Wind and Wind Wave Sediment Transpor t
in Large Lakes and Reservoir s
by
W .G . McDouga l
C .P . Le e
Water Resources Research Institut e
Oregon State Universit y
Corvallis, Orego n
WRRI-99
September 1984
WIND AND WIND WAVE SEDIMENT TRANSPOR T
IN LARGE LAKES AND RESERVOIR S
by
W .G . McDougal and C .P . Lee
Department of Civil Engineerin g
Oregon State University
Final Technical. Completion Repor t
Project Number G864-0 8
Submitted to
United States Department of the Interio r
Geological . Survey
Reston, Virginia 2209 2
Project Sponsored by :
Water Resources Research Institut e
Oregon State University
Corvallis, Oregon 9733 1
The research on which this report is based was financed in part by the Unite d
States Department of the Interior as authorized by the Water Research an d
Development Act of 1978 (P .L . 95-467) .
Contents of this publication do not necessarily reflect the views an d
policies of the United States Department of the Interior ., nor does mention o f
trade names or commercial products constitute their endorsement by the U .S .
Government .
WRRI-99
September 1984
ABSTRAC T
A rather simple model has been proposed to examine the effect o f
wind and wind waves on shoreline erosion . The technique employed fo r
estimating waves is that developed by the Corps of Engineers . Th e
solutions for the combined wind and wave setup is based on radiatio n
stress concepts, as is the generation of the longshore current . Th e
sediment transport model is based on energetics . And finally, th e
shoreline evaluation model is based on conservation of sediment . Th e
methodology and numerical results were produced for the first fou r
tasks . Only the methodology was outlined for the fifth task .
Without the completion of the fifth task, it is difficult to quantify the effects of wind wave erosion on the shoreline . However, it i s
clear that the sediment transport is increased in a narrow band alon g
the water line . Whether this sediment transport is of sufficient magnitude or duration to cause significant erosion remains unanswered .
FOREWORD
The Water Resources Research Institute, located on the Oregon Stat e
University campus, serves the State of Oregon . The Institute fosters ,
encourages, and facilitates water resources research and educatio n
involving all aspects of the quality and quantity of water available fo r
beneficial use . The institute administers and coordinates statewide an d
regional programs of multidisciplinary research in water and relate d
land resources . The Institute provides a necessary communications an d
coordination link between the agencies of local, state, and federa l
government, as well as the private sector, and the broad research community at universities in the state on matters of water-relate d
research . The Institute also coordinates the interdisciplinary progra m
of graduate education in water resources at Oregon State University .
It is Institute policy to make available the results of significan t
water-related research conducted in Oregon's universities and colleges .
The Institute neither endorses nor rejects the findings of the author s
of such research . It does recommend careful consideration of the accumulated facts by those concerned with the solution of water-relate d
problems .
ii
TABLE OF CONTENTS
Pag e
INTRODUCTION
1
METHODOLOGY
2
1.
Wind Waves
2
2.
Water Depth
4
3.
Longshore Current
16
4.
Sediment Transport
19
5.
Shoreline Evolution
23
CONCLUSIONS
24
REFERENCES
25
iii
LIST OF FIGURE S
Figure
Pag e
1.
Effective fetch method
3
2.
Forecasting curves for wave height
5
3.
Forecasting curves for wave period 6
4.
Definition sketches for nearshore domain 8
5.
Wind stress time coefficient 13
6.
Setup profile
15
7.
Longshore current profiles
20
8.
Longshore sediment transport profiles 22
iv
INTRODUCTIO N
There are many large bodies of fresh water in the Pacific North west . Significant wind waves may be generated on these bodies of wate r
which increase sediment transport along the shoreline . This increase d
transport may result in erosion and turbidity . Higher levels of turbidity reduce the environmental and recreational quality of the lake .
Increased erosion may result in a terraced shoreline which has reduce d
recreational and aesthetic value and is more difficult to manage as
a
flood control reservoir .
A model is proposed to address this problem and involves severa l
specific tasks . The first task, estimating wave conditions, employ s
existing methods . Next, an expression for the water depth along th e
shoreline is developed including the combined effects of wind an d
waves . The wind setup is due to a surface stress at the free surfac e
while the wave setup is due to the onshore-onshore radiation stress .
Longshore currents are also calculated for the combined effects of win d
and waves . The forcing mechanisms are similar to those in the setup ,
but the flow is now resisted by a bottom drag rather than a pressur e
gradient . A sediment transport model is presented which is based o n
energetics . This model has been widely used in the marine environmen t
and the empirical transport coefficient has been determined . The fina l
step, determination of the shoreline response due to the longshor e
currents and waves, is discussed but no analytical results ar e
presented .
METHODOLOGY..
The determination of the wind wave effects on sediment transpor t
involves several specific tasks . These include : 1) estimating th e
waves from the wind speed and duration and the lake depth and length ; 2 )
determining the influence of the wind and waves on water depth at th e
shoreline (setup) ; 3) calculating the wind and wave-induced currents ;
4) estimating the sediment transport ; and 5) calculating the change i n
shoreline configuration . Each of these tasks is discussed in greate r
detail in this section .
1.
Wind Wave s
The wind waves which are generated in the lake or reservoir are
a
function of the wind speed and duration, as well as the fetch length an d
depth of the lake or reservoir . The fetch length may be determine d
using the effective fetch method (U .S . Army,, 1962) . The application o f
this method is shown in Fig . 1 .
The generation of wind waves for a given wind speed is eithe r
limited by the duration of the storm event or by the size of the body o f
water . Wave generation is, therefore, referred to as being fetch, o r
duration, limited . For wind speeds of interest (greater than 30 mph),
a
fetch length of 30 statute miles will become fully arisen in a duratio n
of less than 4 hours . A body of water with a fetch length of 10 statut e
miles will require a duration of less than 2 hours . Therefore, i n
smaller bodies of water, it is reasonable to assume that the generatio n
of wind waves is fetch limited . Bretschneider and Reid (1953) propose d
a model considering the effects of bottom friction and percolation in
F"
e
_
.
155 .46
13 .51 2
-II 51 Units
Where based on map scal e
One Unit =1714 fee t
eff =11 .51 X
52Ig~=3.7MIies
X i = Projected radial distanc e
(ie X1 = r Cos-c )
0
I
Fig . 1 .
6000
' 1
Scale In fee t
1200 0
Effective fetch method .
3
the bottom sediments . This technique was modified by Bretschneide r
(1965) and Ijima and Tang (1966) yielding the relationships given i n
Figs . 2 and 3 for wave height and period, respectively . In Figs . 2 an d
3, g is the acceleration due to gravity (ft/s 2 ) ; H is the wave heigh t
(ft) ; U is the wind speed (f t/s) ; F is the fetch length (ft) ; and d i s
the water depth (ft) .
The forecasting technique assumes that the bottom is flat and tha t
the lake is of constant depth . These assumptions may be acceptabl e
because the length scale associated with the wind waves is small wit h
respect to the length scale of the lake . Also, if the water depth i s
more than twice the wave length, the waves are deep water waves and are
insensitive to the depth .
2.
Water Dept h
In the central parts of the lake or reservoir, where the waves ar e
generated, the waves tend to be somewhat insensitive to the botto m
profile . However, as the waves approach the shoreline and ultimatel y
break, they are very sensitive to the bottom profile and water depth .
Because of this sensitivity, special care must be taken when estimatin g
the water depth near the shoreline .
As the wind blows over the surface of the water, a shear stress i s
developed . This stress is balanced by a pressure gradient associate d
with a setup of water at the shoreline . This wind setup, sometime s
referred to as a storm tide, may significantly increase the water dept h
at the shoreline, relative to the wave height .
There is a second mechanism, which is associated with the win d
waves, that also produces a setup of shoreline water levels . This wave
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setup produces a pressure gradient which balances the change in momentu m
flux associated with the waves as they break approaching the shore line . This breaking zone, or surf zone, is the region in which th e
effects of the waves are most important in generating turbulence, cur rents, and sediment transport .
Definition sketches of the nearshore region are shown in Fig . 4, i n
which <n> is the mean displacement of the free surface due to wind an d
waves . The brackets denote time averaging over the wave period . Thi s
is done to remove the fluctuating component of motion at the wave frequency . The MWL (mean water level) is the time-averaged free surfac e
while the SWL (still water level) would be the location of the fre e
surface if no wind or waves were present . The SWL is assumed to b e
known for the design conditions and it is necessary to determine th e
MWL . The still water depth is termed h and the mean water depth (tota l
depth, actual depth) is denoted as d and is given by the sum of th e
still water depth plus the setup .
A general form for the depth- and time-averaged equation of motio n
in the surf zone is (cf . Liu and Mei, 1974 )
pwd
Dt
<u> = <p>Vh - p wgd V <n> + V . (-s + t r + t m )
+
Ts
IVs l + T b
(1 )
in which D/Dt is the material derivative, V is the two-dimensiona l
horizontal gradient operator, V . is the divergence operator, <•> is a
temporal averaging operation over one wave period, <u> is the depth- an d
time-averaged horizontal velocity vector, <p> is the time mean pressure ,
<n> is the time mean displacement of the free surface due to th e
7
a)
C
a)
0
a)
Q)
0
L
c,
X
PLA N
Z
PROF/L E
Fig . 4 . Definition sketches for nearshore domain .
8
presence of the wind and waves, p
w
is the water density, g is th e
acceleration due to gravity, d is the total depth, s is the radiatio n
stress tensor, t r is the integrated Reynolds stress tensor, t m is th e
integrated viscous stress tensor,
Ts
and
Tb
are the surface and botto m
stresses, respectively, and s and b are the material coordinates for th e
surface and bottom, respectively . Seeking steady-state solution s
(8/8t = 0) and assuming no longshore gradients (3/3y = 0), that the flo w
is sufficiently turbulent to neglect the viscous transport of momentum
(t m = 0), that the time mean pressure, bottom gradients, and mean fre e
surface gradients vary slowly (<p>Vh = 0, IVbl = 1, 'Vs' = 1), and tha t
the turbulent stresses are related to the time and depth mean flows b y
an eddy viscosity model (t r. = p e d «u>0 + V<u>J), Eq . (1) reduces t o
P wgd
-dx
dx
S xy
<n> - dx Sxx +
+T by
T sx
+Tsy+dx
= 0
(2a )
(ueddxv) = 0
(2b)
in which S xx is the onshore-onshore components of the radiation stres s
tensor, S xy is the onshore-longshore component,
component of surface stress,
stress,
T by
T sy
T sx
is the onshor e
is the longshore component of surfac e
is the longshore component of bottom stress, ue is an edd y
viscosity, and
v
is the mean longshore current .
The radiation stress terms, the excess flux of momentum due to th e
presence of the waves, were given by Longuet-Higgins and Stewart (1960 )
for a coordinate system relative to the wave . Transforming to coordinates relative to the beach (see Fig . 4), these terms are given by
9
S xx = E [ (2n-1) cos 2 0 + (n-1/2)sin 2 0 ]
(3a )
S xy = -E [nsinOcosO ]
(3b )
in which E is the wave energy density (E = 1/8 pgH 2 ), a is the loca l
wave height, n is the ratio of the group velocity to the local wav e
celerity (n = 1/2[1+2kh/sinh 2kh]) provided that k, the wave number, i s
determined by 2ir/T = (2k tanh kh ) 1/2 , T is the wave period, and 0 is th e
wave angle as defined in Fig . 4 .
Inside the breaker line (x < x B ), the water is assumed to be shallow and, due to refraction, the wave angle is small . It is furthe r
assumed that the breakers are of the spilling type such that the breaking wave height is related to the local water depth by H = icd . Insid e
the breaker line, the radiation stress terms may then be given a s
S xx = 3/16
pwg
S xy = 1/16 p wg
ic 2 d 2
K 2d 2
(4a )
sine
(4b )
To determine the total depth profile, Eq . (2a) must be integrated ,
employing an appropriate value for the surface stress . The only surfac e
stress that will be considered is that due to the wind . This stress i s
taken to be
2
T sx = Spa c a w cosy
(5 )
10
in which p a is the density of air, c a is a wind stress coefficient, w i s
the wind speed, and y is the wind direction relative to the shoreline .
The wind stress coefficient was given by Van Dorn (1953) a s
The
a
1 .1 x 10-6
; w < 16 mph
(6a )
1 .1x1D 6 + (2 .5x10 6 )(1 - 6
w )
; w > 16 mph
(6b )
term accounts for the unsteadiness in the development of the win d
current . In the simplification of the equation of motion it was assume d
that only steady-state solutions would be sought . This is a reasonabl e
assumption for the wave-generated current because it forms rapidly .
However, the wind current develops more slowly . The S term accounts fo r
this development and estimates for S may be determined from the one dimensional equation for a wind-generated current without waves ,
dv_ 1
p w dt
d (TS
(7 )
Tb)
in which v is the mean fluid velocity . The wind stress is given by Eq .
(4) without the S term and a simple, linear bottom stress is given b y
Tb
= ap w c wv
(8 )
in which a is a dimensional linearizing coefficient and c w is the flui d
stress coefficient . As a first approximation, a may be taken as 3% o f
the wind speed . This is based on the observation that the wind drift i s
about this fraction of the wind speed (Wu, 1975) . Substitution of Eqs .
(5) and (8) into (7) and integrating yield s
cw d
p c w2
W
v= p
cosy [1-e
ac t
w
1
d 11
(9 )
11
in which it is assumed that the water is initially at rest . The maximum
current, v., occurs at very large times . The ratio of the time dependent velocity to v . i s
v
v = [1- e
ac t
w
- d
(10 )
This ratio defines the B given in Eq . (5) . Figure 5 shows this term fo r
dimensionless time defined as acw t/d .
With all of the terms in the cross-shore equation of motion speci fied, Eq . (2a) may be writte n
p wg (<rl> + h)
dx
<n> - dX (3/16 p wgK 2 d 2 )
+ p a c a w2 cosy = 0
; x < xB
A solution to this equation for a planar beach is given by
d=ax+b [1n(b+ 1) - 1] + c
(12a )
in which
a =
3
8+3K
b =
2
m
8 p acaw
2
ap wg
(12b )
cosy
m
(12c )
and c is an integration constant .
Seaward of the breaker line, the'shallow water assumptions may no t
be invoked . However, seaward of the breaker line, energy is conserve d
12
rn
U
0 .2
0 .4
0 .6
DI ENSIONLESS
TH E
Fig . 5 . Wind stress time coefficient .
13
0.8
(friction and percolation are small) so that an energy conservatio n
equation may be written . The time-averaged Bernoulli equation i s
P+u 2 +w 2
p wg
2g
<>n
B p a caw cosy
p wg
2
= ~°
2
+ u° 2gw ° + <
n°
>
(13 )
; x xB
in which ( )° denotes a value in deep water . Noting that the waveinduced mean free surface displacement goes to zero in deep water ,
assuming that the wind setup vanishes at great distances offshore, an d
employing linear wave theory yield s
2
ap c w 2 cosy
H k
a a
<n> _ - 8sinh 2kh +
p wg
'
x
xB
(14 )
The integration constant in Eq . (12) may be determined by equatin g
(12) and (14) at the breaker line, x = x B ,
K2
c =
40 - 3K2
168+3K2
hB-b[1n(
(16 - K 2 ) h
B
+1)- 1]
16b
(15 )
Typical setup profiles are given in Fig . 6 . These setup profile s
are very nearly linear . Therefore, assume
d = b x + b2
1
(16a )
in whic h
b 1 =b
(16b )
1/2d B +b
b2=bin(
d B +b
K2
40- 3K 2
~ + 16 8+3K 2
14
(16c )
hB
It is convenient to define a coordinate transformation such that th e
origin is located at the point on the beach where the depth goes t o
zero . This is given by
d=b 1 x
(17a )
x = x + b 1 /b 2
(17b )
where
It is also convenient to express the depth in a dimensionles s
form . Using upper case letters to denote dimensionless variables defin e
D
= d/xB
(18a )
X
= x/x B
(18b )
(18c )
B1 = b 1
The dimensionless depth profile, including wind and wave setup, is give n
by
D= B 1 X
3.
(19 )
Longshore Curren t
The longshore current due to wind and waves is determined from Eq .
(2b) . The divergence of the radiation stress term is determined employing Snell's Law for refraction and assuming conservation of energy i n
the offshore .
16
-
51 6 pwgK2 sine
~d) 1/2 d
B
(d)
x < xB
(20a )
; x > xB
(20b )
dx
0
The wind stress term is the same as for the onshore equation of motio n
except that the sin(y) component is taken rather than cos(y) component .
The bottom stress term may be approximated using a linear shallo w
water, small wave angle relationship for the waves (cf . Liu and
Dalrymple, 1978) and a linear relationship for the wind current (see Eq .
(8)) .
T by
= - - K p w (gd) 1/2 v ap wc wv
(21 )
The eddy viscosity is assumed to be proportional to a characteristi c
velocity, density, and length . The form proposed by Longuet-Higgin s
(1970b) is employed .
u e = P w N(gd) 1/2
x
(22 )
in which N is a numerical constant .
Employing the above relationship in the equation of motion yield s
NTrbl d
"5/2 dv
"1/2 + an
px
) - (x
1/2 K
K cw dx
dx
(gb l )
b
5 Tr
b1 3/ 2
B
)
1/2
sineB
dB x
16 c w K bl(g d
p
c
x ) 1/2 p w cw w
(g b l
2siny
; x > xB
17
v =
p
(gb)
~1
c
1/2 P w cw
w 2 siny x < xb
(23a )
(23b )
Again, it is convenient to define dimensionless variable s
Nit B 1
P1
K
__
P2
V
(24a )
w
a
7t
K
(gB
(24b )
xb ) I/ 2
1
B1
DB
P3
P4
C
=
=
(24c )
a
P ca
01
Bl l/2 P w cw
v/vBL
=
w2
i nY
(24d )
(g xB)vBL s
v/[516 c w
K
(24e )
b l (gd B ) 1/2 sin6 B ]
The term v BL is the longshore current due to waves alone which occurs a t
the breaker line for the case of no turbulent lateral mixing (Longuet Higgins, 1970a) . It is common to use this for scaling longshor e
currents .
In dimensionless variables the equation is written a s
- P 3 X 3/2 - P4
P
d
1
T.(
X5/2 dV
(
757,) _
1/2
(x
; X < 1 (25a )
+ P ) V =
2
P4
; X > 1 (25b )
Boundary conditions are that the longshore current velocity i s
bounded at the shoreline and far offshore, and that the velocity an d
gradient of velocity match at the breaker line . Equation (25) has bee n
integrated analytically in the course of this study . Unfortunately, th e
resulting solution is rather complicated and it is more convenient t o
18
just use a numerical integration . Several example profiles are shown i n
Fig . 7 for different wind and wave conditions .
4.
Sediment Transpor t
Bagnold (1963) proposed a sand transport model which assumes tha t
the sediment is mobilized by wave motion and wave power is expended t o
maintain the motion of the sediment . The presence of a mean curren t
transports the sediments . The immersed-weight transport rate per uni t
width of surf zone is (McDougal and Hudspeth, 1983 )
i = K' dx
(2 pga2Cg)
u
(26 )
m
where K' is a dimensionless coefficient which depends on the degree o f
the transport mechanism developed, C g is the wave group velocity . Th e
coefficient can be treated as a constant for a particular coast from th e
view of long-term climate . Furthermore, it can be considered as a
constant in a season or in a storm (Hou, Lee, and Lin, 1980) .
Assuming linear shallow-wave conditions exist and that the wav e
energy flux and the bottom orbital velocity outside the breaker line ca n
be evaluated at the breaker line, Eq . (26) become s
'
r (d)
pg K d d
v
x < xb
(27a )
; x > xb
(27b )
dx
S
K' pgK [d
d" (d)]dx
v
In terms of nondimensional variables, the immersed-weight transport rat e
becomes
19
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r
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O
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r
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O.
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en en
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I
(~
{i
)
ua
v
KJ.
cV
0
0
0
0
C?
A
20
0
I
0
0
0
D b dX
d
dx rr
(EL) V (x)
X < 1
(28a )
V (X)
; X > 1
(28b )
b X= 1
where
i
I
(28c )
1 BL
5
1BL __ 8 R p s gKd b S b v BL
(28d )
db
Sb
(28e )
xb
and p s is the density of the sediment .
For the linear total depth profile given in Eq . (19), the sedimen t
transport is
XV
X < 1
(29a )
V
X > 1
(29b )
Dimensional transport profiles are shown in Fig . 8 for the longshore current profiles given in Fig . 1 .
The total sediment transport is given b y
co
IT
= 0J
I dX =
f
1
XV dX
+ 1f
co
Vd X
o
This integral is numerically evaluated .
21
(30 )
N
tr)
X
rn
0
N
0
LC)
0
:r1
0
c.)
0
CD
0 c
h')
U-)
0
H
22
0
<
0
0 O
0
5.
Shoreline Evolutio n
Shoreline evolution is based on the net flux of sediment along a
given beach . The shoreline is broken into a number of segments o f
length AS . The immersed weight dynamic transport given in Eq . (30) ma y
be converted to a volume transport rate by dividing by the porosity, n ,
and the immersed weight density .
S =
I
n( ps-pw) g
(31 )
in which S is the volume transport and p s is the density of the sediment . From conservation of the sediment mass, the change sediment i n
depth is given a s
1
aE
al
M
at - n(p s -p w )g ax As h N p
(32 )
in which E is the normal to the bottom and h Np is the depth to the nul l
point . The null point is the maximum depth of wave-induced sedimen t
transport . For lakes and reservoirs, this depth may be on the order o f
five feet . The change in depth also results in a change in the positio n
of the shoreline . This change is given by
aR
at -
m2 + I ) D E
2 ) t
a
m
(33 )
in which R is the shoreline position relative to the initial conditio n
and m is the bottom slope .
The change in shoreline may be computed using the solution obtaine d
for total longshore transport and then solving Eq . (33) numerically . An
implicit finite difference scheme may be used in space while the solution is explicitly marched in time .
23
It should be noted that at each change in shoreline configuration ,
the breaker angle must be recomputed for each segment . This may b e
rapidly, but rather crudely, done employing Snell's Law .
CONCLUSION S
A rather simple model has been proposed to examine the effect o f
wind and wind waves on shoreline erosion . The technique employed fo r
estimating waves is that developed by the Corps of Engineers . Th e
solution for the combined wind and wave setup is based on radiation
stress concepts, as is the generation of the longshore current . Th e
sediment transport model is based on energetics . And, finally, the
shoreline evolution model is based on conservation of sediment . Th e
methodology and numerical results were developed for the first fou r
tasks . Only the methodology was outlined for the fifth task .
Without the completion of the fifth task, it is difficult to quantify the effects of wind wave erosion on the shoreline . However, it i s
clear that the sediment transport is increased in a narrow band along
the water line . Whether this sediment transport is of sufficient magnitude or duration to cause significant erosion remains unanswered .
24
REFERENCE S
Bagnold, R .A . (1963), " Beach and Nearshore Processes, Part 1 : Mechanic s
of Marine Sedimentation ." In : M .N . Hill (Editor), The Sea, pp .
507-528 .
Bretschneider, C .L . and Reid, R .O . (1953), "Change in Wave Height Due t o
Bottom Friction, Percolation and Refraction," 34th Annual Meetin g
of the AGU .
Bretschneider, C .L . (1965), "Generation of Waves by Wind ; State of the
Art," Report No . SN-134-6, National Engineering Science Co . ,
Washington, D.C .
Hou, H .S, Lee, C .P ., and Lin, L .H . (1980), " Relationship Between Alongshore Wave Energy and Littoral Drift in the Mid-West Coast a t
Taiwan," Proc . 17th Coastal Eng ., Chap . 76, pp . 1255-1274 .
Ijima, T . and Tang, F .L .W . (1966), " Numerical Calculation of Wind Wave s
in Shallow Water," Proc . 10th Conf . on Coastal Engeering, pp . 3845 .
Liu, P .L-F and Mei, C .C . (1974), " Effects of a Breakwater on Nearshor e
Currents Due to Breaking Waves," Ralph M . Parson Laboratory ,
Department of Civil Engineering, MIT, Report No . 12 .
Liu, P .L-F and Dalrymple, R .A . (1978), " Bottom Frictional Stresses an d
Longshore Currents Due to Waves with Large Angles of Incidence , " J .
Mar . Res ., 36 :357-375 .
Longuet-Higgins, M .S . and Stewart, R .W. (1960), " Changes in the Form o f
Short Gravity Waves on Long Waves and Tidal Currents , " J . Mar .
Res ., 36 :357-375 .
Longuet-Higgins, M .S . (1970a), " Longshore Currents Generated b y
Obliquely Incident Sea Waves, 1," J . Geophysical Res ., 75 :6778 6789 .
Longuet-Higgins, M .S . (1970b), "Longshore Currents Generated by
Obliquely Incident Sea waves, 2 , " J . Geophysical Res ., 75 : 6790 6801 .
McDougal, W .G . and Hudspeth, R .T . (1983), " Longshore Sediment Transpor t
on Non-Planar Beaches," Coastal Eng ., 7 :119-131 .
U .S . Army, Corps of Engineers (1962), "Waves in Inland Reservoirs ,
(Summary Report on Civil Works Investigation Projects CW-164 an d
CW-165)," TM-132, Beach Erosion Board, Washington, D .C .
Van Dorn, W .C . (1953), "Wind Stress on an Artificial Pond, " Jour . o f
Marine Res ., Vol . 12 .
25
Wu, Jin (1975), "Wind-Induced Drift Currents," J . Fluid Mech ., 68 :49-70 .
26