LONGSHORE CURRENT AND SEDIMENT TRANSPORT
By
EDWARD BENNETT THORNTON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1970
This work is dedicated to my wife, Sandra, to whom I owe the most.
ACKNOWLEDGE IENTS
The author wishes to thank Drs. B. A. Christensen and
R. G. Dean who provided invaluable guidance and inspiration
during the course of this investigation. Dr. Dean was es
pecially helpful in providing insight into the hydromechanics
of the nearshore region, and Dr. Christensen was particularly
helpful in providing assistance in the sediment transport
aspects of this study. The author also wishes to thank
Dr. Per Bruun who initiated the field study.
Thanks are due a great many people who assisted in the
field measurements which were conducted under all weather con
ditions. The author wishes to thank Mrs. Mara Lea Hetherington
who typed the manuscript.
Financial support for this study was supplied by the Depart
ment of Interior, Federal Water Pollution Control Administration,
under contract with the Department of Coastal and Oceanographic
Engineering, University of Florida.
TABLE OF CONTENTS
ACKNOWLEDGMENTS
LIST OF TABLES
LIST OF FIGURES
LIST OF SYMBOLS
ABSTRACT
CHAPTER
I INTRODUCTION
A. Introductory Note
B. Historical Summary
1. Longshore Currents
2. Littoral Drift
C. Purpose and Scope of the Investigation
II CONSERVATION RELATIONSHIPS AND SPECIFICATION
OF WAVE FIELD
A. Introduction
B. Conservation Equations
1. Conservation of Phase
2. Conservation of Mass
3. Conservation of Momentum
4. Conservation of Energy for Fluctuating
Motion
5. Conservation of Energy for the Mean
Motion
C. Description of the Wave Field
1. Waves outside the Surf Zone
2. Breaking Waves
3. Waves inside the Surf Zone
Page
iii
vii
viii
xi
xvi
Page
III LONGSHORE CURRENT THEORY 48
A. Statement of the Problem 48
B. Currents outside the Surf ZoneNeglecting 50
Bottom Friction
1. Mass Transport Velocity 51
2. Distribution of Energy outside the Surf 51
Zone
3. Distribution of Currents outside the 58
Surf Zone
4. Setdown of Mean Water Elevation outside 59
the Surf Zone
C. Currents outside the Surf ZoneIncluding 60
Bottom Friction
1. Bottom Shear Stress Due to Combined 61
Waves and Currents
2. Changes in Wave Height outside the Surf 68
Zone Due to Bottom Friction and Per
colation
3. Distribution of Currents 72
D. Longshore Currents inside the Surf Zone 73
1. Wave Setup inside the Surf Zone 74
2. Velocity DistributionConstant 81
Sloping Bottom
3. Mean Longshore CurrentConstant 87
Sloping Bottom
4. Longshore CurrentGeneral Profile 89
5. Longshore Current Distribution 92
Including Internal Shear Stress and
Bottom Friction
IV LITTORAL TRANSPORTENERGY PRINCIPLE 103
A. Introduction 103
B. Bed Load Transport 104
C. Total Sand Transport outside the Surf Zone 110
D. Total Sand Transport inside the Surf Zone 112
V FIELD EXPERIMENTS 114
A. Description of Experiments 114
1. Sand Transport Measurements
2. Current Measurements
3. Wave Measurements
4. Wind Measurements
5. Tide Record
B. Error Analysis
C. Results and Comparison with Theory
VI CONCLUSIONS
A.
B.
APPENDIX A 
APPENDIX B 
REFERENCES
Longshore Currents
Littoral Transport
BATHYMETRY AT FERNANDINA BEACH, FLORIDA
PREDICTED AND MEASURED DISTRIBUTIONS OF
BED LOAD TRANSPORT
LIST OF TABLES
Table Page
I Longshore Current Formulas 12
II Sediment Transport 130
III Summary of the Littoral Transport Predictive Equation 137
Formulation
IV Longshore Currents 142
vil
LIST OF FIGURES
Figure Page
1 Distribution of Longshore Velocity and Sediment 3
Transport across the Surf Zone (after Zenkovitch [1])
2 Rip Current System 10
3 Uniform Longshore Current System 10
4 Shear Stresses Acting on the Faces of an Elemental 36
Water Column
5 Schematic of the Surf Zone 49
6 Amplification of Wave Energy Density Due to a Shear 57
Current as a Function of the Incident Wave Angle
7 Resolution of Currents and Waves into Respective 66
Velocity and Shear Stress Components
8 Relative Changes in Wave Setup inside the Surf Zone 79
as a Function of the Incident Wave Angle
9 Laboratory Measurements of Wave Setdown and Setup 82
(after Bowen [55])
10 Longshore Velocity Distribution across the Surf Zone 86
for a Constant Sloping Beach
11 Comparison of Predicted and Measured Mean Longshore 88
Currents
12 Longshore Velocity Distribution across the Surf Zone 91
for a Beach Described by an nth Degree Polynomial
13 Comparison of Predicted and Measured Velocity 93
Distributions for Natural Beaches
14 Velocity Distribution across the Surf Zone for Low 98
Velocities Including Internal and Bottom Shear Stresses
15 Velocity Distribution across the Surf Zone for Moderate 100
Wave Conditions Including Internal and Bottom Shear
Stresses
Figure Page
16 Velocity Distribution across the Surf Zone for Field 101
Conditions Including Internal and Bottom Shear
Stresses
17 Schematic Diagram Representing Stresses Acting to 106
Cause Bed Load
18 Plan of Pier Showing Instrument Locations and 116
Typical Bottom Profile at Fernandina Beach, Florida
19 Typical Sand Grain Size Analysis Taken at Fernandina 117
Beach, Florida
20 Bathymetry Adjacent to the Fishing Pier at Fernandina 120
Beach, Florida, December 17, 1968
21 Sand Trap with Doors Open 121
22 Lowering Sand Trap from the Pier, Showing View of 121
Crane
23 Current Meter and Pressure Transducer(Mounted on 124
Tripod)
24 Attaching instruments to Tripod 124
25 Variation of Measured Sediment Transport with Time, 132
Test Number 26
26 Distribution of Bed Load Transport outside the Surf 135
Zone, Test Number 19
27 Distribution of Bed Load Transport across the Surf 136
Zone, Test Number 22
28 Comparison of Measured and Predicted Bed Load 139
Transport outside the Surf Zone
29 Comparison of Measured and Predicted Bed Load 141
Transport inside the Surf Zone
30 Bathymetry, November 22, 1966 150
31 Bathymetry, May 26, 1967 151
32 Bottom Profiles Adjacent to Pier, July 24, 1964 152
April 14, 1965
33 Bottom Profiles Adjacent to Pier, May 13, 1965 153
May 26, 1967
Figure
34
35
36
37
38
39
40
41
42
43
44
45
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Distribution of
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Transport,
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Test Number
Page
155
156
157
158
159
160
161
162
163
164
165
166
LIST OF SYMBOLS
a = wave amplitude
B = proportionality factor associated with bed load transport
B = proportionality factor associated with bed load transport
inside the surf zone
c = wave celerity
cb = wave celerity at wave breaking
c = speed of wave energy propagation
c = wave celerity in deep water
Cf = Chdzy's coefficient
d = mean sand grain size
D = n + h, total depth of water
Db = total depth of water at location of breaking waves
eh = efficiency coefficient associated with bed load transport
e = efficiency coefficient associated with suspended load transport
E = energy density
E = energy density in deep water
f = friction factor
fh = Th/Nh, dynamic friction factor associated with sediment transport
f = friction factor associated with wave motion
w
fwb = friction factor associated with wave motion at breaking
F = energy density flux
g = acceleration due to gravity
h = depth below still water level
h = depth below still water level at wave breaking
h = depth below still water level in deep water
H = wave height
Hb = wave height at wave breaking
Ho = wave height in deep water
H = significant wave height
i = index corresponding to horizontal coordinate in the xdirection
j = index corresponding to horizontal coordinate in the ydirection
k = wave number
K(a) = ratio between setup slope and beach slope
K = refraction coefficient
r
K = shoaling coefficient
L = wave length
L = wave length in deep water
mh = mass of bed load sediments
m = mass of suspended sediments
Ii = Mi + Mi, total mass transport per unit width or total mean
momentum per unit area
M = mass transport per unit width associated with mean motion
M. = mass transport per unit width associated with fluctuating motion
I
n = transmission coefficient
Nh = normal stress at the bed
Ns = normal stress at level of suspended sediment
p = pressure
P = power/unit area
Ph = power/unit area expended on the bed
P = power/unit area available for suspended sediment transport
5
q = net mass transport rate of sediment per unit width
qh = bed load transport rate per unit width
qs = suspended load transport rate per unit width
Qs = total longshore sand transport rate
r = roughness parameter
R. = total average resistance force
1
s = fraction of incoming energy flux available to the longshore
current
S = proportionality factor associated with suspended load transport
S = proportionality factor associated with suspended load transport
inside the surf zone
S.. = excess momentum flux tensor (radiation stress tensor)
t = time
T = wave period
Th = sediment shear stress
T. = net horizontal force per unit area due to slope of free water
surface
u, = shear velocity
uhw = water particle velocity due to wave motion at the bed
u. = total water particle velocity
1
u! = fluctuating water particle velocity component
1
Umh = maximum water particle velocity due to wave motion at the bed
ush = velocity of a sediment particle on the bed
u = water particle velocity due to wave motion
U = mean velocity component normal to the beach
U. = U + U., total mean transport velocity
U. = mean transport velocity associated with mean motion
Ui = mean transport volodity associated with flucttuating motion
v = resultant velocity vector of combined wave and current motion
ziii
V = mean velocity component parallel to the beach
Vb = mean velocity component parallel to the beach at location of
breaking waves
Vsh = mean velocity of bed load transport in the longshore direction
V = mean velocity of suspended load transport in the longshore
direction
w = vertical velocity component
w = fall velocity of sand grain
x = horizontal coordinate perpendicular to the beach
X = width of the surf zone
s
y = horizontal coordinate parallel to the beach
z = vertical coordinate
a = incident wave angle
ab = incident wave angle at breaking
a = incident wave angle in deep water
B = bottom slope
6.. = kronecker delta
At = unit length of the beach
e = energy dissipation function
f = energy dissipation due to bottom friction
E = energy dissipation due to percolation
E = eddy viscosity
S= water surface elevation
= mean water surface elevation
nb = mean water surface elevation at breaking position
n = mean water surface elevation for zero angle of wave incidence
< = ratio between breaking wave height and the depth of water
at breaking
= rip current spacing
V' = mixing length
v = kinematic viscosity
a = wave frequency
S= velocity potential
S= permeability coefficient
p = density of fluid
Ps = density of sediments
a = local wave frequency
r = shear stress
TF = fluid shear stress
Th = shear stress at the bed
hw = shear stress at the bed in the direction of wave motion
Thy = shear stress at the bed in the direction of mean motion
Ts = intergranular fluid shear stress
e = phase lag
X = phase function
an = water particle excursion due to wave motion at the bed
Subscripts
b = breaker line
h = bottom
o = deep water
s = sediment
ss = suspended sediment
w = wave
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
LONGSHORE CURRENT AND SEDIMENT TRANSPORT
By
Edward Bennett Thornton
March, 1970
Chairman: B. A. Christensen
Major Department: Civil Engineering
On the basis of a simplified model, an attempt is made to com
pletely describe the littoral processes including both the longshore
current and sand transport. The basic assumptions are that the con
ditions are steady, the bottom contours are straight and parallel
but allow for an arbitrary bottom profile, the waves are adequately
described by linear theory, and that spilling breakers exist across
the surf zone. The equations of motion arc employed to the second
order in wave amplitude (first order in energy and momentum). Empha
sis in the analysis is placed on formulating usable predictive equations
for engineering practice. The waveinduced currents generated parallel
to shore are investigated first. The littoral sand transport, which is
dependent on the strength of the longshore current and the intensity of
the wave action, is subsequently examined.
Conservation equations of mass, momentum, and energy, separated
into the steady and unsteady components, are used to describe second
order waveinduced phenomena of shoaling waves approaching at an angle
to the beach. An expression for the longshore current is developed,
based on the alongshore component of excess momentum flux due to
the presence of unsteady wave motion. This approach is parallel to
but differs quantitatively from recent work by Bowen. The wave
induced currents are shown to be primarily confined to the area inside
the breaker line, that is, the surf zone. The longshore current ex
pression is investigated for varying bottom profiles to evaluate the
physical significance of including different frictional resistance
terms. Wave setdown and setup have been included in the formulation.
Comparison with experimental results from the laboratory and field show
that if the assumed conditions are approximately fulfilled, the pre
dicted results compare quite favorably.
The waveinduced sand transport alongshore is investigated from
energy considerations in which the quantity of sand transported is
expressed as a function of the energy utilized in bottom friction,
viscous dissipation, and turbulence. Although an energy approach has
been used before, this is the first application to predicting the
distribution across the surf zone of the alongshore sand transport.
Sand transport data were collected in the field using unique bed
load traps. Wave, tide, wind, and current information was collected
simultaneously in order to check the derived predictive equations for
longshore current and sediment transport. Although the absolute values
of sand transport arc not well predicted, quite reasonable predictions
are obtained for the relative distribution of bed load transport both
inside and outside the surf zone, and could be used as a qualitative
predictive relationship for engineering applications.
xvii
CHAPTER I
INTRODUCTION
A. Introductory Note
Waves gather their energy from the wind at sea and then propagate
until eventually reaching a shoreline. It is here that a wave, having
perhaps obtained its energy from the wind over hundreds of miles and
many hours, is dissipated over a very short distance in a matter of
seconds. The area between where the waves first start to break and the
beach is called the surf zone. The incessant dissipation of wave energy
against the shoreline maintains the bottom sand and water within the
surf zone in a constant state of motion, the processes of which are for
ever striving to reach a state of dynamic equilibrium.
Waves in the open ocean (outside storm areas) tend to be fairly
regular and lend themselves to analytical description. As the waves
move into shallower water, they undergo transformation due to shoaling
and refraction, and their heights and wave lengths change. Theories,
which have been developed to describe waves for various wave heights,
periods, and depths, are reasonably valid up to conditions of near
breaking. The waves reach a maximum height at breaking. Upon breaking,
they lose their ordered character, and can no longer be described
analytically. Breaking waves are classified as spilling, plunging, or
surging. The manner in which waves break determines the distribution
of energy dissipation and the energy and momentum processes in the
littoral zone.
2
Waves traveling toward the beach at an angle to the shore have
a momentum flux component directed parallel to shore. This momentum
flux can result in the generation of longshore currents. These currents
have been found to be largely confined to the area inside the surf zone,
with the waveinduced currents seaward of this zone being relatively
weak.
Longshore currents are an agent for transpoorting material already
"loosened" from the bottom by the more intense wave action. The long
shore movement of sand, termed "littoral drift," is then the result of
the combined action of sediment bed agitation by waves, acting primarily
perpendicular to the beach, and transport by the longshore current. The
littoral drift consists of suspended and bed load components. Inside
the surf zone, where the turbulence is great due to breaking waves, both
of these components are significant and vary across the width of the
surf zone. Outside the surf zone, the turbulence and wave agitation are
reduced, and the sand transport is principally in the form of bed load.
The distribution of longshore current and sediment transport across
the surf zone is the subject of this report. Unfortunately, there is
very little published data on this subject. Zenkovich [1] used fluo
rescent sand tracers and collected suspended sand samples across the surf
zone to determine these distributions. Figure 1 shows the variation of
the sand transport as related to the longshore current and bottom profile.
During the test, the waves were relatively high, and the slope of the
beach small so that spilling breakers prevailed. The waves spilled over
the first bar, and then broke again on the inner bar. The longshore
current distribution is fairly uniform within the surf zcnc. Tlie sand
Oulor Brcakcr
Line
Figure 1. Distribution of Longshore Velocity and Sediment
Transport across the Surf Zo\c after r ZOnkcvich [1])
I I_~ I i~~YlOi~~~P~
4
transport is greatest over the bars where the energy dissipation is a
maximum due to the breaking waves.
A knowledge of the variation and extent across the surf zone of
the longshore current and sand transport is important in design con
siderations of structures placed in the littoral area. This kind of
information is particularly important for groins or similar structures
designed to impede sand movement. On the other hand, it is often
desirable to have natural bypassing about jetties constructed for navi
gational purposes at inlets and harbors. The distribution of effluent,
introduced onto beaches and into the littoral zone, is also influenced
by the currents in the surf zone. There is a very real need for a more
complete understanding of the littoral zone so that further improvements
and preservation of our beaches can be based on more rational and con
crete approaches.
This study logically breaks down into considering, first the w3ves
and waveinduced currents separately and, then, the combined action on
the sediments to determine the sand transport distribution. The
following is a review of the literature pertinent to this study, both
to serve the reader as a review of the past accomplishments and also
to gain considerable insight into the problem at hand.
B. Historical Summary
1. Longshore Currents
a. Field experiments
During World War 11, it became apparent there was an urgent need
to better understand the surf zone and litloral environment in order
to improve amphibious landings. This was the inpetus for initiating a
5
thorough study of the mechanics of the surf zone which was later carried
on after the war. Munk and Traylor [2] gathered data available at that
time, conducted additional field experiments, and developed the first
empirical formula relating the longshore current to the incident waves
and bottom topography. They found that the longshore current was a
function of the angle of wave incidence, the height of the breakers, the
wave period, and the slope of the foreshore.
Shepard and colleagues at Scripps Institution of Oceanography con
tinued this work after the war in a series of field measurements along
a 100mile stretch of the Southern California coast. From an accumu
lation of over 1,000 measurements emerged the first rather complete
qualitative picture of the surf zone. The ideas expressed by Shepard [3]
in the original sumnary of the data have changed very little to this day.
The study showed that the most important parameter governing the long
shore current is the angle of wave approach. It was also found that the
strength of the current was related to the breaker height.
All the field experiments showed that the longshore current was
largely confined to the surf zone and that a substantial velocity variation
could exist across the surf zone. Shepard and Sayner [4] compiled the
results of five years (comprising 800 days of measurements) taken at three
locations along the 1,000foot pier at Scripps Institution of Oceanography.
The three measurement locations were at the end of the pier outside the
surf zone, just outside the breakers, and inside the surf zone. They
established that the currents inside and outside the surf zone were
governed by the same mecianjsim, that is, tchy ,se correlated, and that
the currents could be related to the wave height and direction. hlle
currents varied c3s;ideiebly frLm offshore to the beach; Lhe average
6
current velocity inside the surf zone was 1.0 foot per second compared
to an average current just outside the breakers of about 0.2 foot per
second.
Ingle [5] measured the distribution of longshore currents across
the surf zone in conjunction with sand tracing experiments on a number
of beaches along the California coast. All of these beaches, with the
exception of one, were moderately steep. These tests showed that the
maximum longshore current most often occurred midway between the breaker
line and the swash zone.
The surf zone does not always present such a simple picture as
first envisioned. On many occasions, there are strong currents of
variable direction during times when large breakers approach almost
normal to the coast. These currents are often associated with circu
lation cells and attendant rip currents which constitute a definable
system. It has been found that these rip currents are lest common
along straight beaches with parallel contours and best developed in
areas of irregular bottom topography.
Shepard and Inman [6] measured the real variation of currents
over a stretch of bench south of Scripps pier. They found that a rip
current system was prevalent in this area and that the rips were con
trolled,to a large degree,by the wave convergence and divergence
resulting froi refraction of waves over the rather irregular bathymetry
fronting the area. The areas of wave divergence were particularly
evident over the two submarine canyons just offshore of the bench, and
wave convergence occurred over submarine ridges. In areas of wave
convergence, there was an influx of water cross the surf zone, resulting
7
in an accumulation and a hydrostatic head causing a current away from
the area of high breakers.
These currents flow toward areas of divergence and then turn
seaward in the form of rip currents. This outflowing current then
can turn again toward the point of onshore flow, thereby forming a
complete circulation cell. The effect of wave convergence and di
vergence on the littoral circulation was found to be a maximum during
times of long period waves which are more affected by the bottom
topography.
Bowen [7] presented the results of an analytical investigation
of the case of waves normally incident to the beach and substantiated
some of the observed phenomena. He was able to show that regular
variations in the wave height parallel to the beach could induce
regular circulation patterns. He further showed that the gradients
of wave heights could be possibly associated with edge waves or regular
undulations in the bottom topography.
Sonu et al. [8] found a considerable variation in longshore
currents during field experiments along the coast of North Carolina,
and attributed this variation to the undulations in the bottom and
the bartrough system that prevailed. He concluded that the currents
under stable conditions appeared to be generated by the momentum trans
fer from the plunging breakers over the bar and the mass transfer from
the spilling breakers over the shoal. The hydrostatic head potential
arising from the mass transfer into the surf zone is discharged seaward
by rip currents.
Calvin and Savage [9] conducted field e::periments at the same
location and found that,during the four days of testing, a fairly
8
uniform current system prevailed with no evidence of a rip current
system. This demonstrates that the littoral system is not constant
and is governed by the conditions that prevail at the time of obser
vations.
b. Laboratory studies
Much of the laboratory longshore current data have been found
in conjunction with related littoral drift studies. Saville [10]
measured longshore currents on a model sand beach having an initial
slope of 1:10. The measurements were made after the bottom profile
had achieved equilibrium. It was found that the maximum current always
occurred along the nearshore bar where the waves break and was neatly
uniform across the surf zone, decreasing slightly toward shore. The
currents inside the surf zone were found to be approximately five times
the current just outside the breakers which is in very close agreement
with field measurements mentioned earlier.
Galvin and Eagleson [11] and Brobner and Kamphuis [12] performed
very similar experiments on fixed plane beaches with a slope of 1:10.
These studies showed that the maximum velocity is located between the
still water line on the beach and the breaker position, usually closer
to the still water line than to the breaker position. As the current
flowed down the beach, the local maximum had a tendency to migrate
toward the breaker position. It was also found that the currents were
not uniform in the longshore direction. The currents increased down
stream from an initial velocity at the retaining wall to a point beyond
which a uniform distribution prevailed. The occurrence of riplike
currents was not observed in any of these laboratory experiments.
Two rather general types of littoral systems emergeone, in which
the longshore currents are fairly uniform in the longshore direction,
and, the other, in which circulation cells and attendant rip currents pre
dominate. These systems are depicted in Figures 2 and 3.
Where rip currents do not exist, there is a general return of water
to the offshore to compensate for the shoreward mass transport of water
by the incident waves. LonguetHiggins [13] theoretically showed that
for twodimensional flow the incoming waves produce an onshore drift at
the surface and bottom and an offshore flow at intermediate depths.
This distribution was substantiated in the laboratory by Russell and
Osorio [14] and in the field by Miller and Zeigler [15] whose measure
ments included waves breaking close to shore. Measurements taken at
Fernandina Beach using an electromagnetic flowneter further substantiate
the occurrence of offshore and onshore flow simultaneously over depth.
Thus, at one section perpendicular to the beach, transport can occur
simultaneously onshore and offshore so that there is little net trans
port across the breakers.
A complete description of the surf zone requires considering a
system that is threedimensional and unsteady in time and space. The
tenporal variability appears to be small compared to the spatial vari
ability (Sonu et al., oo. cit.) and can be attributed partially to the
nonstationarity and stochastic nature of the waves. The area variation
of longshore currents can often be related to the variations in the near
shore bathymetry and also to coastal structures. Jetties, groins, and
other structures that protrude into the ocean interrupt the normal lon
shore current, and rip currents and large scale eddies are often located
BREAKING WAVE /
RIP FEEDFR
CURREn \
LONG SHORE CURRENT
! ..' . ** ," .V. A H
Figure 2. Rip Current System
Figure 3. Uriifor.n Longshore CurrenL SysLtem
t I _______ CURR
I / VECTOR
o L 0 N G S HO 0 T E
* r C H ",'
C UR RE N T
at the ends of such structures. Bowen (op. cit.) demonstrated that in
the laboratory circulation cells can be induced by edge waves caused
by gradients in the longshore breaker heights. The general complexity
of the surf zone necessitates that simplifying assumptions be made
in order to meaningfully describe particular phenomenon.
c. Theory
Most of the longshore current formulas predict only the mean of
the longshore current velocity. At present, there are at least fifteen
such formulas, none of which satisfactorily predicts all conditions in
nature. This is not too surprising in light of the rather drastic
assumptions that are required to obtain an analytical formulation.
There are four basic approaches to the development of predictive
equations for currents in the surf zo.ie: (1) conservation of mass, (2)
conservation of momentum, (3) conservation of energy, and (L) empirical
correlation. A review of the more familiar equations is presented below,
and Table I lists eleven of these formulas to facilitate co::arison.
Some of the equations have been expressed in slightly different forms
than originally published in order that they all incorporate the same
parameters; however, these changes deal only with the geometric relation
for the breaking depth and slope of the beach and the relationship be
tween the breaking depth and breaking wave height. The changes do not
alter the currents predicted by the equations. Symbols, utilized in all
equations, are listed in the preface.
Momentum considerations.Putnam, :hunk, and Traylor [16] used both
the energy and momentum equations to derive the first rational equations
for describing longshore currents. They considered the flux of mass and
momentum into a control volume of differential length bounded by the
TABLE I. LONGSHORE CURRENT FORMULAS
Authors Mean Longshore Current, V Formulation Eq.
No.
PutnamMunk 2 sin2b 1/3 Energy Conservation, 1.1
Traylor (1949) f T Solitary waves
E3 sinBsinabsin2a 1/2 Momentum Conservation,
Eagleson (1965) [ gePHb ] Asymmetricperiodic 1.2
waves
[(1 + 2.28gHbsin%)1 /2
PutnamMunk Momentum Conservation,
Traylor (1949) tSolitary waves
A = 20.88 cos
GalvinEagleson
(1965)a gTtansin2ab Mass Conservation 1.4
InmanBagnold Mass Conservation,
(1963) 2.31 T cosbsinab Rip currents included
Bruun (1963) C [0.95 3/2 tansin2 Mass Conservation 1.6
f IT Mass Conservation 1.6
Bruun (1963) 2.31 b
T Rip currents included
[( + 2.28gHsinab)12 2
InmanQuinn Empiricalbased on 1.8
(1951) tansHcosab momentum analysis
A = 108.3
T
H 2/3
BrebnerKamphius 8.0 sin /3 3[sin 1.65o Empiricalbased on
(1963) momentum analysis
+ 0.1 sin 3.30a
H 3/4
14.0 sin 1/28 [sin 1.65 a
BrebnerKamphius T1/2 Empiricalbased on .
(1963) energy analysis
+ 0.1 sin 3.30 ao]
0.241 Hb + 0.0318 T + 0.0374 ab Empiricalleast
Harrison (1968) r calleast 1.1
+ 0.0309 tanR 0.170 square analysis
breaker line and the shore. The change in momentum flux across the
breakers directed parallel to shore is balanced by the bottom shear
stress. Solitary wave theory was used to calculate the momentum of
the breaking waves. In this manner, they obtained an expression for
the mean longshore velocity related to the angle of wave incidence ab,
breaking wave height Hb, bottom slope tanB, wave period T, and friction
factor f. Embodied in all of the momentum analyses is a friction factor
that relates the velocity to the bed shear stress and represents an em
pirical coefficient. This equation was subsequently revised by Inman
and Quinn [17] who found that a better fit to the data originally
collected by Putnam et al., and additional field data collected by the
authors, was obtained if the constant friction coefficient in the
original equation was changed to be a function of the velocity.
Eagleson [18], using the same control volume approach, developed
a mathematical model to represent the growth of a longshore current
downstream of a barrier. In the associated laboratory experiments, it
was found that a large percentage of the fluid composing plunging
breakers (most comaon laboratory breaker type) is extracted from the
surf zone. This fluid already has a longshore velocity to which is
added the longshore component of the breaking wave. This argument
provides a mechanism for growth of the longshore current downstream.
The asymptotic solution to the differential equations showed that the
system is stable and that the growth of the currents was bounded.
These results agreed qualitatively with laboratory results and demon
strated that unless there are perturbations inducing gradients in the
wave e!rgy in the longshore direction, tbp current system tends to be
uniform alongshore and stable for stationary wave conditions.
14
Bowen [19], in a very recent investigation, used a conservation
of momentum approach to determine the longshore velocity distribution
across the surf zone for the case of a plane beach. Reasonable results
were obtained when compared to laboratory data. This is the same
approach that will be used in this dissertation to develop equations
describing the longshore current distributions for more general situ
ations. A more complete discussion will be reserved for Chapter III.
Energy considerations.Putnam et al. (on. cit.) also derived a
mean longshore current equation from energy considerations alone. The
derivation equates the changes in energy flux to the frictional energy
losses parallel to the beach. A difficulty, with the resulting equation,
is that it involves two undetermined constants, the friction factor f
and the percentage of the wave energy available to the longshore current s,
which makes the equation very difficult to apply.
Continuity considerations.Bruun [20] and Inman and Bagnold [21]
derived similar expressions using the continuity approach. These formu
lations are based on the fact that the incident waves introduce a mass
flux of water into the surf zone which is then manifested as a spatial
gradient in the longshore current. Both developments consider a plane
beach of infinite length, implying that mass is uniformly introduced into
the surf zone along the beach. The current will grow (sinre mass is
continually being supplied to the surf zone), and, at intervals, it is
necessary that there be outflow from the surf zone unless the current
becomes unbounded. It is postulated that this outflow occurs in the
form of rip currents which are evenly spaced along the coast. Thus, the
equations cuntrin an unknown parameterthe spcing of the rip cii'rets.
Unfortunately, few measurements have been made of rip current spa ings
15
so that the use of these equations requires additional experimental
data.
It should be noted that,due to the mass flux of waves, there
is always transport of fluid into the surf zone and that, in all the
physical models whether considering a mass, energy, or momentum
approach, the mass flux must be accounted for in order to obtain a
bounded solution. Thus, the assumption, that the mass transport is
uniformly returned across the surf zone, is at least implied in all
the developments which do not include concentrated return flow by rip
currents.
Bruun (op. cit.) also considered the case where rip currents are
absent, and the return flow is distributed uniformly over the vertical
plane containing the breaker line. He reasoned that waves breaking at
an angle to the beach contribute mass to the surf zone and locally
raise the mean water level as the breaking wave crest propagates down
the beach. This results in a slope of the water surface between crests
which creates a longshore current. The longshore current is balanced
by bottom shear stress related to the velocity through the Chdzy formula.
Galvin and Eagleson (op. cit.), reasoning from the continuity approach,
equated a hypothetical mass flux across the breaker line proportional to
the mass contained in the longshore current. Using both field and labora
tory data, the two mass fluxes were correlated.
Empirical correlations.Two types of empirical equations have been
developed. The first type employs physical reasoning to determine the
form and grouping of the important parameters which are then correlated
with experimenntal data. Brebner and Kamphuis (on. cit.) used both the
energy and monmentu equations to obtain reasonable groupings by dimensional
16
analysis of the important parameters. Linear regression was then used
to find the best fit for the longshore velocity to a large number of
data that they had measured in the laboratory.
The second type of analysis employs multiple regression techniques.
Sonu et al. (oD. cit.) used this method to weight the various independent
variables collected in their field studies. They found that the most im
portant variable affecting the mean longshore current velocity was the
angle of wave incidence, and the second most important, although much
less, was the wind. These results are conflicting with those of a
similar analysis reported by Harrison and Krumbein [22] who, using data
collected at Virginia Beach, Virginia, found the most important variable
to be the wave period which proved to be insignificant in Sonu's analy
sis. Sonu also performed a multiple quasinonlinear regression analysis
which showed the most important variable affecting the mean longshore
current velocity to be the wave height. In a later study, Harrison [23],
using another set of data collected at Virginia Beach, found the incident
wave angle to be the most important, followed by the wave period, height,
and beach slope, respectively. Harrison points out that the use of such
empirical equations is necessarily limited in application to "similar"
situations; it is not possible to extrapolate to any particular case
with confidence. The problem with using such techniques is that they
are devoid of any physical basis and, as such, can give spurious corre
lation and conflicting results. The physics of the problem predict a
nonlinear combination of the independent variables where linear multiple
regression is inherently a linear combining or additive process.
Evaluation of thori es.Galvin [2] critically reviewed twelve
mean longshore current theories and tested six of them that were
17
applicable to the "best of the published experimental data" from both
the laboratory and field. In this way, he hoped to determine which
equation most accurately represents the experimental measurements.
Galvin concluded that there is still no completely satisfactory pre
dictor of mean longshore currents.
Sonu et al. (op. cit.) conducted field experiments and found poor
correlation when compared to six of the above equations. Their
experiments did point out the importance of the nearshore topography
on the current system, and how this may affect the outcome of such
results. Shepard [4] earlier pointed out that another reason for lack
of agreement between theory and field experiments could be the vari
ation of current across the surf zone; each field data point is usually
based on only a single location in the surf zone.
Many simplifying assumptions are necessary in developing the
theories. Since exact expressions are not available, it is necessary,
inside the surf zone, to select approximate expressions for the wave
speed, wave shape, water particle velocity, partitioning of energy in
the wave field, alongshore variation of waves and currents, and velocity
and energy distributions across the surf zone. It is possible that im
proved theories for longshore currents will require a better under
standing of the highly nonlinear waves in the vicinity of the surf zone.
However, it would seem that an improvement in existing formulations could
result by including the distribution of these quantities across the surf
zone rather than considering only the mean values.
The importance of considering other factors, such as the wind, was
demonstrated by Sonu's empirical correlations. He found the wind to be
18
the second most important variable in his set of field data. This
shows the difficulty in comparing tests, particularly field data
where information concerning the effects of bottom topography and
Winds often is not included. The extrapolation of data from one
particular location to another without accounting for the importance
of these effects can lead to invalid results.
All of the equations involve unknown coefficients to be deter
mined experimentally. Generally, the friction factor in the momentum
and energy equations is evaluated in the same manner as in open
channel hydraulics. The validity of utilizing results from steady flow
situations certainly needs to be investigated further and could hope
fully result in a refinement.
Sonu points out that another possible improvement might derive
from consideration of the dynamic processes of energy dissipation in
the surf zone environment. The difficulty is that the flux of energy
used in generating longshore currents is only a small fraction of the
total available energy and, as such, represents a second order phenome
non (Galvin, o2. cit.). On the other hand, it appears that the mass
flux into the surf zone represents a primary feature of the surf zone
so that conservation of mass might be a better basis for longshore
current theories. One difficulty with using the continuity approach
is that, although it allows a description of the mean current, no
procedure has yet been developed, based on continuity considerations,
which provides a prediction of the variation of the current across
the surf zone.
2. Littoral Drift
a. Field experiments
Field experiments have played a very important role in the study
of littoral processes. One of the difficulties encountered in sedi
ment transport studies is that the similitude laws relating field to
laboratory conditions have not been adequately established and verified.
Thus, field studies are necessary for the understanding of littoral
problems, either to supplement analytical or laboratory work, or as a
means of solution by itself.
The measurement of littoral drift along a particular beach and
the meaningful correlation with the wave environment is an extremely
difficult task. Hence, only very limited data are available. The
formula, most frequently used for determining total littoral drift, is
attributed to Caldwell [25]. Dimensional arguments show that the rate
of sediment transport can be related to the longshore component of wave
power per unit beach length. Caldwell combined the results of field
studies conducted in Anaheim, California, and those of Watts [26] at
South Lake Worth, Florida, and proposed the following formula for the
total longshore transport per unit time due to waves
Qs = C[pgHb2 cb sin2a At] 0.8 (1.12)
where AZ is a unit length of beach, and C is a constant of proportionality.
This relctonrship has been tested against laboratory data by Savage [27]
and shows reasonable correlation.
The earliest published information on the distribution of sand trans
port along rne beach profile for prototype cornirions was by the Beach
Erosion Board [28]. The distribution of longshore current and suspended
20
sand, obtained from water samples, was measured from piers extending
across the surf zone. These measurements showed that the greatest
sand transport occurred at the breaker line, where the turbulence was
a maximum, and decreased shoreward with another peak in the swash zone
another area of high turbulence. Seaward of the breakers, the sand trans
port decreased with increasing depth. There have been surprisingly few
other field experiments of this type. Watts [29] conducted similar
studies using a more elaborate continuous suspended sediment sampler.
The results were qualitatively similar and showed that the amount of
sand in suspension was related to the wave height, or energy, of the
waves for a particular test. In these experiments, and some by Fukushina
and Mizoguchi [30] using suspended samplers made of bamboo poles, the
vertical distribution of suspended sediments was also measured. These
data showed that the amount of suspended sediment in the swash zone and
near the breaker line can be fairly evenly distributed over the vertical
due to the high degree of turbulence throughout the water column. This
is particularly true at the breaker line where a large vertical velocity
component can be present in the case of plunging breakers. The greater
portion of the transport in the surf zone is due to suspended load with
the highest concentration near the bed; outside the suif fore, bed load
is the predominate mode of transport.
Improvements in tracer techniques, particularly using either radio
active or fluorescent sand tracers, have increased the intensity of
littoral drift studies in the field. A great number of studies have been
conducted in recent years anI have been summarized in a book by Ingle
(op. cit_.). Ingle also conducted a rumbec of studies using fluorescent
tracers on several Southern California beaches. The results of these
21
and some conducted by this researcher, and also earlier investigators,
are all generally similar to that given in Figure 1 in describing the
variation of sand transport across the surf zone. The fluorescent sand
grains are found in greatest concentrations along points of high turbu
lence. In a bartrough profile, the sand moves predominantly along the
bar or in the swash zone. There is a minimum of tracer transport in
the trough.
Tracer studies have been conducted using both the Eulerian and
Lagrangian approaches. Host investigators have used the Lagrangian
approach in which the tracer is introduced at a particular location,
and the concentration distribution of the tracer is determined by ob
taining bottom samples at various sample points. The concentration of
tracers is then determined by counting the tracer grains in the sand
samples. The Eulerian approach is to sample in time along a particular
line across the surf zone, traversing the path of the tracers. A stable
platform or other work facility is generally required in this method.
This was the approach used by Zenkovitch, working from a tramway tra
versing the surf zone and by Bruun and Battjes [31], working from a pier.
The experiments by Bruun were continued by this author.
The inherent difficulty of fluorescent tracer studies is that
quantitative measurements require recovery of most of the tracer. Un
fortunately, the recovery level is generally very low, amounting to only
a few percent. This requires that accompanying measureelvents of the
quantity of sand in suspension, or moving on the bed, also be determined.
However, as a too], or aid, for solving engineering problems in which
qualitative information can be extremely important, fluorescent tracer
techniques alone can be of great value.
b. Laboratory studies
A number of laboratory experiments have been conducted to deter
mine the mechanisms causing sand transport in the surf zone. This
discussion is limited to the threedimensional studies simulating
conditions in the prototype. Krumbein [32] conducted one of the first
of these experiments and concluded that the mean littoral drift was a
function of the deep water wave steepness, H /L Subsequent studies
0 o
by Saville (o_. cit.) showed that a maximum transport occurred for a
wave steepness of 0.025, and,for steepnesses greater or less than this
value, the transport was less. This was further verified by Shay and
Johnson [33] who also showed the transport to be a function of the wave
angle. A maximum transport was found to occur for a deep water wave
angle of 30 degrees.
Although these results imply tlat the transport is a function of
wave steepness, the reason for this dependence has not been established.
Galvin [34] conducted a series of experiments of breaking waves on labo
ratory beaches. He developed a classification for determining whether
the waves develop into plunging or spilling breakers as related to wave
steepness and the beach slope. These results, when compared to the
littoral drift studies in the laboratory, indicate that maximum trans
port occurs for a plungingtype breaker and that the rate of transport
may be more a function of the manner in which the waves break than the
wave steepness.
The study by Saville and a subsequent study by Savage [35] also
provide information concerning the distribution of littoral transport
across the surf zone. A series of traps wn. uFod to traverse the down
stream profile of a model beach so that both the total and distribution
23
across the surf zone could be determined. It was found that approxi
mately 90 per cent of the transport occurred inside the surf zone.
The distribution of transport across the surf zone was in qualitative
agreement with field studies.
c. Theory
Although many predictive formulas have been proposed for the
problem of sediment transport in alluvial channels, there have been
relatively few attempts to explain the transport phenomenon due to
combined wave and current action. This is undoubtedly due to the
limited success found using the predictive formulas for unidirectional
flow and also because of the added complexity of dealing with oscil
lating flow.
A study of the mechanics of the forces acting on a sand grain under
the combined action of waves and currents was made by Eagleson and Dean
[36]. This study considered the combined gravity, friction, current and
wave forces acting on an individual spherical particle on the bed. From
this deterministic approach, accompanied by considerable laboratory
studies, conditions for incipient motion were found. The forces on a
suspended particle were also considered in the same study.
For a given sand grain size and density and beach slopes, there
exists an equilibrium position, the socalled "null point," along the
bottom profile where all the forces on the particle averaged over a wave
cycle are balanced. Eagleson et al. [37] extended the study of the
incipient motion condition mentioned above to better define the "null
point" of a particular sand grain and were able to qualitatively explain
sorting of sand along the profile according to size and weight charac
teristics. The forces acting on a sand grain can be elucidated, and
considerable insight gained into the nature of the problem from this
deterministic approach to the sediment transport problem.
Difficulty is encountered in extrapolating from a single particle
analysis based upon deterministic mechanics to the many particle analy
sis which in reality must be considered as a statistical mechanics
problem. Kalkanis [38] used the same arguments as embodied in Einstein's
approach to sediment transport in alluvial channels and extended it to
the case of simple harmonic motion with a superimposed nean current. He
assumed that the distribution of the turbulent water particle fluctu
ations are Gaussian over the wave period and, on this basis, derived a
probabilistic model for the bed load movement. Laboratory experiments
accompanied this study in an attempt to determine the necessary co
efficients in Einstein's development. Unfortunately, the complexity of
the formulation requires the determination of five parameters which must
be based on considerably more data than have been examined to date.
Iwagaki and Sawaragi [39] developed a formula for predicting the
total littoral transport rate applicable to long term average values.
The basic assumption was that the average littoral transport is pro
portional to the shear velocity of the mean longshore current. The mean
longshore current is determined by the Putnam, Munk, and Traylor energy
formula for longshore currents (see Table 1). The formula, presented by
Iwagaki and Sawaragi, in terms of measured quantities at the breaker line,
is
tan8 sin2ab 4/3 cosab 11/3
Q A [ (1.13)
where A is a constant of proportionality, d is the mean grain size, and
the other terms are as previously defined.
25
Le M1hautd and Brebner [40] proposed an equation for the total
mean littoral transport rate based on physical reasoning guided by
empirical information. The equation assumes the form
Qs cbgH2 T sin aab (1.14)
The exponents and constants in this equation were selected to provide
a best fit to the measured data.
A more straightforward approach is given by Bagnold [41]. This
approach is based on considering the work required to move the sedi
ments as related to the available energy of the waves and currents.
He considers both the bed load and suspended sediment transport. Since
this is essentially the method of analysis being employed in this dis
sertation, the formulation is presented completely in Chapter IV.
C. Purpose and Scope of the Investigation
The aim of the present research is to investigate the distribution
of waveinduced longshore currents and sediment transport from offshore,
across the surf zone, to the beach. Where possible, formulas are de
veloped to predict quantitative information. A simplified physical
model is considered which has wide application to nature. The basic
assumptions are that steadystate conditions prevail and that the bottom
contours are straight and parallel to the beach.
The study naturally breaks down into examining, first, the distri
bution of the shear stresses and longshore currents. Having obtained
this information, it Is then possible to describe the distribution of
sediment transport.
Tools for the analysis are developed in Chapter II. A general
set of conservation equations are set down, and the wave field, inside
26
In Chapter III, theoretical models for longshore current and
shear stress distributions are developed governing the region from
deep water onto the beach. A momentum approach is utilized in which
the changes in momentum flux are balanced by frictional forces in
the direction parallel to shore. The resolution of the shear stresses
into wave and current components is demonstrated. The theoretical
models for longshore currents are checked against existing field data.
Chapter IV utilizes the results of the predicted current and
shear stress distributions to theoretically describe the variation of
sediment transport across the surf zone.
Chapter V describes field experiments conducted which tend to
substantiate the theoretical descriptions of the sediment transport
distributions. The results of the field experiments and theory are com
pared and discussed.
CHAPTER II
CONSERVATION RELATIONSHIPS AND
SPECIFICATION OF WAVE FIELD
A. Introduction
Past investigations have attempted to relate the mean longshore
currents to waveinduced momentum, energy, or mass flux into the surf
zone. The distribution of longshore currents can be similarly investi
gated by considering the changes in the momentum, energy, or mass flux
across the surf zone. The present analysis utilizes the momentum
principle to describe the variation of waveinduced longshore currents
from deep water, across the surf zone, to the beach.
It is known that, due to the fluctuating water particle motion of
the waves, there is a momentum flux component. If the waves have a
direction component parallel to shore, a longshore current can be gener
ated due to changes in the longshore momentum flux component of the
shoaling waves. It is also known that there must be a displacement of
the mean water surface elevation to balance the changes in the onshore
momentum flux component of the shoaling waves.
In waves, the momentum flux is the sum of the pressure and the
product of two velocities. It can be shown that the average momentum
flux is nonlinear in wave height. In order to specify the excess
momentum flux of the waves, it therefore becomes necessary to consider
nonlinear, or higher order, effects of the wave motion. Since the
waves are the mechanism responsible for the generation of the longshore
28
currents, much of the ensuing discussion is devoted to describing the
waves.
The treatment of periodic gravity waves is generally developed
by perturbation schemes built on the exact solution for the linearized
equations of motion with specified boundary conditions. Difficulty,
with this type of analysis,is encountered when considering higher order
theory including a variation in the bottom boundary condition. For
waves traveling in water of varying depth, two different approaches can
be employed: the analytical method and the energy flux method. The
analytical method involves solving the boundary value problem accounting
for the slope of the bottom to the desired degree of approximation.
This technique includes the slope,explicitly,in the perturbation expansion.
The solution for the waves on a plane sloping bottom has bccr worked out
to the second order by several authors. Miei et al. [42] have given a
systematic approach for carrying the analysis to higher orders.
The analytical method is generally more consistent because it
accounts for the bottom slope in the bottom boundary condition to the
same order of approximation as is maintained in the free surface boundary
conditions. The smallness of the bottom slope as well as the smallness
of the amplitude is incorporated in the perturbation analysis. Although
more attractive from an analytical point of view, this method has the
inherent difficulty of requiring tedious computations for each particular
case. Also, and more importantly for our needs, the more rigorous
boundary value problem approach is not readily extended across the surf
zone.
Ite energy method consists of solving the wave problem for a hori
zontal bottom and then extending thfse results to a sloping bottom by
29
means of energy flux considerations. Hence, for short distances, in
the region outside the surf zone, it is assumed that waves on a sloping
bottom can be considered the same as on a horizontal bottom. Then,
adjacent increments of distance are connected together by means of the
energy flux conservation equation. This results in a prediction of the
wave height at any location outside the surf zone. Inside the surf zone,
it will be assumed that the wave height is governed by the local water
depth.
The method employed allows for the inclusion of dissipative effects,
such as bottom friction, and the focusing of energy due to refraction by
changes in bathymetry and currents. The accounting for these effects is
often more important than the deformation of the water particle motion
due to the bottom slope as given in the analytical procedure [43].
The validity of the energy method can be proven by comparison with
the results of the exact analytical solution. To the first order of
approximation in wave energy,for the case of small bottom slopes, identi
cal results are obtained for the propagation of energy. This confirms
the use of the energy flux equation to connect the solution for various
depths for small bottom slopes, at least to the first order in wave
energy. In is assumed that the method can then be extended to higher
order nonlinear waves.
The treatment here is to use the energy method approach to describe
the wave field to the first order in energy (second order in wave ampli
tude). This solution is then substituted into generalized conservation
equations to extract the desired information.
30
B. Conservation Equations
A convenient starting point for this analysis is a statement of
the general conservation equations of mass, momentum, and energy fluxes
applicable to unsteady flow. The analysis is not concerned with the
internal flow structure of the fluid; hence, the derivation can be
simplified by integrating the conservation equations over depth. Conser
vation equations which have already been developed by Phillips [44]
will be used and are presented below.
The conservation equations will be applied to wave motion, but
they are equally applicable to general turbulent motion. The unsteady
velocity field of the wave motion can be expressed in the same manner
as in the treatment of turbulent motion as the sum of its mean and
fluctuating parts
u = (U (x,y,t) + u!(x,y,z,t), w(x,y,z,t)) i = 1,2 (2.1)
where (1,2) refer to the horizontal coordinates (x,y), respectively,
and z is the vertical coordinate. The tensor notation is used only for
horizontal components of water particle motion. The mean current is
assumed uniform over depth for simplicity. The pressure term can be
stated similarly. These expressions can be substituted into the mass,
momentum, and energy equations, and the mean and fluctuating contri
butions identified.
The conservation equations are to be averaged over depth and time
(one can consider averaging over a few wave periods). For the case of
waves superposed on a mean current, all the wave motion would be identi
fied with the fluctuating quantity which, when integrated over the total
depth, can contain a mean contribution due to the waves. The time
averaging of the equations for a general davilopment, being over a short
31
interval compared to the total time, does not preclude long term un
steadiness in the mean motion.
The utility of the conservation equations derived by Phillips is
that the terms involving the mean and fluctuating quantities have been
separated. This facilitates the understanding of the effect of the un
steady wave motion on the total flow phenomenon. They are a parti
cularly useful aid in gaining physical insight into the complicated
mechanisms taking place in the surf zone.
In addition to the "normally considered" conserved quantities of
mass, momentum, and energy, a fourth condition is available to aid this
analysisconservation of phase, or more simply, conservation of waves.
This condition, as applied to gravity waves, was first stated by
Whitham [45].
1. Conservation of Phase
The conservation of phase will be derived for simplicity in relation
to simple harmonic motion. The expressions to be derived are also appli
cable to more general wave motion which can be formed by the superposition
of individual Fourier components. The expression for the water surface
profile for harmonic motion is given by
= (xi) cos(kixi Wt) i = 1,2 (2.2)
where (xi) is the amplitude, and the quantity (k.x. at) is called
2 i
the phase function X. The wavenumber k. and frequency u can be defined
in terms of the phase function, such that
k = x_ (2.3)
I
u= aX (2.4)
at
An important property of the wavenumber can be seen immediately by
vector identity
V x k = V x VX L 0 (2.5)
so that the wavenumber is irrotational. From Equations (2.3) and
(2.4), the kinematical conservation equation for the wavenumber can
be written
ak.
S = 0 (2.6)
at ax.
For a single wave component, the conservation of phase says that the
rate of increase of the number of waves in a fixed length is balanced
by the net inward flux of waves per unit time. If a can be expressed
as a function of k. and possibly x, from local arguments (such as
assuming a uniform simple harmonic wave train locally),and further
>
allowing a mean current U, then,
a = o(k,xi) + kiUi (2.7)
a is the local wave frequency and is the apparent frequency to an ob
server moving with the current. Substituting into Equation (2.6),
Whitham (2op. cit.) shown
ak. ak. ak.
+ c + U. = G. (2.8)
at g x. j ax.
where
au
CG  k 1
i ax. j ax.
1 1
and
c 30 (2.9)
gj 3k.
J
c is the speed at which the energy or values of k. are propagated,
gJ J
commonly called the group velocity. The individual crests propagate
with the local phase velocity c = The notation shall be adopted
where, if c or k appear without vector notation, it is understood that
they represent the modulus of the vector quantity.
Equation (2.8) says that the rate of change of the wavenumber
ki, following a point moving with the combined group and convective
velocity, is equal to G.. Changes in G. are due to variations in the
1 1
mean current and bottom configuration. These geometrical equations,
which state the kinematical conservation for the wavenumber, hold for
any kind of wave motion (Phillips, op. cit.).
2. Conservation of Mass
The general conservation of total mass per unit area can be ex
pressed
p D + I = 0 i = 1,2 (2.10)
at 3x. 1
1
D is the total averaged depth of water which can include a mean elevation
n, above (or below) the still water depth h, so that
D(x,y,t) = (n + h) (2.11)
The overbar shall be used to signify time averages. The total mass flux
M. can be partitioned into its mean and fluctuating components
1
M. = H. + i.
1 1 1
(2.12)
34
The mass flux per unit width of the mean flow is
n
~i pUi dz = pDUi (2.13)
h
The mass transport of the wave motion is
M. pu' dz (2.14)
h
3. Conservation of Momentum
The equation defining the conservation of horizontal momentum is
derived by integrating the momentum equation, including shear stresses,
over depth and averaging in time. The balance of total momentum per
unit area can be expressed
at ( + S..) = T. + R. (2.15)
Here M. denotes the total horizontal momentum per unit area. Hence,
1
the first term on the left represents the rate of change of the total
mean momentum per unit area which includes both the current momentum
and wave momentum.
The total mean transport velocity can be expressed in terms of the
mass flux per unit area
M. M.
U U +  (2.16)
i pD i pD
The second term on the left of Equation (2.15) expresses the momcntur
flux of a stL:dy stream having the same mass flux M. and mean transport
velocity IU. as the actual flow together with an express momentum flux
1
term S.. arising from the superposed wave motion, where
1]
n
=fM.M.
Si (puu.u + P6j) dz pgD26 (2.17)
h
and 6.. is the Kronecker delta.
iJ
The term T. is given by
1
T. =  pg(n + h) (2.18)
1
and represents the net horizontal force per unit area due to the slope
of the free water surface.
R. is the time mean averaged shear stress which must be included
1
in any realistic treatment of the surf zone where dissipative effects
occur. The form of the shear stress was not explicitly stated by
Phillips in his derivation so this term will be expanded in detail.
Also, insight can be gained into the manner in which the momentum
equation was integrated over depth in arriving at the generalized
momentum conservation equation. Considering the shear stresses on a
column of water (see Figure 4) and integrating over the depth, R. can
1
be expressed
P aT. f PT.
dz + i dz i,j = 1,2 (2.19)
1 ;x. az
h h
where T.. includes the combined stresses of waves and currents. The
shear stress on the horizontal plane can be integrated over depth, such
that
z
y x
L N.
ix
.: ,*.' ...' .. 
rFtrc 4I. Shear SLresses Acting on the raceM of an
.E cmntal .ater Col :,in
L .... _____ _ ."_ ." __"_" __"_' " " : v .
Szdz = T.i hi (2.20)
h
where the subscripts refer to surface and bottom.
The Leibnitz rule of integration must be employed to integrate
over depth the stresses occurring on the vertical faces of the water
column. Evaluating the shear stress on the vertical faces and taking
the time average of the terms
11 n
f ^* a f an 3 h
d dz = T .dz .. (2.21)
dx. x 31 n 31xn i. h ax.
h h
Thus, the general shear stress term is given by
a an h 
ST..dz T +  (2.22)
i 3x. n jin 3x. jih ax. ni hi
h i,j = 1,2
4. Conservation of Energy for Fluctuating Mlotion
The equation for the conservation of total energy can be partitioned
into the energy contributions from the mean and fluctuating parts with the
aid of the conservation of mass and momentum equations. In the following
development, it is convenient to work with the energy balance for the
fluctuating motion alone, which can be stated
M. 2 2U.M.2 ;U.
(E 2 (UE + F ) + S = (2.23)
at 2Dh ax. i i 2pD Sij x.
1 1
In separating tle mean and fluctuating contributions, the mean energy
density has been represented as the total energy density of an "equivalent"
38
uniform flow with the same depth and mass flux as the actual flow, plus
the energy density of the fluctuating motion, minus an additional term
representing the difference between the energy density of the actual
mean motion and that of the equivalent uniform flow. Then, the first
term in Equation (2.23) is the rate of change of the energy density of
the fluctuating motion minus the correction term. The second term
represents the convection of the fluctuating energy E, by the mean flow,
the energy flux of the fluctuating motion F, minus the correction term.
The transport of energy by the fluctuating motion is given by
S[(u12 + w'2) + g(z ) + P] dz (2.24)
1j J p
h
which represents the rate of work done by fluctuating water particle
motion (for turbulence, this would be the work done by the Reynolds
stresses) throughout the interior region of the flow, plus the work
done by the pressure and gravity forces. The last term on the left of
Equation (2.23) represents the rate of working by the fluctuating motion
against the mean rate of shear. Dissipative effects of turbulence have
also been allowed where e is the rate of energy dissipation per unit area.
5. Conservation of Energy for the Mean Motion
The energy budget for the mean motion is given by
[! UiMi + g(n2 h2) + i 2 + g) + = .
2t ax. x. I I
1 i
(2.25)
The terms of this equation represent, respectively, the rate of change
of the kiretic and potential energy of the mean motion, the transport
39
of the total mean energy, and the rate of work by mean motion on the
fluctuating motion and bottom shear stresses.
In the development of the conservation equations, no restrictions
were placed on the wave slopes or amplitudes. Also, no restrictions
were placed on the fluctuating motion so that the equations are equally
applicable to wave or turbulent motion.
C. Description of the Wave Field
1. Waves outside the Surf Zone
The descriptive equations for the wave field are derived by, first,
solving the linearized boundary value problem over a horizontal bed. The
solution can, then, be extended to a higher order by perturbation tech
niques. The development presented here will retain terms to the second
order in amplitude (first order in energy and momentum) and neglect all
higher order terms. The wave solution can then be substituted directly
into the conservation equations providing a means for describing the
waveinduced mean motions. In making this substitution and dropping all
terms of orders higher than the second, only knowledge of the first order
(linear) wave water particle velocities and surface elevation is neces
sary. This is because, in expanding and then averaging over the period,
the terms involving higher order quantities in velocity and surface
elevation go to zero. The pressure must be known to the second order in
wave height; however, the average second order pressure component can be
determined from the first order water particle velocities and surface
elevation terms. Thus, only the linear wave solution is required.
In utilizing linear theory, it is assumed thar the motion is irro
tational and that the fluid is incompressible and inviscid. These have
been shown to be good assumptions, due to the fact that viscosity plays
only a very minor role in determining the hydrodynamics of the wave
field, and, thus, the vorticity is very weak [44]. The fact that the
linear theory is a good approximation is demonstrated by its success in 
describing many observed phenomena.
Assuming simple harmonic motion, the surface elevation is restated
as in Equation (2.2)
H cos(k.x. ot) i = 1,2
where the wavenumber components are
k = k eosa
x
(2.26)
k = k sina
y
The arbitrary angle of wave incidence a is measured between a line
parallel to the contours and the wave crests. The velocity potential
for the first order solution is given by
4  O_ csosh h sin(K.x. ot) (2.27)
2 ck cosh 1h a
The velocity is related to the gradient of the velocity potential
u.= V4 (2.28)
k
H L i cosh k(h + z)
u. = cosh kfh ) cos(k.x. ot) (2.29)
i 2 c k cosh kh i 1i
The frequency equation relates the frequency to the local wavenumber
anl water depth
o2 = gk tanlh 1h
(2.30)
41
The pressure can be determined as a function of the depth to the second
order by integrating Euler's vertical momentum equation over the depth
and retaining terms of second order. Only knowledge of the first order
water particle velocities and surface elevation is required to deter
mine the mean second order pressure term. The timemean pressure is
given as
p g 1 H2 H2 sinh2 k(z + h) (2.31)
p = sinh2 kh (2.31)
where the first term is the hydrostaric contribution, and the second
term is the second order mean dynamical pressure component.
The group velocity of the waves can be determined from Equation
(2.9)
k k.
i do o i
(c ) = n = c.n (2.32)
gi k dk k k i
where the transmission coefficient is
1 2kh
n (1 + 2(2.33)
2 sinh 2kh
1
For deep water (infinite depth), n = and, in shallow water, n
approaches 1.
The energy of the waves is proportional to the wave height
squared and is equally partitioned between the potential and kinetic
energy, such that the total energy is
E pg2 (2.34)
Recalling that the group velocity expresses the speed of energy propa
gation, the energy flux of the wave motion is
F. = Ec .
l gl
(2.35)
42
Assuming negligible wave reflection and a mildly sloping bottom,
Equations (2.32) and (2.34) can be substituted directly into
Equation (2.35) to obtain the flux of wave energy.
The mass transport of the waves can be determined by recalling
Equation (2.14) in which the fluctuating velocity was integrated from
the bottom to the surface. Since the water surface elevation is un
known, the integrand is expanded in a Taylor series about the mean
water surface level n
n n
M. = pu dz + p[u.(n) + z + ...] dz (2.36)
h n
where the time mean average of the mass transport below the trough is
zero (first integral), and only terms of second order in amplitude
will be retained. Substituting for the wave profile and water particle
velocity of the wave motion, Equations (2.2) and (2.29), the mass
transport of the waves is
E i
.E = p u (2.37)
i ui c k
Simple harmonic motion has been selected to describe the waves because
experience has shown that this solution gives fairly good results for
the deep water case. Inside the surf zone, however, the approximation
is not good, but the assumption of simple harmonic wave trains will be
retained with the exception of certain modifications on the celerity
and amplitude. The wave field could just as easily (in principle, anyway)
have been specified using the cnoidal or other higher order wave theories.
It is felt that, for the present investigation, the accuracy gained would
not justify the much increased complexity of the resulting equations.
2. Breaking Waves
The seaward edge of the surf zone is usually delineated by the
point where the waves first start to break. Inside the surf zone,
the waves are unstable, and the fluid motion tends to lose its ordered
character. Waves break in different ways, depending primarily upon
the wave steepness and slope of the beach. The manner in which they
break has a very definite influence on the hydrodynamics inside the
surf zone which, in turn, affects such quantities as the sediment
transport, longshore currents, and wave runup.
Based on observations by Wiegel [46], breaking waves are usually
classified as spilling, plunging, or surging. Spilling occurs when
the wave crests become unstable, curl over slightly at the top, creating
a foamy advancing face. Plunging occurs on steeper beaches when the
wave becomes very asymmetric; the crest curls over, falling forward of
the face resulting in the creation of considerable turbulence, after
which a borelike wave front develops. Surging occurs when the wave
crest remains unbroken while the base of the front face of the wave,
with minor turbulence generated, advances up the beach.
There is a continuous gradation in the type of breaking, and
Galvin [34], in a more recent classification, found it convenient to
add a fourth category of collapsing to describe a type intermediate to
plunging and surging. He performed extensive laboratory investigations
to quantitatively classify breaking waves according to the wave and
beach characteristics. Combining his results with earlier works, he
grouped the breaker type depending on beach slope tanS, wave period T,
and either deep water or breaker height H. The breaker type can be
sorted by either of two dimensionless combinations of these variables,
44
an offshore parameter, H /(L tan2B), or an inshore parameter,
Rb/(gT2 tanS). As either of these parameters increases, breaker
type changes from surging to collapsing to plunging to spilling.
Spilling breakers are associated with steep, relatively short period
waves and flat beaches; plunging breakers are associated with waves
of intermediate steepness and the steeper beaches; and surging breakers
are associated with waves of small height and steep beaches.
On natural beaches, breakers classed as spilling are most commonly
observed, followed in decreasing order of frequency by plunging, col
lapsing, and surging. In the laboratory tests, spilling breakers are
relatively rare compared with collapsing and surging breakers, because
slopes used in laboratory tests are usually steeper than slopes commonly
found in nature due to the physical limitations of space.
The breaking index curve provides a relationship between the breaking
depth Db, the breaking wave height Hb, and the wave period T. In shallow
water, the relationship simplifies to Hb/Db = K, a constant. Reid and
Bretschneider [47] compiled breaking wave data from several sources in
cluding both laboratory and field data. They found a fairly good corre
lation for breaking waves as compared to the breaking wave criteria
predicted by the solitary wave theory. The solitary wave theory pre
dicts a value for < of 0.78, while other theories predict slightly
different values. Theoretical values range from 0.73, found by Laitone
[48], for enoidal wave theory, to a value of 1.0, found by Dean [49]
using a numerical stream function theory. Experiments on steeper
laboratory beaches show that the value of < can be much larger. All
the theoretical values have been calculated, assuming a flat bottom,
and correspond to beaches with very gentle slopes. The important point
is that the bre;ilcr height is governed by the depth of water.
3. Waves inside the Surf Zone
Inside the surf zone, energy is dissipated due to the generation
of turbulence in wave breaking, bottom friction, percolation, and
viscosity. The waves in the surf zone constitute a nonconservative
system in which the use of potential flow theory is no longer valid.
In fact, there is no analytical description available for the waves
in the surf zone. Hence, one is required to make rather gross as
sumptions and then to test these assumptions experimentally. The
linear wave theory will be retained as the input to the conservation
equations, but with modification to the wave amplitude and speed. The
wave height inside the surf zone is controlled by the depth and is of
the same order of magnitude. Thus, even the second order theory in
wave height is a rather poor assumption, but seems to agree surprisingly
well with measurements of some phenomena.
Spilling breakers lend themselves to a physical treatment since
the potential energy and momentum flux of the waves inside the surf
zone can be expressed approximately in analytical form. If the beach
slope is very gentle, the spilling breakers lose energy gradually, and
the height of the breaking wave approximately follows the breaking
index curve [50]. The height of the wave is then a function of the depth.
In the breaking process, plunging breaker heights may also be inter
mittently described by the breaking index; however, due to rapid dissi
pation of energy in breaking, the height may fall below the breaking
index curve, and the residual wave energy may later reform and then break
again in shallower water. This was generally noted for moderately gentle
slopes in laboratory experiments by Nakamura et al. [51].
46
In the present analysis, it will be assumed that the waves act
v
as spilling breakers inside the surf zone and that they follow the
breaking index, K = 0.78, as predicted by the solitary wave theory.
The wave height inside the surf zone is then given by
H = KD (2.38)
It is further assumed that the kinetic and potential energy are
equally partitioned so that the total wave energy can be described in
terms of the wave height which is a function of the depth
E = pgK2D2 (2.39)
8
This is a nonconservative statement of the energy distribution within
the surf zone.
The waves inside the surf zone are assumed to retain their simple
harmonic character so that the wave profile and water particle velocity
are described by
II
n = cos(k xi ot)
2 ii
k (2.40)
u = c k cos(kixi ct)
The expression for the horizontal water particle velocity is based on
the Airy wave theory and has been simplified for shallow water.
In very shallow water, the waves are nondispersive with the wave
speed being only a function of the depth. It has been found experi
mentally that a reasonable approximation to the wave speed in the surf
zone is that predicted by the solitary wave theory [52].
c ,= i + K) = (i'+ <) .n
(2.41)
47
The wave field has been completely specified. These results may
now be substituted into the general conservation equations to describe
waveinduced phenomena inside and outside the surf zone.
CHAPTER III
LONGSHORE CURRENT THEORY
A. Statement of the Problem
The study of the area in and about the surf zone presents a diffi
cult problem due to its very complex nature. A proper treatment of the
surf zone must consider a threedimensional problem of unsteady fluid
motion and is further complicated by moving interfaces at the upper and
lower boundaries, that is, at the water surface and sediment bottom.
It was also noted earlier that more than one longshore current system
could occur for seemingly similar conditions. Thus, it is necessary to
make definite and simplifying assumptions in order to make the problem
tenable to a theoretical approach.
This analysis considers the steadystate distribution of quantities
on a line normal to the shoreline. A schematic of the surf zone area is
shown in Figure 5. The analysis is restricted to the case of an arbitrary
bottom profile with straight and parallel contours in the ydirection
(parallel to the beach). The water depth is then a function of the
xdirection only (perpendicular to the beach). Since the distribution of
mean properties of the wave field is a function of the depth, this elimi
nates any ydependence.
An exception to this was found by Bowcen [7]. Using the fact that
incident waves can excite transversal waver, commonly called edge waves,
he showed that if these ,'raves are standing a\.e';, or only slowly pro
gressive, gradients in the mean water surf, c can be developed in the
48
BREAKER LINE
WAVE CREST
LONGSHORE
CURRENT
DISTRIBUTION
Figure 5. Scheme of the Surf Zone
50
longshore direction which in turn can result in circulation cells. Thus,
a more exact formulation has to assume suitable spatial averaging in
the longshore direction so as to preclude the effects of any transversal
waves.
It is assumed that wave reflection is negligible. This assumption
is justifiable outside the surf zone for gently sloping bottoms. The
present analysis is shown to be most valid for spilling breakers which
implies a gently sloping bottom. The wave reflection is least for this
type of breaker condition and is assumed negligible inside the surf zone
as well.
Shear stresses at the surface due to the wind will be neglected.
Summarizing, this is a steadystate problem (except for periodic
wave motion) with no ydependence such that
3h D_ E V U (3.1)
= 0 (3.1)
Sy 3y 3y ay
The notation U = U and U = V will be used henceforth. No restrictions
x y
will be placed on the direction of wave approach for, indeed, it is the
wave motion in the ydirection that is the driving mechanism in the
equations of motion.
The problem can be conveniently discussed by considering separately
the areas outside and inside the surf zone. Since conditions inside the
surf zone are dependent on the incoming waves, the case seaward of the
breakers will be examined first.
B. Currents outside the Surf ZoneNeelecting Bottom Friction
A determination cf the distribution of mass transport and energy
of the waves is frsat necessary in oicer to solve for the waveinduced
currents outside the surf zone.
1. Mass Transport Velocity
Due to the absence of any ydependence, the mass conservation
Equation (2.10) reduces to
3M
ax
X=0 (3.2)
Integration gives
M = constant = 0 (3.3)
which must be equal to zero since the beach forms a boundary in the
xdirection. This then says
M E
U= cos = cosa (3.4)
pD pcD
which states that there is a mean reverse current balancing the mass
transport onshore due to the wave motion. This must be true everywhere,
both inside and outside the surf zone, to ensure that there is no accumu
lation of mass or growth of currents in the ydirection in order to
maintain steadystate conditions in accordance with the original as
sumptions.
2. Distribution of Energy outside the Surf Zone
The determination of the energy distribution is necessary as a means
of relating the wave heights at various depths. The energy, in turn,
must be related to the local angle of wave incidence. The angle of wave
incidence is affected by refraction which can occur since the waves are
allowed to approach at an arbitrary angle of incidence to the bottom
contours. The problem is further complicated since a shear flow is
allowed which can also produce wave refraction.
52
The general statement of the phase relation gives an expression
relating the frequency, mean current motion, and wave angle. Equation
(2.7) can be expanded to give
w = o + Vk sina + Uk cosa (3.5)
The assumption of steadystate conditions requires that a be uniform
and constant.
The wavenumber was shown to be irrotational
3k 3k
x = 0 (3.6)
3x 3y
Since the wave length and amplitude of the waves are independent of the
ydirection, the gradient of the local wavenumber in the ydirection
is zero.
ak 3k
S= 0 =  (3.7)
'y ax
Integrating the gradient of k in the xdirection gives
k sina = constant = k sina (3.8)
o o
where the subscript "o" refers to conditions in deep water. This is a
statement that, for straight and parallel contours, the projection of
the wavenumber on the beach is a constant. Substituting into Equation
(3.5), a general expression for the celerity of the waves can be de
rived where
S= k( + V sina + U cosa) (3.9)
or util]inng LquatL.ion (3.8)
c
0
c = ( _ V) sina U cosa (3.10)
sina
o
For U = V = 0, this expression simplifies to Snell's law for wave re
fraction.
The conservation of energy for the fluctuating motion, Equation
(2.23), can be expanded for steadystate conditions to give
U 1. 2 U 1 2
a (EU +Ec + SE + S (3.11
Dx x gx 2pD 2pD xy ax xx x (3.11)
where the gradients in the ydirection are zero. This equation states
that the change in energy flux, due to currents and waves plus the work
done by the excess momentum flux on the straining motion, is equal to
the energy dissipated by turbulence and work done on the bottom. The
product of a stress times rate of strain is a quantity that can be
associated with power per unit volume. The last two terms on the left
of the energy equation can be interpreted in this context where the
excess momentum flux tensor then represents a stress and the velocity
shear a rate of strain. LonguetHiggins and Stewart [53] were the first
to take cognizance of this, and named the excess momentum flux tensor
a radiation stress.
The excess momentum flux tensor can be determined by substituting
the wave expressions into Equation (2.17). In general terms of energy,
group velocity, and wave speed, an expression applicable to both inside
and outside the surf zone is given by
r c 2c c
E  cos2a + (s 1) E  sin2a
c 2 c 2 c .
S 1___
ij pD
C g sin2a E & sin2e + E (2 1
Sc 2 c(3.12)
54
The effects of turbulence and surface tension have not been included.
Referring to Equations (3.4) and (3.12), it can be seen that U and
S are of order E. The first and last terms of Equation (3.11) in
xx
volving the product of these terms are of order (E2) and, hence, will
be ignored in this analysis. From the result of the conservation of
mass, Equation (3.4), Ux = 0, so that the third and fourth terms of the
energy equation are zero. Substituting for Sxy and F and retaining
only terms of first order in energy, the energy equation reduces to
3 sin2a 3V
(Ecn cosa) + En = E (3.13)
ax 2 3x
where the substitution, c = en, has been made. Substituting for the
g
wave celerity,as given by Equation (3.10), yields
En [(0 V) sina cosa U cos2a] + En cosa sina V 
Dx sine ;x
(3.14)
Again recalling that U is of order E, and that all higher order terms
involving the product of U can be neglected in this analysis, the
energy equation can be written
c
En (s V) sina cosa + En cosa sina = E (3.15)
3x sina ax
Expanding and cancelling terms, gives
c
o _
S  V) En sina cosa = E (3.16)
sina 3x
Far outside the surf zone, energy dissipation due to bottom effects
can be ignored, and the energy losses due to turbulence can be assumed
negligible. Since the term in the brackets of Equation (3.16) is
55
nonzero, the result of the energy equation outside the surf zone,
assuming no energy dissipation, is
En sin2a = constant (3.17)
The relative amplification of the wave energy is then given by
n sin2a
E o o (3.18)
o
E n sin2a (3.18)
0
Since the term involving U in the celerity equation resulted in the
product of higher order terms that are not included, Equation (3.10)
can be written, consistent with this analysis
c
c = (sin V) sina (3.19)
0
or in terms of the wave angle
c 1
sin = sina [ ] (3.20)
o 1  sina
c O
o
Equations (3.16) and (3.18) give a complete description of the wave
amplitude and direction outside the surf zone.
For no current, Equation (3.10) reduces to Snell's law for wave
refraction and can be combined with Equation (3.18) to give
c
E c cos = E cosa = const (3.21)
g o2 o
where the local quantities are related to the conditions in deep water.
The changes in energy density E, or wave height, can be determined from
this equation as a function of the local wavenumber and water depth.
For decreasing depth as the waves approach the shore, the local wave
length and the angle of incidence decrease. The effect of shoaling is
determined by the group velocity. The group velocity initially in
creases slightly so the energy density decreases; the group velocity
4
then decreases resulting in a continual increase in the energy density
towards shore. A maximum wave height occurs at breaking. Due to the
change in wave angle, which is the result of refraction, the wave crests
become more nearly parallel to the beach. The energy density is less
for waves approaching at an angle to a constant sloping beach than for
waves whose orthogonals are normal to the beach because wave refraction
results in divergence of wave energy.
The effect of a shear current on the energy density can be illus
trated most simply by considering the case of waves in deep water where
the depth is noc involved as a parameter. The relative amplification of
the energy density as a function of V/c is shown graphically in Figure 6
for various values of the initial angle a It can be seen that for the
case of currents with a component in the direction of wave propagation
(positive ratio of V/c ) there is a decrease in energy density. For
angles greater than zero, there is a point at which the amplification
suddenly becomes very large and tends to infinity. As a t 90 degrees,
the infinity occurs for all currents in the positive direction. These
infinities represent caustics, or crossings of the wave rays. In reality,
the theory is no longer valid near such points.
As has been shown, there is an excess of momentum flux due to the
presence of the waves. For conservation of momentum flux, there must be
a force exerted in the opposite direction such as a hydrostatic pressure
force or bottom shear stress to b:'lance this excess momentum flux. It
will be prCusentco that oul.si.d the smuf zole the component of excess
mormentum flux directed perpendicuiair to the contours is balanced by mean
waPtr levell setdown.
1.5
1.4
1.3
20
.2 
0.8 
1.0 0.8 O.G 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0
00
Figure 6. Amrnlification of Wave Energy Density Due to a
Shear Current as a Function of the Incident
Wave Angle
58
3. Distribution of Currents outside the Surf Zone
There is a component of excess momentum flux directed parallel to
the shore due to the oblique wave approach. The question of whether
this radiation stress can induce a current can be investigated by con
sidering the general ymomentun equation. Neglecting changes in the
slope of the free water surface, Equation (2.15) can be written
T (Mx + S = R (3.22)
ax x xy y
where the gradients in the ydirection are zero. The conservation of
mass equation showed that Mx = 0; hence, the first term is zero. Com
parison of the energy equation solution, Equation (3.17), and the excess
momentum flux tensor, Equation (3.12), shows that
c
S = E _ sina cosa = constant (3.23)
xy c
Therefore, the change of momentum flux due to the waves and mean motion
in the ydirection is zerothere is no driving force for generating a
current outside the surf zone. The only waveinduced current far out
side the surf zone is then due to the mass transport velocity.
It shall be assumed V = 0 since there is no driving force for
generating currents. The mean current in the ydirection then reduces
to the mass transport velocity
M
U E sin (3.24)
y pD pcD
The nmas transport velocity generally decreases as the depth in
creases so that the longshore currents in deep water are very weak. This
is in ag'rement w'ith obsc vantons outside the surf zone. However, very
near the surf zone, the nmacu transport velocity tends to becou e first
59
order in amplitude and is no longer negligible; but, at the same time,
bottom effects become important making the derivation inaccurate.
Thus, very near the surf zone, dissipation of energy must be considered.
This will be discussed in Section C.
4. Setdown of Mean Water Elevation outside the Surf Zone
It is interesting to investigate the balance of momentum perpen
dicular to the beach. The xmomentum flux equation outside the surf
zone is given by
aS
Sxx a
x= pg(h + n) (3.25)
where dissipative effects have been neglected. The excess momentum
tensor was shown in Equation (3.12) to be a function of the wave energy.
As waves shoal, the amplitude, and hence the energy, increases (except
over a short distance in shallow water); thus, the excess momentum flux
tensor must also increase. The changes in the flux of excess momentum
is balanced by a slope of the mean water level. It can be seen from
Equation (3.25) that a positive gradient of the excess momentum flux
(increasing) results in a negative gradient of the mean water surface.
This results in an increasing setdown as the water becomes shallower.
Longuetlliggins and Stewart [54] have solved Equation (3.25) for the
case of waves approaching perpendicular to an arbitrary plane bottom.
They found
H2 k
n 8 nh 2 (3.26)
8 sinh 2kb
where it is seen that the setdown is a function of the local conditions
only. Thus, for the case of waves approaching at an angle, if the curva
ture of the wave rays is small, one would expect the same solution for
60
the local setdown. Mei (o_. cit.) solved the problem of waves shoaling
over a plane bottom by means of a perturbation expansion, taking into
account the sloping bottom boundary condition. He derived an expression
relating the local setdown to the deep water conditions
H 2 cosa
2 o o k coth kh
8 cosa (2kh + sinh 2kh (3.27)
This can be expressed by the local conditions using the solution to the
energy equation which relates deep water conditions to shallow water
conditions. Equation (3.27) then reduces to the same expression as
Equation (3.26) for the local setdown.
This setdown outside the surf zone has been verified experimentally
by Bowen et al. [55] in the laboratory; although close to breaking,the
experimental values for setdown were less than the predicted values. As
breaking is approached, and the wave height becomes larger, the second
order theory becomes less valid and would be expected to give less accurate
results. No energy dissipation has been accounted for, which can become
important in shallower water. The effect of including energy dissipation
would be to decrease the energy density as the waves shoal, which pre
sumably would decrease the energy density gradient. This would result
in a decrease in the excess momentum gradient, decreasing the setdown.
This would help explain the discrepancy between theory and e:perim;nt.
C. Currents uutside the Surf ZoneIncluding Bottom Friction
The energy equation has been used to relate the changes in wave
height dun to changes in the wave direction, current velocity, and depth
at a prrtLJculal location. Having determined the wave height distribution,
the morarLtu and continuity equations were used to solve for currants
61
generated parallel to the shore. Dissipative effects were neglected in
the above derivation of the longshore currents, and the evaluation of
the energy equation was rather straightforward. Near the surf zone, in
shallower water, the bottom effects can no longer be neglected, and the
dissipation term must be included in the derivation. It will be assumed
that all the energy dissipation outside the surf zone is due entirely to
bottom friction and percolation. Turbulent and viscous energy dissi
pation above the boundary layer is very small and will be neglected.
Energy utilized in transporting sediment, either along the bed or in
suspension, will be assumed to be included in the bottom friction loss
term and will not be considered separately. This aspect of the problem
will be considered in Chapter IV.
The lnss of mechanical energy is due to work done by the turbulent
shear stresses acting on the bottom. To evaluate the dissipation due to
bottom friction, it is first necessary to investigate the shear stresses
occurring at the bottom.
1. Bottom Shear Stress Due to Combined Waves and Currents
The determination of the bottom bed shear stress for uniform steady
flow has been fairly well established. For oscillatory flow, and particu
larly combined waves and currents, the bed shear stress is not so well
formulated. This is due primarily to the lack of good empirical data.
Jonsson [56J compiled the available laboratory data and conducted ad
ditional experiments dealing with turbulent boundary layers in oscilla
tory flow. From this study, he developed a classification of the flow
regimes similar to that for steady flow. The classification is based on
a roughness parameter and characteristic Peynolds number. It is found
that the wave boundary layer in nature can always be considered in the
"hydrniauli cally rough" Uinrhulcnt regime.
62
Unlike the boundary layer in open channel flow, which essentially
extends over the entire depth of flow, the boundary layer under wave
motion constitutes only a very small fraction of the vertical velocity
distribution. This is because the boundary layer does not have an
opportunity to develop under the unsteady velocity field of the wave
action. Above the boundary layer, the free stream is well described
by the potential flow theory (at least for small waves).
Jonsson used an oscillating flow tunnel to simulate prototype wave
conditions in the laboratory. In the experiments, he was able to measure
the vertical velocity profile in the boundary layer and determine the
bottom shear stress. He found that for simple harmonic motion the velo
city profile in the boundary layer could be approximated by a logarithmic
distribution and that the instantaneous bottom shear stress was related
to the velocity by the quadratic shear stress formula
hw fw 2 uj hl (3.28)
hw w 2 nih
;hw = fw tUth2 cos( t 8)Cos(U t 0) I (3.29)
where uh is the velocity amplitude of the oscillating flow just above
the boundary layer, 6 is a phase lag, and f is a friction factor
associated with the wave motion.
He found, further, that the friction factor was practically constant
over an oscillation period. The c.nrtancy of the frictJon factor ior
particular flow conditions is an important result which allow for a
better analytical determination of the corbinrcd shear stIess dWu to w'aves
and currents. Userng the availahbl datt front several source s, he found
that the friction factor for wave motion alone for rough turbulent
boundary layers could tentatively be represented by
1 1 h
+ log = 0.08 log  (3.30)
4Wr
w w
where r is a measure of roughness, and 4h is the maximum water particle
excursion amplitude of the fluid motion at the bottom as predicted by
linear wave theory
H 1
C H 1 (3.31)
'h 2 sjnh kh (3.31)
Equation (3.30) is based on the roughness parameter r being a
measure of the ripple height. The wave friction factor is seen to be
a function of the wave characteristics. This is because, for granular
beds consisting of a particular grain size, the ripples adjust their
dimensions according to the wave motion, and it is the ripple geometry
that determines the effective roughness.
Unfortunately, there is only, at best, a qualitative understanding
of the ripple geometry as related to sand wave characteristics. Generally,
the ripples are much more symmetrical in shape and much longer crested as
compared to those found in alluvial channels. Inman [57] collected a
large number of observations of ripple geometry and wave conditions from
Southern California beaches. These observations extended from a depth of
170 feet to the shore. Since the wave and sand characteristics vary from
deep water to the beach, the ripple geometry would be expected to vary
also. These observations showed that the size of the sand is the most
important factor in determining the geometry of the ripple. In general,
the coarser the sand, the lager the ripples. Also, there was a general
64
correspondence of decreasing ripple height with decreasing water depth.
The ripples were smallest in the surf zone where the higher orbital
velocities of the waves tended to plane the ripples off; the ripples
were almost nonexistent for surf zones with fine sands. The ripple
wave length was related to the orbital excursion. As the bottom orbital
velocity of the shoaling waves increased, the ripple length decreased,
increasing the effective roughness, but, at the same time, the ripple
height is decreased, decreasing the effective roughness. Since quanti
tative relationships among ripple geometry, sand grain size, and wave
characteristics have not been determined, one is forced to rely on ob
servations to determine the effective roughness, and, for this reason,
Inman's data are especially relevant.
It is now desired to superpose a mean current on the wave notion
and determine the total shear stress due to wavas and currents combined.
The difficulty in using the quadratic shear stress formula is in stipu
lating the friction factors. The wave friction factor was seen to be a
function of the wave properties for a deformable bed, that is, the fluid
motion; whereas, the friction factor for steady currents for rough turbu
lent boundary layers is only a function of the system geometry. It would
seem reasonable to expect that for weak currents, as compared to the water
particle motion of the waves, that the wave dynamics would dominate the
hydrodynamical system. For this reason, it is desirable to derive the
combined bottom shear stress in terms of the wave friction factor alone,
even though it is less well defined than the friction factor for steady
cul rents.
I: is assumed that the Loral instancan'oius bed shlar stress for
combined wives aid currents is re]aled to the velocity by
Th = p 2 V V (3.32)
where v is the resultant instantaneous velocity vector of the combined
wave and current motion, and f is the friction factor. Since the problem
has been formulated as a combination of wave and current motion, it
proves convenient to resolve the shear stresses into components in the
direction of the wave and current components. To do so, the velocity
is first resolved into components in the direction of the current and
wave motion
)(3.33)
vl= (u + V2 + 2u V sin) (333
w w
where u is the instantaneous velocity of the wave motion measured just
w
above the frictional boundary layer near the bottom. V is the mean
motion which was assumed uniform over the depth and in the longshore
direction only. Resolving the component shear stresses in the direction
of the velocity vectors results in the shear stress and velocity vectors
being proportional to each other as can be seen from the geometrical
representation in Figure 7. The shear stress component for the wave
motion can then be written
>
 > w f i i
I T= P IV U (3.34)
hw h 2 wU
v
and, similarly, for the shear stress in the direction of the mean current
y = p P J V (3.35)
v
If friction factors for the wave and current motion are now defined,
f = f I (3.36)
w wl
\2 \
S+1....
Figure 7. Resolution of Currents and Waves into
Respective Velocity and Shear Stress
Components
f = f (3.37)
y
the shear stress components can then be written
f
hw = u (3.38)
hw 2 w w
f
S = p V2 (3.39)
hy 2
The shear stress for waves, as defined in Equation (3.38), is in the
same form as given by Jonsson (op. cit.) for which information of the
4 friction factor was found experimentally. Knowing f and the velocity
of the waves and currents, the other friction factors and, hence, shear
stresses can be determined. The total shear stress, in terms of the
shear stress components, is
h= ( w2 + by2 + 2Thw hy sina)2 (3.40)
l h' hw bhy hw hy
Combining Equations (3.32) and (3.36), the total shear stress can also
be written
Sf
Th= p vU (3.41)
or substituting Equation (3.33)
f 2
S U (uw2 I V2 + 2u V sina) (3.42)
Since only the steadystate conditions are being considered, it
is desired to find the time nean values. It will be assumed a prior
that near and inside the surf zone the mean currents are much less than
v
the maximum velocity of the wave motion. Only the case of just outside
the surf zone is considered here, hut it will be shown later that the
68
equations developed are valid for inside the surf zone as well. The
time mean value of the absolute value of the water particle velocity at
the bottom due to the waves is found by integrating over time Equation
(2.29) to give
2 H gk (3.43)
uwl n 2 o cosh kh
Assuming relatively small incident wave angles a, and that V << lu,
as a first approximation
v = I (3.44)
The total bed shear stress is then given by
 wi*; (3.45)
Th = p 2 .u (3.45)
b 2P wjw
This result will be used later in calculating frictional energy dissi
pation at the bottom.
2. Changes in Wave Height outside the Surf Zone Due to Bottom
Friction and Percolation
The energy equation, Equation (3.11), can now be written to include
the dissipation term. Some of the higher order terms previously neglected
become first order in energy in shallow water and might be included.
Collectively, the contribution of these higher ordc.r terms is only approxi
mately 4 per cent at the breaker line and, considering the. inaccuracies
in calculating the energy dissipation terms, can be neglected. This leads
to a simplified energy equation
(Ecn coha) = (c + e ) = e (3.4b)
where Ff is the mlean rate of energy di!;sipnt.on doe to bottom fiction,
and eq represents the mean rate energy dissipation due to percolation.
It is assumed that the instantaneous rate at which energy is dissi
pated per unit bed area due to bottom friction is given by
4
Ef = 'h'vh (3.47)
The dissipation function is to be substituted into the energy equation
I
concerned only with the fluctuating motion, so that v = u + The
w pD
second velocity term being of second order is much less than the mean
of the absolute value of the velocity of wave motion and will be
neglected. The bottom shear stress is given by Equation (3.45). Jonsson
(oo. cit.) evaluated the energy losses experimentally using the dissi
pation function as given above. He found that the phase shift in the
shear stress term could be ignored, and the instantaneous dissipation
term evaluated in terms of the wave particle motion at the bottom.
f
f = p I u mh3 s3c wt (3.48)
The energy friction factor f would not necessarily be expected to be
equal to the wave shear stress friction factor f since a phase shift
was introduced in the shear stress term. Indeed, it was found experi
mentally by Jonsson that they were not equal for laminar flow. However,
for rough turbulent boundary layers, f and f were, for all practical
purposes, the same and constant over the wave period for the experiments
conducted. Thus, the energy friction factor can also be given by Equation
(3.30).
Since the frictional energy dissipation is defined in terms of the
absolute value of the bottio velocity, and theL velocityshear stress
product is a symmetrical oddfunction, the time mean value is equal to
70
the average of Equation (3.48) over onehalf the wave cycle. The mean
frictional energy dissipation is then given by
Pfw oH 3 3
Cf =12s (sinh kh) (3
This is the same dissipation term found earlier by Putnam and Johnson
[58] who made similar assumptions in their derivation.
Putnam [59] also theoretically examined the energy losses due to
percolation. If the bed consists of permeable material, pressure
variations due to the wave motion induce currents in the permeable 
layer, and these currents will dissipate some of the mechanical energy
of the wave.
For a permeable bed whose depth is greater than approximately one
third the wave length L, the amount of mean energy dissipated,egby
viscous forces in the permeable bed per unit area of the bottom per unit
time is given by
c 2 pk112 ( )
I 2v (cosh kh)2 (3.50)
where jis the permeability coefficient of Darcy's law, and v is the
kinematic viscosity of the water. If the depth of the permeable bed is
less than 0.3 L, a more complicated expression must be used so that it
will be assumed here that the bed layer is always greater than this
depth.
It is necessary to evaluate Equation (3.46) numerically due to the
nonlinear dissipation terms. This can be written in terms of the wave
height in a difference form
I = H K K [1 8 n Ax]2 (3.51)
n n1 s r p. (112c cosca)> .
g
where
K (C ) n (3.52)
s (c )n
gn
is the shoaling coefficient reflecting the changes in the group velocity,
and
cosa
K = ( n1) (3.53)
r cosa
n
is the refraction coefficient for a constant sloping bottom. The term
in the brackets of Equation (3.51) includes the effect of energy dissi
pation. This equation is equivalent to Equation (3.18) for no dissi
pation, E = 0. The wave height can then be evaluated by starting at a
known value and proceeding to any desired position where the changes
in the wave height compared to the initial wave height are found by
integrating the dissipation term.
Savage [60] conducted laboratory studies to test the above equations
and found reasonable agreement for the changes in wave height due to
frictional dissipation. He further found that the energy losses by perco
lation were less than the theoretical values by a factor of between 4 and
10. The tests indicated that for sand sizes less than 0.5 millimeters,
there is practically no loss due to percolation. On the other hand, for
sand grain sizes larger than 2 millimeters, the percolation losses could
become very large.
Bretschneider [61] tested the above equations with field data from
the Gulf of Mexico. Percolation losses were not considered in his analy
sis since the bottom consisted of relatively impermeable fine sediments.
Comparison of field measurements showed that using friction factors that
72
were close to those predicted by Equation (3.30) gave results in reason
able agreement with the theory. Thus, it is assumed that the use of
Equation (3.51) for predicting the wave height at a particular position
is valid and will be used in this analysis.
3. Distribution of Currents
Having determined the wave height as a function of the incident wave
angle, wave period, and local depth, the waveinduced currents parallel
to the shoreline can be determined from the ymomentum equation
as
xy = L 1 (
x = ( pgH n cosa sina) = hy (354)
This equation states that the change of the ycomponent of momentum in
the xdirection is balanced by the bottom shear stress directed in the
ydirection. Only the shear stress acting on the bottom is taken into
account. Since the wave height could not be expressed in an explicit
form, it becomes necessary to solve for the changes in the excess
momentum flux numerically. The longshore velocity is then found from
the calculated bottom shear stress as related by the quadratic shear
stress formula
V2
S= pf 2 (3.55)
hy y 2
where the time nean value of f can be determined from Equations (3.36),
y
(3.37) and (3.43)
"h fw 1 fw H (3.56)
fy V w V o cosh kh (3.
and f is determined by Equation (3.30). Therefore, the velocity outside
the surf zone, taking bottom friction into account, is
73
V o cosh kh 3 (H2n sin2a) (3.57)
8 f kH ax
w
This gives the velocity in terms of measurable quantities.
D. Longshore Currents inside the Surf Zone
It has been shown that longshore currents could be induced by
changes in the momentum flux of the waves directed parallel to the
shore outside the surf zone. The wave height inside the surf zone
and, hence, the water particle motion of the waves decreases due to
intense energy dissipation. It might be expected that, due to the more
rapid changes in momentum flux inside the surf zone, faster currents
can be generated.
A simplified approach will be investigated first, and, then,
additional refinements will be included to illustrate various aspects
of the problem and to obtain more accurate results. A uniformly
sloping bottom is initially assumed so that the physical principles in
volved are not lost in the algebraic details. A more general profile
is then considered later. Also, the first solution considers the re
sisting force parallel to shore to be only due to shear on the bottom,
and internal shear stresses are neglected.
Again, it is first necessary to specify the energy distribution
across the surf zone in order to determine the distribution of momentum
flux. The energy, as related to the wave height, was shown to be a
function of the local depth. Outside the surf zone, it was shown that,
due to the excess momentum flux directed perpendicular to shore, there
was a setdown of the mean water surface. The changes in the mean water
surface elevation inside the surf zone are investigated next.
74
1. Wave Setup inside the Surf Zone
Both the wave height and speed are a function of the total local
depth of water. Since there is an unknown change in the mean water
level to balance the excess momentum flux of the waves, the total
depth of water is unknown and must be derived to specify these quanti
ties.
Conservation of momentum flux in the xdirection states
x(U xx + S ) = Tx (3.58)
ax xx xx x
where there are no gradients in the ydirection. It was explicitly
assumed that the net mass flux perpendicular to the beach was zero so
that there is no contribution from the mean motion to the momentum
flux perpendicular to shore. This is necessary to comply with the
steadystate assumption. If a net mass transport were allowed into
the surf zone and there were no rip currents or other offshore return,
the currents would grow unbounded. Dissipative effects, such as bottom
friction and turbulence, have not been explicitly stated but are
accounted for in the potential energy decay. Equation (3.58) reduces to
S = T = pgD (3.59)
ax xx x ax
which states that the change of excess momentum flux due to wave action
is balanced by a change in the mean water level.
It is assumed that the excess momentum flux tensor inside the surf
zone can be expressed in terms of the energy and wave speed in the same
form as in shallow water. This assumption implies that even under the
breaking waves, water particle motion retains much of its organized
character as described by linear wave theory.
75
Experiments conducted in the field by Miller and Zeigler (op. cit.)
show that this assumption is not as extreme as one might first expect.
Velocity meters were used in these experiments to measure the internal
velocity fields in breaking waves. The results showed that the internal
velocity field of the near breaking and breaking waves, corresponding to
spillingtype breakers, could be qualitatively described by Stokes'
third order wave theory. The tests also showed that different types of
breakers had different internal velocity fields as would be expected,
and that the spillingtype breaker was the most organized.
From Equation (3.12)
S = E(cos2a + ) cos2a (3.60)
xx 2 pc2D
where S is a function of the energy density which decreases inside the
xx
surf zone. Hence, both the momentum flux and the balancing force have a
negative gradient so that the change in the mean water level inside the
surf zone will be positive, representing a wave setup.
The phase relation, as given by Equation (3.20), relates the velo
city and wave speed to the incident angle relative to the conditions at
the breaker line, and can be written
(Vb V)
sin = sinb [1 +  sinab] (3.61)
cb Cb
In this investigation, waves having a range of periods of approximately
418 seconds are considered. Waves of these periods are greatly re
fracted before reaching the depth at which they break. Thus, the angle
of wave incidence at the edge of the surf zone is generally small
usually much les;n than 30 degrees. Assuming the maximum value of the
76
longshore current velocity would be approximately equal to the component
of the wave speed in the longshore direction, then
IVb V < V < cb sinab (3.62)
so that
Ivb vl (cb sinab)
cVb sinab < cb i sinab << 1 (3.63)
cb b c b
Thus, an approximation for Equation (3.61), using the binomial expansion,
can be conveniently employed.
(V V)
sina sin [1  sino + ...] (3.64)
bb Cb
The cos2a term is given by
2 (\(V V)
cos2 = 1  sin2ab [1 2 b sinab + ...] (3.65)
Cb b cb b
or as first approximations
since = c sin (3.66)
cosa = 1 ( ) sin2 (3.67)
b
Putnam et al. [16] used the same assumption as in Equation (3.63)
and verified it in the laboratory and field. le found that the long
shore current never exceeded the longshore component of the wave
celerity.
Galvin [24] makes a special point to take exception with this
assumption, pod, thus, it is necessary to justify its use hore. In
his laboraLory experiments, he found that Lhe moan longshore velocity
often exceeded the component of the wave speed in the longshore di
rection. This was also found in a similar laboratory study by Brebner
and Kanphuis (og. cit.). Galvin points out that this is because much
of the water composing the breaking waves has been extracted from the
surf zone and that this water already has a longshore velocity which
is added to the breaking wave component.
This seeming discrepancy in experimental results is clarified by
examining the kinematics of breaking waves. The waves broke by plunging
in Galvin's experiments; whereas, in the experiments of Putnam et al.,
it is implied the waves broke by spilling. Iversen [62] conducted labo
ratory studies of breaking waves on various beach slopes. He found that
the backwash velocity in spilling breakers was much less than in plunging
breakers so that the fluid contribution to the spilling breaker is rela
tively less. Also, surf zones with spilling breakers are much wider with
the backwash having much less effect. Galvin's argument does not apply
to spilling breakers on flat beaches which derive little fluid from the
surf zone at incipient breaking. Thus, the assumption given by Equation
(3.63) appears to be valid for spilling waves.
Changes in the total water depth can now be solved by substituting
Equations (2.39), (2.41), (3.60) and (3.67) into (3.59), resulting in
 1 2 i_) 1 3
SpgK2D2 [( 8( ) (1 sin ) + = ogD (D h)
(3.68)
Expanding and combining terms, this eq oation can be easily inte
grated to give
2 22 sin2:
[3  (1  .) +1 D U2 ii bD = h + constant
[3 (1 Tb
12(1 + ) S 1 (+ )9) 2D
(3.69)
78
Imposing the boundary condition that Db = hb + b at the edge of the
surf zone, and using the equivalent approximate form of Equation (3.66),
Db
sin2 = sin2ab b (3.70)
The total depth variation is
D = ( K() Db + (1 K(a)) (h hb) (3.71)
(1 K(ab))
where K(a) is a function of the local wave angle such that
K(1)=1 1 (3.72)
12(1 + <) (1 + <) 2
This form corresponds to an earlier formulation by LonguetHiggins and
Stewart [54]. It should be noted that this equation is applicable to
an arbitrary bottom profile and that the wave setup depends only on
the local still water depth. The only restriction on the bottom profile
is that it monotonically decreases shoreward in order that the assumption
of spilling breakers be fulfilled across the surf zone, that is, that the
wave height must always be a function of the local depth.
The effect of oblique wave incidence on the setup can be investi
gated most simply by assuming a constant sloping beach. Figure 8 shows
the relative changes in the local wave setup for various angles of wave
Y
incidence. It can be seen that the wave setup decreases for increasing
angles, as would be expected.
For all practical angles of wave incidence, as shown in Figure 8,
the change in the total local depth is less than 2 per cent. The setup
is ipiportant in this analysis for calculations of the wave height. Re
calling that the wave height i; assumed proportional to the total depth,
0.05
0.03
0.03
D
0.02
0.01
201'
. 10 
1.0 0.8 0.G 0. 0.2 0.0
Figure E. Relative Changes in Wave Setup insiJe the Surf Zone
as a Function of the Incident Wave Angle
_~ _~_ __ ~
80
an error of less than 2 per cent in the wave height calculations is
introduced by neglecting the angle of wave incidence. This approximation
is certainly within the accuracy of the basic assumption that the wave
height is proportional to the depth. Hence, for purposes of this
analysis, the effect of oblique wave incidence on the wave setup in
side the surf zone will be neglected to simplify calculations. The
equation for wave setup then reduces to the case of waves normal to
the beach. This is essentially the same solution obtained by Longuet
Higgins and Stewart ibidd.) where they considered the case of waves
normal to a beach only. The difference in the solution given here is
due to the inclusion of the momentum flux of the mean wave motion in
the radiation stress term.
The depth inside the surf zone simplifies to
D = (1 K) h + Khb + b (3.73)
and the wave setup inside ihe surf zone is given by
n = K(a) (h. h) + 0b (3.74)
where the setdown at the breaker line nb can be determined from
Equation (3.26).
Bowen et al. [55] conducted laboratory studies to verify the
theory predicting the change. in mnan sea level for waves normally
incident to the shoreline. In his experiments, a plane beach was used
with a slope of 1:]2. The. brciain", Iv'vcs wc:e of the plunging type,
and, after breaking, the residual energy propagated shoreward in the
torm of a bore.
81
Well outside the breaking point, the prediction of the setdown
by Equation (3.26) and the measured values compared very well. Figure 9
shows a typical observation from Bowen's experiments of the wave height
envelope and mean water level.
Just outside the breaking point, the assumptions of linear theory
are no longer justified, and the measured values of the setdown are
less than predicted by the theory. Between where the waves start to
break and the plunging point, the setdown was found experimentally to be
rather constant. The measurements showed that inside the plunging point
the wave height is very nearly proportional to the mean water depth.
This supports the basic assumption of this analysis that H = KD. As
Bowen points out, it is surprising that the setup is so well described
by theory using the approximation for the wave momentum flux since linear
theory is essentially assumed valid inside the surf zone.
2. Velocity DistributionConstant Sloping Bottom
With the mean water profile and energy distribution specified, the
variation of the longshore current across the surf zone can be deter
mined. The ymomentum equation inside the surf zone can be written
similarly to that outside the surf zone.
aS
= R (3.75)
@x y
Again, recall that the net mass transport perpendicular to the beach is
zero, and that there is no ydependence. This equation says that the
change in the momentum flux parallel to the shore due to the waves is
balanced by the force required to overcome bottom friction and internal
lateral shear stresses which must maintain a longshore current. A loss
WAVE PERIOD 1.14 sees.
WAVE HEIGHT H,06.45 cms. Hb
BEACH SLOPE Tan = 0.082
8.55 ciis.
MEAN WATER LEVEL, 7)
THEORY
S .W.L.
EXPERT II!ENT
ENVELOPE OF WAVE HEIGHT
WAVE CREST
.nsta.tW'arg.rar'
r .
400 300 200
DISTANCE FRO' STILL IATER!
100 0
LINE O01 PEAhl. x (ims).
Figure 9. Larboratory :leasure%,..nts 0o !1V'..c Setdow,. and Cetun
(after Bower. [55])
BEACH 
0
0
o
o
o
POINT
PLUNGE
POINT
2.0
1.5
c rms.
1.0
0.5
L0.
4
2
Z ces.
 
. m I I XK KI( X I
  p0
83
of momentum due to potential energy decay across the zone is also implied.
It is assumed that the changes in momentum flux are balanced solely
by the bottom shear stress parallel to shore, R = Thy Substituting
for S from Equation (3.12) gives
xy
a E2
x [E sina cosa 7 coso sinal = Thy (3.76)
ax pc D hy (3.76)
The height, celerity, refracted angle of the waves, and, hence,
changes in momentum flux inside the surf zone can be expressed in terms
of the local depth of water. Substituting Equations (2.38), (2.41),
(3.66), and (3.67) into Equation (3.76) gives
3 1 2 1 <2 sin.b 5/2 D sin2 b
x[8 pg <2 (1 D1)+] (1
ix 8 8 U( + <) Db 2 hy
(3.77)
which can be written
3/2 D aD
AD3/2( 0.7 sin2b) n= hy (3.78)
Db b x hy
where
S2 sinab
A pg .2 (I
16 1 + )
This is a general equation expressing the changes in momentum flu:: across
the surf zone in terms of the total local depth of water. Included in
the formulation is the setup of water and the effects of wave refraction
inside the surf zone. This equation is subject to the restriction that
the waves be described as spilling breakers, and, hence, the depth con
tinuously decreases shoreward from the breaker line. Thus, the momentum
flux decreases monotonically inside the sulf zone since both the energy
and wEve angle decrease with doerea; ;ing dep.h.
