
that the major portion of this transport is confined to the surf
zone.. Outside the turbulent wave breaking region the wave
induced bottom stresses exerted on the sand grains decrease
rapidly as does the mean longshoree) current (Thornton, 1969).
We expect, therefore, for the sediment transport rate to decay
rapidly seaward of the breakerline. Furthermore, LonguetHiggins '
formulation tends to overestimate the magnitude of lateral friction
in the seaward zone (due to the artificially high mixing coef
ficient) as well as the role of bottom friction (since shallow
water theory magnifies the true orbital velocities in this
region). Multiplying (1) by x* and cos Ob and integrating
across the surf zone, O
? orei. pr unit time
If, as with LeBlond (1972), we assume that the volume transport
of sand which accompanies this longshore flow is simply a :fraction
of the total water transport then we have
Qj C (3).
II
The determination .of T in terms of meaningful physical quantities
measured in the laboratory and field is discussed in section III.
2. Shoreline Movement
There is no rigid boundary separating the beach from the.
ocean. Where they meet at any moment defines the instantaneous
shoreline. A function of space and time this line undulates in
response to wave runup, the presence of edge waves, wind and
waveinduced setup, the tidal cycle, surge, etc. For.our pur
poses we consider it to be the mean water level with respect to
the local tidal conditions. Mathematically at any time, .t, this
line can be described by an equation of the form (see Figure 2)
F(.y,0,) (4)
We can establish the following relations for the local normal,
A
n, and tangential, t, unit vectors:
ae .o'r n (
', '" y' c
A
and since n and t are orthogonal (A t 0)
/ Z /OF ' (6)
1 };j
A
S XL
S7 A
Figure 2.
Definition of relation between fixed and local
coordinate systems.
13
The movement of the shoreline normal to itself will be
a function of both the magnitude and the longshore variability
of the littoral drift. The beach.will prograde if more sand
is deposited in an area than removed over some time interval
and will retreat if the sand extracted exceeds .that supplied.
In other words, local erosion and accretion depend on the sign
of the longshore divergence of the sand transport; this quantity
is expressed in terms of the.longshore coordinate, y, as
d V d X dL K ,.
P(; 3JFJ 7 > (7)
L t' .,; ^ .i /
The displacement of the shoreline in the xdirection
depends solely on the variation of Q in the ydirection and any
displacement in the ydirection depends only on the xcoimponent
oF vith reference to the righthand coordinate system of
Fiurce 2 .e see that th e projection of on the y axis is
Uy
given by
a) it xrF
/ Y / Id F I Z V2
/'_. L, [fL / 8
and its xcomponent resolution is
JF
CY i ,~\ y1 F jf '. (9)
[\C / (y/ j
where Ois positive counterclockwise.
14
Therefore we can express the temporal change in the
horizontal coordinates of any beach point (y,x) as a balance
.. jF'j)) C ) dx a
JrJ
We follow the approach of LeBlond (1972) and cast (10) and (11)
into forms more appropriate for application to arbitrarilyshaped
shorelines. Referring to the vertical crosssection of the
planesloping beach in Figure 3 it is assumed that the profile
remains unchanged in time, i.e., the slope at a particular point
along the beach is constant. In response to erosion or accretion
the entire profile shifts laterally inward or outward, respec
tively. This is a convenience and implies that either: (1) Thce
distribution of sand transport capacity across the surf zone is
such as to maintain the profile, or (2) there may be a smoothing
effect normal to shore due to waves reworking the sediment into
an 'equilibrium' profile. Neither of these hypothesized factors
is considered explicitly here. The amount of sand gained or
lost is proportional to the area of the parallelogram EFGH.
'D' represents the depth beyond which there is little or no sand
transport (in this study, D is the depth at which waves begin to
break). The initial plan shape of the beach is specified by a
set of discrete points (see Figure 4) whose movement in the
set of discrete point's (see Figure 4) whose movement in the
< .
Crosssectional profile of lateral movement of a beach
point.
Figure 3.
f'o'S .i f i 1 h.ihA
r^r Y
4: _~~
dfZ
A;~ ) li
J41 ;.'
Figure 4.
Representation of strandline in LeBlond (1972).
17
horizontal plane isdetermined by the netamount of sediment
transported into or out of the control volumes. These control
volumes are bounded at the mean shoreline by line s:;n.:t.:
joining adjacent beach points, by the planesloping bottom, and
by a line parallel to shore at an offshore depth, D. A simple
continuity equation relates the translation of the beach points,
normal to the local shoreline, to the longshore .divergence of
sand transport, i.e.,
+  (12.)
In terms of the fixed coordinate system (y,x) and in
view of (10) and (11), (12) becomes
oiv c,_<; 0 G.
"; P J. (13a)
J t o.s't aL '
where D is a function of position along the beadc, i )e., D = D()
[Note: There is a typographical error in LeBlond's (1972) equation
14b.] Equation (13) is valid for. a righthand coordinate system
with 0 positive counterclockwise or a.lefthand coordinate system
with 6 positive clockwise.
It is obvious that there will be seaward discharcres of
water and sediment (e.g., in rip currents) interrupting the
lonq.hore flow. In a strict 'control volume' approach these trais
ports have to .be accounted for to' satisfy .mass balances. We are
18
assuming that on large spatial and time scales their contri
bution is minor.
III. Longshore Tjai.sport of..Sediment
In and near the surf zone the waves provide a.large
part of the stress required to dislodge sand particles: an.d .make
them available for transport by' the mean currents. There are
two modes of sediment movement which can result, ''suspended'
transport or 'bedload' transport. Suspension of sand particles
in the fluid column can occur in response to the turbulent
action of the breaking waves and the presence bf a small cur
rent is sufficient to advect these sand grains. 'Bedload'
motion is the treep of sediment particles in constant or
intermittent contact with the bed and requires a threshold
shear stress .to overcome static friction and initiate motion.
The dominance of one mode over the other is largely dependent
on incident wave type and to a smaller extent on the sand
characteristics. Suspended material is more likely to be
associated w ith plunging breakers reas ereas bedload movement
often predominates when the breakers are spilling or surging.
Spilling breakers invariably occur when large waves break on
a mild slope. As the incident wave height decreases and/or
t"eh, heAci slope increases in a continuous fashiCion the .spillinrg
breaker evolves successively into a plunging, collapsing, and
surging breaker. Galvin (1972) has provided both descriptive
and parametric classifications for that's breaker types. The
transition in the: direction spillingplungi ngcollaspsing
surgin. is a inverse Ifunction of the .deep.:ater wa.e steepness
(or breaker height) and.a. direct function if the beach slope.
For.some time it has been recognized that.a parameter
critical to the question of whether sand is moved onshore or
offshore (resulting in a 'summerswell' profile or a 'winterstorm'
profile, respectively) is the deepwater wave steepness, H /L
where Ho is the deepwater wave heightand L is the deep
water wavelength. Laboratory experiments.by Johnson (1949),
Rector (1954), Scott (1954), Saville (1959) point to a value of
H /Lo = 0.025 as marking the transition between winter and
summer profiles. Values greater than this correspond to erosion
and values less than this to deposition, although this is not
a stringent rule. Saville's (1950) experiments suggest that
the suspended mode of sand transport dominates over the bedload
mode for large wave steepness and that this relation is
reversed when the wave steepness is low. However, it cannot
be stated that there exists general agreement as to which mode
is predominant in the surf zone.
Presently there are no definitive experimental studies
relating deepwater wave steepness and breaker type to the sand
transport mode and its direction.
We are restricting our attention to the littoral drift
component. Dean (1973) has formulated a relation between the
longshore transport of suspended material and the longshore
component of energy flux. By assuming that a fraction (empirical)
of the energy flux is consumed by the falling sand grains,
determining a volumetric suspended concentration and using.
LonguetHiggins' (1970a) expression for the average longshore
velocity he obtains
o D f ,
(14)
where Q
CD
s
ps w
w
Hb
b
E
a
= volume transport rate of. sand
= bottom drag coefficient
= beach slope
=,density of sand and water respectively
= fall velocity of sand grains which is a function
of grain diameter
Save height at breaking
=wave angle at breaking
= longshore flux of wave energy = C bEbsin
where Eb is the wave energy density and Cgb the
group velocity at the breaker line.
In contrast, by considering only bedload motion, Komar
and Inm.an (1970) followed Inman and Bagnold (1963) and expressed
lte onshoree transport rate as an immersed weightI transport.
1 ,'
(15)
Q Q
where g = gravitational constant
a = correction factor for pore space
Laboratory and field studies indicated that I could be set
proportional to the product FbCb where Fb = lateral wave thrust
at breaker line = b (see LonguetHiggins (1972));
 sin 29b
cb = phase velocity at breaker line, :i.e.,
o (16)
22
where o is the empirically determined nondimensional constant
of proportionality. If we rearrange (15) we get an expression
for Q
Qa. ( (17)
If we expand equations (14) and. (17) and use the
theoreticalempirical breaking criterion
b/ 'h (18)
where hb = breaking depth
y =0.8
we obtain, respectively,
~ GHC,,s
G1 S )(19)
and QG G(, ,. )20)
Equation (19) in comparison to (20) contains the additional
parametric dependencies on cD, s and w and therefore offers the
possibility of modeling the effect of these parameters. There
is also to be noted the difference in the exponents of H It
is instructive to compare the expression for Q given.in
section II, equation (3), with the above results. Equation (3)
can be rewritten as
/ C / (21)
where
7 Z I
We note that the functional dependence on the breaker.height,.
Hb, and angleof breaking,
Furthermore, the product Fbcb, contained in (20), is the quantity
against which many laboratory and field observations are taken
(see Shore Protection Manual Vol. 1, U.S. Army Coastal
Engineering Research Center, 1973). Equating (20) and (21) we
obtain an expression for our unknown proportionality coefficient,
T, i.e.,
5 P ; a(22)
If we insert typical values
cD = 0.01
K = 0.5 'see Das (1972))
py = 1.02 g/cmr
3
s = 2.65. g/cm (quartz)
a = 0.6 (packed sand)
S= 0.2 corresponding to P = 0.13
we get T = 0.0026. Physically this means that for each cubic
Smelter of water transported across .a plane perpendicular to the
shoreline 0.002.6 cubic'meters of sand will accompany it. There
is considerable scatter.in :the data and, therefore, in the
estimates of :K ::easurements have been made by various means
O
24
under a variety of test conditions (e.g., differing wave 'spectra,
beach bathymetry, duration of record, instrumentation, in
terpretation, etc.). Das (1971, 1972) has described several of
the methods employed in the lab and field for determining the
rate of sediment transport and summarized much of the data on
Ko. Noda (1971) has reviewed the techniques presently available
for measuring littoral drift in the field.
In view of the orderofmagnitude uncertainty in the
value of K we treat this quantity as a control variable subject
to adjustment over a reasonable range. It seems unlikely that'
K will assume a single value appropriate for all beaches'since
there are beach parameters (such as w. in (19)) whose significance
has not been guaged.
IV. Stability Analysis of Governing Equations
It is customary and worthwhile to determine if one's
Working equations are subject to any intrinsic instabilities
for some range of the parameters involved. If the instabilities
of the analytic form of the equations can be identified, then,
spurious results appearing in their.numerical integration can
be labeled and/or avoided. We consider, again, equations (13)
in the coordinate system defined in Figure 4. To avoid unneces
sary and lengthy computations we postulate that the original
(unperturbed) shoreline is straight and lies parallel with' the
horizontal axis. By superposing small perturbations on this
configuration and examining'under what conditions these
25
disturbances grow, decay, or .remain unchanged we can detr;rmin
!when our equations will behave peculiarly, i.e., admit
oscillatory solutions that "blow up".
We express Q, the transport rate, as
 ( C^(^7 ) ^"(23)
where B contains implicitly all empirical constants, as well'as
the functional form of Hb which is assumed independent of the
longshore coordinate; ~o =. b where :"o is some constant initial
value for b adjusted due to changes in beach orientation (6).
Upon expansion this becomes
r = i /SSv ( ,,, 6 t e4 ; E)s(r 'Q *
Q0L = r ( (24)
We note that (see Figure 2)
l st, d (25)
= C~'Id. :0 (26)
.y
so that we can write the lovngshore derivative ,:of Q as
 (27)
where
oY o "d
y 0 (28)
Cs)S I S ra (29)
Now, let y be a function of the original (t = 0) arc
length 's' (linear) and the time 't', i.e.,
7,
Then we can evaluate the derivatives in (27 ) as
SY /) s
y) / o
and
2
9K
^j
d d 2 )'!
( __J_
(31)
92y ^Y/ d{ ^
y Ss Y/,
0 .^.J.,_~._o~. s._ / ~ .__
Jsl '* ^k
In view of the folloing relations
S7
d 3 / S S
^s^ ^..s
(32)
(33)
we can rewrite (30) and (31) as
)
)x a^ 7 y.4
(34)
(30)
Sdx7d 5
d?; f;:(X/
* _ J/S 0
C)Y Y/,
LI i~ ~4 .~ i
i"~~)L~,
3s,.: Js, I
i Js 35 z `7 3
it jir
3 3 i
S2/ d y J .
Y/ 5 o., Z
1 ~
Our original governing equations (13a) (13b) become
 cJ 41
.ot
*o~
where 3 yQ/y..is defined by (27), (28), (29), (34) and (35).
We now introduce expansions of the form
x f b ot . ing elana to 0 (.)
a(l ~os,b l so cy,:t, ), o l w,.n v i,, :.) +....
ana establish the following relations valid to O(E)
c)? : _
dsir .4
0 0
.1J
c2" ,**J" ,
,^ *. .......
< : & 
and
Zy (d y)
***.1 <._
 ~~ [
(3,5)
(36)
(37)
(3;8)
(39)
~____ iliL_____~___
d~s
"Y
dZo \
(e),
28
Substituting (39) into the x equation, (36), we obtain
C / d 5 J'i o (
0 L
(40)
The 0(c) equation extracted is
ta P 0%
or
d 'S (41)
This is of the form of a onedimensional heat (diffusion) equation
which is a wellstudied linear secondorder partial differential
equation. A fundamental property of this equation is that initial
value information can only be propagated in one direction, i.e.,
it is not possible to integrate this equation backwards in time
to determine the initial distribution of xl. For this reason the
coefficient on the righthand side of (41) must always be ,positive...
We identify the regions of stability and instability according
to
/ i (0 Z o .i .
1, .,. (42)
I .0, / ; 5
 ( I 2 U<12d. < 0
29
Thus, for breaker angles greater than +450 the shoreline will
be unstable to perturbations of all wavelengths and will undergo
oscillations of increasing amplitude. This has been confirmed
numerically. A small disturbance of arbitrary wavelength is
imposed on an initially straight beach such that b assumes values
greater than +450. It is found that the shoreline is stable to
the disturbance if gb does not exceed +450 and is unstable
otherwise.
It is interesting to speculate whether such an instability
occurs in the field. Bowen (personal communication) has noted
that windgenerated waves in small lakes can break on the beach
at very acute angles and cause the shoreline to deform in a wave
like manner, i.e., be responsible for periodically spaced shore
line protuberances. Aerial photographs of coastal areas
frequently show undulations of the shoreline with.definite
wavelengths. However, Dolan (1970,71) and Vincent (1973) have
correla ed the existcnc. of these wanders with inner and outer
submarine bar rhythms. These bar systems may represent in
themselves an instability of the submarine bed to longshore
currents as suggested by Sonu (1972) and theorized by Barci.lon
and Lau (1973).
In our model we exclude angles, #b, which fall within the
unstable range of (42). It is conceivable that a secondorder
term in Q added to the righthand side of (36) and (37) might
damp the growing oscillation. .Rowever, thephysical justifi
cation of such an "artificial viscosity" term is not clear. The
30
omission of  b >45 is not considered serious. It is usually
true, especially for swell waves on mild slopes, that refraction
will limit the breaking angle to the stable regime. This may
not be the case for local sea on steep slopes. An analysis of
our study results indicates that over 95% of the %b's generated
satisfied the stability criterion.
V. Finite Difference Form of Equations
We wish to express our governing equations in a form
appropriate for numerical integration. Equations (13a) and
(13b) are discretized according to LeBlond (1972) as
t J 73 v A (43a)
d \ /
S(43b)
where the superscript n denotes the time level and the subscript j
the space level. Referring to Figure (4) we make the following
comments:
1. 0. is the orientation of the beach segment, j, in
the fixed coordinate system (y,x). The O's, of course, are
altered as the beach points migrate.
31
2. B. is the 'effective' (averaged) angle at a point, j,
given by (Qo._. Q_)
3. A. is the sum of the distances between point j and
adjacent points j1 and j+1, i.e.,
4. The transports Q are evaluated at midsegment points
and are characteristic of a segment (not a point).
At endpoints these definitions of B, A, and Q need to be modified.
The general class of integration schemes we adopt is
'predictorcorrector'. A.predictorcorrector method represents
an iterative approximation to a fully implicit scheme.
Kurihara (1965), Lilly (1965), and Baer and Simons (1970) have.
discussed the performance (e.g., stability, conservation
properties, accuracy, phase errors, etc.) of several of the
more widely used predictorcorrector schemes (leapfrogtrape
zoidal, AdamsMoulton, Mlilne). The advantage of such multi
step methods lies in their ease of application.and speed
(provided the proper step size, At, is chosen). We employ
Hamming's (1962) predictorcorrector method which consists of
the fourthorder Milne predictor and Hamming corrector. The
Hamming corrector is favored over more traditional correctors
(Milne, Moulton) because it exhibits stronger stability,
although at the price of an increase in the magnitude of the
truncation error.
32
"+' " h fJ
Predictor: 4 (44)
< *' ? .7 . C ]
Corrector: 7 (45)
where z = (y,x) and f represents the righthand side of (43a,b).
It is obvious that, in addition tothe initial datum, 3 values
of z and the corresponding f's are required at the n1, n2,
and n3 time levels. Since these are not available a special
method is required to generate them. We revert to a numerical
method based on a Lagrangian interpolation formula (Ralston, 1965,
p. 191) which yields estimates for zl,z2,z3, given zo, namely
Z /Y ^fJ tf ) (46)
Z o 2 / (47)
The error term is 0(At5). We guess values for zl1z2,z3, calculate
the corresponding flf2,f3 and use (46), (47), and (48) to com
pute new values of zl,z2,z3. This procedure is then iterated to
convergence. (An alternative method for furnishing starting values
is the RungeKutta scheme.) Ralston (1965) has provided a careful
analysis of the properties of predictorcorrector methods as well
as their merit in relation to other schemes.
The step size, At, must satisy several criteria. In a
physical sense.it is controlled by the spatial increment Ayj
33
(the distance between neighboring beach points) and by the
average speed of the sand particles, vs (a function of the long
shore. current strength and the grain {characteristics).. Linear
computational stability requires that
nt a v A/ : (49)
A rough estimate for vs can be had by noting that the triangular
wedge through which the longshore current flows has a cross
sectional area of 1/2 Dbxb and therefore the sand transport
rate equals z
v = zTV/SX (50)
A.more rigorous requirement than (49) is that the increment,
At, be small enough to meet the convergence condition on the
corrector equation, (45), preferably small enough so as to
achieve convergnc_ inn one or two interations; it must also.be
sufficiently small .to satisfy any restrictions on the najgnitude
of the local truncation error which is given approximately by
(Ralston, p. 189)
(51)
where z + is the predicted value and Zn+l the corrected value.
In addition, the step size should be large enough so that round
.off errors and the number of derivative evaluations is minimized;
otherwise, the multistep method loses its chief advantage,
34
namely, speed. Ideally one would like to adjust At so that
only one application of the corrector equation is necessary.
Equation (51) is helpful in two ways: 1. knowledge of E, as
the integration proceeds, can suggest in which direction At
should be adjusted for efficiency; 2. E can be used to actually
modify the solution of the corrector equation. The proper
choice of the step size is a function of the geometry of a beach
site and the incident wave energy levels.
VI. Computer Program
Structure
The program is divided into 11 sectionsa core and ten
subroutines. We note below the designation and function of each
part:
1. Main Program
a. Read input parameters (which run program)
b. Establish shoreline
c. Call working subroutines
d. Execute Hamming predictorcorrector (repeat)
2. Subroutine EMPIRCL
a. Set value of constants appearing in expression for
longshore current
b. Compute coefficient, T, the ratio between the sand
and water transport rates
3. Subroutine ADJUST
a. Read in values of breaker height, angle, .and dura
tion (fractional) of a particular wave type for
each beach segment; compute transport rates
b. We expect the angle of wave attack to change as
the beach orientation is altered. An adjustmentt"
angle, the difference between the old and new beach
angles, is added to the original (b and a revised
transport figure is calculated. This is done at
time intervals chosen by the user. Any accompany
ing refractive modification of wave height is
considered secondary and is neglected.
4. Subroutine INITL
Generate all necessary starting values for use by the
Hamming :scheme as outlined.in section V.
5. Subroutine DERIV
a. Given the beach coordinates compute the beach seg
ment angles and the spacing between adjacent points.
b. Given the volume transport rates of sand along the
beach compute the incremental change in position of
each beach point :over a time interval, At. A Fortran
ENTRY statement links DERIV with that part of sub
routine ADJUST that recomputes the incident angles
on some regular basis because of the reshaping of
the shoreline.
6. Subroutine AREA
a. The surface area 'df the beach is an important
quantity. Its change can be monitored by co.p:uting
the area difference between two successive strand
lines.. In Figure 4a the calculation is straightfor
ward since the y coordinate of each endpoint remains
constant. Figure 4b represents the more general
case wherein the endpoints are allowed to move
freely. A rough estimate of the net areal change
(additions due to accretion minus depletions due
to erosion) can be had in the following way: (i)
connect the endpoints A, B and E, F as shown, (ii)
compute area under curves AF and BE (summations
over a series of trapezoids); these are the exact
areas under a discrete beach which is itself an
approximation to the real strandline, (iii) compute
the areas of trapezoids ABCD and EFGH, (iv) sub
tract the two numbers in (ii) and, then, from this
result subtract the areas computed in (iii); this
number represents crudely the increase or decrease
in beach area. If the positive or negative con
tribution near an endpoint is desired we can esti
mate this at the left end to be ABA' where A' is
the point on curve AF at.which a line dropped from
B parallel to the vertical axis intersects. The
x coordinate of point A' is determined by linearly
interpolating between the beach points on either
side. The area, then, is just the area under AA'
minus the area ABCD. Similarly, the area EFE' can
be computed.
b. Approximate volumetric changes can be obtained by
multiplying the discrete trapezoidal areas by the
local value of Db.(see Figure 2).
7. Sub utine .iSC
f, for some reason, i 'is desirable to have ;he spacing
between neighboring beach points more or less equal, it is possible
to reset the beach points to accomplish this. The circumstances
which might dictate this action aremany: (i) a more rational con
trol over the size of At would result; (ii) a few beach points may
be moving at an anomalous rate compared to their neighbors (e.g.,
the point.at the tip of a rapidly expanding spit); (iii) eqqal
increments might be more compatible with the longshore resolution
of the wave field, etc.
With reference to Figure 4c we shift.only interior points;
endpoints must retain their positions if the beach shape is not to
be disturbed. All or only part of the shoreline can be reset.
A 4,vAT Tf
LAND
Figure 4.a. The difference. in area between successive strandclines
whose endpoints have the same abscissal coordinate.
WATER .
L AN D
,I
S' 1
Figure 4b. The difference in area between successive strandlines
of arbitrary shape and orientation.
I ,
^^ fxecl
Figure 4c. Schematic diagram defining beach points and their
movement in the 'resetting' process.
39
Over that portion of the beach which is to be redefined the lengths
of the discrete longshore segments are summed over and divided by
the total number of segments to yield an "average" spatial incre
ment. The first interior point is moved along the segment immediately
to its left (like a bead on a string) until the distance between it
and the fixed point on its left side is the "average" increment.
This interior point now becomes the fixed point for the next interior
point, i.e., the second interior point is moved along the line seg
ment joining it to the new fixed point until the distance between
them is, again, the average increment. (note: movement along .these
segments can be forward or backward). This process is repeated
until the fixed point on the right hand.side is reached (either
the right endpoint or the point defining the right boundary of that
portion of beach to be reset). Because the right boundary point is
not allowed to move'this procedure must be iterated 4 or 5 times
before all the beach increments converge toward one value. This
method must be applied thoughtfully; otherwise, the resultant shore
line mayv deviate too much from its former shape.
8. Subroutine RESULTS
Display results of computationsin printout form
9. Subroutine PLOTTER
Use the Florida State University plotting package (Fortran
callable, calcomplike routines) for displaying the shoreline evo
lution graphically.
10. Subroutine ROTATE
Rotate the NS, EW axes if beach points are desired in a
new coordinate system. This is .used for plotting purposes, i.e.,
to show:direction of maximum beach change.
40
11. Subroutine ERROR
Calculate error between actual and predicted quantities.
Below we provide a listing of the program with accompanying
comments
PROGRAM SiORLIN(TAPEi,INPUT,OUTPUT,TAPE5=INPUT,TAPES=OUTPUT,PLOT)
fr** THIS PROGRAM MONITORS THE CHANGE IN THE PLAN SHAPE OF A SHORELINE
r DUE TO THE DIFFERENTIAL LONGSHORE TRANSPORT OF SELIENT INDUCED BY
r WAVES BREAKING AT A ANGLE TO THE SHORE.NECESSARY INPUTS ARE THE
r HAVE CHARACTERISTT.S (H IGHT,ANGLE OF INCIDENCE DURATION)AT THE
C 3REAKER LINE AS A JN3TION OF LONGSHORE POSITION AND THE ORIZONT&L
( COORDINATES OF THE POTNTS WHICH DEFINE THE SHORELINE AS RECORDED
f AT VARIOUS TIMES.
r THESE QUANTITIES ARE ASSUMED TO HAVE 3EEN GENERATED IN ANOTHER
C PROGRAM.IN ADDITION IT IS ASSUMED THAT THE BREAKER ANGLES WHOSE
r ABSOLUTE VALUE IS SRE TER THAN 45 DEGREES HAVE BFEN EXCLUDED.
f AB HAS UNITS OF DE0SEES.
f
C
C THIS PROGRAM IS WRITTEN IN FORTRAN IV LANGUAGE .FOR THE CDC 6500
r COMPU.TER.SAMPLE V0LJES FOR MANY iF THE PROGRAM PARAMETERS APE USED.
C PLOTTING ROUTINES ARE WRITTEN FOR THE GOULD PLOTTER.
r
r THE LETTERS A,..,I.T. AS THEY APPEAR THROUGHOUT STAND FOR THE PHRASE
r EAS DEFINED IN TEXT..
C
DIMENSION FRAC(50)
DIMENSION OXX(90),DYY(90),HI(90),H2(90),Zl(90),Z2(9 ) ,X(90),Y(90)
COMMON DEL(0) ,ErA(90) THETA(9"0)
COMMON X0(9 ) 0),X1(90),X2(90)X3(90),Y (90),YI(90) Y2(90),Y3(90)
COMMON DOY (90) ,DX3(90). ,OY (90),rXi(90),DY?290),DX2(u) ,oDY3(90),
TDX3(93)
CMMON/ LOC/fI (5 30) '
C MM ON/ L LK/ ( 50 )
C 0 ;. , '/ '' L ; C 1
COMMON/d 1/KOUNT. Th117,(50)
COM'ON/3LK5/L (10) M( IC)
COMMON/BLOCO/II
COMMDN/3LOCC/DI
COMMON/3LOC2/LIMITLI 1,LIM2
COMMON/3LOC3/CV PT V"T2
COMMON/3L0C/OIF DIFV, DIF1, 0IFVi
COMMON/3LOC7/IVT Dr
COMMONM/LOC11/COEFF
50 FORMAT(IX,5F .2)
1I FORMAT(2I 2213,I5, =5 0)
52 FORHAT(iH ,2(2X,TI), X,2(2X,I3),?XI5,2X,F4.0)
<5 FORMAT(1X,10T3)
5n FORMAT(2X,7(3XTI) ,5X, F5.l)
61 FORMAT(3XI4)
C+*Y THE PLOTTING MODE IS ENTERED INTIPLOTLIS AND GOULD LIBRARY
r ROUTINES ARE CALLED.
CALL PLOTS( 0. 0,0.0, 4HPLOT,P)
r** THE INTEGER VARIABLES L AND M ARE USED TN SUBROUTTNE ADJUST
r TO PICK OUT THE BEASH POINTS OF INTEREST IN COMPUTING NET
C TRANSPORT RATES FR VARTABLE STRETCHES OF SHORELINE.
REAO(5,35) (L(I) M( I) I= 1.0)
0 3OEFF IS USED. IN SUI3PUTINE ADJUST.
COEFF=0.5
L ... R P.cA D ROGRAir PAPA'1ETEDS:NM1 iT THE_ iJ'i*JM E Or WAVE DATA S.TS,
N2 TS T 3 N!"U:B;E? OF TTIES THE CORRErCTOR EQUATO1 S. AR APPLLIEO
SLI' TS THEi NUi'!H EF OF DOTNiS W'HIG4 1AKE UP THE BAS: S'HOrELTNE,
M MAX IS THE MBEF C TNT~ S WrICT MAKe UPI THE SHOELTNE AS
Cr 0E.zr.'VD AT A LATER TI1M'tIrNT IS THE TI!TERVAL(EXP ESSO s.
r NUIA3FP OF TTM'". ST=PS) IT WHICH RESULTS A.E DISPLAYr.P3T T IS THE
C TIME STE>P.
rE rU( C, si) N1,M2, TM, M)IN, IFI T, DT
t .I T (o, Z) i i, 2 LTI MA IP''L IT TT
r '4 SO;F Wr.;KINS DEFI"7ITIOTn S:
LI IT=LIM41
LI 1= L T 1 i
LiM 2= LTM 2
C :" FRAC IS THE FRACTION OF Tr1 TOTAL TTME OF RECORD(EXPRESSED AS
r THE UNIT 1) DURING W1HTH A PARTICULAR i;4PWAE ACTS.H AND A3 aE THE.
r YRE.AKEr< HEIHT AN" AtGL,,RSPECTIVELYFOP THAT WAVJE AS
C CnlPUTED FOi< EACH SEC4 SEGMENT.
READ( ) (F aC(I) =1, Ni .. .
READ(l) ( ((H3(II J),3J), J), II=1,Ni) ,J=i, LIMI)
r
Cp"* THE DURATION OF A PARTICULAR tWA9E TYPE OVER THE PERIOD OF A YEAR
C IS COMPJTED TN TFM3S OF NUMBER 0; TIME STEPS. IF THE I TEGRATION
.( PROCEEDS LESS THAN A YER.THEN TIS CALCULATION MUST Sc3 '10DIFIEl.
.0DO t I .i,N
GU N41C(I)=FRC(I) E03e. ? ./lT
ry.x THE' TOTAL NU'4.ER 3F TIME STEPS IS COMPUTED.YFAR IS THE NU3E, O'
V ARS OVER WHICri TTH NTEG.
YE ,R=, .
NUMnT=YEAR36:>" 2+. /CT
C ,fllFINE FACTORS FOP O0\NVERTIN5 FROM RADIANS TO DEGREES AND. ACGKY
PI =,. ATAN( .)
CVRT1=PIT/180.
CVRT2=i./CVPTi
C REAi INITIAL VFRTICtALHCORiZnTAL COORDINATES OF :EACr POINTS.
S ALSO RE\D THE COPr.DINTTES :S REOROE3O AT SOME LATER TIME IF THESE
r ARE AjAILA.rL.E(FOP C3 ARISCN WIT PREDICTED VALUES).
FirEAD (5k ) (XO (T) ,YC(I) I=1, LIM)
i A:i ( c^ n'0 ) X(YX(i) ,YY(T r T= ':'AX,
0 C V: CONVERTS OUTSTAN3DI IS UNITS TO METERSIN THIS CASE,FEET TO
C ME TER S .
CV=1. /3.28
DO i I=1,.LIM
XO (I) = V^X' r(I)
YO (I)= CVlYC(I)
r, THE INITIAL VERTIIA ANN HORIZONTAL COORDINATES ARE DESIGNATED
c X AND Y,RESPECTTVELY.
X( I) =X0(I)
1 Y(T)=YO(I)
DO 12 I=1,M ,X
YX(I)=cM*XX(i)
12 YY(I) =C YY(T
r*A THE ATAN2 FUNCTION aLLOS FOP UNAMBIGUOUS EVALUATION OF
r THE ,NGULIP ORIE T4TION OF EACH SEGMENTSS IPPESPECTiV' OF
(t UADRANT LOCATTn'.T HIT TA AND Bc.TA ARE A...I.T.
00 17 I=1,LIM1
17 THETA(T)=ATAV (X(1+1)X(I)., Y +1)Y(T) )
S,, THETA.(0) .,TH: T (LTi') t:RE DFFFIMIP AS S.THEY .!ERE FOR THE JU.PITEP
r ISLAND RD'J::GT,A Z C SU i' r.PT IS PEtL TTTED IF A PROPER STGOAGE
r LO.ATIO.N IS A&.LLOTFT TE ZEROSU3SCRiPT .' AR.TCLE.
THE T ( TTA,.TA( I ) (T HE A (2) T r T ( 1))
THLT T( LIK) :HF (Ll l ('T ET, (LI 2)THE't LIM ))
DO 18 I=1,LTl "
18 BETA (I)= (THE:TA I) +T c)TA(T1) )/2.
' TE EHT."IR.Ijr SEPTM:T T'RPORT OEFFI iENMT IS COMniTED
CALL EF IPCGL
r
C"' tIE CALL AOJUST TO ESTA3LISh THE FIRST SET OF BREAKER H.ETGHTS. AN n
( AJG ESL .
r 4 I TS T41 V D TA ST NUJY E,R.
IT=1
CALL AOJUST(LT)
r
K' N KOUN IS. A FLAG USF3 IN SU2B OUTINr. E F;I !
KOIIN 1
r
E7 FPRIV CO(PUTES Th CHANGE IN THE (Y,X) COOROINATES ')73VE 1 TTIE ST P.
CALL DERId (Xo, YCfn X ,. nYO 'LT );')
KOU T 0
r', ALL REQJTP.ED STARTING VALUES FOR THE COORDINA TS (Y,X) ARE
C GENFATED IN INTTL,
CALL INITL( LT., IT D)
C
rC" IND IS AM ITNICATOR CF CONVERGEN"E OP L.CK OF CON'ERSENCE. TN TNTTL.
IF(IND.EOD.)30 TO 10 C
WRITE (. 2 C0)
230 FORMAT( x A CrON IER EN"E*)
Gn TO 102
10, CALL DEIV(X1,Y1,3YXt1, 1,LTM)
CALL DERIV (X? Y2, ,X? LI M)
CALL DERII (X3, 3,DO3,DY3,LI;)
COPUTE THE' DIFFERENCCF I rLAN AREA OF THE EACH AS OcSERJEO
r INITIALLY AND AS O3SFRVEO AT SOME LATER TIME (OIFA1) AS ELL AS
S H V THN E T P I,,LU1%TNIC F HN TF \i )
SLL. A ( Y X IY, LT, Y NXX, MAX)
DIFAi=nIFA
DIF iD=OTFV .
WHtTE (o,67) DIFA1, 0IF l
57 FOPMAT(9X,TH OFERIVE CHANGc I\ BEACH ARFA OVEP StME TIME
SINTERVAL=*,Fl.l//2'?,"THE 3SERVED ChANGE IN BEACH VOLUME OVER
SOME TIME INTFRVA_=l,F!0.1)
C'**. INITIALTZE THE TIIE STED COUNTERS III,IV:I III REGULATES THE
;AlTTIO i'TWFEl iA.'F D 3AT*. SETS IV M''lITORS THE TOTT!. NUiJER
rLL T E TEPfE
Sf T T = "
G' TO 3
2 II=1I+1
CALL ADJUST (LTIM)
r
f ~ IF ALL ,DATA, SETS IAIF E3EN USLc~n~, RETURN TO DATA SET NO,..
TF(IT..E ,ML 1) II=I
III=0
3 III=1II+1
IVrTV+I
IF(IV.GEoNUMDT)GO TO3
IF(III.EQ. Nd(II)+1) GO TO 2
r
rx IP OFTE1;NFES THE INTERVALS AT HHTCH RESULTS ARE PRINTED OUT AND
r PLOTS MADE.
TP=(IV/IPRINT) *IPRI'ITIV
IF(IP) )S, 4,5
h CALL AREA(Y,X,LIM,Z1,Z2,LTM)
WRIT= (,ol6) DIFA PF'!
5 FO P'lT(td *3HAlN.G I7 TdE SURFACE EREr. OF THF SUBAERI'AL BEACH"I,
iF10. //ZX, VD3LtI'JtM PIC CHANGE IN THE SIj.AERIAL 9EACH:=,Fi..:F1)
CALL RESULTS(X, 7,2,Z1 i,LITH~ 'AX)
LHL PLOTTER(X,Y,Z2,ZlLTM)
IF AN INSPECTION OF THE RESULTS LEADS TO THE CONCLUSION TIAT
THiE BEACH POINTS S3ULL0 E RESET THr, Wz ST THE INODTCATOR IN01
EQUAL TO 1. J' ANI J? ARE THE LEFT ANO EIGHTT O3UNDARY POTNTS FOR
TH SECTION OF EAOC TO BE RESET.
Ji=1E
T N 1 1
J2 = [ (
iF(TN i.n D 1) CAL_ RFSF, (Z2,Z1,Z J J .i)
IF(TIV ."'f. 'In1 ) G. TO 102
r
C
i
r THr PRFDIrfli EIOUATIrN"7 ACi APPLTnq
b O ..J=l),'1
Pi. (J)= YO(J +'./3. 3TT (2. *nY (J)DYZ (J)+2.*Yv l(J))
7 H (J) XJ) )++G./3, 7T (2. ~' X3 J)XDX2 ).2.t X .(J))
CALL,
CALL Li .Iv (Hi i XX,tuYYt LIM)
S" LCOUiNT IS A CQOUiTr W4ICK ,PECTFTESc NUM:'ER OF TIMES TiHE CnohE3T
., 5PQUA I O S ACE APPFLIE .
r'
Lo C U iN\T 3
r THLP THL CPOC TC. ErUSTITCNS A3K APPLITL
6 DO c J=L, LIri
71. (J)= 1./ .8 .*Y3( (J)+3. /8, OTT (OYY (J)+2 ."DY3 (J) .Y2( J) )
8 Z2( J) = 93./B8.XZ7(J ./8. x!(J)+3./8.0T* (XX(J)+2.IOX(J) rX2(J))
LCOU E T=LCOC!I)T+ G C i
TF(LCOU!JT.Ef.N?)G3 Tr0 i
9 CALL DERIV( Z2 ,Z.i, XX ,'nY,LiM)
GO TO 5
r
L.X" AS ThE ITTNTRATICN IS ADVANCED 1 TIME STEP ALL PERTINENT VaRIrLES
c ARE UPDATED.
10 DO 11 J=1iLIM
DYOfJ)=OYi(J)
DX. (J) =0Xi(J)
DY1(J) =5Y2 (J)
DX1 (J) =9X2(J)
OY2(J) =9Y.3(J)
DX2(J)=DX3(J)
KO (J) Xl(J)
0 (J)..=Y1 )f J)
XI (J) =X2(J)
Y (J).=Y2(J)
Y2(.J) =Y3(J)
X2 (J) =X3(J)
SI )U =71(J)
CALL DERIv (Z2 Ziy,33J 3 lY ,LTM)
GO TO 3
1I2 PRINT 103
i 3 FORPAT(1H ,*THIS TS THE END OF THE rURRNT RUNS)
END
>U .O TIN 1 ,'i CL
" PIDCLS ASSTSINS VALJS ALL P.AR'CETcRS WhiCH APPEAR IN TtHE EXPRESSICi;
r FOR THE S,_'IMENT TRAPF'F.ORT COEFFICIENT* *
REAL KON
COM. N/ L 'OCn / I
CO H i'!ON/3LL.C9/T9 T *V
1 H .; i N\13i / 3 L C r''/ A iA
r"DlEFITNITIONS, A=ALLOLaACF FOR OPRE SPACE
r C~j= 30TTOM C.RA' COFFICIENT
SKO=_3Oi'cTAT Op PROPORTIONALITY BETWEEN THE IMMERSED
r NETGi1T TRANSPORT RATE ANO THE QUANTITY F~)C(3) AS
r OEFI'JED Tr' TnE TEXT
SN=A MFNSUP 'F T. EFFECT IVENESS OF LATER L MI'YTNG
G= RAV IT".TI ONAL O.NST IT (MKS)
r G"AMIMA=ATT) 3ET'EcN WAVE HEIGHT AND MEAN JEPTi :AT BREAKING
r ALPHA=PATIJ 3FThEEN WAVE 4AMPLITUnE AND LOCAL DEPTr TM THE
r SURF ZO3L. ALPHAAS SUPPORTED. BY MEASUREH ENTS,IS
r USU_.LY SET EOUAL TO) G.MM/2.
r S=AN A.JERAE S.LOP_ FOR THE COASTAL SIT.E 3EING CONSTOERcD
C RHOWRHOS=DcNSITY OF WATER AN) SAND :ESPErTIVELY
r Tv=TNAPOTT CFFcTET FOR THE IATER MOTION
t=SEDIM.N TANSPORT COEFFICIENT
C P, 7l2, 2, ST ARE A.,.I.T.
wC o. 3
N 0.01 
K0=0.3
N=J.0 :3
GjAMMA 78
ALPHA=30.3
S= u 2
P=SPI N /( A.ALPhA kP)
Pi=, 75tSORT(9,/15, +1./ )
P2=. 73 SQPT( 3. /15 .T 1./P)
P. T= (1./ (P ( l+2. )'(P PI 2) (Pi )) +2./( ( P2.) 3.))
P OW=1. 02;]
PHOS=2. ;1
TV =PSI'5.*"PT*uLP: r. SO M T(G) L) G/ ( c n MM *5./2.))
T = KO u MMA D" P 0 / ( P I PSI* A ALH 5. ( HC S 3 ))
:""NOTE: IF,IN THE COO "31 NV T :" SYSTEM ESTA L LTSHEn.,THF OE PTH Gr.AdIENT TS
C NEGATIVE, THE THE SIGN OF TV AN3 T IS REVERSE
WRITE( ,1) NP,PSITV,T
1 FORiIAT(2X,TN= ,F7. //2X, p=,F 7.4//2X,'oSI= *,F7.4//2 +TV= *,Flu .3
2//2X, T=:+, FI/.')
PE TU I','
END n
SU6?OUTINE AOJUST(N)
r'xw ADJUST SERVES T.FEE PUP3SES:
C (1) IT UrOATES TIE INCIDENT WAVF CLIMATE.
( (2) AS THE S0RELIINM FVLVES IT ADJUSTS THF INPJT BREAKER
C ANGLeS ANu RE COMDUJTES TF LONG cPOKiE TRANSPORT )RATFS.
r (3) A NET TPNISPOT FIGURE IS CALCULATED FOR PRESPECIIFIE
STRETCHES 3F SHORLIN.FT.HIS HAS SIGNIFICANCE WITH RESPECT
STO A DETRMIAETHATION OF WHETHER 1HE SEDIMENT FL.OU TS
T CON INt T3 MORE O3 LESS CLOSED CELLS oR IS A CONTINUOUS
r STREAM INTERRUPTED ONLY OCCASIONALLY.
DIMENSION CHI(9C) ,O FT(25)
COMMONi DEL(93 ) ETAt 2r ) ,T.HETA(90)
CO'M ON XG 90),41(o ) X2(99 1,X3(9n) Y (9Y) ,Yi(9)),Y?(90),Y3(90)
COMMON aYO (90) , ? (9 ) ,DY1(93) ,D 1 (0qoI r1v2(96) ,0X2(0O) ,nY3(00),
*D13(90)
COMinN/BLOC/i (5l,90)
COMMON/3LK/A' (5 Cy, C)
CJdM imO",/3LOC ,/ l I
CO' M ON/3LOC / I/LIMIT ,L Mi,! LTi!M2
C Q'; N/ i Ott 3 C V KXTir I T L T t'
CCn.CN/dL C /VOVf "9 5
CO 0 N/1LO 10G / GA 'Mi A
COM,0 0N3/3L OC 1/C 0 FFF
L.nMMON/3LK /PHIO ~I (90)
COC'1lON!/3LK3 1/D30, ,C(C)
COMMi ? N/ LKr iL ( 1, ) M( 1 )
f (.1)
DO 7 J=I,LTITi
C3(J) =:B(I11 J)/3AMMA
7 PH T J) =VRT 1iA1 (II,J)
C+*i' WE MAj E THE, FOLLOWI0., ASSUMPTIONS;
03(C) =03(i)
D3 (LIM)=D03(LT 41)
PHI(C,)=PHI(i)(PHI (?)1FI ) 1))
PHI(LIM)=PHI( LIMi ) (FHI(LiM2) PI (LTM1))
r*'XoTE;IF TH. ,ND LOUkNOA0v CONDTTTtnS WE''ERE C=O ., WE WOULD) SET .PHI(0)
r. PiI(LIM) =PIT//., Cr.PFSPGONDIIN'G TO THE INCIDENT WA.VE CRESTS BEING
r PcRPrENDIULA: TO THE SHORtLINE.
p T!jMnN
FENT0Y Ti.ANSPT
)CUFFF TS A SO'F'A'.T A IT lrAEILY CHOSE FrACTIONAL .0=FFICT::NT
H'TICH i ANSLA TSF CH'AN',. IN L3CAL '3.EAC Or NT'ATI ON INTO A
CHANG IN LqCL 3F.AkR A NiLELE'3LONJ il972) SET IT EOUAL TO 1.i
no 3 J= LIM:IT
I=J1 i
3 PHI (I) =o I( .) +CCLrF (GTI(L 3T( T 3 ))
P A ~u 3F r'AUTTON. TH' AjOV\ ADJISTMLNT OC .THE BREKE< AN3; L. UST F
ADE 1THOU 'IH TIF LL' I TlE S:'3ELN EAHIRI.TS Si.HAR i.T TAL t'U .T
AN IS A RG1ON C" rA'rI CHANE. TrHEN TH APPLICATION OF .fH'S AF'0jL'A
tMAY. L[~ TO iNST A3lL TTS.TN TIS CAS TT IS 3ETTEP TO oD OMrPUT THE
N':.iK
ALTO.GFT ET .
'..* t ANn I ARE A.n. .,lT.
DO 1 j=,LI Tr
I J1
V(T) TV (GAHA' OP(f ) ) ) (5 ./2,) ST N(2.,PHI(T))
1 0CI) =T* (I)
4L SET HI EtQU'L TO PFSN~,IT VLLUE OF FBT .
00 J=1,,L.IMIT
I=J1
.'* C' I1.1I)=JE! tA(1I)
L(T) AND M(I) KEP.FS E"T THE LEFT ANO EIGHTT .dOUNCA7Y POINTS
ON AN I.lTERVAL D. WMHTCH HE WISH TO CuiMtPUTE A .NET TRAiNsORT. RATE.
POINT C
DO t; I=1,10
'iFT ( T) = .
L1=L (I)
Mi= M(T)
DO 6 J=L1,M
ONET (I)=QNET( I)+ 4( J) (.J1))
COiT I NUZ
W.RITF (8,8)LlI I",CMFT.(T)
CONTINN.lU"
FORP A ( XIb T !? ( 'iu 1 XF7. 1)
FORMAT(1H leftET "D3I!T',5Sx, RIGHT POINT .,5X, NET RATEr)
RETURN
END
SU3ROUTINz INITL(N,INn)
T N TTL TS A S LF < If.A. TOr
s NT L'iS F T_ C iORJINATES, Y,V) ,F THF 3.7: POINTS FO USE:
c;r .IEPt F o TOui r S .i If Q ATIONS. ri NEHO 1 SC T D I E TTzXEL
i S i N X>1 (90 N X 2(9Y) Y 3(9 C)L YYi{9C) YY2(9C) ,YY3( 9 )
CO"i,' N 'EI(3 US) .3ET.( 9') T.FTAC('O)
CO 'L MO C X ( 1( r) ,X2(90) 3 ( .u) (9 ) Y1(9.) ,Y2 (1 ) ,V3.(90)
.C OMM DYl (9u) (9 1 OYi(' ) ,0X (9") YYT(90) ,aX2 O) ,Y3(93) ,
OUMMU,/3LOC2/LIMITT ,LIM1,LTM2.
.CO iMON/ L3C2/ TIV D .
...* IMA IS .lT, MaXIHJM NUI m.FR OF. ITERATIONS ALLOWED
T MA X= 2 .
r*> ED S 1S TH, L`'SSEST PFPPPITTED OIFFEF'ENuE bETWErCN 000OTDINITE VALUES
r GE NERATr0 ON SUCLESSIIVE ITLRATIONS CC'ONVcRGENCE CpITER'ION)q.
SEPS =' ~
r, G cSSS IS A A R .APiTTA
Cr CO ORDIN TE V1, LU ES.
GUESS= .3
CALL DE'IV( 1 A ,1 C,3YG 0 Y 3, N)
PO0 2 T= N ,
SYl.( T.).=YO (I) i S U E' S
2 X1(I) =XO(T) JESS
CALL nERj ( X1 ,Yi. 1,DY 1,N)
PO z I=t, N
Y ) .( Y I f. !).3 U ES'F
:=3 X I I I f lT)) UE SS
C r; LL ''4 1)X/. Y X. Y? ,N) .
0 0 i ,1 I, .N
Y3.(. )=Y? (T) UFS I1
'/ 4 XJ(I>=X2(I)E, SS
CALL %E1I'f X3, Y'.JX3 YY ,N)
IF .F.T= T .^ 1 A
.*IFr..T F.i.E., O I 'iAX) S3 .T 1.5 .. "
If
U.:IJt ,l ) ,Jr; A [1 ;
r
00 . T=1,N
XXI(I) = I< ( I)
XY 2(T) ( 2 ( T
XXs() =X3( I)
YY1(I )=YilfI
SY (I ) =Y 1 (I)
6 YY AR(I)= Y3(I)
DO "7 '=1
Xi (I) =XJ (T) IT/2L'.* (9,"X^( I) +19.c DX1(TC 53.*ODX (I)+XD 3fI))
Yi (I)=Y3 (I)+ T/C?t, '( vC( ) +i19. D' Yl (I)5. DY2( I) +3Y3(T))
X? (I) = < (I) +rf/3. ,(' (T) +u." nx (I) UO, (I))
Y2 (I) =Y1 (1) DoT/. + (" Y. ( I)+. Y1( I)+DY2( I))
X3 (I) =X'(I) D3iT/ .'( 3,+U.
7 Y3(I)=Yj(1) t rT/8. ^(3. DY (I) +9.*)Yil(T_ 9.*OY2(I)+3.40'3(T))
8 I=1
9 IF(AR.SC(Xx (1I)Xi(If;) LE, EPS.Ai.D.A3S(.YilC)Yl (I) ) .LE.EPS)GO TO 1'
GO TO 13
10 IF(AQS(XX2(I)X2(I)) )LE.EPS.ANJ.4i3S(YY?vI)'2(I)).LE.FPS)30 TO 11
,0 TO 13
11 I (AbS(XX3 ) X3(I) ).LF.EPS.AN,.ABS( YY3(I)7r3(I)) LE.EPS)GO TO 12
GO TO 13
12 I. EQ.LIMIT)GO TO '3
rO TO 9
1.3 CALL DERIV(Xli,Y1,JX1, Y1l,N)
CALL DERI/V (X2,Y2, XZ2, yY ,N)
CALL DERIR(X ,Y73,3X3 ,DY3,)
GO TO 5
1 TND=1
0S TO 17
I IND=0
270 FORMAT(1H ,8(4X,FiC.3))
17 RE TURN
END
SUBROUTINE DERIV(XYyOX,DY,LIM)
,E ORIV COMPUTES THE INCPEMENTAL CHANGFS9OY AND DX,IN THE (Y,X)
COORDINATECS SIVEN BECM SEGMENT ANGLESTHE DISTANCE BETWEEN ADJACENT
EACH POINTSGiEL./?, THE SPEAKER DEPTHS AND THE T:?ANSPORT RATS
ODISENSIC N DX(j0) Y( 9?)) ,X(9 ) ,Y19.0),
ODINMNSION FACTOR(9 )
COMMON X0(90),X1(90),X2(90),X3( 90) Y.(90), Y(90) Y2 (90) Y3(90)
COMMON DYO(90),DX0 (90) ,5Yi(90)0o 1(90O ,DY2(90),OX2(13),DY3(90),
.0X3(90)
COMMON ED L(90) ,ErT(9C),THETA(90)
COMMON/3LOCi/PT
COMiiHON/3LOC 2/LIMIT ILI 1,LIM2
COMMON/BLOC5/Q ,Q(90)
COMMON/3LOC'/IVDT
COMMON/3LK3/OB0q03 (0)
COMMON/3LK,/KOUNTIII, NDC(50)
DO 1 I=~jLIMI
i THETA(I)=ATAN2(X(Iii)X(I) Y(I+i)Y() )
THETA(G)=THETA ()(THETA(2)THET(1) )
THETA(LIM)TA(LIALI) ((LI)(THETA(LIM2)THETA(LIMi))
00 2 I=iLTM
2 BETA(I)=(THETA(I)f tETA(I1))/2.
.>$ IF WE ARE AT THE ST4PT OF THE PROGRAM(KOUNT=1) OP THE WAVE DATA
SET IS CHANGING W= .ENTER SUBROUTINE ADJUST TO RECOMoUTE TRANSPORT
FIGURES.
IF(KOUNT.EQ.i.OR.III.EQ.NDC(II)) CALL TRANSPT.LI')
47
4 1 i T;?
3 i)L L .i 1. S .r ( (i t ) X ( +1 ) ) + (Y(I+ Y y( T I+1) ) * 2)+S 0 r (.( y (.IT +1 I). 
'X(T)) ?+ (Y.( +I )Y (T)) 2)
... PEL(1) AN1 uEL(Ll ) Ab nO FIN TN A SPECIAL W AY.
[ZL ( ) SO T ( ( ( ) X ( 1 )) + (Y )Y (.)) ".?) 2
[ I L( I ) =SOr 'T A ( (L l' L) X t L.l ) + (Y(LIM) Y(L.I l) )" 2) (Y .
DO 3 i1=,LIT
FACT OR (I)=2 ./13 (T) ( ( T) 0(1 ) ) /D L (I:)
DY(I)= STN( t _TA(T)) FA.CTOr(r T)
r: DX(i) COrS(?ET.A(3 ) 'F CTOR(I)
r r^ O :IF, TIN TA (O Rn I'JT!ATL SYSTEM CIOI tUSci T DEPTH GtADITN IS
r NE ATTV THLi! Tu IGNS PR)SELLIN THE SIN AN COS2 FUNiCTIONS
SAOVc RE E.'d.>F FrFcD.H itWVV: THE SIGN OF V (EOUI I LENJTLY,T) ALSO
G CHANGES. THF 'NT EFFECT S TO LEAVE TF.QEOUJATTONS .U4NCAhLSE..
OF TJ r' .
rNr
SU 9"ROUTINE APF EA (, Y, LTM,A, 3,N)
r
r I GIVEN HE( COORRiN1TES OF THE oFA.i POINTS ?T 2 DIFFERENT TIES TiHE
r CHANGE IN SUBAFPIAL SURFACE AREA OJER THAT TIME TNTERVAL IS COMPUTED.
r COOi
c COORDTNATES (A, ) REFFF TO THE MQC RE RECENT POSI Tni LC.URVE 2).
r
r AN APPROXIM'ATF CA_DUL'ATION IS PERF3R"^ED ALSO FOR THE CHANGEE .IN THE
S. VOLUME OF MATERIAL.
f NOT' .TH'T .ABSOLUT L'AULURTS OF ADEA AND VOLUME APE NOT..C O'UJ TED .UT
C RATHER THE RELATIVE C NGE IN EACH.
r THIS SUBROUTINE C.,N BE MODIFIED TO COtiPUTE..CHANGES OVER A\Pt DS.CRETE
C LENGTH OF BEACH.
T ,
C THE SIGN CONVENTIONS I.STABLTSiED IN THIS SUoROUTTNE ARE FJO A
C COORDiN'ATE SYSTEM I1 WHTC THE DEPTH GRADIENT IS POSIT VE.
( TF THE.JEPTH GRPUTENT IS NEAT:IVE.T4HE RESULTS CITED .3ELOR NEED
r TO 3E MOOIFIn.
DIMIEN'SION YDEL ( 9 ) (9 ) ,Y (90) ,9(90) ,A(9q ).
L3GICAL A1,A2,A3,A. bi,32,5s 3 .B
r0 1C'N .DEL ( 9j) BET.A( ) ,THTA T A(9i)
COMiMOi N XJ(9i.) 1(90) ,X2(90),X3(9G),YO(90) ,Y (9 . ) Y;(
COM ON iJYr ( ) DX'!( 9r) Y1 ( 90) X 1 ( 90) Y 2(9 ) ,X2 ( ) 0 ODY3(9 ) ,
SDX ( 3 [ 1)
Qii * J
VO LI =1 .
no 1 1i=i LT '1,I
ARE 1 =AR E +.r' (Y1+ 1+1) Y(T) ) ( I ) + ( I) ) ( D
1 VOL1...VOLi+. 5 (Y (I+ 1 f1) ) (X {I+ 1) +X(I) .D9(1)
COJMoUTE AREA UNDJR CUPVE 
Ni =NI
,&RE.".. 2=0. .
VO L 2 = G ,
nO 2 T=1, N1
AREA2= Ao~A2+.t5(A(I+1) A(I)) ( l+1) +3(T))
2 VOL2=.''OL2+. (A(I+1) . ( i ) ( ( ;(I)I1) +9 (.I) ) n3 (I)
rOVDUTTE THE AEA OF THE FAP LEFT TRAPE70IO A..T,.T.,
To A 1=A3S (. ("(1)A (1) )" (X (1) (.1) ) )
SVOLS3=TPAPIfrn( 1)
r' PIITE T HE H \RE' F0 TH Fi P. PIGHT T r.. ZOE A ,. D. I .T .
TRA P2 =A3S ** (Y(. TM) A ( ')) Y"'(X (LIMF)'3 (N ) )
VOL4 TRAP20 r3 (LI M)
rOMPUT. THE CONTRI3'JTIC I U. TO TiH EX'T NSF TnN OF THE EN: IN.I T LEPT
IF ( Y S T, J. ( oM . ). 'To. 3
D0O .I=1,2
.F ("Y ,' I l), T. .1 ) Yr:3" TOn 5
l C') IT I % )_
5 KI +1
r *' A .N'E X(K) IS DEFINEDl (SOLELY :`F TH.E "PUPPOSES O0C THIS
r S.U3ROUUTINE) 3v LT\!EAR INTERPOLATYIN. A NeW Y.(K) IS ALSO DEFINED.
THE OLO VALUES AFE STORE.
KY=X (K)
X(K)=X(K) +(A(KI)X(K)) (A(1) Y(K) /(Y ('<!)Y(K))
YK=Yf K)
Y(.K)=A(1)
Kl=KI
ARE A=0.
p tLA=0.
O 6 T=1I, K
REA3=ARES3+.5 (Y( I+1) v(I)) x(X(I+1) +X(T)
F VOL inL +. (Y (I+1) (T)) (:< (1+1) +X(I) )'*DB(I)
Xt K) =XK
Y( )= YK
3 00. T = ,2?
IF(A ( 1). 3T.Y(1) SO TO 8
COUNT NU
8 K I+
rI* A 1EW 3(K+ IS DEFINED "Y LINER INTEPPOLATION.A MEH 1 (K) IS ALSO
c DLFINEO.THe OLD VALJES ARE STORED.
K=B3(K)
b(K.)= (K) (9=(Ki) (K) ) (Y(i)A(K))/(A(Kl)A(K))
A(K) =Y (1)
K1=K
AoEA3=0.
VO L= 0.
DO 9 I=1 K1l
AR .A3=A A3+.5 (A(I+1) A(I))* (3(I+1) + (T))
a VOL5 = VOL5+... (A I+1) A (I)) (3 T+1) +3(1 )) 09(i)
B(K) =9K
A(K) = AK
COMPUTE THE CONTRIBUTION DUE TO THE PXTENSTOiN OF THE ENrDOOTMTS(RIGHT).
IF(Y(LI5 ).GT. A(N))30 TC 11
00 12 I=1,2
IF(A(N1).LT.Y(LTM)) GO TO 13
12 CONTT NU
13 K=N I
PvU N, NEW K(K) IS DEFINED "Y LINEAR INTERPOLATION.A NEW ;(K) IS ALSO
C nEFINEO,.TE OLO VALJES ARE STOPEB.
cK=B( K)
3(K)=r3(K) +(='(K+i)3(K)) (Y(LI M) A (K))/ (A(K+1) A (K))
AK=A(K)
A(K)=Y(LIK)
AREA=O,.
VOL6=0.
0 19 T=KN1
AREA4 AEA t (A( +)A( ) A(I)) (1i+1)+5(I))
!"+ VOLTA:VOLS'o5: [A(T+1.)8tl)) (3(I i )+J! ))x03(Ir)
( K ) =: K
.(K) =,K
11 DO 15 I=1,23
IF( '(LI;I)).LT.A(,M)) GO T0 16
1i CONTITNUI
16 K=LIM I
fr* A NW X(EK) S OFIN F PY INEAR FNTERPOLATION.A NEW Y(K) IS ALSO
r DCFTNED .FH OLD 4 ItS AL J STOPED.
XK=X (K)
X(K)=X(K) (X(K+ ) X(K ) ) (^(N)Y(K))/(Y(K+l)Y(K))
YKY (K)
VOL= 0.
DO 17. =K,LTLMI
AR: A = AREA4+. 5 (Y(I i)Y (I) ) I (X(I 1) +Y( I))
17 VOL =VOLo+.5 (Y(T1+1) Y( I)) (X If t) +X(I) ) D (I)
X(K>=XK
T(. K =YK
r
L A.RL A2REA1E
VOL VOL J OL1
"' WE .NOW CO'PUTL THE A(.AtL (PIFA) ANn VOLUPETRIC (DTF/) C.IANGES.
. x Al ,A A,AL ,j ?,I P ,3 A:E LCOG, ''L V iA.R LES 'HiC DEFTN F HE
POSSIL :r'':L NATION; "HI r 'L1 ,i TE N IlPDTINTb OF 2 ST RA N LINES
A1=A ( i) .G T. Y i ) .A AP ) T iL 1)
A2=,(1) G". Y(1) .A4? ') (N) L1.. Y' (LTCI
A3=, (1) .L f .v( 1) AN t ( ) LT.Y(LIM)
A A( l) .LE.Yl I).AMP. A(N) .GT .,Y(LIM)
b[ =3 (1) GT.X( 1 ) X \PU, ,. r ( N ) T .X (LTI)
I2=: (1) .GT. X(1) A.f r, M (N) .L X (LIM)
i3=' (I).LToX i) 1 P. (N).LT.Y(LIM)
94 =Jt ) .LE. X( ) .ANDi. F (M) .GT.X (LIM)
r 4. TH T FINNITIONS C7 PTFA ANC. nFiJ 'ED'tn .n W I' CH COM31 I;',TION
r OF THiL 1~VE I I2J._ ST'TEMENIS IS TRUE.
" 
TF( A. AIU .9?)
IF (A 2 A24U. 92)
IF (A I.A ,i )
IF (A AND. D r)
IF( A ANn. 31)
IF (AL. AND. Bl)
IF (A2 AND. 3)
IF (A2 4 ND. 3)
IF(A.! ANr.84)
TF(AI .ANn. 4)
IF (A3. VNO.32)
IF (A3.AND.. 02)
IF(A1 .AND. 1)
IF(A1 ANDO. !)
IF (A3 .ANO. 3)
IF (A .ANO, 3)
TF(A A Ni .8 2)
IF(AL.AND. 2)
IF (A 2 ANO. )
IF(A2 ANOD. )
IF( A3 AND l)
iF(A3. AND. B1)
IF(Ai. AND. t7)
IF(A1 AND. 3)
TF A1 .AN.. 2)
IF(A AN 32))
IF (A3. AND. P4)
IF(A3.AND..31)
IF(AL. AN B3)
IF (AL AND. 3)
iF(AT ' 3.Si)
L :. 4"Ld a s
DIFA=C
DIF:/=VOL
nTFA=t
DIFd = VOL
DTFA=ATRAOTTRAP2
niF FJ=J OLY OL3 V OL;
nIFA=i, +TR Pl+T AP2
0IFJ = CL+VOL3+ OL4
DIFA=4+TRAPlTAP2
DIFV=VCL+ JOL3 OL4
DIF=ATRAP1+TRtP2
DIFV=VCLJOL.j+4OL4
DIFA=4+?. "AREA3TR4PiTRAP2
01FV= /O L+ 2. 0 L 5V DL3VOL
DIF =^.2 .AoE(3+TR.n4i+TR^P2
DITF 4=L2. VJOL5L+VOL 3VOL
DTFA= 2. "AREAATa P 1 TPAP
I FJ =J OL ?. V OL 6VOL 3+V\OLb
DIF A +Z.. AREALA+TRPl TR'VP2
DIF J= CL+2. VOL5 +O L3VOL,
DIFA=A+?. *ARPATATPPTRAP2
DIFJ= OL+ ?. OL6VOL3VOL4
DIFA=42. AREA4+TK l+ TRP..P2
DIFi=VOL2. VOLotV3L 3VOL.
DIFA=42. ABrCAA+2.*AREA3TRAPi+ToAP2
nIFO=JOL2,* OL6+Z.*VOLSJOL3+VOL4
JFA+2. 'AAR"AST2.*AREA3+TRAPiTpAP?
DIF!=VOL+2. *VOL2. *VOL +VOL3VOL4
DTFA=?2. ArkA3 2.*AR A. + TRAPi+TRAP2
DIFV=JCL2., JOL 52, 'VOL6+VGL3+VOLL
., i v A : I,A 1rR ..r P2
FND
SU30OUTINE RESET x:,Yy. L, J)
r r ESET REAP AN GES THE EAC H POiNTS FOR ANY SECTION. OF THE SHORELINE
C OP ALONG THE ENTIRE SHnRELINE SUCH THAT 1HE SPACING .BETWEEN ADJACENT
r POINTS IS MADE MrF:E OP LESS EQUALANO.,AT THE SAME. TIME,DISTORTITON
r OF THE oAlGH SrHAPE IS MINIMTZED.
C L IS THE FAR RIST FIXED POINT A.D.T.T.
r J IS THE FAR LEFT F7TXE POINT A.D.I.T.
OItcNSION X(90) (90),n ELXY(90)
COMMON OEL(90) ,bETA(9" ),THETA(90)
r+x 'JICOUNT IS A 30UNTR FOP THE NUMBER OF TTFES THE FOLLOWING PR.OCEt'.IRE
C IS ITEfAT0D.
IC OU NT =
C'+W. NJ'IT. IS NUiMB3R OF "ERMITTED ITERATIONS.
NUMIT=5
Ji= J+ 1
L2`= L2
r THE PEACH A 'EShNtT A4ISL.S AND THE OISTa N.I 3ET ELTN SJOCE SSrIVE :
C POINTS TS CC'1PIlTF D.
11 DO 1 I=J,L1
THFTA(T)=ATAN?(Y((T+i)XfT),Y(I.+i)Y(T))
1 DELXY (L)=SO(T((X(I+1) v (I)) 2+(v (I+l)fv (Ji))"2)
[,vMPUTE A MEAN DISTANCE BETWEEN EACH POINTS.
r
(
(
(.
C.
C'J i' J
n0 2 I=J,L..1
2 SUM=SU.i ' LYY I)
DE LME L SUM/( L J+1 )
T=J
*" EXECUTE TTE PROCEDULF AS D7SCRIEUD IN Ti;F TEXT FOR THL
TRA!4SLATTON O TH 3FArP POINTS E. i O!r THE FIKRT POINT
ALONG THE 3I~:H SESIENT TO ITS LEFT 'JUTTL THE DISTANCE 8ETWcFEN
I.T AND THE P*Tt!T JN ITS LEFT IS D~3 LNLW, cOEFTIN Ti:. SEGMENT
c.TWICEN TiTq NEW POSITION AND Tri PTDO NT I'.:.IEDIATELr TJ ITS FIP mT
AND NOV= THIS SCOND POINTT ALO'G TH NE' SEGMNT... TC.
?3 IF(OFLNEW ELY (T) )1CI ?0 ,3
i1 X(I+1) =<( T+ ) C (CF NE DELAY(I)) SIN( 'H TA (I))
Y(I+1)= (T+1) ( OFLNE OD ELY ( I ) 0 S(TnETA(I))
GO TO ?I
30 Y(X =:X I+i 1) F(DEL nNEt n LXY (I)) SIN( TiETFA(T+l))
Y( I+ =Y(t +1) + (EL N i FLF Yf ) fStTOS~T Tf, I+ 1)
1 DELXY (I f =S1 RT(( .rT+>?)X(It)) F+^( I+2)) +L))2)
20 I.=I+
TF(T.EO.L) GO TO 22
GO TO 23
22 1CoJNT=ICO.UN!T+
IF(TiCOUNT.NE.NUMTT) :0 TO 11
*CK TO SEE IF TE' DISTANCE STWEEN ADJACENT POINTS IS APPROXIMATELY
WRITE(i,25) ( ELXY(I) I= ,Li)
25 FORMAT (2X,5 (3X 13.1 ))
RETURN
END
SUBROUTINE RESULT (X Y Z2, ZI L MAX)
C"*,* RESULTS.PRINTS OUT SOME OF THE MORE IMPORTANT NUMBERS GENERATED
C BY SHORLTN.
DIMENSION ZI(90),Z2(90),X(90),Y(90)
COMMON OEL(90) ,BETA(90) ,TETA(9'0)
COMM3N X0(90),Ki(90),X2(90),X3(90) Yf(90) Yi(90),Y2(9U),Y3(90)
COMMON OY0(99) ,0X3 (0) ,Y1(90) DX1(90),O0Y2(96) ,X2(90) tY'3(90),
;DX3(90)
COMMON/3L
COMMON/BLOC21LI MITLI 1 LTM2
COMMON/3LOCS/QO,Q(90)
COMMON/3LOC7'/IVDT
COMMON/lLK2/PIO,iT (0)
TIME=IV*DT
DAYS=TIME/24.
YEAOS=OAYS/355.
PRINT ivIV D,TIMEo.DAYS,YEARS
1 FOPMAT(3X,*D O.TT^ STEPS(TV)=,h4,5X,4TIME INOPEHENT (T)=.,F4,~I
~iy HRS. ;Rf EL SED7 T E( I4XUTi= Fil.1,ix, HRS ^,F3.:i x,
'(0 D AYS= '" F 2, 1X4 '*Y EA', 7 )
PRINT 2
2 FO RMAT(//SX POINT13 X X(TI ME=0) ,10X,*Y{TTME=0)*,10X,X (TII E=
'TVXOT)* itOX, Y(T14E=YTVXDT)P,5XT*X( PRESENT) ,5X, Y(PRESENT) /17X,.
ZMEIERS*,13X, *METERS*,3X, METERS ,1X,*METERS*,15X,METRS*,
RiOX, M METERS) *
IF(MAX.GT.LI A) K=MA <
IF(MAX.LE.LIM) K=LIM.
DO 4 .I=i,K
4 WRITE(5 3)I,X(T),Y(I),Z2(I) ,Z (I),XX(I),YY(I)
3 FORMAT(X,1i I3,3X,3(7X, Fi.2),16X,F1i.2,8X,F1G.?,5X,F1a.2)
PRINT q
5 FOPMAT(//X,*POINT*,i0X,3EACH ANGLEIOX'0XWAVE ANGLE*,0X,*9EACH
POINT SEPARATION* ,SX,'TRANSPORTS*/17X,^*DEGREES*,l4X,.*OEGRES*,13X
*METERS*~22X *UB3I METERS/R*)
r OONVETT. FROM RADINS TO DEGREES.
DO 8 1=1,LIMIT
J=Ii
PHI(J)=CVCT2*PHI(J)
8 THETA(J) =VRT2*THETA(J)
DO 6 I=1,LIMIT
J=II
b WRITE(6,7)J,THETA(J),PHI(J),OEL(J),Q(J)
**** N RI=1RDLM GPEE& 9ACK TO RADIANS.
J=I1
PHI(J) =CVRTiPHI(J)
10 THETA(J)=CVRTI*THETA(J)
7 FORMAT(iH y ?XI3 79,FO.2,9XFI0.2,13XFit0.2,25X,F13,2)
CALL. ERROR (X, Y,ZZL,LIM,MAX)
RFET'RN
ENO
rSU OU 1TIN E OI Z 2 Zi, LI ,M4X)
r+4 rRPn, 304N '3E 'nL MiNFF "FDSIATILE THAN IT IS NOW. T PPESET TT
r COMPUTES THE FoC:NT L_ .OR OF THE 'REOTCTED AREAL ANDJ VOLUMETRIC
C CHANGES AS CuMA"PF.j TO THO.) OSERkIn. IT ALSC CAN CO'PARE OBSEnED
r ANn PRFiCTE.0 SHL KLT4E PUJSITIONS POINT Y POTNT.
DIMENSION Zi T'),"Y(9 ), 3 LTT (3O) xaLT y (30)x(t nlv y(o)
CO_,lM N rEiL(3),) FTA( .G ),TH7TA(90)
k4 IM N 9 q
COHMOr/3LK 1/ X (Qr) ,Y .)
COMMU N/ LOC(/ DIFA, DI FJ nIFr., TF'
r' *" SHORELINE 3N c ,O.1^RE1 ITHT TFTSELF LT VARIOUS STAGES IF THE
c THE NIi',I3ER OF POTIlTS HHICH CONSTITIUT; THE SHORELINE IS A4LAAYS THE
r SAH(iHIS CthLO 6: FFCTEO 3Y A SUROUJTINE HICH D[ECREASES OR
INCREcASS THq NU 3 OF POINTS WILE RETAINING THE _EACH SAPE
S SUCH A SU30QJTINE 13 NOT PPOVIDED HERETN).
IF(LIM.NE.AX) GO TD
PO 1 I=1, LIi
.DELTAX(T)=?Z (I) y (I)
1 DELTA.Y (I) =Z1 I) YY (I)
WRITE(b,2)
2 FORMAT(IH ,'OELTAr (T) ~" ' LTAy(T),)
WRITE (6,4) ( I.,9 O LTAX(T) DELT 7Y (I) I=i, LIM)
FORM1AT (3RX, TL.,2 ?X, Fl.E 5 P9. 2)
5 PCTI= A3S((DIFADIF' )/OIF*10 0.)
PCT2= A3S(fO.IFVCIF/1)/OIFv100o.)
WRITe(6to) PCT1,PT2?
6 FO~MAT(iH ,PER C.ENT E"OR IN REAL CHANGE PREDICTION= FF4 1//IX
*PEP. CENT El.ROR IN JOLIUMETRIS CHANGE PREDISTIOiN=,Ft.)
FE TU P N
FNn
SUEBOUTINE PLOTTE?((,Y,Z2,Z1,LIM)
^l PLOTTER CAN MAKF AS IMAY SEPARATE PLOTS AS DESIRED ON A SINGLE
F FRAME.
r., I TITLE,LA:U,LA9V ARE LLErnTh LA3ELS.
?: ,i'rN iTLF (.') LAdUL(?) LA8V(2)
r U AN2 AF E THEI LOCATIONS FOP ALL THE. HORIZONTAL AND3
r VERTICAL COO INAT ,PESPECTIVELYTO BE USED IN A SINGLE FRAME.
DI'MENSTON U(120), (10)
CO:4 MMON O ( 90) ET ( ) 3 ),Y ( IVY2(9 ) ,Y3(90)
COlON YO(90)Y (9) ,DYi(93),DXi (90) r2(913) X2()(9 ;,3(9 ),
tOX3 (0C)
SCOM ON/3LKt/ XX( X 36) ,vOY( AS) ", .....)
C"'" ,.QP IS I LESS THAJ rh NU iER OF PLOTS ON A SINLE F
IUSS(I) SPrIFIS TiE POINTS N THE AiRPAYS U AND W AT WHICH THE
ONLOING PLOT IS T PMNATED AND A NEW PLOT IS BEGUN,
I FOP THE SAKE OF C3MLTENESS WE INCLUDE THE OTHER SiUROUTINES
r THAT COMPLEMENT PLCTT~R,NAMEL EASY, SCALING GRAPH ANO O3ORDEP 0
S THESE SU3ROUTINFSIN TURNINTRFACF WITH ROUTINES(EG, PLOTSYMBOL,
S Nti.iBiEPETC. ) ROM TIE PLOTLI3 AN@ GOULD LI3rARIES.
A SAMPLE FOLLOWS.
COMiMON/IU3S/IUBS(I) ,TP
TJ 3'S (2 )=5
TUL'(: 2)=15
ITrTTLE(I) =oHi ST.
ITI TL () = H rH
IT TLE(3) ISL N
ITTTLE() =7H 1i73,
ITTTLE(5) =7 1970
LAU( 1) ='HY 3O0 0 ,
LA L U(2) ="H ()
LABt(1)=7HX OCORO
LA (2) = ri (M)
PO 1 T=z iL
SW(I) =X(I)
1 U(I)=Y(I)
DO 2 T=1,1.
h(I+f~4)=XX( T)
2 (I+i ) =YY(I)
CALL POTATE(WU,28)
C LL EASY(U, ,28,ITITL.FL 3U,L. 3V, 0)
CALL PLOT( 0,0.999)
100 IQ =*
IUPS(3)=19
DO 3 .T=1,1,
W(I)=Z2(I)
3 U(I =Zi>(I)
DO I=1, 14
W( I+1 ) =X( I)
L U(I+i4)=Y(I)
CALL EIASY(0,q,2ATTITLELA3LAV0)
CALL PLOT(C,0,9999)
PETUN .
END
SUPROUTITNE ROTATE(K,Y,M)
r
C~**ROTATE REEVALUqWES THF BEACH POINT COORDINATES TN A SYSTEM THAT
r IS ROTATED THROUGH N AN3LE N; WITH RESPCT TO THE OLD AXES
C OEF MOTIVATION FOR THIF OPERATION TS TO MAKE CLEARER THE DIRECTION
r AND MAGNITUDE OF SHORELINE CHANGE I.E. ENHANCE VISUALLY THTiS CHANGE
DIMENSION X(N),Y(I)
COMMON/ 3LOC3/C VRT RT 1, VRT
rfl*ANG TS THE POTATION AlGLF IN RADIANS POSITIVE COUNTERCLOCKWISE
ANG=2u. z CVoT1
COSA=COS ANG)
SIN =SIN (ANS)
o \ T = K )
v0 i T=ieN!
Y( i)=Y ( ) COSA+X(I) INA
1 X(i)=YSAVExSINAX'(I) *OSZ 0
FETUPN
EN
SUO1,OUi NE AA.;Y (U,V,N,I i i TL ,LA3U AilAiV,IDASH)
C THI S aU ROUTINE PLOTS A STN'LE G. RAPh ON A EACKGROUNC THAT IS
C LAR TESAN. X I[ THE HORIZONTAL ARRAY, Y THE VERTICAL ARRAY,. N THE
C OF POINiS TO BE fLOT TED, TITLE A hOLLkERITh TITLE L A5X A HOLLERITH
C LADiL FOR X, AN9 LABY A HOLLERii H LABEL FOR Y. NAME IS A FILE
C NAME IN HIOLLERIi;H FOi;,MAf
C A.JY AUJ3OUTItN4E  CHECK .ON CALL FORVHOLLERITH VARIABLE
u..ilNSION U(N).,V(N> ,ITI LE(5 ,.LABUl2) ,LABV(2)
U MAX U ()
U Mi NU (1)
VA ,AX=V( 1)
V iIN=V 1)
DO lu I=2,N
if (U(I) GT1.UMAX) UMAX=U(I)
IF(U(I) .LT.U'IIN) UHIN=U(I)
IF(V(I) GT.VMAX) VMAX=V(I)
IFI ( (1.L) .LT .VMIN) VMIN=V I)
lG CONi NUt
SIFFX=UMAXUMIN
01FFY=VMAXVMIN
UMAAX=+UAX+J .03 DIFFX
U..UNI =Ut N0, .03ODIFFX
jVMAX =VMNAX+0 ..03* OIFFY
S/VMIN=VH ilT; 0 3 OIFFY
DX=DIFFX/2C
XX=ALOGC' (DX) +10.
KX=XX
KX=KXluO
RX=OX/(,dJ.U **KX)
.P X 1i= '
IF(RX.GT.1..4) PX=2.0
IF(RX.GT.3.3) PX=5.0
IF(RX.GT.7.1) PX=10.0
DELU=PX'*OX/iX .
LAGEL=5
IFiRX.G T.3 .3.AN.O.RX,LE.7.1) LABEL=4
OY=DIFFY/. .
YY=ALOb 0i DY) +1 0.
KY=YY
KY=KYIGJ
RY=0Y/( 10. 0**KY)
PY=1.0
IF(RYY.GT.i .4) PY=2.G
IF(RYCbT.3.3) PY=5.0
IF (RY.GT.7 .1) PY==10.b
DOLV=PY*DY/RY
D L V P Y 0 i.< / Y
CALL SCALNG(UiINUMAX, VMIN,.VMAX)
CALL eO :EU.IT" U'AXVMIHn VMAX ,DLL bOELV ITTTLE, LAlULABV.LARE..
; AL GL L S G L;N1 i I N U A V M I .MIT A X, ;)
END
SUBROUTINE SCAL.NG(UMINI, UMAX,VMINV, VAX)
CO H ON/ CALE/A ,7C Oly, Y, XR,XL
C iIS SUEROUUrNE SES U! SE CALI;i. BEr'EEN (U,V) AND (X,Y)
C fUV ) ArLE (ABSC I2SSA,ORI JTJTE) AND 0(XY) :ARE IN INCHES ON CRT.
C THE RLAIIONS ARE XA*U+U, Y.=C*V+D,
C EQUAL SALE FACTORS (A AND C) ARE USEO IF THE GRAPH IS NEARLY SQUARE.
A=6.5/ (UM AXUMIN)
C=4.75/tMAXVMIN)
R=A/C
IF(R.GT.1.5.0R .R.LT.L..7) GO. TO 10
A=AMIN1IA,C)
C=A
10 CONTINUE
.B=4.0G .5*A*(,U AX+'UMI N)
D=2.9CG,.5*(VMAX+VM'IN)
Y i=C.VM.AX+D
Y0V=CVMIN+q
XL A U.lIN' +
X 1( MUHAX+B
RETUr t N
END
bU1ROUU INL E RAP kU,V,N,IDAS h)
COMMON/IUtj S/U S(iO, [(
COMMON/CA LE/A, C CDXR, XL
DIMENSION U,N) ,,V(N) (6) (66
X=A VU(1)+B
Y=C v (1) +0
CALL PLOTir ,Y,3)
JO =S,=IA3S ( IASH)
IF(JuASHiLE .9) GO TO 2C0
C COUNT NUfINBE, OF DIGITe IN 1LASH
NbIGIf=6
IF (JOASfH.LE99999 NDIGIT=5
I JDAS,i.LE.9999) NDIIlz=4
If (JDAsi.LE. 999) NDIGiT=3
IF(JOASrH.LE.99) NCIGIT=2
C OECOMPOSE INTO FOUR INTEGERS
NJEN= 1LU3 0
NUM=JD AH
00 99 K=1,o
I K)= NUHI/NDEN
NU;l=NUMI( K) NDEN
NhEN=NOFN/11
99 CuNTINUC
C MINIMUM SPACING IS 1/128 INCH
00 100 K=1,6
IF(I(K) .GT.7) i (K)=7
100 CONTINUE
NFIRST=7NOIGIT
DO 101 NI=NFIRST,6
H (Ni)=,1./ (2.0 I(NI))
101 CONTINUE
C INITIALIZE
LEVEL=NFIRST
EXCESS=r(NFIRST)
00 103 K=2,N
C DRAW JASHEO LINE TO NEXT POINT IN ARRAYS.
XOLO=A*U(K1) +8
YOLO=C*V(K1)+D
XNE=AMU(K) +B
YNE:=C*V(K) +0
X CIFF=ANEtXOLO
YDIFF=Y NEWYOD
TOTAL=SQRT(XOIFF XDIFF+YOIFF YOIFF+3 .00001)
ACOS=XDIFF/TOTAL
YSIN=YOIFF/TOTAL
TCGO=TOTAL
X=XOLD
Y=YOLD
C NcXT SECTION OF LINE
10' CONTINUE
ITE S T.=LEVEL+NDIGIT
IPEN=2
IF(( (ITEST/.2) *2) .EQ.ITEST) IPEN=3
P=AMiN1 (EXCESS TOGO)
X=X+P*XCOS
Y=Y+P*YSIN
CALL PLOT(X,Y,IPEN)
TOGO=TOGOP
*EXCESS=EXCESSP
F (EXCESS.LT.O. 0J5) LEVEL=LEVEL +1
IF(LEVEL.Gr.6) LEVEL=NFiRST
IF(EXCESS LT.0.005) EXCESS=H(LEVEL)
IF(TOGO.GT .0,C5) bO TO 102
C OTHtRWISE GO TO NEX[ POINT IN ARRAY AN3 CONTINUE
1l3 CONTINUE.
RETURN
C SOLID LINE GRAPH WHEN IOASH HAS ONLY ONE DIGIT.
2u0 CONTINUE
IQB=1
00 10 K=2,N '
X=A*U(K)+6;
Y=C*V(K)+D
IF(K.EQ.IU 3 (IQB)) 1C00 1001
10C 'CALL PLOT(X,Y,3)
IQ3=IQu+1
bO TO iJ
101 CONTINUE
CALL PLCT(X,Y,2)
10 CONTINUE
RETURN
END
SUiROUl NE or 00RDE (UlINI,lMAX,VMIN,.VMAX,;U,DV, TITLE LAUUtLAB:RVINC)
LIM N 0NS I 'L,. A (5 ,LA.U.(2) LAf3BV 2)
*CCMMON/SCA E/A, r_,C,'J, r 1,YBA R,X L
C THIS .SUROUTINE DRA;3 A kEC ANGU L R OR ER T IC. MARKS AT ..DI. V
C INCREMt t AlI ALON. ArJCIS A, ORDINATE, iHEiN LABELS EVERY ING T tCK Ifi KS.
C lIT IIIT S ,RITfEN ALONu T.E TOP, LAEU,LABV ALONG THE ABSCISSA,
C UxUDINATl AFcIS. THC FIRS FOJU A..GJMLNT ARE 1RANSFERED 'TO SET.
CALL aCALNG(UINI, uMAX,VMIN, V1AX)
U A LL r i L N G U ii I N 11. AX I VM I V :'l A X N
ULAE=OU*INC
VLA3=DvINC
FLAG=
C OPAi 'LO:.uER (XR,Yd) iO (XPYi) TO (XL YT) (XLXL,YB) .TO (XR,Y3)
CALL PLOT(AR,YB,3)
CALL PLOT( XR,YT,2J
CALL PLOr( XL,YT, 2)
CALL PLOT(XLYB, 2)
..CALL PLOT( .ARYB,2)
C NUMBER OF FIGURES TO RIGHT OF DECIMAL POINT
N;xU=1.5ALOG10 (ULAb)
NRI=1.5AL OG10 (VLAB)
IEU=0
IEV=u
IF(NRU.GE.4) IE U=2NRJ
IF NhKVr.GE.. 4) IEV=2NRV
IF(NPU,.GE.4) NRU=2
IF(NPV .E. 4) NRV=2
QU=AMAXI(A3S (UMAX) ,A3 (UMIN))
MU= ALOG 10i (.U)
IF (MU.GE.L4) IEU=;iU
IF(U.U.GE.4) NRU=2
Q =AMAX1(ABS( 'VMAX),ABS (HMIN)
MV=ALOolO(QVJ
IF(MV.GE.4) IEV=MV
IF(MtV.GE.4) NRV=2
C SCALING FAGTORa FOR U AND \ .
FU=1O0.0J*IEU
.,! *F =v'1.F=i .6**IEV
N CHU=N MRG U
IF(NRU.LT. ) NCHU=I
N CH V =Nr, v
IF (NRV.LT. 0) NCH =
C TIC KMARKS ALO.U A oCi,:;A
U:U+ , LQJ J51'D
20 CO i INUE
U=UcU
I=1 1
IF(U.LE.UM'IN) GO TO 30
C OTf:i HIE uKiA C TICK MARKS
X=A*U+B
C .
T ICK= 0.084
IF ( ((I/INC IN C) .EQ. I TICK=2. 5*TICK
Y B= Y +Ti ICK
CALL PLGCT(XYB, 3)
CALL PLOT(X,Yu5,2)
C DRAW .LIN' F:F'O; X X,Y ) TO (X,YT0.1)
Si =YT 'K
CALL PLOT(X.YT 3)
CALL PL:O.T(XYTtI,2)
IF ( l/IiNC) >:INC) .NE.I) ,0 TO 20
C O HEKwISL, h RiTE U BE.LOi TICK MNArK
C NUMBER OF FIGURES IC LE.F OF DECIMAL POINT
: iL.U= 1 *
SUIFU
I (A:I.;(UU) ..LT..OG]l) GO 10 25
NLU= LO 10( ABS(UU)) 1
25 C.nrt r JJlln
IF(NLU.LE. ) NLU=I
NT=N LU +NCI U
XLA=XJ. 07*NT
YLAJ=Y0.2
IF(NCHU..GT .i.ANJ.NChU.LT. 1) NCHU=G
CALL NUH3ER(XLAB, LAB, .14,UU, .,NCiiU)
GO 1 0 2
30 CuNTINUE
IF(XLAB.LT.XL) IFLAG=1
C TICK MARKS ALON ORDINATE
J= (MAX+OV)/DV
S=J*CV
V=/j+3 .i000IODV
4l CONTINUE
J=J1
V=VOV
IF(V.LE.VMIN) GO TO 50
C OTdERWISE DRAW TICK MARKS
Y=C V+0
C ULRA'W LINE FROM (XL,Y) TO (XL+0.1,Y)
TICK=:. 04
IF(((J/INC) *INC> EQ.J) TICK=2. 5TICK
XLL=XL+TICK
CALL PLOT(XL,Y3)
CALL PLOT(XLLY,2)
C ORAr LINE FROM (XR,Y) TO (XR0.1,Y)
XRR=XRTICK
CALL PLOT(XRY,3)
CALL PLOT(XRR,Y,2)
IF(((J/INC)*INC).Nrc.J) GO TO 40
C OTHERWISE, RITE V NEXT TO TICK MARK.
C NUMBER OF FIGURES TO LEFT OF DECIMAL POINT
NLV=1
Vv='//FV
IFtABS(vV) .LT.G.j1) GO 10 26
NLV=ALOGil (ABS(VJ))+1
26 CONI NUE
IF(NLV.LE.1) NLV=1
NT=NLV+N1V
XLAB=XL0.06
YLAd=Yu.07NT
C ELIMINATE LAST LABEL NHEN OVERLAP CCULD OCCUR AT CORNER.
IF(IFLAG.EQ.I.ANJ.YLAB.LT.YB) GO TO 50
SF(NCHV.GT .l.ANO.NCHV.LT.1) NCFV=0
CALL NU;'1BER(XLA=!YLAD G14, V,9G.,NCHV)
GO TO L40
C i IN''I FLE ON TOP BORDE~, AESCISSA AND CROINATE LABELS
HT=0j.14
XJ=3.4
Y=YT 0. 2
CALL SYM3OL X,Y ,HT,IITLE 0. ,50)
ISCL=8H( X10 )
XXL+0 4
Y=zYG.5
CALL SYMBOL(X,Y,HTLABU,0.,20)
IF(IEU.EQ.0') GO TO 60
C OTHERWISE, WRITE SCALING FACTOR
X=XL+3.2
CALL SYM30L (X ,Y ,HT,ISCL,G. ,8)
X=XL+3.8
Y=Y+0 .1
Q=IEU
CALL NUMBER(XY,3.07,Q,C.,i)
60 CONTINUE
X=XLG.3
Y=YB+0.2
GALL SYMBOL. (X,Y, HTLABV,90. ,2J)
IF(IEV.EQ.0) GO rO 6.1
C OHLR ISE, I RITE SCALING FACTOR
Y=:fz+3. J
CALL SYI30L (X ,Y,HT,ISCL,90. ,8)
y =A+. 1
.Y Y+0.6
Q=IEV
CALL NUMBER (X,Y ,G. 7,Q,9 ,1)
61 CUNilINUE
RETURN
VII. Apalachicola Bay
1. Nature of Inputs
St. George Island is part of a barrier island chain in
the Apalachicola Bay region of northwest Florida (see Figure 5).
Because of its location it is subject on the average to low
tomoderate wave energy levels. Waves propagating from deep
water toward the island pass over a broad, shallow continental
shelf region and experience bottom friction damping, the degree
of which depends directly on the wave height and period
(equivalently, wavelength) i.e., the higher, longer waves are
attenuated more rapidly. The net energy loss can be substantial
when integrated over the total travel time from intermediate
depth water to the point of incipient breaking. Other means by
which energy.can be subtracted from a wave train are the pre
s.nc of adverse winds and shear currents, and nonlinear wave
ay inleractionc (including dissipation due co capillary wves)
In shallow water the'wave energy density.increases, competing
effectively against bottom friction to enhance the wave height
and induce breaking. An exception may occur on verymild slopes
where bottom damping is sufficient to extinguish the wave.
Depending upon the bathymetry,refraction and diffraction ,can
augmentt or reduce the local wave height.
Walton "(1973) in a study on the distribution of littoral
drift,along the entire Florida shoreline considered deepwater
,wave .data. ,as shis soure .of wave information and incorporated .in
Figure 5. The Apalachicola Bay region.
/6
G** *L
9Y
I5
his model the influences of bottom friction, shoaling and
refraction. We .have chosen to utilize Walton's model on wave
modification in shallow water.as the means by which we generate
the breaker data essential to our model, b(y) and H, b(), the
breaker angle and height as functions of longshore position.
We will not detail Walton's work, since he has provided a
thorough explanation of his methodology, but rather outline his
general approach and the changes we introduce.
The wave data source is the U.S. Naval Weather Service.
Command, Summary of Synoptic Meterological Observations available
from the NOAA Environmental Data Service, National Climatic
Center, Asheville, N.C. These are shipboard observations of
meteorological and sea conditions made by ships.in passage. The
drawbacks inherent in such data are many (we shall not en,ulmerate)
but they represent the best general compilation of marine data
at present. The record extends through the years 18651971 with
eic ty percent of the observations occurring during the period
19541971. The .pert inent annually averaged tables are Table 18,
which gives the percent frequency of wind direction versus sea
heights, and Table 19, which gives the percent frequency :of wave
height versus wave period. Using these tables several bits of
information are computed. The frequency of occurrence of a wave
of a given height, period and direction of propagation is de
termined and expressed as'a fraction of the total time of record
and is subsequently adjusted according to the following formula:
the geographical oceanic region which is assumed to contribute,.
waves to a specific coastal area is divided into "data squares";
this necessitates that data from adjacent squares be weighted.
Walton used 2040'data squares as shown in Figure 6 to blanket
the Florida coastline and linearly interpolated the wave climate
between adjacent squares. We chose a set of finer resolution 10
squares in the Gulf of Mexico (Figure 7) because of the high
density of data in each square and the coverage of the Florida
Panhandle. Our method of weighting, somewhat different from
Walton's, is illustrated in Figure 8. A reference line is drawn
due south of.St. George Island. Additional lines are drawn to
the center of each 10 square and the angle, 6, between these lines
and the reference line is measured. A weighting term, d,,with
respect to.62, is determined from the formula
2 .i 1 (52)
where 6. 283, 8., 17, 36>, 51., 58, 6 i ,
so that d = 0.417. The individual weighting factors,
are then applied to each wave type in the respective squares to
ascertain the contribution from each square to the mean frequency
(the fraction of time over which a specific wave endures):. At
this point a set of deepwater input data has been established.
The next step is to track each wave component into shore
monitoring its change in direction due to refraction and its
change in amplitude due to shoaling, refraction, bottom friction
and percolation through the sand grains. Walton's numerical
model to accomplish this has the following structure:
Data 'sauarcs used in Walton (1973).
* *
Figure 6.
.I '
; I 3 Q I i 2 *. i
__ I __I 1
..... " " 
4 ...i i _.__ _
I i
I
.. .. .i
 . s .. i r 
S I u d i n i
Figure 7. Data squares used in present study.
g I
S3
tc~*re c
Figure 8.
Geometric weighting of annuallyaveraged data from
each square.
1. The orientation of the wave fronts as they approach
shore is computed using Snell's law of refraction for a bottom,
topography composed of straight and parallel bottom contours.
The refraction and shoaling coefficients are calculated con
comitantly.
2. The computation of the coefficients of bottom friction
and percolation follows the work of Bretschneider and Reid (1954).
Required inputs are the lengths and slopes of a series of bottom
sections comprising a bottom profile normal to the stretch of
shoreline being considered (rather than the true profile over
which the waves pass).. In our model we consider seven profiles
coincident with the seven lines in Figure 8 and weight the
results in the same manner as before.
The product of these calculations is the breaker height,
the breaker angle and the fractional duration of each deepwater
wave type for a segment of beach. Repeating this process for
each beach segment of interest we obtain a longshore distribu
tion of breaker heights and angles, each of these quantities
contributing independently to the magnitude of the longshore
current, e.g., a decreasing angle of incidence longshore could
be offset by an increasing wave height and viceversa. The
reader is referred to Walton (1973) for a more complete discus
sion of the assumptions, approximations and limitations under
lying the above data reduction and analyses.
The application of our numerical model requires that we
discretize the strandline in a coordinate system established
with reference to some semipermanent landmrk. A feature which
is present on all the bathymetric sheets.of the U.S. Hdrographic
Office that we have used (which provide us with a. progressive
history of the St. George shoreline) is the St. George' light
house on Cape St. George (see Figure 5). Stapor (1971) has
indicated that the lighthouse, constructed in 1847, has a margin
of error associated with its position on the charts which falls
within accepted map standards. The lighthouse is the origin of
our coordinate system which, for convenience, has its ordinate
running due noxth and its abscissa due east. The shoreline of
1873 as depicted on smooth sheet No. 1184 is divided.into 57
segments; the northwest tip of the island is beach point 1, the
northeast tip beach point 58. The points are irregularly
spaced, being packed more closely where the.beach exhibits
large horizontal curvature; the maximum spatial increment is
840 m (in the mildly concave middle section), the minimum about
320 m (in the area of Cape St. George). Resolving each beach
point into vertical (x) and horizontal (y) coordinates, the
separation between points and the angular orientation of each
segment is straightforward to calculate. This information, when
fed into the wave modification program previously discussed,
ultimately determines the longshore variation of breaker height
and angle.. Supplementary information on the Apalachicola Bay
region is provided by smooth sheets:H1265 (1974), H57945 (1935),
H5319 (1935)., 2265 (1896) 6788 (1943). The 1873 .strandline
serves as the baseline for the predictive model. If the
present (circa 1970) strandline can be generated, even
qualitatively, then the model could be used, albeit cautiously,
for future projections.
2. Longtime Integration of Predictive Equations
The integration of (43a,b), as they apply to St. George
Island, cannot be accomplished blindly. One must be aware of
any special features that contribute to the dynamic balance of
the island.
St. George Island in 1873 was composed of threeparts
separated by two hurricanecut inlets (the bay side of the island
is marked by hurricarwashover deposits). In Figure 9 is a
schematic diagram identifying the major sections and the inlets.
The stability of tidal inlets is a complex.problem which we do
not treat here (see O'Brien and Dean, 1972; Dean and Walton, 1975).
The littoral drift past an inlet can be interrupted and sand
deposited, leading to closure of the inlet. The question of
closure rests on knowledge of the inlet crosssectional .area,
the tidal velocities, the wave climate, the magnitude of bottom
and side friction and the level of littoral drift. We assume
that the rate of sediment transport across the inlet is reduced
relative to its upstream value by some fraction. Since Sand Island
Pass and New Inlet Pass both eventually close, an estimate of the
volume of material contained in these inlets, the period overwhich
closure progresses (assumed to be unidirectional) and an average
upstream littoral drift rate can yield a value for this fraction.,
An alternative method for determining the rate at which the longshore
SA; AC HI C" .
"1. .
5AAYf
i SLe A5D
...J .
\ c
Sy/D of' r~ .xico
Figure 9, The southwest portion of st, Geirge sand, 1873.
drift is trapped in the vicinity of an inlet is to obtain an inde
pendent measure of the growth rate of the shoals insideand outside
the channel. In the absence of such information we simply.assume
that the transport rate across the mouth of the inlets is'the
average of the upstream and downstream values. This is an unwar
ranted assumption if the inlet does not bypass a substantial por
tion of the longshore sediment load for, then, the downstream
shore is likely to be cut back due to sand deprivation.
Due south of Cape St. George is an extensive series of
shoals projecting some 8 kilometers into the Gulf of Mexico. These
shoals are focus areas for incoming wave energy and, consequently,
thebreaker energy expended on the shore to move sediment is re
duced. This submarine relief will attenuate, redirect, or even
block waves propagating toward the Cape. We expect that the level
of wave activity in the vicinity of the Cape, as computed previously,
will tend to be an overestimate, at least, in relation to the energy
levels at connticuous portions of the beach. We, therefore, ~reduc
the energy input to this region, due to waves from the south and
southwest, by about 25%; this figure is arrived at by considering
the degree of wave damping over this special bottom relief and the
percentage of waves that are likely to break far from shore.
Off the northwest tip of Sand Island is a rather permanent
shallow, submarine feature, the East Bank. The tidal ebb flow
through West Pass has transported local material seaward'and the
wave levels have been too low to reverse this trend and confine
the sediment to the littoral zone. This shoal, which sweeps to
the south and west, almost attaches itself to the shoreline. With
the Cape St. George Shoals intercepting waves from the southeast
and the East Bank doing the same for waves from the southwest, it
is anticipated that the levels and periods of wave activity in the
.area of Sand.Island will be diminished compared to those values
computed in disregard of these prominent shoals. As Iefore, we
reduce the transport figures, accordingly. In addition, much of
the longshore drift toward West Pass is likely to be diverted to
the East Bank by tidal currents in the presence of low incident
wave levels, i.e., only small quantities of sand will be deposited
at the tip of Sand Island. This situation will prescribe the boun
dary condition at the northwest end of St. George Island in our
model.
It should be noted that there is an overall bias toward
low wave energy in this study. Since ships tend to avoid bad
weather our deepwater wave observations are on the low'side.
Also, major storms, such as hurricanes, can cause rapid and marked
fluctuations in a beach system. We are presuming that over a long
period (e.g., greater than 50 years) there is a "smoothing" effect
line responds to longterm forcing (e.g., hurricane breaches in
a barrier island on a tidal sea are usually repaired on a relatively
short time scale).
To be specific, the integration of the governing equations
for St. George Island over the period 18731970 was carried
out according to the following procedure:
I. The wave characteristics at the breaker line (height,
angle, fraction of a year over which a particular wave acts), as
generated by Walton's program,' serve as input to our model. A
cumulative frequency of occurrence of all waves yields the time,
expressed as a fraction of a year, during which onshore waves
are expected, e.g., for St.. George Island it was found that
onshore (breaking) waves are present about half the time, the
exact number being 0.51. Thus over a 98 year interval relatively
calm periods prevailed for approximately 48 years. The sets
of breaker data are inserted in random order into the model to
compute the forcing function for the longshore motions. Equations
(23a,b) are integrated for a number of time steps equivalent.to
1 year. This process is repeated for as many years.as desired,
i.e., one year does not differ from any other year inasmuch as
the deepwater wave climate remains unchanged (although the
breaker angles do change in response to the evolving beach
shape).
2. In (24a,b) it is found by trial that choosing a
step size of At 50 hours and applying the corrector twice is
the most efficient compromise, i.e., the truncation error is
kept small and the integration proceeds fairly rapidly.
3. As mentioned in section V special care must be taken
at endpoints. We see in Figure 4 that for point jr Qj_' j
and A. must be defined differently than the same quantities as
they apply to the interior points. By endpointss" we mean those
points at the extremities at or near which the longshore trans
port approaches zero (i.e., Q = 0). This definition is offered
in lieu of more detailed information about the tidal, wave, ,and
current dynamics in these areas. A more formal consideration
of the sediment flux at the tips of St. George Island would
entail finer wave refraction and diffraction computations, a
knowledge of the magnitude of the tidal streams and of the
leakage of sand from the island to offshore shoals. By refer
ring to the boat sheets and noting where theshoreline beiains
to curve.inward away from the. predominant wave direction we
choose the endpoint positions and measure manually the effective
beach angles,, 0, at these points. These angles are important
in that they determine the direction in which the ends pro
grade or recede; they are adjusted as the integration proceeds.
The'initial at the northeast tip of.the island is 2T,/3 (radians),
*at the northwest tip it is 31/2. The endpoint A is twide the
distance between the endpoint and. its .neighboring point.
4. Other parameters in the model assume the following
values:
K = 0.4
.N = 0.05
It was found that the magnitude of the sand transport was not
particularly sensitive to variations in N.
Results are displayed in Figures 1015. St. George
Island is viewed in 3 sections: 1) the Cape St. George region,
2) the long, arclike middle section, and 3) the elongating
northeast tip. bapporting evidence is proided by Stapor (1971)
who determined the areas of erosion and accretion for shorelines
in the Apalachicola Bay .region. He considered the redistribution
of the bathvmetric contours to be indicative of.the direction and
Figure 10. Southwest St. George Island as recorded in 1873 and
1970.
' ,, E
r:JV i E
.T .:2 L ,I U
. The letter
has under
*s 'A' and 'E' signify that the more recent of the
once local accretion or erosion,respectively.
two strandlines
A
N AN!) (a ~AS9
R*
\ \
N
\% "% fF7
OCEAN
C,
n r,
,. ~I..
II '4 I
V r3L ? 
\ '\
Ai 70 \ 
( ^ *; \...7< >
\ '; r^ .*
A LA^
.. "*^
0 1. ric__
( .',
lc / 0
t
E. .
: I
!_ j
,,
L; i
Piguid 11. A comp. rion of southwest St. George Island in 1873
and 1970 (predicted).
r * .
". .
'ANP IS. PASS
A
OC EA J
,o.
_I.. '^ . '. _
1'/13
A
( g c. ),
1I~.~I:
A. l i
' C
Figure 12.
The middle section of St. George Island as recorded
in 1873 and 1970.
r. 
D i
I L.} I N L
(c t0
L
L
r
"i
1
AX
~
'i
lII
}pL~
,~~~~1
1 7 1/ 9
' OCEAN
c.
~~I~~ 

  ,  n~ f    
o}"7 ,
I/
,.. )
Figure 13. A comparisor of the middle. sectich of St. George Island
in 1873 aid 197C predictedd).
T  
L u 
r r
A,'
h 
: (73
E ."
I
I
I
* 
A
S flNEW INLET
C R'D
*Y CYGORD
~1 .Z4L
1. Ff
l XI ij

r.
mL
~3Lfl
OCEAN
I ,
L
1.
Figure 14. Norlihcast St. George Island as recorded in 1873 and
1970.
0 r' r
U
Ir
r7 ti
I' r
J.~~~~~ ~~~ ..; ,1...~ l~.~.: ,_~__ J
I.
OCEAN
.
iI
 ,*; 1
A
K~ /7 
/F3
A ,:.
/ /;
//
t ./
.__J.. _. ._L.
I
L ,r C:
ti r'i
Figure 15. A comrari::on of northeast St. George Island in 1873
and 1970 (predicted).
j 9 
r, i c ,, M
' 0 ,/
/

C I
C
i "
J
2.. 
Z;L
3 20
O2.8 3 OQ4
iF73
4' '
vY C,0 0IRD
.L.
" i
x
I
%O
1
 ~~~~ 1
...L.
( M.
( X I 1
and volume of sediment transport. By subtracting isobaths as
given on old and new boat sheets and contouring the results
Stapor was able to compute and. approximate sand budget for
St. George Island and adjacent areas over a 7080 year period.
The salient differences between the actual plan profiles
of 1970 and those of 1873 (Figures 10,12,14) are:
1. The Cape St. George area undergoes a lateral shift
of its strandline to the north and west, i.e., there is
erosion to the east of the'Cape and accretion to the west. The
northwest tip of the island is cut back slightly. Sand Island
Pass is closed. These features are confirmed by Stapor's
computations as seen in Figures 16 and 17, where, in addition,
sand deposition is observed on the East Bank and the Cape St.
George Shoals. Our predictions (Figure 11) show the same trends.
as Figure 10 with the exception of the inlet closure. (note: since
we have not modelled thee inlet dynamics it is not expected that
the mou t of the inlet will enlarge, diminish, or migrate
appreciably.
2. The long, middle section of the island, over most
of its length, experiences erosion (Figure 12). Stapor (Figure 18)
indicates a sand transport direction away from the concave
3 3 3 3
middle: 76 x 10 m3/yr to the southwest and 60 x 10 m./yr to
the northeast. By expanding our concept of 'control volume' to
include any length of shoreline we can compute a net volume drift
rate for a particular stretch of beach acted on by a particular
wave climate. By summing over the difference.between successive
transport..rates, i.e.,
Cape St. Ceorge
ST. JOSEPH'S
POINT
1 8 74. 
l3 V
0 3 10 .( 70 &a
C_=^<'Ki~~Ipl~
CAPE SAN
S1B74
5 DOG ISLAhD
ICi3
C? // Je
1' E ST. GECF
ISLAND
. lighthouse
1 0 1 2 3 l.
, WEST PASS AREA
S1GE 6 
S1942
> U C^ ," 1 90 ?  
Shoreline changes, Apalachicola Bay (Stapor 1971).
/.
Figure 16.
i/
... .
/ \ \ ,.L .\C .
*. \" ... .[{ "
'
Bathym.try, 1942, 6 foot C.. \
Shoreline, 1942 \
Deposit;oaal sitea '
Erosioaal site C.L
Ligihtlh. e Aad dat, t1'
LiliI~~:e ~ CAPE
ST. GCEO6
SHO AL .
f t i
Figure 17. Erosion and deposition patterns, West Pass Area
(Stapor 1971).
" . X 7" *' .
.. .,f tU , .{
isyiU 55
.' 18. T fig e ..'Gr Is and aart
Figure 18. Transport figures for St. George Island and adjacent
coastline (Stapor 1.971)
(53)
(see SUBROUTINE ADJUST in computer program), for our range of
wave parameters, we obtain figures, corresponding to Stapcr's,
3 3 3 3
of 425 m /hour (approximately 35 x 10 220 x 10 m3/ye.r).
Figure 13 displays an.overall erosive pattern. but differs from
Figure 12 in the degree of shoreline retreat.and the position
of maximum change.
3. According to Figures 14 and 16 the northeast end of
the island has been growing (a spitlike feature). Figure 14
indicates :that areas adjacent to the expanding tip have advanced
seaward interrupted only by a. few smaller pockets of erosion.
Figure 15 predicts a substantial growth at the endpoint although
not the eventual sharp veering to the north (away from the
dominant direction of wave approach) seen in Figure 14. In
addition the prediction shows the shoreline being cut back
along the entire stretch of beach upstream of the endpoint,
i.e., the tip of the island is being fed sand from nearby
beaches as well as from the island's middle section. This con
trasts with Figure 14 which shows the middle section to be the
major contributor of sand.
Because the wave data is averaged on an annual basis and
extraordinary wave conditions (e.g., hurricaneproduced) are
filtered ;out it is not expected that future trends will differ
significantly from those occurring in the past. Using the
observed strandline of 1970 the nearshore wave field is recomputed
and a 20year projection is made. The results (Figures 19,20,21)
exhibit patterns similar to those computed before,such as the
erosion of the concave middle)which makes it susceptible to
breaching. The exception is the northeast tip of the island
which now turns inward as well as prograding northward, i.e.,
the beach reacts to the incident waves in such a way as to
minimize their erosive effects. In the presence of an ample
sand supply and low wave energy it is unlikely that the end of
the island will curl in appreciably; rather, its excursions in
a westerly direction should be intermittent, being counteracted
by sand deposits sufficient to direct the tip's advance to the
north and east. Figure 21 again points to erosion immediately
upstream of the tip with accretion farther west.
The foregoing predictions in most cases are in reasonable
agreement qualitatively with observed strandline changes. For
the results to be more pleasing quantitatively certain improve
ments are obvious: (1) the quality of the wave observations could
be enhanced by in situ recording of swell and local sea (shallow
water wave generation is.not accounted for in this study);
(2) rather than considering the depth contours to be straight
and parallel offshore of each beach segment, a formal refraction
analysis using the actual bathymetry should lead to better fore
casting of nearshore wave conditions; .(3) an attempt could be.
made to model the effects of severe storms (hurricanes).' Strictly
speaking, this falls outside the outline of the present report
Figure 19. Southwest St. George Island, 1970 and 1990 (predicted)
r*
C CE
S L rI i i
*
~N
"*. "\ /970 (oCeirv ef! )
A \
\ "\
'S
.e dcoe 4
5AND 35.
PASS
1J I
'
!
V I
^ L
Sr
l E70
OCEAN
4j'.., ,'i.J L I
~~~~~T7 T 1 i ~7
LL ~i~.
1 '^ *,r
Figure 20. Middle section of St. George Island, 1970 and 1990
(predicted)
7' 71 V
.I L R Li
.I C'3
j/I .^ '
.r
.' y :
! ._
/
/!rcd;'''d '
L^
/r
1 /"
/'
/
42
A
A
p ~
OCC AN
1 '
v, ' f
I F 
i', '
i .
I. .. I
f s'frSt F*P 
1T
'm.'
Li
i
I
.. 1
._ J id 
.A I
r
'/ .i
i i L
I*~~C~I (Il~VI*R ~ m"Vv C"~
Figure, 21. Northeast St. George Island, 1970 and 1990 (predicted).
,T'; ) A
(pred;c^cZ ed
_. r, E ...."
.i ^: .... '. .
2 .... 2 8 3 O 0 '
.S i .Mi X*C 
 ,
. ... : .OC E
"" ,. 0 :
1 '
'*,~ z "s ~ J"
C 0 R )( X t .
because longshore drift would then only be one of the important
components of motionthe offshoreonshore movement of sediment
associated with high water levels and high waves would be very
significant. The present model could be used to monitor a beach
after it has incurred heavy damage to ascertain what contribution
the longshore transport makes to restoration.
In addition, for a coastal region with complex endpoint
boundary conditions (e.g., islands, spits, capes, penisulas) a
packing of thebeach points at the ends is advantageous if it is
accompanied by a finer resolution of the wave field. Since
nearshore wave measurements are usually lacking this would have
to be accomplished by more careful refraction and diffraction
analyses. The diffraction analysis could be based on experi
mental data or be an approximation to the existing mathematical
theory for ideally shaped barriers. Furthermore, in such regions
ere there is sharp curvature of the wavefront the longshore
gradients in the breaker wave amplitude and incident angle may
drive a nonnegligible longshore current. Shoreline change is
often manifested most dramatically at endpoints; this dictates
that we treat theseboundaries in special ways.
VIII. Jupiter Island
The Jupiter Island phase of this research was undertaken
in cooperation with the Coastal Engineering Laboratory of the
University of Florida. Jupiter Island, about 15 miles north of
West Palm Beach, is the site of a recent beach restoration project.
During its duration wave, wind, and beach.profile data were
recorded by a University of Florida contingent. Figure 22 shows
the island.bordered on the north by :St. Lucie Inlet and on the
south by Jupiter Inlet. The project limits are marked. Figure
23 is a photograph of the construction site and Figure 24 a
diagram of the beach nourishment area showing the location of the
sand fill which is placed on alternate sides of the public beach.
Its movement and redistribution within the project limits is
monitored by beach profiling (a sample is shown in Figure 25).
The wave height and period are recorded at only one point
along the beach in 20 feet of water. The breaker angle, in the
absence of more than one wave pressure sensor for directional
resolution, .is logged visually at approximately the same longshore
site. Shorenormal profiles of the beach are taken before the
placement of the fill, at the time of fill, and from 6 months to
1 year afterward at varying points along the beach.
In this study we have the benefit of in situ \wave
observations, albeit at one point, and finer bathymetric data.
In order to generate the necessary longshore breaker data we
employ a refraction program (Dobson, 1967) and, .by working out
ward from the position of the one wave guage, a deepwater wave
climate is established. The details of such an approach are.
contained in Mogel,. et al. (1970). Suffice it to say, a fan.of
wave rays of different periods is trackedseaward from this one
nearshore point across a bathymetry which is represented by a
fine inner grid of depth points and :a larger coarse outer grid.
The inner grid width is determined by the longshore distance
rN
~:A
figure 22. Juoi ter Is;land.
SjuCh tJ~r~fr*it;.h L\~It
' ~ U.44 .. LI .~  .
.L_ 4 i ri r _.L _...: .4
:ij
I1
*
r:
a'... 1. *.4 a C 4t444 .44 *4 44 : 4 4
rl ~Figure 23; An aer;L~ai phlOtograpLh of the f&1J. area, Jupiter Isizndi.
~ 
)~
I
i_
c .~~ ~
I
il_:
I .
4` LC ~ ~
1Lj~
.1
fi.
r
1
I' .
i
i.
r ..:..':
4 4..
.3~
/ '4,
,~ .4.j
Sj r 7 ~ 'c,
~~. .44... K
X~~ ... '
4/ .
.119
4. .4 ,,
i :i:v~
~~~Srr~  

