JOURNAL
OF GEOPHYSICAL
RESEARCH, VOL. 90, NO. C6, PAGES 11,939-11,944, NOVEMBER
20, 1985
Wave Energy Saturation on a Natural Beachof Variable Slope
ASBURYH. SALLENGER,
JR.•
Branchof PacificMarine Geology,U.S. Geologic.al
Survey,Menlo Park, California
ROBERT A. HOLMAN
Collegeof Oceanography,
OregonState University,Corvallis,Oregon
Time seriesof flow weremeasuredacrossthe inner surfzoneduringa storm.Thesedata wereusedto
quantifythe dependence
of waveheight(transformed
frommeasured
flow)and velocityon localslope
a.nddepth.Similarto previousstudies,
asincidentwavesbrokeandpropagated
into the surfzone,wave
energy
beca.rne
saturated,
andwaveheight
wasstrongly
dependent
ondepth.
However,
theratioofrms
waveheightto localdepth(•rms)
wasfoundnot to be constant
but to varybetween
0.29and0.55'•rrns
increased
withlocalslopeandwasindependent
of deepwater
wavesteepness.
Thu•thesurfzonesimi•
larityparameter
(theratioofslopeto thesquare
rootof steepness)
didnotadequately
parameterize
•rms'
INTRODUCTION
1965; Iverson, 1952]. For monochromatic wa•ves•,
Bowen,et al.
Wave energy in the inner surf zone is thought to become
[1968] and Battjes [1974] useda surf zone similarity parame-
of3'onslope
andsteepness
saturated
sothatthelocalwaveheight
H isa linearfunction ter•0 to showthedependence
of the local depth h,
tan fl
H= 3'h
•o=(Ho/Lo)•/2
(1)
where3',accordingto somephysicalmodel studies,varieswith
bottom slope and wave steepness.
The significanceof (1) is
thatwaveheightis described
everywhere
in•thesurfzonewithout having to determinethe details of energydissipationthat
resultsfrom breakingwaves.Equation (1) is a critical element
of theories of longshore currents [Bowen, 1969' LonguetHiggins, 1970; Thornton, 1971], wave setup [Bowen et al.,
1968], and other surfzone processes.
Althoughthe dependence
of 3'on slopeand wave steepness
has been extensivelyinvestigatedfor monochromatic waves,
few data are available that quantify the variability of 3' for
randomwavesin the field.In the presentstudywe measure
d
time seriesof flow acrossthe surf zone during a storm. We
transformedthe measurementsof flow to wave heights and
determined3'acrossa profile whoseslopevaried,both in time
and space,nearly an order of magnitude.The data were collected under a wide range of incident wave conditions from
steepsea waves during a storm to long-period swell waves.
The data set was variedenoughto allow the dependence
of 3'
on bottom slopeand wave steepness
to be thoroughlyexam-
(2)
where fl is the slope of the bottom profile, H0 is deepwater
wave height,and Lo is deepwaterwavelength.For most model
data, 3'increasedwith increasing•0-
Most field studieshave determined3' by usingfilms that
followindividualwavecrestsacrossthe surfzone [Sverdr•p
and Munk, 1946; Suhayda and Petrigrew, 1977; Weishat and
Byrne, 1978]. Weishat and Byrne [1978] found 3' to have a
mean value close to the solitary wave theoretical result, although they found 3' to vary with breaker type, breaker type
being a functionof fl and Ho/Lo. Sverdrupand Munk reported similar results.However, thesefilm resultsprobably do not
reflectthe total distribution
of waves.For randomwavesat
anygivenmeandepthin the surfzonetherewili be distributions
of bothbreaking
andunbroken
waveheights
•Thorn-
ton and Guza, 1983]. As a result of following single-breaking
wave crests,presumablythe best developedand largestbreakers,the measured3'Of a film studymay be significantlylarger
than 3' determined from the entire wave distribution, which
includesboth breaking and unbroken waves.
Using an extensivearray of instruments,Thorntonand Guza
ined.
•
[1982] obtained time seriesof flow acrossa surf zone. They
In the followingsectionwe reviewtheory and the previous transformedthese flow data to surfaceelevation speqtraand
modelgndfieldstudieson waveenergysaturation.Then,after found 3'rms
(3'basedon rms wave height)for the inner surf zone
disc•Ussing
th• design
of theexperiment
anditssetting,
we
to be 0.42, a factor of 2 or more less than the monochromatic
presentr½sul.ts
on how3'variedwith slopeandwavesteepness. wave
results.
Since
thebeach
onwl•ich
theyworked
wasof
LITERATURE
REVIEW
nearly uniform slope, they were unable to determine whether
there was a dependenceof 3' on fl. Wright et al. [1982] and
Downing [1983] also found 3'for random waves of the field to
be significantly lower than what was found for monochro,
Using solitary wave theory, McGowan [1891] showedtheo-
reticallythat 3'shouldbe a constantequal to 0.78. However,
physicalmodel studiesemployingmonochromaticwaveshave
shownthat 3'dependson beachslopeand/or wave steepness,
matic waves.
varyingroughlybetween0.7 and 1.2 [Galvin and Ea•tleson,
EXPERIMENT
• Now at U.S. Geological
Survey,NationalCenter,Reston,Virginia.
Copyright1985by theAmericanGeophysical
Union.
Paper number 5C0601.
0 ! 48-0227/85/005C-0601$05.00
During October 1982, a field experimentwas conductedat
theFieldResearch
Facility(FRF) of the U.S.ArmyEngineer's
Waterways Experiment Station in Duck, North Carolina. The
overall objectiveof the experimentwas to determinethe processesthat cause nearshoremorphology to change during
11,939
11,940
SALLENGER
ANDHOLMAN'WAVEENERGY
SATURATION
=,
-'
,
LEGEND
CM - CURRENT METER
P$-
PRESSURE
WG
FRFPIER
SENSOR
WG - WAVE
GAUGE
MC - MOVIE
CAMERA
$$ - SUSPENDED SEDIMENT SENSOR
AREA OF
CM
WG
CM
WG
CRAB
SURVEYS
USGSSLED
I
$Y,STEM
'
¾' •
WG
'5'ss
'2'
M•C
;
C M_._.•
CM
I
500
-- SHORELINE
i
i'
400
300
DISTANCE
WG
WG
CM
MC•
I
i
200
100
NORTH
(m)
Fig. 1. Plane view of the USGS sledsystemand the FRF pier.
storms. Some initial results from this experiment are reported
in $allengeret al. [1985] and Holman and Sallenger[1985].
The FRF is located on a long straight beach of a barrier
island that has a single well-developednearshore bar. The
foreshore is composed of coarse sand (mean diameter: 1-2
mm), whereasseaward of the trough of the bar, sand is relatively fine (mean: 0.18 mm).
Most of the data discussedin this paper were obtainedwith
the USGS sledsystem,which consistsof an instrumentedsled
towed along the bottom, both offshore and onshore,with a
double-drum winch and triangular line arrangement [$al-
During October 1979, a similar, but smaller-scale,experiment was conductedon the shore of Monterey Bay, Califor-.
nia, at Ford Ord. Flow and pressure data were obtained
acrossthis high-energysurf zone by using the USGS sled
system.
RESULTS
Most of the data discussed here were obtained
at the FRF
during a 3-day storm, October 10-12, 1982. Representative
spectraof surfaceelevation•usingrecordsfrom the offshore
wave .rider,are shownin Figure 2. During this high-energy
lenger et al., 1983]. On the sled, 4-cm-diameterelectro- period, wave steepness(calculatedby using peak period and
magnetic current meters (Marsh-McBirney model 512) were tabulated on Figure 2) and width of the incident spectrum
mounted in a vertical array at 0.5 m, 1.0 m, and 1.75 m above varied considerably.On October 10, strong northeastwinds,
the bottom. Flow data used in this report were measuredwith which reached 13 m/s, generatedsteep waves with a broad
the 1.0-m sensor,although 7rmswas independentof depth of incidentspectrum.By October 12 the storm had moved offmeasurement.Also mounted on the sled was a pressuresensor,
which was used to determine local depths. Data were telemetered to a shore receivingstation where they were digitally
recorded at a rate of 2 Hz. All record lengths were 34.1 min.
As the sled moved along the shore-normal transect,the nearshore profile was measuredby using an infrared rangefinder
on the beach, sighting on optical prisms mounted on top of
the sled'smast. The sledsystemwas set up 500 m north of the
FRF pier (Figure 1) to avoid effectsthe pier has on local flows
and sedimenttransport [Miller et al., 1983].
Additional wave data discussedin this report were measured with a pressuresensorlocated severalhundred meters
seawardof the sled transectand a wave-rider buoy moored in
20 m of water 3 km offshore of the FRF. The pressuresensor
data were collectedby the Coastal Data Information Program
[ScrippsInstituteof Oceanography,
1982].The wave-riderdata
were telemeteredto shore and processedby the FRF [Coastal
EngineeringResearchCenter, 1982].
40
-
Oct
30
-
z
< 20
E
>
•
DATE
12
iI
iI
I!
I'•
TIME
Holm) Tp(s) Ho/Lo
Oct 10
1600
2.25
6.9
0.03
Oct 11
1700
1,96
14.2
0,006
Oct 12
1600
2.40
16.8
0.005
oc••
/oc,,0
, / ,,-I, /\
10
0
0.1
FREQUENCY
0.2
0.3
(Hz)
Fig. 2. Representative
spectrafor the 3 days of the storm,using
data &om the offshore wave-rider buoy. Inset tabulates significant
wave heights(Ho), peak periods (Tp), :hod wave steepnesses
(no/•0).
SALLENGER
AND HOLMAN: WAVE ENERGYSATURATION
4
STATION
-
-
- --f
10
7
2.6
LOCATIONS
11
8
•, Oct11t 1V'/' 1V4
11,941
2.4
13
1V8 2V0
2.2
19
'7
--
0.42
2.0
ß
1.8
,-,
•-2
E
LU-3
•
W-4
-
--oc,,
Oct
----
ßt•11
ßß
ß
1.6
ß
1.4
II
• 1.2
10(11)
Oct 12
I
-,5
,--,
t
I
100
I
I
200
DISTANCE
I
300
ß
1.0
I
ß
ß
.11'
400
0.8
OFFSHORE
I I
ß
0.6
Fig. 3. Profiles and sled station locations for the experiment at
Duck, North Carolina. The order in which the stationswere occupied
is indicated by each station number. Profiles for the 10th and 11th
were very similar; only the profile of the 10th is shown.
ß
ß
0.4
0.32
ß
ß
0.2
I
I
I
I
4
5
6
7
0
3
0
8
h
shore,and local waves were light; waves were principally of
Fig. 5. Pressuredata measuredwith Sxy gauge 580-m offshore
the swell type and, relative to October 10, the incident spec(Figure 1). The data were obtained from the Coastal Data Infortrum had significantlydecreasedin width (Figure 2). Through- mation Program. Data were taken at 6-hour intervalsfor the periods
out the 3-day period, the sled transectwas within the inner October 10-13 and October 23-25, 1982.
50% of the surf zone. On October 12, significant breaker
height of nearly 4 m resultedin a surf zone width of roughly
sinh I klh
900 m, the breaker zone occurring 400 m seaward of the end
H(f) =
(4)
o cosh (I k I(h + z))
of the pier (Figure 1).
Prior to the storm, a relatively small longshore bar was
where k is wave number and z is the instrument depth measurveyed(Figure 3). Additionalsurveysto the north and south
suredpositivelyupward from the still water surface.Hrmswas
of the profile showedthe bar at this time to be reasonably calculated over the incident wave band of 0.05-0.33 Hz. The
linear and shoreparallel [Sallengeret al., 1985]. During the 3
cutoff of 0.05 Hz was chosento remove low-frequency waves,
days of the storm, the bar became better developed,and it
such as surf beat, that would not be expectedto break in the
migrated offshore about 57 m (Figure 3). Unfortunately,
surf zone. Thornton and Guza showed that (3) was a reasonduring this high-energyperiod, we were unable to adequately
able approximationof H .... evenin shallowwater. Guza and
determinethe three-dimensionalmorphology of the bar. ObThornton[1980] showedthat Hrmscalculatedby using(3) and
servationsby Short [1979] and others suggestthat bars are
(4) comparedwithin 20% with H•m•data calculatedfrom both
linear when incident wavesare large.
pressuresensorand surface-piercingwave staff records.For
For each day of the storm, flow records were obtained
the presentdata we found H•mscalculatedfrom flow data was
acrossthe nearshoreprofile (Figure 3). (Sinceone would not
roughly 10% belowthat calculatedfrom pressure.It shouldbe
expect waves to be saturatedin the trough of the bar, we do
emphasizedthat H•m• of (3) incorporatesboth breaking and
not discussany trough data). Following Thornton and Guza
unbroken waves.
[1982], we transformed flow spectrato surfaceelevation specHrm• is plottedagainstdepth in Figure 4. Again, all of the
tra by usinglinear theory, and we calculated Hrmsas
data were obtained from the inner surf zone where depth was
shoal enough for most waves in the incident spectrum to
mrms
'- 2N//•
Im(f)l2 [Gu(f)+Gv(f)]df
(3) break. Thus energysaturationwould be expected.The 7rm•=
do.os
0.42 relationshipfor the inner surf zone [Thornton and Guza,
1982] doesnot fit the majority of data. For most of the data,
where Gu(f)is the cross-shoreflow spectrum,Gv(f) is the long= 0.32 is indicated.Note that the data that seemto agree
shore flow spectrum, and H(f) is the linear transformation 7rms
with the 0.32 trend were all obtained on the relatively gentle
functiongiven by
slope,well seawardof the bar crest(comparerun numbersof
Figure 4 to station locationsof Figure 3). The data that are
2.0 •
closer to the 0.42 trend were obtained
from the much more
steeplyslopingpart of the profileimmediatelyseawardof the
bar crest.
/
,-., 1.2-
E
'--'
1
1
• 1.0 --
.E
27
0.4
28
-
.////•:
411
7'rm
s =0.32
I
ß
28
-I- 0.80.6
ß%
0.2-
0.4
'7 =
0.32
0.2
0.1
0
•
0
1
2
•
i
i
i
3
4
5
6
h (m)
-
0
0
Fig. 4. Sled data from the storm (October 10-12, 1982). Station
numbers are used as data points. Refer to Figure 3 for station locations.
1
2
3
4
5
6
Ho
Fig. 6.
7rmsversusH o for pressure
sensordata shownin Figure5.
11,942
SALLENGERAND HOLMAN' WAVE ENERGY SATURATION
0.6
6
7'rms
= 3.2 tan • + 0.30
0.5
-
7
109
r2--0.70
Z•
•81
0112
9
_o•10
<C•11
0.4
.•rl• ©©oo
_
UJ<( 13
LEGEND
ß
Duck.
NC
14
_
_
:• Fort Ord. Ca.
0.2 -
20
4
I
I
0.02
0.4
(Downing.1983)
112
I
110
0.04
I
0.05
I
0.06
I
I
120
SEAWARD
140
I
160
I
180
(m}
111
Twin Harbors Beach, Wa.
I
0.03
810100I
113
(Wright
et al.,1982}
0.01
I0 60I
DISTANCE
(Thornton and Guza, 1982}
o
0
I
0
Goolwa Beach, Australia
0.1 0
A.
I
ß Torrey
Pines
Beach,
Ca.
n
113
I
_
LU'Z 12
0.3
111
I
I
0.07
109
0.3108
tan)•
B.
Fig. 7. 7rms
versustan fi, wherefi is the slopeof the bottomprofile
at the appropriatesled station. The data point from Thorntonand
Guza [1982] representsa maximum 7rms.Thus this datum is not
directlycomparableto the other data and is not usedin calculating
the trend.
0.2
I
'
0
0.01
I
0.02
I
0.03
0.04
I
0.05
I
0.06
tan )•
Fig. 8. (a) Stationlocationsfor Fort Ord, California.(b) 7rnas
versus
fi for the Fort Ord data.
Well seawardof the sledline wherethe slopewas about the
same as the gentle slopediscussedabove,wave energyalso
appearedto saturatewith 7rms--0.32.Plotted in Figure 5 are
data from the pressuresensorlocated 560 m offshore.Low-
frequencywaveswere includedin calculatingthesepublished
wave heights.However, the pressuresensorwas far enough
offshoreso that the wave heightdata werenot seriouslycontaminated. Included in the plot are data obtained every 6
hours through two storms. One was the storm discussed
above in regard to the sled data, and the other was a storm 2
weeks later, during which offshoresignificantwave heights
reachednearly 6 m. Since the pressuresensorwas in much
deeper water than depths occupiedby the sled and sincewe
includedlow-energydata precedingand followingeachstorm,
not all of the data plotted in Figure 5 would be expectedto be
saturated.Thus for the roughly constantdepth of the pressure
sensor (varying somewhat with tides), wave height ranged
from relatively small to a maximum saturatedvalue governed
by equation (1) with 7rmsequal to about 0.32 (Figure 5). This
apparent saturation is confirmed in Figure 6, where using the
samedata, we plot 7rms
againstH0, deepwatersignificantwave
height (approximated by significant height at the wave-rider
buoy).For H0 lessthan about 2.3 m, 7rm•simplyincreasdwith
H0, indicating the waveswere not saturated.However, for H0
greater than about 2.3 m, wave energy appeared to saturate,
with 7rm•becomingroughly a constantat about 0.32, irrespective of H0.
The dependenceof 7rmson tan /? is shown in Figure 7.
Slopesplotted on Figure 7 and presentedin Table 1 are local
slopesat the measurementlocation evaluated over horizontal
distances of 10-15 m. The solid circles indicate
manner as described above. Unlike the Duck data, for which
TABLE 1. Wave Height and Profile Parameters
Date
October 10, 1982
Station
Number
7
Location
Duck, North Carolina
8
10
11
13
October 11, 1982
14
Duck, North Carolina
17
18
19
20
October 12, 1982
21
Duck, North Carolina
24
26
27
28
October 18, 1979
108
the sled data
discussed
thusfar; a rough trend is apparentwith 7rm•increasing with tan
As discussedabove, additional sled data were gatheredat
Ford Ord, California. These data were analyzed in the same
Fort Ord California
109
110
111
112
113
*Not includedin trend calculationsof Figure 7 (seetext).
Hrms,
m
Urnas
,
m/s
h,
•rnas
1.29
0.85
4.11
1.44
1.10
1.20
1.43
0.92
1.02
0.81
0.91
4.51
1.20
0.85
!.05
1.36
1.64
1.39
1.03
0.92
1.02
0.96
1.71
1.04
1.05
1.26
0.92
0.89
0.78
0.91
1.06
0.71
0.79
0.73
1.00
0.74
1.09
0.64
1.37
0.95
0.77
1.04
0.74
1.17
2.51
4.03
4.51
3.83
2.29
4.14
4.87
4.03
5.19
3.61
3.81
1.73
3.23
2.6
2.5
2.1
2.5
1.7
2.9
0.31
0.32
0.44
0.30
0.32
0.31
0.46
0.33
0.34
0.34
0.33
0.29
0.33
0.53
0.28
0.30*
0.40
0.35*
0.44
0.38
0.47
tan fi
0.014
0.007
0.052
0.004
0.010
0.012
0.049
0.004
0.010
0.008
0.010
0.016
0.007
0.061
0.004
0.0
0.011
0.0
0.021
0.0
0.05
SALLENGER
AND HOLMAN' WAVE ENERGYSATURATION
0.4
-
0.3
0% x x
•
0.2
LEGEND
0.1
ß Sxy gauge
x
0
0
i
t
t
i
I
0.01
I
I
,
i
I
0.02
t
I
Sled data
I
,
I
0.03
,
ß
Ho/Lo
Fig. 9. Yrms
versusHo/L o for the sleddata obtainedon the gentle
slopes(Figure 3) and the eight apparentlysaturateddata points from
the offshore pressuresensor (see Figure 6) that were obtained on
about the sameslopeas the sleddata.
11,943
was saturated [Thornton and Guza, 1983]. The data from
Duck indicated a landward increasein 7 that we have related
to an increasein/•. Another interpretationwould be that the
landward increasein 7 resultedfrom the distribution of wave
heightsin the incidentwavefield.However,this interpretation
seemsunlikely,sincethe Duck data were obtainedwithin the
inner surf zone, where we assume most waves would have
broken. In any casethe importanceof slopeis confirmedby
the Ft. Ord data, which showed7 decreasinglandward with/•,
oppositeto what would be predicted by distribution of wave
heights.In contrast to the inner surf zone data discussedhere,
distribution of wave heightswill likely have a significantcontrol of 7 in the outer surf zone, where some waves are unbroken.
For sedimenttransport applicationsit is of interest to predict the wave-inducedvelocitiesdirectly. Using linear theory
and shallow-water approximations,wave-inducedoscillatory
7rmsincreasedwith decreasingdepth (compareFigures3 and
4), •rmsfor the Ford Ord data decreased
with decreasing
depth velocity
isgivenby
(Figure 8). Note that the part of the Fort Ord profile of inu = 0.5(ah)ø-57
(6)
terestwasconvexup (Figure 8a), whereasthe part of the Duck
profile of interestwas concaveup (Figure 3). Becauseof these whereg is accelerationof gravity. Substitutingfrom (5) yields
differencesin profile shape,Yrmsfor both the Fort Ord and
Duck data increasedwith tan/• (Figure 8b). Most of the Fort
Ord data (Figure 8b) are plotted along with the Duck data in
Figure 7. Not plotted are data from stations 110 and 108
(Figure 8), which are on the part of the profile with zero slope
and are apparentlynot saturated.On this part of the profile,
wave heightdecreasedas a resultof continuedbreaking,even
though depth was constant (Figure 8). The Ford Ord and
Duck data, along with some additional data from the literature, show a reasonabletrend, although there is significant
scatterwith Yrms
for the Ford Ord data plotting higher than
•rmsfor the Duck data. Linear regressiongives
7.,,s- 3.2 tan/• + 0.30
with r 2 - 0.70.Standarderrorsfor slopeand offsetare 0.5 and
0.01,respectively.
The 99% confidencelevelthat 7,,,sis dependent on/• is r 2 > 0.26.
The parameter7•m•does not appear to depend on wave
steepness.
Sleddata, obtainedon the seawardpart of the profile where bottom slope was roughly constant(Figure 3), are
plotted againsttto/L o in Figure 9. Also plotted are the offshore pressuresensor data, which appear to be saturated
(Figure 6) and were obtainedon about the sameslopeas the
sleddata. Figure 9 showsno dependence
of 7,,,• on Ho/Lo.
DISCUSSION
Based on early monochromaticstudies, equation (1) has
commonlybeen usedwith 7 equal to a constantof 1. In this
study,•rmswas found not to be constantbut to vary significantly with slope.Furthermore, similar to resultsof some earlier studies,we found that 7 = 1 for random waves overestimates7,,,, by a factorof 2 or more for mostnatural slopes.
For laboratory data the surf similarity parameter •0 (equation 2) appearsto parameterizemany different surf zone processes,including setup, runup, breaker type, and saturation
[Bowen et al., 1968; Batties, 1974]. Furthermore, runup data
obtained in the field can be parameterizedby •0 [Holman and
Sallenger,1985]. However, in the presentfield study we find •
to be independentof Ho/Lo and thus •0.
In the field there is a distribution of wave heights, and the
largest waves will break farther offshore than the smaller
waves, causing a landward increase in •; • would continue
increasinglandward until all waves were broken and energy
U,m.•
= 1.6tan/•+ 0.15
(ah)ø.•
(7)
Using measuredU .... calculatedfrom measuredflow spectra,
resultsin essentiallythe samerelationshipas (7).
CONCLUSIONS
Flow measurements, which were transformed into measures
of wave height, were obtained acrossthe storm surf zone. The
data set was appropriate to quantify the dependenceof wave
height on local slopeand depth. As incidentwavesbroke and
propagated into the inner surf zone, wave heights became
strongly depth dependentand could be describedby equation
(1); 7rms
increasedwith increasingslopeand was independent
of wave steepness.
Thus the surf similarity parameter-the ratio
of tan//to the squareroot of steepness-does
not adequately
parameterizesaturation.
Acknowledgments.We thank Jeff List, Tom Reiss, Bruce Richmond, Bruce Jaffe, and Peter Howd for setting up and operating the
sledsystemand Curt Mason and the FRF crew for generalsupport of
the experiment.Jeff List did the programmingfor most of the calculations used in this paper. This work was funded jointly by the U.S.
Geological Survey and the U.S. Army Engineer Waterways Experiment Station, Coastal EngineeringResearchCenter, Vicksburg, Mississippi.Additional support for R. A. H. was provided by Office of
Naval Research,Coastal SciencesBranch contract NR 388-168.
REFERENCES
Batties, J. A., Surf similarity, in Proceedingsof the 1,1thConferenceon
Coastal Engineering,vol. 1, pp. 466-480, American Societyof Civil
Engineers,New York, 1974.
Bowen,A. J., The generationof longshorecurrentson a plane beach,
or.Mar. Res., 37, 206-215, 1969.
Bowen, A. J., D. L. Inman, and V. P. Simmons, Wave setdown and
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(ReceivedFebruary 26, 1985;
acceptedApril 22, 1985.)