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MR: TT -10
U.S-ArmY
Caast-Eng. Res Ctr.

MR 77-10
CAD-A04T7 641)

Mathematical Modeling of
Shoreline Evolution

by
Bernard Le Mehaute and Mills Soldate

MISCELLANEOUS REPORT NO. 77-10
~ OCTOBER 1977

‘ DOCUMENT
\ COLLECTION -
oe ae

Prepared for

U.S. ARMY, CORPS OF ENGINEERS
COASTAL ENGINEERING
RESEARCH CENTER

Kingman Building
Fort Belvoir, Va. 22060

Reprint or republication of any of this material shall give appropriate
credit to’the U.S. Army Coastal Engineering Research Center.

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BEFORE COMPLETING FORM

1. REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER
MR 77-10

4. TITLE (and Subtitle) 5. TYPE OF REPORT & PERIOD COVERED

{ATHEMATICAL MODELING OF SHORELINE EVOLUTION Miscellaneous Report
6. PERFORMING ORG. REPORT NUMBER
Report No. TC-831

7. AUTHOR(S) 8. CONTRACT OR GRANT NUMBER(s)

Bernard Le Mehaute
Mills Soldate DACW72-7T-C-0002

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK

11.

DD , 5 On", 1473 Evition oF 1 Nov 65 1s OBSOLETE

AREA & WORK UNIT NUMBERS
Tetra Tech, Inc.

630 North Rosemead Boulevard FSIS 51
Pasadena, California 91107

CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

Department of the Army October 1977

Coastal Engineering Research Center (CEREN) Os eee aia

Kingman Building, Fort Belvoir, Virginia 22060 4 FZ

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Approved for public release; distribution unlimited.

DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if different from Report)
SUPPLEMENTARY NOTES

KEY WORDS (Continue on reverse side if necessary and identify by block number)

Beach slope Shoreline evolution
Coastal engineering Waves

Mathematical modeling

ABSTRACT (Continue on reverse side if necesaary and identify by block number)

A critical literature survey on mathematical modeling of shoreline
evolution is presented. The emphasis is on long-term evolution rather
than seasonal or evolution taking place during a storm. The one-line
theory of Pelnard-Considere (1956) is developed along with a number of
applications. Refinements to the theory are introduced by considering
changes of beach slope, wave diffraction effects, wave variation, and
variation of sea level. The case of hooked bays is also reviewed.

inued

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It’ is concluded that a finite-difference mathematical scheme could be
For the large

developed for engineering purposes for a small wave angle.
wave angle, shoreline instability does not permit use of a reliable

mathematical model at this time.

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PREFACE

This report is published to provide coastal engineers with a litera-
ture survey on mathematical modeling of shoreline evolution, which it is
hoped will lead the way in establishing a flexible and practical numerical
method suitable for predicting shoreline evolution resulting from the
construction of navigation and shore structures. The work was carried out
under the coastal structures program of the Coastal Engineering Research
Center (CERG)):

The report was prepared by Bernard Le Mehaute, senior vice president,
and Mills pon AEE Tetra Tech, Inc., Pasadena, California, under CERC
Contract No. DACW72-7T-C-0002. Funds for the preparation of this litera-
ture review pare of the contract were provided by the,U. -S. Army Engineer
Division, North Central, Chicago, Illinois.

The authors acknowledge the assistance of Dr. J.R. Weggel, CERC, and
Mr. C. Johnson, U.S. Army Engineer District, Chicago, in providing a list
of papers on the subject matter, along with pertinent comments relevant
to the situation in the Great Lakes.

Dr. Weggel was the CERC contract monitor for the report, under the
general supervision of G.M. Watts, Chief, Engineering Development Division.

Comments on this publication are invited.

Approved for publication in accordance with Public Law 166, 79th
Congress, approved 31 July 1945, as supplemented by Public Law 172,
88th Congress, approved 7 November 1963.

OHN H. COUSINS
Colonel, Corps of Engineers
Commander and Director

VI1

CONTENTS

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI).
SYMBOLS AND DEFINITIONS.
INTRODUCTION .
THE FIRST MODEL (PELNARD-CONSIDERE)
1. Refinement and Extensions of the Peimerde Coniideie Model
2. Example of Shoreline Evolution .
THE TWO-LINE THEORY OF BAKKER
THE EFFECT OF WAVE DIFFRACTION .
SPIRAL BEACHES .
PROTOTYPE APPLICATIONS
CONCLUSIONS.
LITERATURE CITED .
TABLES
u versus ¢ (u).
Summary of mathematical models for shoreline evolution
FIGURES
Beach depth definition

Successive beach profiles updrift of a long groin
before bypassing . pala.

Successive beach profiles updrift of a groin after
after bypassing

Matching transition between solutions 1 and 2
Sand bypassing long groin as a function of time

Comparison between experimental and theoretical shore-
line evolution .

Comparison between experimental and theoretical sand
bypassing discharge.

Page

19
ILE)

CONTENTS

FIGURES-- continued

Spreading of sand along a shoreline due to instantaneous
dumping at a point .

Sand dumping along a finite stretch of beach
Equilibrium profile between two headlands

Two theoretical forms of shoreline equilibrium of
river deltas

Differences on shoreline configuration due to onshore-
offshore transport near a groin

Notation for the two-line theory

Evolution of shoreline and offshore beach limit near a
groin

Effect of wave diffraction

Hooked beaches

Indentation ratio for a range of wave obliquity .
Orthogonal arrays for numerical scheme of hooked bay
Orthogonal arrays for numerical scheme of hooked bay
Semilogarithmic profiles

Relationship between shoreline retreat and change in
mean water level .

Page

25
25

38
41
43
43
45
46

CONVERSION FACTORS, U.S. CUSTOMARY TO METRIC (SI)
UNITS OF MEASUREMENT

U.S. customary units of measurement used in this report can be converted
to metric (SI) units as follows:

Multiply by To obtain
inches 25.4 millimeters
2.54 centimeters
Square inches 6.452 square centimeters
cubic inches NOs SY cubic centimeters
feet 30.48 centimeters
0.3048 meters
square feet 0.0929 square meters
cubic feet 0.0283 cubic meters
yards 0.9144 meters
square yards 0.836 square meters
cubic yards 0.7646 cubic meters
miles 1.6093 kilometers
square miles 259.0 hectares
knots Lo BSDZ kilometers per hour
acres 0.4047 hectares
foot-pounds 1.3558 newton meters
millibars 10197 = 107 ® kilograms per square centimeter
ounces 28.35 grams
pounds 453.6 grams
0.4536 ki lograms
ton, long 1.0160 metric tons
ton, short 0.9072 metric tons
degrees (angle) 0.1745 radians
Fahrenheit degrees 5/9 Celsius degrees or Kelvins?

= ———

Ito obtain Celsius
use formula:

== =

Ge G/9) E +82)
To obtain Kelvin (K) readings, use formula:

3 (G/2) CF 252) = 27/5015.

E(u)

SYMBOLS AND DEFINITIONS

time

horizontal axis at S WL parallel to the (initial) beach profile

horizontal axis at S W L perpendicular to the (initial) beach
profile

beach depth (depth beyond which sediment transport is negligible)

wave angle with beach profile

wave angle with beach profile at infinity

longshore transport (littoral drift) discharge

constant = Ep
~ DD de
Qa
O
parameter u =
~(4Kt)®
ie 2
: 2 -u-
Fresnel integral = E(u) = = e du

length of groin
time for the beach profile to reach the end of the groin
transform time t, = 0.62t,

sinusoidal beach amplitude (at time t = 0)

parameter related to beach wavelength [|:, = (72) K

parametric value of x defining volume of beach dumping
parametric value of y defining volume of beach dumping
parameter used to define hypocycloid beach profile between

headlands

COSinriELEMcES WSOGl sin Tlie WaliceOrail Chemie scoramullal co
characterize the effect of wave angle

breaking wave height
water depth at inception of wave breaking

group velocity

Intictconeil GCheisee Comsteume 6.42 > Ome

distance of shoreline from a horizontal axis parallel to the
initial beach profile

distance of the offshore beach limit from a horizontal axis
parallel to the initial beach profile

equilibrium distance y, - yy

onshore-offshore transport per unit length of beach
onshore-offshore transport parameter (dimension iT)

longshore sand transport discharge in shallow water
longshore sand transport in deeper water

eer? 2

distance of the beach profile to a spiral center

angle parameter in mathematical description of hooked bays

spiral angle in mathematical description of hooked bays

depth of hooked bays

distance between headlands

MATHEMATICAL MODELING OF SHORELINE EVOLUTION

by
Bernard Le Mehaute and Mills Soldate

I. INTRODUCTION

This interim report presents a critical literature survey on the
subject of mathematical modeling of shoreline evolution. Hopefully,
this review will lead the way in establishing a flexible and practical
numerical method suitable to predict shoreline evolution, resulting
from the construction of navigation and shore protection structures in
the Great Lakes.

To focus attention on the most pertinent literature, the subject
under consideration is limited to long-term shoreline evolution as
defined below.

Three time scales of shoreline evolution can be distinguished:
(a) Geological evolution taking place over centuries;
(b) long-term evolution from year-to-year or decade; and

Associated with these time scales are distances or ranges of influ-
ence over which changes occur. The geological time scale deals, for
instance, with the entire area of the Great Lakes. The long-term
evolution deals with a more limited stretch of shoreline and range of
influence; e.g., between two headlands or between two harbor entrances.
The short-term evolution deals with the intricacies of the surf zone
circulation; e.g., summer profile-winter profile, bar, rhythmic beach
Paeeerisp meee.

For the problem under consideration, long-term evolution is of pri-
mary importance, the short-term evolution appearing as a superimposed per-
turbation on the general beach profile. Evolution of the coastline is
characterized by low monotone variations or trends on which are super-
imposed short bursts of rapid development associated with storms.

The primary cause of long-term evolution is water waves or wave-
generated currents. Three phenomena intervene in the action which
waves have on shoreline evolution:

(a) Erosion of beach material by short period seas versus
accretion by longer period swells;

(b) Pp etkect om) CMake) level changes toniverosion yan
(c) effect of breakwaters, groins, and other structures.

Even though mathematical modeling of shoreline evolution has in-
spired some research, it has received only limited attention from
practicing engineers. The present methodology is based mainly on

(a) the local experience of engineers who have a deep knowledge
of their sectors, understand littoral process, and have an inherent
intuition of what should happen; and

(b) movable-bed scale models that require extensive field data
for their calibration.

In the past, theorists have been dealing with idealized situations,
rarely encountered in engineering practice. It seems that mathematical
modelers have long been discouraged by the inherent complexity of the
phenomena encountered in coastal morphology. The lack of well-accepted
laws of sediment transport, offshore-onshore movement, and poor wave
climate statistics have made the task of calibrating mathematical
models very difficult.

Considering, on one hand, the importance of the subject of deter-
mining the effect of construction of long groins and navigation
structures and on the other, the progress which has been made in
determining wave climate and littoral drift, it now appears that a
mathematical approach could be useful.

The complexity of beach phenomena could, to a large extent, be
taken into account by means of numerical mathematical scheme, (instead
of in closed-form solutions), dividing space and time intervals into
small elements, in which the inherent complexity of the morphology
could be taken into account.

Furthermore, better knowledge of the wave climate, a necessary in-
put, will allow a better calibration of coastal constants such as
found in the littoral drift formula.

This study emphasizes the relative importance of various reports and
reviews the most important ones. Conclusions based on this review are
presented, pointing out the deficiencies of the state-of-the-art. (Sub-
sequent investigators should attempt to bridge the remaining gaps. )

The reports are presented individually, primarily in chronological
order. Two milestone developments from this survey are reports by
Pelnard-Considere (1956) and by Bakker (1968b). Others are extensions
and refinements, experimental verifications, support papers, numerical
procedures, and side issues, including the latest developments on
"hooked beaches" or crenulate-shaped bays.

Il. THE FIRST MODEL (PELNARD-CONSIDERE)

The idea of mathematically formulating shoreline evolution is attri-
buted by Bakker (1968a) to Bossen, but no reference to Bossen is given.
The first report which appears in the literature, on mathematical model-
ing of shoreline evolution, is by Pelnard-Considere (1956). His
theoretical developments were substantiated by laboratory experiments
made at Sogreah (Grenoble), France. The experimental results fit the
theoretical results very well. It is surprising that such relatively
simple theory has not been more frequently applied to prototype cases by
the profession (at least as it would appear from the open literature), a
fact which may be attributed to the lack of knowledge of wave climates.

Pelnard-Considere assumed that:

(a) The beach profile remains similar and determined by
the equilibrium profile. Therefore, all contour lines are
parallel. This assumption permits him to consider the problem to
be solved for one contour line only.

(b) The wave direction is constant and makes a small angle
with the shoreline (<20°).

(©) Mae lomgsinore tremsporct, ©@ , 28 linearlky wellaced co ene
EAMGSME Ore IS emeile Gre wmeiclencSs @ C(@ | se(@), s(@)) = ein w).

(dd) the beach has ‘a fixed Gili=derined)) depth, DD) (Fussy le
D is a factor relating erosion retreat to volume removed from
profile, which could be defined by the threshold velocity of
sand under wave action. A practical method of determination
Ot D IS wiyeEm wm Sectiom WIL,

Despite the crudeness of these approximations, the Pelnard-Considere
model can be considered as a milestone in demonstrating the feasibility
of mathematical modeling of long-term shoreline evolution. For this
GeAasSOlpeetiasEsmyudced use tol descrmbel in=someudetally hiss theoretical
development.

Consider an axis, ox , parallel to the main coastal direction and
an axis, Oy , perpendicular seawards (Fig. 2). The angle the deepwater
wave makes with the axis, ox , is a. The angle of the wave with the
shoreline a at any location is asSumed to be small; therefore,

S A = chy dy dy
a = a tan ae . (1)

xe (0) ox fe) OX

{le
ge
I
(o}
Le}
ie)
|
e
i}
I

(y = £(x,t) gives the form of the shoreline as function of time t).
The littoral drift Q is a function of angle incidence a and can be
put into a Taylor series:

Figure 1. Beach depth definition.

DBS SWAB MWw AAAS

Figure 2. Successive beach profiles updrift of a long groin before
bypassing (from LeMehaute and Brebner, 1961).

(@=@_ j} & os 5 (2)

oO
Ci = @
oO

in which Q denotes the transport, Q , when the angle of the wave

incidence is ao: Substituting equation (1) into equation (2) yields:
#3 3Q oy
Ue Qos E a= a OX : (3)
fo)
During the interval of time, dt , the shoreline recedes (or accretes) by
a quantity dy . Therefore, the volume of sand which is removed (or

deposited) over a length of beach, dx , is D dx dy . The quantity jus
equal to the difference of longshore transport during time, dt , between

x @imGl x 4 GS Moo,

ONderancd ands (GON a8 ax ) oe 3

io@e's
9Q
ane dt
Therefore,
dQ oy Il @Q)
D = — pe eS
dx dy ae Gbxalig =, © ve Dox - (4)
Substituting the expression for Q , a being small, and defining
, _ L dQ
<= 0 do ©)
a= a
(o)
yield:
2
Oo _ oy
cas TENE (6)
OX

which is the well-known diffusion or heat-flow equation.

K is approximately constant at a given site. Bakker (1968a) found

K equal to 0.4 x 10° cubic meters per meter depth per year, at an exposed
site along the coast of the Netherlands. Equation (6) demonstrates that the

rate of accretion or (erosion), 2, is linearly related to the curvature of

the coast, the derivative of the longshore transport rate with respect

to the angle of the wave incidence, a , and inversely propor-

tional to the beach depth, D .

The above equation will be recognized as the well-known diffusion
equation. A number of classical solutions of mathematical physics are
applicable to the diffusion equation when boundary conditions are
specified. Pelnard-Considere (1956) applied his theory to the case of
a littoral barrier or long groin. This case is reviewed below:

The longshore transport rate along a straight, long beach is sudden-
ly stopped by the construction of a long groin built perpendicular to
the beach (see Fig. 2). The boundary conditions are:

(a) y= oO or alll x Wine c= © Wine Cheracceirizes em
initial straight shoreline.

o which is

(b) At the groin, the longshore transport rate Q =
1.e€., when

realized when the waves approach the shore normally;

@
<S
Il

-tan Oo AL XS © 5

(©) O% =O aie a Mancye cisicemes UpCiniire (€ x) 5 ancl Qe OR

Q is the steady-state longshore transport along a straight beach
for the given wave conditions. The solution for the given boundary
conditions is:

tan e D
y = V4Kt exp (u) - x Vo E (u) ' (7)
V 1

where u = (aay > 927 i (Ww) 1S the Birssnel inteerrelll ,,

E Z te
(u) = = e du (8)
u
Values of E (u) or more frequently, @¢ (u) = 1 - E (u) , can be found

in tabulated form as given in Table 1:

Table 1. u versus ¢ (u).

~¢(@) @.112 0.225 0.528 0.428 0.520 ©.067 O.799 O-9iO O.995 i

Fig. 2 illustrates the shoreline evolution as defined by equation (8).
It 1s interesting that these curves are homothetic with respect to the
OUAIVIO HO wBEHER Sar

oA = oB = oC

ssieie — GieSo os
oA~ oB~ oC

The horizontal lengths grow with t , and in particular,

tan e
oy = ———— 2 Ke 2

Vie

A tangent to the shoreline at the groin intersects the initial shoreline
defined by y=o0 at a point a distance of 2 VkKt/t updrift from
the groin.

The ratio of the area of sand accumulation, such as is in OYX, 5 to
the area of sand contained in the triangular fillet, oyx , is 1.56 and

the distance OX, = 2.7 ox . This ratio permits rapid assessment of

the total amount of sand accumulated updrift from a single measurement
of the angle as > and determination of D as shown in Section IV.

The end of the groin of length, oy = 2 , is reached when

2
e =) = (10)
4K tan oa
1e)

When t ae , the boundary conditions must be modified since the groin
no longer traps all the sand but bypasses some of it.
If the same theory is applied to the beach downdrift of the groin

and if assumed that the wave diffraction effects are negligible, the
beach is eroded in a form symmetric with the updrift accretion.

When t = t,, the end of the groin is reached by the shoreline and
sand begins to be bypassed around the groin.

The boundary condition at the groin becomes oy = & (constant) for
anor The solution then becomes (Fig. 3):

WV. 1B x ° (11)
V4Kt

The curves representing the shoreline become homothetic with respect to
ie. AUS OY B 1.80%

The area between the shoreline and the ox axis (oy x) is given by:

5 18 5 Vikt

The area of triangular fillet, OY 6X
Hence,

OV 26°

= Sh eS Ee vee ee oe (12)
OY 9X 1 1
. Vikt

and

Ox = 2x

The shoreline as described by equation (7) at time t = ty is slightly
different from the shoreline defined by equation (11) at iene | AS

shown in Fig. 4.

The volume of sand defined by both curves is equal when the time ty of

in equation (11) in such a way

equation (7) is replaced by the time ty

that

2
ee

cot
[—) 5

LESS Les US) Onze : (13)

ct
(SH

Figure 3. Successive beach profiles updrift of a groin after sand
bypassing (from Le Mehaute and Brebner, 1961).

Eq.11(t’, = 0.62t, )

Eg: 7, (t=t,))

Figure 4. Matching transition between solutions 1 and 2.

Therefore, the shoreline evolves initially as represented by equation

(then whent its — t , the shoreline keeps evolving as given by

equation (11) as at the time were t = 0.38t, . Then’, the sediment dis-

charge, Q , bypassing the groin is equal to the incoming. discharge
Q minus the volume of sand which accumulates per unit of time.

KD2

Oe) =) Qin ; (4)
a [pKce-9- 584) | M2
i.e.,
Oey) = @ Ge : (15)
S tana [ rK(t-0.38t,) | ye
O il
or again
0.638
QE is NO Tt 12 (16)

[‘e/ep Z 0-38 |

In dimensionless terms, the following values are obtained for equation
(lo), (See Rie, 5) 2

DO Re Re eB
ion

S- S|) So SS ©
WG
oO
~S

Bs
j=)
Orv
OV
ial

It takes a long time before the value of Q approaches initial dis-
charge, Qo , downdrift of the groin.

"(1961 ‘Louqerg pue syNeYyey, oT wWorz)
OWT} FO UOTIOUNF e& Se UTOIS BuO, BsuLsseddq pues

v € : 2

NOISNAdSNS NI LYOdSNVUL
HLIM 3AYND WOlLoWud——“~ ee

JAYND 1WW9ILIYOSHL

*¢g oinstTy

y3.1.VMDIvaHE
|} Ad GaddOLS—>

GNVS T1V

00/0

The shoreline may be deduced at any time, t , by a homothetic trans-
formation about the oy axis from the knowledge of the shoreline at a
given time, ts > and also by applying the simple relationship (see Fig.3):

AD i AC
ie a 1/2
Z 0.38t, | [*2 f 0.58, |

The theory of Pelnard-Considere has been verified in laboratory ex-
periments with fairly good accuracy. The steady-state littoral drift,
Q, , was obtained experimentally from preliminary calibration over a
straight shoreline. The results of these experiments are shown in Figs.
6 and 7. However, the shoreline predicted by theory is not expected
to be valid downdrift of the groin because of the influence of wave
diffraction around the groin tip. Some sand begins to bypass the groin
by suspension before t =t (see Fig. 5). Also, different boundary
conditions apply to different contour lines since the deeper contour
lines reach the end of the groin before the contour lines which are near
the shoreline, which implies the one-dimensional theory is no longer
entirely satisfactory.

(17)

Subsequently, Lepetit (1972) also conducted laboratory experiments
which verify the results of a numerical scheme based on the theory of

Pelnard-Considere. He used the law, Q=Gsimk oP VEOS a) >.) Mepetaiit ls exe
periments were carried out with a very small angle between wave crest
and shoreline.

1. Refinement and Extensions of the Pelnard-Considere Model.

After Pelnard-Considere's contribution, the mathematical formula-
tion of shoreline evolution has proceeded at a slow pace. The first
refinements came in improving the longshore transport rate (littoral
drift) formula, in particular, modifying the expression relating sedi-
ment transport to incident wave angle.

Based on results from laboratory experiments performed by Sauvage
: : 3 er)
and Vincent (1954), Larras (1957) introduced the function f(a) = sin rie

also used by Le Mehaute and Brebner (1961). New theoretical forms of
shoreline evolution are determined as solutions of the diffusion equa-

: : : , ail, (ON ts
tion. Introduction of the relationship f(a) = sin = instead of tana,
allows obtention of solutions valid for larger wave angles.
Of particular interest are the cases of shoreline undulations, since
assuming linear superposition, any form of shoreline may be approximated

by a Fourier series. The solution of the diffusion equation is then of
the form:

“(QS61 ‘eLepTsUuoj-pxreUuteg WorZ) edLeyOstp BSutsseddq

pues [eoTJeLO09Yy} pue [ejUoUTIodxe usemzeq uostszedwoj) °*/ oansTy

“(9S61 ‘SLOpTsuoD-preuyted WoIZ) UOT NTOAS
QUTTOELOYS [TBOTELO9Y} pue [eJUoWTLedx9 useMIEeq UOSTIeduOD

A¥YO3HL
LNSWIYSdxXZ ——-— —

20/0

"9 oinsTy

31VOSs

(23)

shin eae y<
y =ube cos K éae x5) (18)

which indicates that shoreline undulations tend to decay exponentially
and disappear with time. B defines the beach undulation amplitude at
eune, tc = © , zinc YN WS sellencecl co whe welyellemeenl, bf Or dais winewila=
tion through the relationship:

2n \2 19)
Ls 2) K ; (
16

Shoreline evolution due to the sudden dumping of material at a given
point may be represented by:

rs 4K

Viel

Equation (20) gives the spreading of the sand along the shoreline since
the integration J ydx, which expresses the conservation of sedi-

MEME Win tne SySeOm,) 1S) 2 Comswehre' (SES Rie, 5 Wnts solution was also
mentioned by Pelnard-Considere.

It is interesting that much later, Noda (personal communication,
1974) investigated the same problem by taking an initial condition for
sand dumping.

Y = constant when |x| <x
Vy = sb. @)) =

9 when [x{ >x

as shown on Fig. 9. Using the functional relationship now commonly
accepted, f(a) = sin 2a , Noda found that the solution: to the diffusion
equation to be:

- erf a ° (21)

Figure 8. Spreading of sand along a shoreline due to instantaneous
dumping at a point.

Figure 9. Sand dumping along a finite stretch of beach
(initial condition).

Even though the initial condition is different from the previous one, the
solutions tend to be similar as time increases and are, therefore, both
applicable to the problem of shoreline sand dumping.

Also of interest is the solution, proposed by Larras (1957), of a
beach equilibrium shape between two headlands or groins described by the
equation:

where s is the distance along the shoreline. This indicates no sand
transport along shoreline configuration and, therefore, yields an
equilibrium to obtain:

ds = L cos ue da (where L is a proportionality constant),

which gives
4 5 4 (22)

YS ols [eos um aR gOS =|

Equation (22) defines a hypocycloidal form as might be found between two
headlands (see Fig. 10). R is a parameter which is related to the
relative curvature of the shoreline. When R-+o , a straight shoreline
solution is obtained.

Another family of solutions was given by Grijm (1960, 1964). In
these two publications, Grijm used the most commonly accepted expression
for dependence of longshore transport on angle, f(a) = sin 2a , and
applied the theory to cases where the angle of incidence, a , is not
necessarily small. Subsequently, he established the kind of shoreline
which can exist mathematically under steady-state conditions.

Even though the theoretical approach obeys the same physical assump-
tion as the previous theory (except for the allowable range for the
angle of incidence), his mathematical formulation is not as simple. The
shoreline is defined with respect to a polar coordinate axis. The con-
tinuity equation is solved in parametric form, which is integrated
either by computer or by graphical methods. Details of Grijm's compu-
tations are not available.

WAVE

acer

HYPOCYCLOID

HEADLAND 34° HEADLAND

Figure 10. Equilibrium profile between two headlands.

The main interest of the report lies in the results. When the long-
shore transport rate reaches a maximum value (a = 45°) , the shoreline
tends! £0. commun cusped: ec jedmCapema Si SNOW mlm kslsern lulls

Also of interest is Grijm's (1964) mathematical formulation for
different forms of river deltas for which he finds two possible solu-
tions, one with an angle of wave incidence everywhere less than 45°, and
another with the angle of incidence greater than 45°. The shoreline
curvature also depends upon the angle a as shown on Fig. 11. The
problem remains indefinite since it is unknown which solution is valid.

The formulation of Grijm does not lend conveniently to numerical
adaptation.

Bakkere and Edelman (1964) also studied the form of river deltas, but

instead of,using f(0) = sin 20 , as Grijm, they used the linear approxi.
mation as given by Pelnard-Considere; i.e., f(a) = k, tana for
o <tana < 1.23 . They also investigated the case of large angle of
approach using the function:
9
f(a) = PERG One | 32S < tame < ©

Bakker and Edelman's (1964) solutions are similar to that of Grijm;
however, they also found a periodic solution as Larras (1957) did:

1
y = exp ae Qa = on D COS JEx . (23)

Equation (23) represents a sinusoidal shoreline for which the ampli-

: : : : dQ ne :
tude of the undulations decreases with time if S is OSMwiyVe (Ase. .
OMe , d é :
for small angles of wave incidence), but increases when 2 1s negative

(j.e., for large angle of wave incidence). The shoreline 1s thus un-
stable and the amplitude of the undulations increases. It can be
deduced that Grijm's solution for large angles of incidence is not
naturally found, since they are unstable and will be destroyed as small
perturbations trigger large deviations.

Bakker (1968a) implies that Grijm did not discover this instability
because he confined himself to solutions growing linearily with t in
all directions, while the exponential solution in t also exists.

Komar (1973) also applies a numerical scheme based on the Pelnard-
Considere approximation to the problem of delta growth. He found shore-
line shapes identical to Grijm in the case of a small angle of approach.

From these investigations, it is remembered that the Pelnard-
Considere approach is very powerful to predict shoreline evolution under
small angle of incidence. But under large angle of incidence, instabili-
ty of the shoreline makes it very difficult. Furthermore, the

SMALL ANGLE

VO
a LZ ZZ

(unstable)

Figure 11. Two theoretical forms of shoreline equilibrium of
river deltas.

phenomenology of interaction between wave and shoreline is not accurately
defined mathematically.

2. Example of Shoreline Evolution.

Because of its importance, an example application of the theory of
shoreline evolution is presented. However, the example is slightly
modified to account for the generally accepted longshore transport rate
formula:

k 2 5
Q= 76 Hy Ce sin 20h (24)

Q = longshore transport rate cubic feet per second
= wave breaking angle

= breaking wave height

= wave group velocity at breaking

= 4 comstame 2 ©,42 x 108

og = specific weight of seawater.

For the case of a groin perpendicular to shore, consider the average
beach conditions:

Hh = 5 feet
qd. =~ 6.4 feet

Ce = tot
O

De 20 xe8c

Oe = V gd, = 14.4 feet per second
Thus,
2 JK 2
a 2
K D 16 °& Ht Ce cos 2a,

2 2

VG te Ga. So Giddy

= = OF 92
8 x 20

Substituting into equation (10), yields:

ain

Spin eae epomeaas Mer ibd ak Or. days
4K tan a
fe)
In tabular form for various groin lengths,
eetate 50 100 200 500
ty days 3 NS 52 325
Check:
For 2 = 50 feet Area Oxy = 1.56 x Area oxy
2 2
oh gL _ O78 2 = 22,400 square feet
= 1.56 ——— = —
2 tana tana
fe) e)
Volume = (Area oxy) (dD) = 4.5 x 10° cubic feet
KD ;
Q = 7 tan 20 Sa lnOmcubalc ree taperEsecond
AS XK 10° 5
ty = ia We A AT 258 x 10 seconds = 3 cays.

III. THE TWO-LINE THEORY OF BAKKER

One limitation of the solutions of Pelnard-Considere is the assump-
tion of parallel depth contours. Bakker (1968a) realized that the one-
line theory of Pelnard-Considere and its subsequent development may, at
times, lead to some inaccuracy, since beach slope variations along the
shore were not considered. Beach slope variations with respect to time
(summer-winter profiles) are not important in the long-term shoreline
evolution. Nevertheless, if an adequate onshore-offshore profile
response model was available, a suitable mathematical representation of
It could be developed (Dean, 11973; Swart, 1974).

Near coastal structures, the deviations of the model from prototype
conditions can be considerable. Pelnard-Considere finds that the accre-
tion and erosion patterns are symmetrical with respect to the groin as
shown on Fig. 12. However, in reality, the updrift profile becomes
steeper than the equilibrium profile and the sand moves seaward. The
downdrift profile is flatter than the equilibrium profile and the sand

Coastline with Parallel Contour lines

<p FS Arrows show direction

of movement

a ee

More Reasonable Approximation

Figure 12. Differences on shoreline configuration due to
onshore-offshore transport near a groin (from
Bakker, 1968b).

is pushed shoreward by the waves. To reproduce the onshore-offshore
movement in a mathematical model, it is necessary to schematize the
coast by two or more contour lines instead of one.

Bakker's (1968b) two-contour-line model is not easily applied to

- practical engineering problems encountered by designers, due to lack of
knowledge about onshore-offshore transport. However, his contribution
toward establishing a realistic mathematical model of shoreline evolu-
tion is of sufficient importance to deserve detailed review.

Bakker (1968b) assumes that the profile is divided into two parts
(FIG, 1S). Une wpe perres extencing CO a ceo, Dy 7 ane atteicred by,

the groin, the part below D, extends offshore to a depth of D, + D,

1 1
which is the assumed practical seaward limit of material movement.
The “equilibrium distance", w , is defined by a distance (Y5 - y,)

corresponding to an equilibrium profile under normal conditions; i.e.
far away from the groins.

The onshore-offshore transport is defined by:

OA OR yest MON C3 (25)

where a, is a proportionality constant (dimension We). When

Y, - Y5 + w) iS positive, the transport is seaward; when negative, it

is shoreward. q. has been found by Bakker for a part of the Dutch
coast equal to 1 to 10 meters per year for a depth Dy = 3 meters.

Leen 7 = 4 = 5 Ene, q = Gy, Oa = TD.

Now, following Pelnard-Considere; i.e., developing the expression
for the longshore transport rate Q ina Taylor series in terms of a

O21) Ae See. (26)

which gives in linear approximations:

S dQ dy
Gs Ge E =| ; eh

Figure 13. Notation for the two-line theory.

Defining

d
aa [32 | ] (28)
Co = @& ’
)
then,
ea)
Vis Qs gre ae

This equation is now applied to both lines, y, Co and y,():

oy :
2 etl (29)
US Ron Shy tee

oy
» a ; (30)
= Wan = Sp aE

The equation of continuity,

CO ye aN , ew)

is modified by the term a due to onshore-offshore transport so that

3Q, yy
See Corer Wi oe (2)
3Q oy
D sf 2 “
pabaacrs Re =) Bre ()
Substituting equations (1), (2), and (3) for Q> Q,5 oF gives:
2
ay dy
1 B 1 34
SO is, Os 3) il Be oP

2
: fn Go ie yh 2D 22 ‘ (35)
CD ORD ge een ay Om le CE
ox

Adding both equations yields:

2 DD Z
MCLE eee Raley eghnes n(n (nena ae
2
DB p? ah ae ax° of
in which
Gla ar Gl 1
UPR YE Ae wes 37
eee ne eee mp ee, Pale (S71)
leh ha,
2 49
For simplicity, Bakker (1968b) assumes De ee ee which implies that
1 2
derivatives of the littoral drift transport with respect to
@8 are proportional to depth D . Then, dividing
a = a
O

equation (6) by Dy and D, respectively, and subtracting, yield :

oO.) q_D JOY.)
ES Dep OM a) aaeen

(38)
aR eee

jew)

jes)
<

where y, = ee wo Oe which is the equation for the offshore-onshore
transport Ys . It is interesting that the offshore-onshore transport

is independent of the longshore transport.

Using the auxiliary variable,

FS ye exg0 ao , (39)

the diffusion equation is still obtained:

ony 8Y 6 (40)

2 dt

Bakker has applied his theory to a number of idealized cases, in-
cluding the behavior of a sand beach near a groin, assuming

DBD. = D
2
Tenant (41)
The boundary conditions are:
gle Ihaicngnll Comebiietom (ic = ©))e Yau Dore Or O<<o emGl ct = ©

be athens whens to:

(1) Yada Oop’ for xem and o<t< (which implies an equilibri-
um profile)

(e) a ie Or * = ©
(3) Os = tana = IOP xX S ©

The results are expressed in terms of lengthy power series, and are
represented graphically in Fig. 14.

The case of equilibrium beach profiles between groins was also
investigated by Bakker (1970).

Despite the complex refinement of the two-line theory, as initially
developed by Bakker, a number of phenomena that have significant in-
fluence on the beach profile are still neglected. Among these are:

a. The influence of rip current near the groins is twofold: rip

currents transport material from beach to the offshore and cause wave
refraction.

LONGSHORE TRANSPORT

t= 0.04 T N 2:0
P \groin
\
N 2.0 4.0 6.0
6.0 4.0 2.0 \ : %

°
WILLS 7d

N 2.0
t= Up N :
\groin
N 2.0 4.0 6.0
6.0 4.0 2.0 Ny

STSS7¢

2.0

ad
i}
8

f
(he)
KR
[e)
He
=}

N 2.0 4.0 6.0
6.0 4.0 2.0 N a aaa

i)
é
(=)

Figure 14. Evolution of shoreline and offshore
beach limit near a groin
(from Bakker, 1968b).

b. The influence of diffraction on the leeward side of groins which
has an effect in the immediate vicinity of the shoreline.

Ge The effect of changing wave direction caused by refraction changes
the magnitude of longshore transport rate and the boundary conditions.

Ge The nonlinearity in the transport equation is of minor importance
for small angles of incidence (for a> 45° , the cvastline becomes
unstable as previously mentioned).

The two-line theory has been verified experimentally (Hulsbergen,
Van Bochove, and Bakker, 1976), and shows a trend identical to the ex-
perimental results. There are some differences at a small scale due to
secondary currents, breaking wave type, changes of wave height due to
small changes in morphology, etc. These, however,*%are short-term
rather than long-term evolution phenomena.

IV. THE EFFECT OF WAVE DIFFRACTION

The effect of wave diffraction was subsequently taken into account
by Bakker (1970). Initially, this was done for the one-line theory of
Pelnard-Considere and later for Bakker's two-line theory.

Pelnard-Considere's equations,

Oy a“
Q=9W-ae 2 , aw =—% (42)
a = a
fo)
and
oy . _ Geo) 29) (43)
at D dX
Still apply. OR and oq) vare now, tuncerons som soy Sumce) both the

incident wave height and angle of approach vary along the shore with
x , because of wave diffraction.

Inserting the expression for Q in the continuity equation, yields:

tO (44)

Shs)

It is assumed that the longshore transport rate, Qo , 1S proportional
to the angle of wave incidence, (a, - ax » and the square of the rela-
tive wave height. The variation of wave height with x is given by the
diffraction theory of Putnam and Arthur (1948). The modification of

wave diffraction by wave refraction is neglected.

A similar approach has been proposed by Price, Tomlinson, and Willis
(1972), who assume that Q = Wee. E sin 20 , where E is the trans—-
S
Mikecedvenernay, whach ass alsionalstunctelon Ole xq asmas) » oun (anid ie 1S cle

submerged density of the beach material). Price, Tomlinson, and Willis
then obtain the one-line theory equation:

ORS : dE da Ca
Y sin 2a ae 2E cos 2a ae ap ID) aa ae (45)

which is solved numerically with

MOVE (46)

Laboratory experiments were performed with crushed coal by Price,
Tomlinson, and, Wallis ((1972))> * theytheory, eivange, thevette cel ots waive
diffraction was verified by the experiments at the beginning of the
test. After a 3-hour test which may correspond to a prototype storm
duration, it is stated that the wave refraction pattern invalidates the
input wave data and a complex boundary condition developed at the up-
drift end of the wave basin.

Bakker's (1970) consideration of wave diffraction has been included
in his two-line theory where,

oy)

i Tacorle dani rosa Gn
oy 5

Bo = Boo Glo. ie )

Neither the deepwater line, defined by y.(x,t), nor qd, and Qo2 5 us

affected by diffraction.) Filg. 15 presients typical results lobjtained trom
this theory for the case of beach evolution near a groin and between two
groins.

Wave incidence

Offshore Contours

16At

16At
sAt
Shoreline Contours

Accretion and erosion near a groin, numerical solution
with diffraction (two-line theory)
Wave incidence

Offshore Contours

BAt
16At

Shoreline Contours

16At

4At
8At 8At
16At 4At

Behavior of beach and inshore between two groins
(two-line theory)

Figure 15. Effect of wave diffraction
(from Bakker, 1970).

V. SPIRAL BEACHES

Hooklike beaches (Fig. 16) are common along exposed coasts and are
formed by the long-term combined effects of refraction and diffraction
around headlands. Yasso (1965) discovered that the planimetric shape of
many of these beaches could be fitted very closely by a segment of log-
arithmic spiral; the distance, r , from the beach to the center of the
Spiral increasing with the angle 0 according to

Eiryts Exp E cot 6 | (49)

in which gB is the spiral angle.

Bremmer (1970) has shown the logarithmic spiral to give an excellent
fit for the profile of a recessed beach between two headlands.

The evolution of spiral beaches belongs to the geographical time-
scale domain (Sylvester and Ho, 1972). However, similar evolution has
also been observed over smaller time scales in consonance with the
definition of long-term shoreline evolution adopted in this study.

So far, only empirical rather than theoretical mathematical repre-
sentations of spiral beaches are available. The empirical approach has
been fruitful in providing the spiral coefficients 8 as function of
wave angle, a , with the headland alinement (Fig. 16) (Sylvester and
Ho, 1972). The "indentation ratio" (depth of the bay to width of open-
ing) also depends upon a and, in most cases, varies between 0.3 and
0.5 (Fig. 17).

There have been many attempts to explain this peculiar beach forma-
tion (Leblond, 1972; Rea and Komar, 1975). Leblond assumed that the
rate of sediment transport is proportional to the longshore currents as
given by the theory of Longuet-Higgins (1975). He also assumed that the
beach profile is not modified by erosion or accretion so that the con-
tinuity equation from the one-line theory can be used in a two-dimen-
sional coordinate system.

Thus, the variation in longshore current iE ens tty with wave angle
Will yalelid he alte) or erosionwor accretion:

Difficulties arise in expressing this variation of longshore current
in areas subjected to wave diffraction. Leblond (1972) points out that
classical wave diffraction theories are too complicated to be used in
his theoretical scheme. Another difficulty arises from the fact that
the barrier (headland) is not thin as it is assumed in the theory of
diffraction of Putnam and Arthur (1948). To account for this effect,
Leblond introduces an empirical correction coefficient to the theory
of Putnam and Arthur over a two-dimensional network. The results of

HEADLAND

Figure 16. Hooked beaches.

Figure 17.

CRENULATE- SHAPED BAYS

LEGEND
e@ Vichetpan Experiment
O Ho Experiment
X prototype bays
® typical bay
A Bedock Singapore

Indentation ratio for a range of wave
obliquity (from Sylvester and Ho, 1972).

such a complex scheme, which is plagued with numerical instabilities,
are shown in Fig. 18. Even though the results show how oblique waves
initiate an erosion pattern that might eventually lead to the formation
of hooklike beaches, they do not show that the beaches represent a good
fit to segments of a logarithmic spiral.

Rea and Komar (1975) developed an approach to overcome the numeri-
cal instability encountered by Leblond. They combined two orthogonal,
one-dimensional arrays as shown on Fig. 19. In this way, deformation of
the beach can proceed in two directions without the necessity of a two-
dimensional array. The wave configuration in the shadow zone was
described by various simple empirical functions which resulted in beach
configurations fairly approximated by a logarithmic, spiral.

5 -*

The main interest in the work of Rea and Komar (1975) is that they show
the lack of sensitivity of the shoreline evolution in the shadow zone to
the actual pattern of incident waves used. Also, the sensitivity of the
beach shape to the energy distribution seems to be small.

VI. PROTOTYPE APPLICATIONS

The application of mathematical models of shoreline evolution to pro-
totype conditions is not very well documented in the literature. It is
certain that, at least in its simplified form such as given by Pelnard-
Considere, the method has been used by practicing engineers and designers.
It has been reported in unpublished reports but very little has appeared
in the open literature.

Weggel (1976) has formulated a numerical approach to coastal process-
es which is particularly adapted to prototype situations. In particular,
it includes:

a. A method for determining the water depth beyond which the onshore-
offshore sediment transport is negligible. This information is particu-
larly useful in determining the quantity D used in Pelnard-Considere's
theory and others. It is also useful in determining the effect of a
change of sea level. Beach profile data are plotted on semilog paper

and the base elevation of the most seaward point varied until an approxi-
Mate straight line is obtained (see Fig. 20). He found D = 70 feet at
Pt. Mugu, California.

b. The effect of a change in sea level, a situation pertinent to the
Great Lakes, is also taken into account in a way proposed by Bruun
(1962). Using the principle of similarity of shoreline profile, the
shoreline recession Ay is related to the change of water level a _ by
the relationship (Fig. 21):

ab
Oe Guat Ge

f WAVE DIRECTION

Figure 18. Orthogonal arrays for numerical scheme of hooked bay
(from Leblond, 1972).

HEADLAND

Figure 19. Orthogonal arrays for numerical scheme of hooked bay
(from Rea and Komar, 1975).

ELEVATION ABOVE SEAWARDMOST POINT (ft)

100

PROFILES AT PT. MUGU,
CALIFORNIA (STATION 1467 + 60)

3rd approx.

2nd approx.

1st approx. Ne

0 1000 2000 3000 4000 5000 6000
DISTANCE OFFSHORE (feet)

Figure 20. Semilogarithmic profiles (from Weggel, 1976).

“TOAST 103eM
uvoWl UT O8UeYD pUe }e91}9I SUTTOIOYS Usemjeq dtysuoT\epoy

“TZ 9an3ty

ec. A numerical scheme in which the effect of wave diffraction could be
included.

d. A statistical characterization of wave climate and longshore energy
iF LU Re

Examples of recent prototype analysis and prediction of shoreline
evolution by mathematical modeling are Apalachicola Bay by Miller (1975)
and the Oregon coastline by Komar, Lizarraga-Arciniega, and Terich (1976).
Both studies are based on numerical schemes related to the Pelnard-
Considere (one-line) formulation.

VII. CONCLUSIONS

There are two methods of approach to the problems related to littoral
processes. The first one, typified by the previously discussed reports,
consists of analyzing global effects. The method essentially based on
establishing ''coastal constants" for a model by correlation between
long-term evolution and wave statistics and subsequently, to use the
model for predicting future effects. It appears that this method is the
most promising for engineering purposes and could be termed the macro-
scopic view. The main results are summarized in Table 2.

The second approach, the microscopic view of the problem, consists
in analyzing sediment transport, step-by-step, on a rational Newtonian
approach, starting with wave motion, threshold velocity for sand trans-
port, equilibrium profiles of beaches, etc., until the individual com-
ponents can be combined into an overall model to predict shoreline
evolution. The second method or scientific approach has not progressed
to the point where it can be applied to engineering problems in the
foreseeable future.

However, much progress has been made in the last 5 years toward
understanding the hydrodynamics of the surf zone through application of
the "radiation-stress'" concept. In theory, establishing a reliable
mathematical model of surf zone circulation should permit a determina-
tion of the resulting sediment transport. Practically, however, inter-
action between a movable bed and the surf zone circulation, and the
inherent instability of longshore currents limit this approach to the
realm of research. Among the problems that make this approach difficult
are the refraction and diffraction of water waves, uncertainty in pre-
dicting rip current spacing, and the effect of free turbulence generated
by breaking waves on the rate of sediment suspension.

Finally, the complexity of mathematical formulation, based on the
radiation-stress concept, makes it difficult to use as a predictive tool
when dealing with forcing functions expressed by statistical multi-
directional sea spectra. This method is promising in explaining local
effects (e.g., near groins), rhythmic topography, beach cusps, and short-
term evolution due to unidirectional sea states. All these effects are

Table 2.

Sediment transport

Experimental

Summary of mathematical models for shoreline evolution.

Application to ideal

Date Author alongshore Validity onshore-offshore Theoretical developments, verification cases
1956 Pelnard- Griencean é ay
Considere i) ax Very small angle No Diffusion equation Laboratory Groins
closed-form solution with pumice
sin 2 asa. - 2% °
1957 Larras 4 ° ax Small angle (<2S ) No Diffusion equation, No Groins-sudden dump
closed-form solution sinusoidol undulation,
equilibrium shape
between groins

1960 Grijm sin 2a Sin all angles. In No Nonlinear differential No Forms of deltas

case of large angle equation

an inconsistency in

the assumption

ay small

1961 Le Mehaute, Ja
Brebner Sera Diffusion equation No Groins-sudden dump
closed-form solution sinusoidol undulation,

equilibrium shape
between groins

1964 Grijm sin 2a implied Small and large No Cylindrical system of No Forms of deltas

angle coordinates-numerical
or graphical method
1964 Bakker, kK, tan Small angle No Nonlinear differential No Forms of deltas
Edelman ko tan ale octana<l.23 equations,
2 large angle closed-form solutions
ae kK. = 1.23 1.23<tana =
K 2
1
1968 Bakker tan a Very small angle Yes System of linear differ- No Groins and combina-
(two-line theory) ential equations tions of groins
1) power series solution 3} simple groins
2) closed-form solution 2 stationary shore-
3) closed-form lines
3) sand-wave pro-
pagation
1970 Bakker, sin 2a Small angle Numerical method No Groins
Breteler,
Roos,
1972 Price, sin 2a Small angle No Numerical method based Laboratory Groin
Tomlinson, “2 on Pelnard-Considere with crushed
Willis a =a _-tan wey (1956) coal
° ax
1972 Lepetit sina cosa Small angle No Numerical method based Laboratory Groin (updrift
on Pelnard-Considere with Bakelite and downdrift)
(1956)
1972 Sylvester, Empirical fit Yes Crenulated-shaped
Ho bay or spiral
beaches
1972 Leblond Proportional to Small angle No Numerical method Spiral beaches
longshore current
(radiatiom stress)

1973 Komar sin 2a Small angle No Numerical method No Growth of deltas,
reorientation of
beaches between
two headlands

1975 Komar sin 2a Any angle Numerical method based Hooked beaches

on empirical model re- (spiral beaches)
fraction-diffraction

1976 tulsbergen, sin 2a Small angle Yes Application of the two- Laboratory Groin

Van Bochove, line theory of Bakker with dune sand
Bakker
1976 Weggel sin 2a Small angle Yes Mathematical and No Groin

numerical formulation

Table 2. Summary of mathematical models for shoreline evolution.--continued

Variation of beach Modification Variable
slope taken into Effect of by wave Variation of Effect of Application to
account Diffraction refraction direction sea level rip currents rototype cases
No No No No No No No
{one-line theory)
No No No No No No No
No No No No No No No
No No No No No No No
No No No No No No No
No No No No No No No
Implicitly (through No No No No No No
the two-line theory)
Implicitly Yes No Yes No No No
(periodic wave)
(constant depth)
No Yes, but not com- No No No No No
pletely formulated,
not applied
No No No No No No No
Combined effect, Yes No No Fit with
refraction- prototype
diffraction cases
No Yes, but umsuccess- Yes No No No No
fully (numerical
instability)
No No No No No No No
Combined refrac-
tion and diffrac-
tion
Implicitly No No. No No No No
: Yes Yes No Yes Yes No No
(in principle) (in principle) (statistically)

Main Conclusions

First significant milestone of intro-
duction of mathematica] modelivg to
the study of shoreline evolution

Extension of the method of Pelnard-
Considere to other idealized cases

Two forms of solution:
One with concave shoreline (small angle)
One with convex shoreline (large angle)

Same as Larras (1957)

Two forms of solutions as in 1960
applied to a number of deltas,
idealized cases

Instability of shoreline under large
angle (even a straight shoreline)

The most Significant contribution since
1956 demonstrating the influence of
beach slope

Influence of diffraction changing wave
condition leading to stable shoreline
near groin

Experimental verification for more
diffracting waves, i-e., updrift,
fairly satisfactory

Good experimenta] verification

Combined effect of refraction & dif-
fraction in protected areas; affected
also to geological time evolution

Qualitative fit only, unsuccessful; two
large distances between data point.
Complexity of combined refraction-
diffraction effect.

Numerical application of Pelnard-
Considere

Empirical development

Fairly good experimental verification
except near groin

Aimed at investigating real cases
on Great Lakes

superimposed on the long-term evolution for which an analysis can be
done independently.

Among the significant recent reports leading toward understanding of
surf zone circulation and related bottom topography are: Bowen and
Inman (1969) who advocate the presence of edge waves as a cause of rip
currents and beach cusps; Hino (1974) who states that rip currents are
the result of mobility of the sand bed and hydrodynamic instability;
Sonu (1972) and Noda (1972) demonstrated that a perturbation on bottom
topography causing waves to refract and have varying intensity along the
shore induces a variation in radiation stress which ,in turn enhances rip
currents; finally, Liu and Mei (1976) applied the radiation-stress
concept to a groin perpendicular to shore and to an offshore breakwater.

These investigations offer at least partial answers to a number of
important problems, important in understanding shoreline processes. It
definitely indicates that the radiation-stress approach holds the poten-
tial key to understanding many types of nearshore currents, heretofore
unexplored. It is also evident that the study of surf zone hydrodynamics
will rapidly reach a plateau if sand-water interaction problems are not
mastered, and at this stage, these can only be considered empirically.
Determinism leaves off with the inception of turbulence.

Even though the dynamics of nearshore currents hold the key to
understanding of beach processes, application of the methodology based
on radiation stress to investigate shoreline evolution mathematically is
still beyond the state-of-the-art.

Both approaches could be pursued in parallel and the results of the
scientific approach could slowly be incorporated into a practical
engineering model.

Conclusions based on the literature survey, as summarized in
Table 2, are:

a. There is sufficient laboratory verification to give credibility to
a mathematical approach to the study of shoreline evolution for small
angles of wave approach.

b. For large angles of incidence, there is a lesser chance at arriving
at a successful formulation since shorelines are then unstable and the
resulting shoreline evolution could not be predicted without the
initiation of more basic research beyond the present state of knowledge.

c. Even though no field measurements subsequent to mathematical pre-
dictions have been found in the literature, many practicing engineers
have applied the theory of Pelnard-Considere (1956) to predict shore
evolution by taking into account variable wave climate. The method is
easy to apply and provides valuable information.

d. Engineering applications to prototype cases based on more sophisti-
cated approaches such as given by the two-line theory of Bakker (1968b)
are not known. These more sophisticated approaches can be currently
considered as belonging to the realm of research rather than of engineer-
ing practice.

e.~ Local effects, diffraction, rip currents, wave refraction and inter-
action between these effects are, at present, still not so conveniently
formulated to be used by practicing engineers. Introduction of these
effects, if and when important in the mathematical formulation, is
feasible but will require further investigation.

f. A simple numerical scheme that can be used by design engineers and
planners and which includes theoretical or empirically all important
effects could be developed. Effects that should be included in the
mathematical model are wave diffraction, loss of sand by rip currents
along groins, sea (lake) level variation, and beach slope variation
near groins.

g. The introduction of the concept of radiation stress in the mathe-
matical formulation is not recommended at this time, but research
related to this approach should be pursued in view of the eventual
input that subsequent results could have on then existing operational
mathematical models.

ae)

LITERATURE CITED

BAKKER, W.T., and EDELMAN, T., "The Coastline of River-Deltas," Coastal
Engineering, 1964, pp. 199-218.

BAKKER, W.T., "The Coastal Dynamics of Sand Waves and The Experience of
Breakwaters and Groynes," R'JKSWATERSTAAT, 1968a.

BAKKER, W.T., ''The Dynamics of a Coast with a Groyne System," 11th
Conference on Coastal Engineering, 1968b, pp. 492-517.

BAKKER, W.T., ''The Influence of Diffraction Near a Harbor Made on the
Coastal Shape," Study Report R'JKSWATERSTAAT, WWK 70-2, 1970.

BAKKER, W.T., KLEIN BRETELER, E.H.J., and ROOS, A., "The Dynamics of a
Coast with a Groyne System," 12th Conference on Coastal Engineering,
Washington, D.C., 1970, pp. 1001-1020.

BOWEN, A.J., and INMAN, D.L., "Rip Currents," Journal of Geophygicas
Research, Vol. 74, No. 23, ‘Oct. 1969, pp. 5479-5490.

BREMMER, J.M., ''The Geology of Wreck Bay,'’ M.S! Thesis, University of
British Columbia, Vancouver, 1970.

BRUUN, P., "Sea Level Rise as a Cause of Shore Erosion," ASCE Waterways
and Harbors Diviston, No. 88, 117, 1962.

DEAN, R.G., "Heuristic Models of Sand Transport in the Surf Zone,"'
Proceedings of Conference on Engineering Dynamics tn the Surf Zone,
May 1973, pp. 208-215.

GRIJM, W., "Theoretical Forms of Shorelines," 7th Conference on Coastal
Engtneertng, 1960, pp. 197-202.

GRIJM, W., "Theoretical Forms of Shorelines," 9th Conference on Coastal
Engineertng, 1964, pp. 219-235.

HINC, M., "Theory on Formation of Rip Currents and Cuspidal Coast,"
14th Conference on Coastal Engtneering, June 1974, pp. 901-919.

HULSBERGEN, C., VAN BOCHOVE, H.G., and BAKKER, W.T., "Experimental
Verification of Groyne Theory," Conference on Coastal Engineering,
Hawaii, 1976..

KOMAR, P.D., "Computer Models of Delta Growth Due to Sediment Input From
Waves and Longshore Transport," Geological Society of American Bulle-
tin, Vol. 84, July 1973, pp. 2217-2226.

KOMAR, P.D., LIZARRAGA-ARCINIEGA, J.R., and TERICH, T.A., "Oregon Coast

Shoreline Changes Due to Jetties,"' Journal of Waterways, Harbors, and
Coastal Engineering Diviston, Feb. 1976, pp. 13-30.

LARRAS, J. "Plage et cotes de sables," Collection du laboratotire d'
Hydraulique, Eyrolles, Paris, 1957.

LEBLOND, P.H., "On the Formation of Spiral Beaches,'' 13th Conference on
Coastal Engineering, 1972, pp. 1331-1345.

LE MEHAUTE, B., and BREBNER, A., "An Introduction to Coastal Morphology
and Littoral Processes," Civil Engineering, No. 14, Queen's University,
Canada, Jan. 1961.

LEPETIT, J.P., "Transport Littoral: Essais et Calculs," Conference on
Coastal Engtneering, 1972, pp. 971-984.

LIU, P., and MEI, C.C., "Effects of a Breakwater on Nearshore Currents
Due to Breaking Waves,'' TM-57, U.S. Army, Corps of Engineers, Coastal
Engineering Research Center, Fort Belvoir, Va., Nov. 1976.

LONGUET-HIGGINS, M.S., "Longshore Currents Generated by Obliquely
Incident Sea Waves,'' Journal of Geophystcal Research, 1975, pp. 6778-
6789 and 6790-6801.

MILLER, C.D., "The Numerical Prediction of Shoreline Changes Due to Wave
Induced Longshore Sediment Transport,'' Geophysical Fluid Dynamics
Institute, Florida State University, Tallahassee, Fla., 1975.

NODA, E.K., "Rip Currents," Proceedings of the 15th Conference on Coastal
Engineering, 1972, pp. 653-668.

PELNARD-CONSIDERE, R., "Essai de Theorie de 1'Evolution des Formes de
Rivage en Plages de Sable et de Galets," 4th Journees de l'Hydraultque,
Les Energtes de la Mer, Question III, Rapport No. 1, 1956.

PRICE, W.A., TOMLINSON, K.W., and WILLIS, O.H., "Predicting Changes in
the Plan Shapé of Beaches," 13th Conference on Coastal Engineering,
IQ72, (Do USP U2o)-

PUTNAM, J.R, and ARTHUR, R.S., "Diffraction of Water Waves by Breakwaters,"'
Transactions of Amertcan Geophystcal Union, Vol. 29, Aug. 1948.

REA, C.C., and KOMAR, P.D., "Computer Simulation Models of a Hooked
Beach Shoreline Configuration," Journal of Sedimentary Petrology,
Vol. 45, No. 4, Dec. 1975, pp. 866-872.

SAUVAGE, de ST. M., and VINCENT, J., "Transport Littoral," ''Formation
des Fliches et Tombolos," 5th Conference on Coastal Engineering,
Grenoble, France, 1954.

SONU, C., ''Field Observations of Nearshore Circulation and Meandering

Currents,'"' Journal of Geophystcal Research, Vol. 77, No. 13, July
1972, pp. 3232-3247.

SWART, D.H., "A Schematization of Onshore-Offshore Transport," 14th
Conference on Coastal Engineering, 1974, pp. 884-900.

SYLVESTER, R., and HO, S.K., "Use of Cremulated Shaped Bays to Stabilize
Coasts,'' 13th Conference on Coastal Engineering, 1972, pp. 1347-1365.

WEGGEL, J.R., ''On Numerically Modeling Coastal Processes," U.S. Army,
Corps of Engineers, Coastal Engineering Research Center, Fort Belvoir,
Va., unpublished, 1976.

YASSO, W.E., "Plan Geometry of Headland Bay Beaches," Journal of Geophy-
Seah imaseciael,, Vols 155 Red. UNS; as MOPS INS.

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