JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 102, NO. C4, PAGES 8663-8679, APRIL 15, 1997
Generation of edge waves in shallow water
T. C. Lippmann
Centerfor CoastalStudies,ScrippsInstitutionof Oceanography,
La Jolla,Califomia
R. A. Holman
Collegeof OceanandAtmospheric
Sciences,OregonStateUniversity,Corvallis
A. J. Bowen
Departmentof Oceanography,
DalhousieUniversity,Halifax, Nova Scotia,Canada
Abstract. Theoreticalgrowthratesfor resonantlydrivenedgewavesin the nearshoreare
estimmedfrom the forced,shallowwaterequationsof motionfor the caseof a plane slopingbed.
The forcingmechanism
arisesfrom spatialandtemporalvariationsin radiationstressgradients
inducedby a modulatingincidentwavefield. Only thecaseof exactresonance
is considered,
where
thedifference
frequencies
,andwavenumbers
satisfytheedgewavedispersion
relation(thespecific
carrierfrequencies
arenotimportant,onlytheforceddifferencevalues).The forcingis ex,'unined
in
theregionseawardof thebreakpoint
andalsowithinthefluctuatingregionof surfzonewidth. In
eachregion,theforcingis dominatedby the cross-shore
gradientof onshoredirectedmomentum
flux, exceptfor largeanglesof incidenceandthe lowestedgewavemodes.Outsidethe surfzone,
thespatialandtemporalvariationof theforcingis determined
by considering
theinteractionof
two progressive
shallowwaterwavesapproaching
thebeachobliquely.In the surfzone,incident
waveamplitudesareassumed
to be proportionalto the waterdepth. Thusinsidethebreakpoint,
radiationstressgradientsareconstantandno forcingoccurs.However,at the breakpoint,gradients
arisingfrombreakingandnonbreaking
wavesareturnedon andoff (like a wavemaker)with
ti•nescales
and lengthscalesdeterminedby the modulationof the breakerposition. The tbrcingin
thisregionis stronger,with inviscidgrowthratesresultingin edgewavesgrowingto the sizeof
the incidentwavesof the orderof about10 edgewaveperiods,a factorof 2-10 timeslargerthanin
the offshoreregion. Usinga si•nplepar,-uneterization
for frictionaldamping,edgewave
equilibrium
mnplitudes
arelbundtodepend
linearlyontheratiotan]•/Ca , wherefl isthebeach
slopeandCais a bouomdragcoefficient.
For tan•/Ca about3-10,equilibrium
amplitudes
can
be as •nuchas 75% of the incidentwavesover most of the infragravityportionof the spectrum.
Whentheforcingis turnedoff, thesedissipation
ratesresultin a half-lifedecaytimescaleof the
orderof 10-30 edgewave periods.
Introduction
1985; Oltman-Shay and Guza, 1987; Howd et al., 1991;
Since the initial observations of Munk [19491 and Tucker
[1950] much effort has been aimed at understandingthe origin
and importance of low-frequency (relative to wind waves)
surfacegravity wavesin shallow water. Field data obtainedon
natural
beaches
haveshown
thatlong-period
(O(102-10
3 s))
infragravity motions often dominate power spectra in the
inner surf zone and swash, particularly during storms when
incident wave heightsin shallow water are severelylimited by
breaking [Huntley, 1976; Huntley et al., 1981; Holman,
1981; Thornton and Guza, 1982; Holman and Sallenger, 1985;
Sullenget and Holman, 1987]. The data also indicate that
infragravity surface gravity waves are predominantly
composed of edge waves, longshore progressive waves
trappedto the shorelineby refraction,together with a smaller
componentof leaky waves, normally reflected waves which
escapefrom the nearshoreinto deepwater [Guza and Thornton,
Copyright1997 by the AmericanGeophysicalUnion.
Papernumber96JC03722.
0148-0227/97/96JC-03722509.00
8663
Herbets et al., 1995b].
The existenceof edge waves in nature has been known for
some time. Bowen and (;uza [1978] and Holman [1981]
suggest that the growth of edge wave amplitudes, an, is
determinedby the coupling between the nonlinear forcing,
F(a1, a2, x), arisingfrom an interacting
incidentwavefidld
andthecross-shore
edgewavewaveform,On(x),
(1)
rYe
f F(al,a2,x)On(x)dx
g
o
where al and a2 are incidentwave amplitudes,rYeis the edge
wave radian frequency, g is gravity, t is time, and x is the
cross-shorecoordinate. Growth occurs when the forcing
pattern is not orthogonalto the edge wave waveform. The
details of the forcing are describedby the spatially varying
form of F. In this work we derivean analyticexpressionfor F
and numerically integrate the coupling integral to obtain an
estimatefor the growth rate.
The principalforcingis believedto be derivedfrom pairsof
incident wind waves, which produce spatial and temporal
8664
LIPPMANN
ET AL.: GENERATION
OF EDGE WAVES IN SHALLOW
variations in radiation stress gradients of the same timescale
and lengthscaleas the edge waves. Outsidethe surf zone in
intermediate and shallow depths, forcing occurs through a
typical second-ordernonlinearinteraction[Gallagher, 1971;
WATER
and Mei [1981], they assumeconstantbreakingcriteria so that
modulations in incident wave heights are manifested in
fluctuations in the width of the surf zone.
However, their
in specifying the forcing functions all the way to the
shoreline. Here we assume that amplitude variations are
eliminated by saturationand no oscillatory forcing occurs.
However,thereis a time dependentmomentumflux inducedby
the modulation in breakpoint position associatedwith the
variation in incident wave height [Foda and Mei, 1981;
Symondset al., 1982; Symondsand Bowen, 1984; Schaffer,
model is limited to only two dimensions (no inclusion of
longshore variability), precluding the possibility of edge
wave forcing. Nevertheless, the results of Symondset al.
suggestthat edge wave generationby temporal variations in
surf zone width is possible.
Schaffer and Svendsen[1988] retain the ideasof Foda and
Mei [1981] and Symonds et al. [1982], by allowing
fluctuations in both breakpoint positions and incident wave
energy inside the breakpoint, although they limit their
1993].
discussion to the two-dimensional case. List [1992] extends
Bowen and Guza, 1978]. Inside the surf zone, difficulties arise
Outsidethe surf zone (referredto herein as offshoreforcing)
this mechanism in a numerical model to include arbitrary
the generationmechanismevolves from the second-order topography and further includes incoming bound waves,
forced (or bound) wave generated by wave groups in crudely•nodeledempiricallyusingfield data,as do Schafferand
intermediatewater depths[Biesel, 1952; Longuet-Higginsand Svendsen [1988]. They forego a more theoreticalapproach
Stewart, 1962, 1964; Hasselmann, 1962]. In two dimensions,
the bound wave is, in principal, released at breaking and
reflected at the shoreline; thus the incoming and outgoing
waves form a complex pattern in the cross-shore,however,
without the longshorecomponentnecessaryfor edge wave
since it is not clear what the correct boundaryconditionsare
for boundwave dynamicsin the shoalingand breakingregion,
although it is expected the forced motions become nearly
resonant in shallow water [Longuet-Higgins and Stewart,
1964; Okihiro et al., 1992].
Schaffer [1994] usesa WKB (optics) approachto describe
generation. This idea was extendedto three dimensions
by
Gallagher [1971], who showed that nonlinearinteractions the incident waves and drives the edge waves with radiation
between incident wave pairs could producelow-frequency stressmodulationsdeterminedby fluctuating amplitudes(i.e.,
oscillationswith a nonzero alongshorewavenumber. Bowen
and Guza [1978] generalized Gallagher's model to include
forcingof all modesand showedwith laboratoryexperiments
that resonantresponsewas greaterthan forcedresponse.This
same conclusionwas reachedby Foda and Mei [1981] in a
fourth-orderWKB expansionof the momentumequations.
In the surf zone, forcing occurs because an interacting
incidentwave field producesspatial and temporalvariationsin
locations of the position at which a wave breaks (herein
referred to as surf zone forcing). Momentum is transferred
from incidentto lower frequenciesthroughwave breaking,and
the interactionsagain lead to fluctuations in the flow field
with timescalesand spacescalesof the order of wave groups.
This idea was first explored as a generatingmechanismfor
long waves by Foda and Mei [1981] and Symondset al.
[1982]. As will be shown later, there is also a contribution
from nonbreakingwaves within the region of fluctuatingsurf
zone width.
In the work by Foda and Mei [1981], low-frequencywaves
are generatedby wave-wave interaction assuminga fixed
breakpoint position for all waves. This allows incident
modulationsto progressto the shoreline,and edge waves are
forced everywhere. In nature,however,the initial breakpoint
of individual waves is not constant through time, nor is the
breaker line uniform along the beach. Under most conditions,
temporal and spatial variations in the width of the surf zone
occur on large timescalesand space scales,of the order of
infragravity scaling [Symondset al., 1982]. Although the
basic physicsinshore from the breakpointis doubtful, Foda
and Mei do consider resonant edge wave growth, also
consideredhere, and thus a brief comparisonwith their results
is made later.
Symondset al. [1982] considerthe problemof long wave
forcing by time modulationsin surf zone width for the caseof
a planeslopingbed and later includedinteractionswith barred
topography[Symondsand Bowen, 1984]. In contrastto Foda
the two incidentwaves are assumedto approachfrom the same
direction). Numerical solutionsare considerablycomplicated
by an angularbottomwhich necessitates
a matchingcondition
at a corner in the profile. Furthermore, Schaffer's offshore
profile is horizontal; thus waves propagatingoffshore beyond
this "shelf" are no longer refracted, thus limiting the possible
edgewaveswhich can exist. Low-frequencyresponseis found
through numerical calculation, with resonant modes
determinedcritically by the single-incidentwave angle and the
offshoreprofile configuration. This resultsin a very specific
set of results only valid for theserestrictedset of conditions.
In this paper we present a mechanism for the resonant
forcingof edgewavesin shallowwater throughmodulationsin
radiation stress gradients. We consider two interacting
shallow
water
incident
waves whose difference
wavenumbers
and frequenciessatisfythe edge wave dispersionrelation. The
total forcing is found by integratingthe growth rate equation
froin the shoreline to deep water (relative to wind waves).
Contributionsto the forcing from inside and outside the surf
zone are examined by separatingthe forcing integral at the
breakpoint. We considerthe simplestcaseof a planar beach
profile, where forcing in the surf zone is determinedby a time
and spacedependentfluctuationin the width of the surf zone.
In the next sectionwe review the theoryfor edgewaveson a
plane sloping beach beginning with the forced shallow water
(depth integrated),linearized equationsof motion, leading to
an analytic expressionfor the initial (undamped) edge wave
growthrate. Model resultsare then presented,comparingthe
relative contributionsto the forcing from the componentsof
the radiation stressand also the strengthof inviscid growth
rates in the surf zone and offshore regions. Resultsare then
discussed in terms of the validity of model assumptions,
parameter sensitivity, and implications in field situations
(spectralforcing). The effect of a linearizeddissipationterm
on edge wave growth is discussed. In Appendix B the
extensionof the calculationsto an incident wave spectrumis
briefly discussed.
LIPPMANN ET AL.: GENERA•ON
OF EDGE WAVES IN SHALLOW WATER
where L n is the Laguerre polynomial of order n. The
approximate shallow water edge wave dispersionrelation is
given by [Eckart, 1951]
Model
Edge Wave Theory
The forced, shallow water (depth integrated), linearized
equationsof motionincludinga linearizeddissipationterm are
[Phillips, 1977]
2
cye- gke(2n+1)tanfi
Growth
m +
3t X-b--fxph
-
Sxx+
•+
Sxy+
-
a, gTfy ph
- •,u
3y xy
Tyy
s"
-•v
(2)
3t
(h.)+
(8)
Rates
We seek an expressionfor the time rate of change of edge
wavemodalamplitudes,
or growthrate, 3an/3t. Substituting
(3)
(6) into (5) and allowing an to be a slowly varying function of
time give
2a
n 3a
n -anZ(qJn)]e
-ire
-F
and the continuityequation
--+
8665
(9)
3--'•qn
+i2tYe
3y
hu)-0
(4)
where
g-3xlh3•-•l+(CYe2-ghke
where x andy are the cross-shoreand alongshorehorizontal
Cartesiancoordinamswith x positiveseaward(for a right-hand
systemwith z positiveupward),u andv are the corresponding
life- key- Getß
horizontalcomponentsof velocity, • is sea surfaceelevation,
• is a friction coefficient (discussedlater), h is the still water
In (9), and in subsequentequationsfor the growth rate, we
depth,andp is the densi• of water. The equationshavebeen have droppedthe complex conjugatefor brevity, thus later we
averaged over an incident wave period so that the time will take the real part to obtain the magnitudeand phase. The
dependence
is on infragravi
W andlongerscales.Spqare the function Z(cpn) is the homogeneousequation for shallow
radiation stresses of short (incident) waves introduced by
water waves and vanishes for the case of resonance considered
Longuet-Higginsand Stewart [1962, 1964] and describethe
flux of pth directedmomentumin the qth direction.
Combining (2)-(4) and temporarily dropping the damping
term yield a single second-order(inviscid) equation in sea
here. If we assumethat the edge wave growth rate is slow
relative to its period, then the first term describing the
accelerationin growth can be neglected. Thus an equationfor
the initial edge wave growth rate is
surface elevation [Mei and Benmoussa,1984],
i2o'•,c-•--c),•e-iV•'
-F
(10)
c-)271
c-)t
2
p•
ax+ ay +•yy(ax+ ay
(5)
(describedlater), (ai)o, the normalizedgrowthrate becomes
= Fxx+ Fyx+ Fxy+ Fyy= F
whereFxx,Fyx,Fxy,andFyyarethecomponents
of thetotal
forcing, F, correspondingto the four secondderivativeson
the right-handside. The forcingand frictionaldissipationare
assumedto be of secondorder, and we ignore their effects on
both the wave solutions and the dispersionrelation (which
couldbe significantif the forcingand friction are large).
For a planebeach,h= xtanfl, where/8 is the beachslope,
the homogenous
case(free waves)is satisfiedby edgewavesof
the form [Eckart, 1951]
rln(x,y,t) - •-anc)n
(x)e
-i(l•y-cr•t)
+(*)
1 e3a•
=(ai)ogtanfl(2n+l)
-2rri I F0,•dx
(ai)of
•gt
e-iv/•
(11)
0
in
which
thenormalizing
factor
[•cpn2dx
- 1/2k
ehas
been
used.Thecoupling
integralexpress4s
therateat whichenergy
is transferred
from
(6)
complex
conjugate
of theprevious
term,andi-4Z-f. The
cross-shorestructureof the edge wave waveform, qS,•(x), is
given by
(7)
the
incident
waves
to resonant
lower-
frequencymotions. The inverseof the magnitudeof (11) is the
number of edge wave periods necessaryfor the edge wave to
grow to the size of the primary incidentwave.
Coupling Integral
where CYe=2nfeand ke=27r/Le are the edge wave radian
frequencyand alongshorewavenumberOreandL e are the edge
wave frequency and wavelength), n is the edge wave mode
number,an are the complexmodal amplitudes(which can be
resolved into a magnitude and phase), (*) indicates the
c),•
(x)- e-k•XL,•
(2kex)
The cross-shoredependenceis eliminated by multiplying by
qS,•
and integratingfrom the shorelineto infinity in the crossshore direction, as in (1). Incorporating the dispersion
relation and dividing by the primary incident wave amplitude
and Incident
Wave Amplitudes
In deep water, resonant interactions are quartic and only
forced waves are produced by triad interactions. In shallower
water, resonant triad interactions can occur, producing
variation in radiation stresswhich can match the edge wave
dispersion relation. This spatial and temporal modulation
gives rise to the possibility of edge wave growth. Although
the modulationscalesremain the same, the forcing function F,
which dependson wave amplitude, varies as waves shoal in
intermediatewater and eventuallybreak in the surf zone. As a
consequence,there are two distinct forcing regions separated
by the contourof the breakerline, xb(y,t) (shown graphically
in Figure 1), and the coupling integral can be written
8666
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
OffshoreForcing
Breakpoint
•b '4'6•b
I
No Forcing
Shoreline
!•Ie ---keY-- (Je
t
Figure 1. Graphical representationof the edge wave forcing regionsrelative to an oscillatingbreaker line.
Thevertical
axisis Z= cre2x/gtan,B
andthehorizontal
axisis gre=key-Get.Theoffshore
forcing
region
is
defined as seawardof the •nostseawardbreakpoint. Inside the minimum breakpoint,no forcing occurs. The
rangeof forcingby a modulatingbreakpointpositionis a functiono.__f
incidentwave modulation,8. The
regionswhere waves are alternatelyshoalingand breaking(between Zb and the oscillatingbreakerline) are
labeled I and II, respectively.
Go
x!,(y,t )
the mean wave amplitudeand the modulation(aboutthe mean)
is determinedfrom a2 [Symondset al., 1982].
Becauseof sharpdifferencesin the shoalingand breaking
regions, the amplitudes of the incident waves are treated
differently. In the offshoreregion the amplitudesare described
by progressive shallow water waves over a plane sloping
bottom; thus an interactingwave field has modulationsall the
way to the breakpoint. Wave amplitudes in this region are
referenced to a convenient location (e.g., the breakpoint).
Inside the surf zone, wave amplitudesare stronglyattenuated
shorewardby breakingand are describedas a linear functionof
local depth,
IF(al,a2,x)cPnCx)dxIFsz(al,a2,x)cpn(x)dx
o
o
+ IFoff(al,a2,x)cpn(x)dx
(12)
xt,(y,t)
whereFszandFor
f aretheforcinginsideandoutside
the
breakpoint, respectively.
The forcing functions are
determinedby the temporal fluctuations of radiation stress
gradients,which are in turn determinedby •nodulationsin
incident wave amplitude.
It is the slowly varying breakerline, xb(y,t ), that creates
first-order fluctuations in the surf zone coupling integral. If
we assume that the breakpoint varies sinusoidally with gre
tilne and lengthscales(describedlater) andfurtherseparatethe
couplingintegral at •'b and expandin a Taylor'sseriesabout
1
ai =•-Th
x <xb
(15)
where ), is a constantof O(1) [e.g., Thorntonand Guza, 1982].
For a plane beach,referenceamplitudescan then be given by
'•t,, thento first order,(12) canbe written(AppendixA)
1
(ai)o=• ?xb
tan/3
(16)
where x•, is the position of the breakerline, a function of y
0
and t.
+IFoff(ala2,x)cPn(x)dx
Once the incident waves break, •nodulations in incident
(13) amplitudes
x/,
Thus the first-order contributionsto edge wave growth occur
through primary interactions from both breaking and
nonbreakingwaves within the fluctuatingregion of surf zone
width and through cross-interactionsof a pair of waves
seawardof the average breakpointposition.
In the following, the wave field is considered to be
co•nposedof two waves,a primary wave with amplitudea I and
longshorewavenmnberand frequency(kl, fl) and a secondary
wave with mnplitudea2 and slightlydifferentwavenumberand
frequency(k2, f2)- If we choose
a2 = •a I
vanish and the forcing becomes constant
shoreward. The possibility for edge wave forcing occurs
because the breakpoint position varies on timescales and
spacescalesassociatedwith the incident beat. Thus the limits
of integration in (12) for the surf zone integral will have
modulationswith edge wave timescalesand spacescalesthat
are deterIninedby the fluctuatingbreakpointposition(Figure
2). The •nagnitudeof the modulationis determinedby the
maximum
and •ninimum breakpoint
amplitudes,
ama
x = aI + 8aI and amin = aI -c•al, respectively,so that the
cross-shore
rangeover which forcing occursis definedby
(14)
where • << 1 is a constant,then the primary wave determines whereZ'b is the meanbreakerposition.
(17)
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
8667
within the fluctuating region of the surf zone, which will
the same frequency and wavenumber as the
.'''''' I ''""""'f(ao(l+•))
oscillate at
fluctuatingsurf zone boundary(given by the tiine varying
a=Th/2
•
z
/
f(ao)
I
contour of the breaker line).
We choosethe velocity potential for progressiveshallow
water waves over a plane sloping bottmn after Stoker [1947]
I....... i ....... f(ao(1-fi))
•
x•(l+fi)
msl
,.x
Figure 2. Graphical representationof the envelop of
incident wave ainplitudes over a plane sloping bed as a
functionof cross-shore
distance.Surf similarity( a = 7h/2) is
assumed once waves are breaking so that inodulations in
incidentamplitudesproducevariationsin surf zone width. The
vertical (elevation) and horizontal (cross-shore distance)
diinensionsare arbitrary.
(I)i --aig[Jo(Xi)cosyt'i+Yo(Xi)sin
I//i] (22)
where
Xi_2(
cri2x
)112
gtanfi
IIIi - (kiy- crit
)
and Jo and Yo are the zero-order bessel functions of the first
and secondkind, respectively,ki is the alongshore
cmnponent
of the incidentwavenumbervector, and the subscripti is an
indexwhichrefersto individualincidentwaves. For Xi >3
(equivalent
to H>-•(Ttanfi)2/2,so validfor nearlyall
Parameterizing
the Forcing
oceaniccases),the besselfunctionsinay be approximated
by
Following Phillips [1977], the general forin of the
radiation stressis given by
,q'pq
--[3__jh
C_)W
3X
q (S
pq Z--
[Stoker, 1947]
Cos(X/--•)(23a)
Jø
(Xi)= '•i
2/¾2
•
sin(X
i-•)
(18)
where x is the horizontal spacevector in which subscriptsp
and q denote either horizontal coordinate, (I) is the velocity
potential of the incident waves, fi is the mean pressure,and
Introducingcomplexnotationyieldsthe followingform for (I)i
(hi--•
aig(--2)l/2ei(X'-%+
cr
i •,•rx
i
+(*) (24)
5pqis theKronecker
delta.Themomentum
balance
hasbeen
spatially averagedin the directionof the wave crests(allowing
separation of the turbulent and wave terms) and temporally
averaged over the incident wave period (allowing mean or
slowly varying properties to be evaluated), denoted with the
overbarsin (18). In shallow water •--pgz
and assuming
7]= 0 at secondorder,
(23b)
The form of our velocity potentialis only strictly valid for
waves approachingthe beachshorenormally. However, Guza
and Bowen [1975] show that nonnormalanglesof incidence
have only small effectson the forin of the velocity potential
due to refraction decreasingthe angle of incidence and show
that where shallow water solutions are valid, the solutions for
Spq
•ph•x•
•x•+
•pq
• •
We consider
the bichromatic
case where
(19)
the incident
wave
field is composedof two discrete wave trains with slightly
differentwavenumberand frequency,
• - •
+ •2
shore-normal
and obliqueincidenceangleare nearlythe same.
Substitutionof (24) into (21) producesterms with sum and
differencewavenumbers
andfrequencies.The sumterms(highfrequencyforcedwaves) are unimportantto the generationof
infragravity motions [Bowen and Guza, 1978] and are not
consideredfurther. The difference terms describethe long
timescalesand space scalesof the incident wave modulation
and allow for the possibilityfor edge waves at infragravity
(20)
where the subscripts1 and 2 refer to the two incident waves.
frequencies
Substituting• into (19) produces
Spq
-Spq(*l*l)+2Spq(*l*2)+Spq(*2*2)(21)
The first and last terms represent the self-self interactions
which generate•nean flows and harmonics. The harmonicsdo
not contribute to low-frequency forcing in either region and
are therefore neglected. Letting the amplitude of a primary
wave •i be much larger than any secondary waves •2
(equation (14)), then •1•1 terms are of O(1), while cross-
interaction
terms•1•2 areO(8) and•2•2 termsareO(82).
The stressarising from •i•i
interactionswill generate a setdown outsidethe breakpointand a setupinside the breakpoint
Of = CY1-CY
2
(25a)
[kfl-Ikl
-k21
(25b)
and
where k! and k2 are the longshorewavenumbersof the incident
wavesand the subscriptf indicatesthe forced differencevalues
associated with the incident wave inodulation.
For resonance
to occur,(25a) and (25b) inustsatisfythe edgewave dispersion
relation (8).
Incident wave angles are asstunedto follow Snell's law for
wave refraction and are chosenat the breakpointwhere the
8668
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
linear shallow water phasevelocity is simply a function of the
local
depth,
c-•,
_
sothat
ki •yi2
'
g
=
sin O•o =
tYisin(o•i)o (26)
X4o
X•
5 e
+sina
3(2n+
1)X•
222sin
3-Xe
-i3X
l+sina
12(2n+
1)2Xe
o•
1sin
a2+1)(29)
ß
where
ooindicates
thedeep
water
condition
and(o•
i)oarethe
wave angles of the individual waves taken at some reference
depth, ho. If we further referenceai to the breakpoint,where
ho = hi,, thenproductsala2 are givenfrom (24) by
=7r
(Sei(Xe-U/e)
p (gtanfi
o
The three terms are derived from the componentsof the
interaction
radiation
stress
fromFxx, 2Fxy,andFyy. The
three componentsof the interactionforcing are compared
later. Contributions
to edge wave forcingby nonbreaking
wavesseawardof the breakpointbut within the fluctuating
]1/2
where the subscripto refers to values at the referenceposition
region of surf zone width are consideredin the next section.
Inserting(29) into the couplingintegralin (13) resultsin a
complexexpression
for the undmnped
response
of edgewaves
andevaluating
the appropriate
derivatives
give an analytic by oscillatingforcingseawardof the •neanbreakpoint. After
(e.g., the breakp.oint).
Now substituting (24) into (19), incorporating(26)-(27),
form of the radiation
taking the real part, expressionsfor the normalized initial
stress
growth
ratemagnitude,
Goff,andphase,
Ooff,
aregivenby
(28)
Spq
=•
(7off=
1 •an=
3•'•-•
1/2(30)
fe(al)o
at 4(2n
+1)•(•bb)3(pn2+Qn2)
where
3 2a•
I
lpq= 2a• 2al2+1
where
Pn- I (32cos
Xe- B1sin
Xe)q),•XedXe
JPq+a2 2ala
2+1
XeA
-•(Xl
0•1X20•2)
Kpq=
•(Xlal
_X2a2)
Qn
= I(BicosXe
+B2sinXe)q)nXedXe
Xb
0
1
B1 =
3-Xe
2+sino•1q-sina 2 - 2sina• sina 2 + 1
X•5
3(2n+l)Xe
2 12(2n
+1)2
X•
3
XiX2
O.e2
x/¾2
Xe=2 gtan•
B2- Xe
4
Surf Zone Response From Breaking and
Nonbreaking
Waves
The cross-shoreintegral parameterizingthe surf zone
contributionto the total edge wave forcing rangesfrom the
ß 1• interactions,O(•5) termsare from •2
interactions, shorelineto the breakpoint,0_<x_<xb. FollowingSymonds
et al. [1982], we do not allowmodulations
in waveamplitude
andO(•52)
terms
arefrmn•2•2 interactions.
insidethe mini•num,•nostshoreward
breakpoint
by choosing
Using typical values found in nature, f• =f2 =0.1 Hz,
xb--lOOm, and tanfi=0.02, we find A is O(10 -3) at the ¾ to be constantfor all waves. Thus for a plane beach,
•nodulationsin breakpointamplitudesgeneratespatial and
breakpoint, and therefore terms in (28) containing A
contributenegligibly to the interaction and are neglectedin temporalvariationsin the width of the surf zone. Symondset
long wave
the following. This implies that althoughinteractionsof the al. usedthis as the basisfor their two-dimensional
incidentwavesdeterminethe responsefrequency,the specific model.
Wave mnplitudesare expressedin terms of cross-shore
carrierfrequenciesare not important.
andsin(o•!
)oandsin(o•
2)ohave
been
abbreviated
o•!ando•2,
respectively,for simplicity. In (28), O(1) terms are a result of
Response Outside the Surf Zone
The forcing function outsidethe mean breakpointposition
is found directly by insertingterms of order •5in (28) (arising
from cross-interactionradiation stresses)into the right-hand
side of (5) and evaluatingthe secondspatialderivatives,
breakpointposition using (16). The surf zone forcing
functionis then foundby inserting(28) into the right-hand
sideof (5) andevaluatingthe spatialderivatives.Sincewe are
only interestedin first-orderapproximation,
we needonly
retainthe pri•naryself-interaction
terms(AppendixA). Inside
the breakpointfirst-order radiation, stressgradientsfor
breakingandnonbreaking
wavesaregivenby
LIPPMANN
ET AL.: GENERATION
3Bo.•t,(Xe
)o3
•Sxx
breaking
-- 2Xe
3x
OF EDGE WAVES IN SHALLOW
3x- 4Bøx
sin
ctl
Ibreaking
0Syy
=0
3y
integrationare symmetric;hence z'l =-z' 2 --z'.
Cm=
(32)
m=l
where
• 'r2
iTM)
35pq
e_imu/,
r!(y,t)
3Xp
1
where z'1 and z'2 define the interval over which forcing occurs
and (*) indicatesthe complex conjugateof the previousterms
under the summation. The form of (32) is given for the
breaking waves only; nonbreaking waves in the fluctuating
region of surf zone width are rc out of phase with (32).
Symondset al. presentan extensivediscussionfor finding the
Fourier limits of integrationfor the two-di•nensionalcase. We
extend their ideas to three dimensionsin the following. The
mean breakpointmnplitudeis defined by the larger wave, and
the fluctuation about the mean is defined by the amplitude
lnodulation,B, suchthat by usinglinear superposition
of two
sinusoidal
38pqsinmz
3p
m'r
(36a)
re=l,
2....
(36b)
The first term in the series(Co) representsthe mean value
over all •e space and subsequentlyvanishes when second
derivatives are taken.
For m > 0, the series consists of a
primarymodulation(m = 1) and its harmonics(m > 1). Thusto
lowest order (consideringonly forcing contributionsfrom the
primary) the total forcing from breaking and nonbreaking
waves in the fluctuating region of surf zone width is found
(after evaluatingthe secondspatialderivatives)
rsz--- --123ø
ei(-'l•'•-IF")
p
(37)
1(3)sinx
sintzlXt,2(cos•:
sin•:
sinx)}
waves.
Becauseradiation stressgradientsare discontinuousat the
breakpoint, we represent (31) as a co•nplex Fourier Series.
Extending Symondset al. [1982],
Using (35),
are
Co = const
(31d)
depth is eliminated using (26) (the incident alongshore
wavenumberis conservedacrossthe nearshoreregion).
Since we are taking T to be constant,all modulationsinside
the initial breakpoint vanish, and each component of (31) is
constant. In essence,variationsin the forcing are defined by a
wave maker type problem, where the forcing is turned on and
off at the initial breakpoint(as in the work by Symondset al.
[1982]). Periodic fluctuations in radiation stress gradients
arise from fluctuations in the position of the initial
breakpointon timescalesand spacescalesof the modulation.
Alongshore modulations allow for the possibility of edge
X--' Xb
coefficients
(31c)
where
Bo=pg(ytan
fl)2/8. Thedependence
ofwaveangle
on
3Spq
+(*)
3Xp =Cø
+ZCmeiU/
(35)
Symonds et al. [1982] show that for small • the limits of
(3lb)
the Fourier
•)S
xy
8669
=cosx/,-xt,
=
(31a)
nonbreaking
3Syx
=0
WATER
ß 1+• z' 6(2n+1)z' z'2 •-i
The first andsecondtermsarisefrom Fxx forcing by breaking
and nonbreakingwaves, respectively. The relnaining three
termsarisefrownFxy forcingby breaking
wavesonly. The
componentsof the forcing are comparedlater.
The initial growth rate is found by inserting (37) into the
first coupling integral and evaluatingin the salne manner as in
the offshore region, except that the limits of integration are
now given by (17). Taking the real part gives expressionsfor
the magnitude, G sz, and phase, Osz,of the initial edge wave
growth rate in the fluctuatingregion of surf zone width
Gs
z= 1 3a,•
_ 127
2 2
fe(t,,'••o
3,- (2n
+1)•t,
2(Mn
+Nn
)1/2(38)
where
X2
Mtl •
waves
f (DlCO
S•-•-•b
- D2
sin•'•-•b
)cp
nXt,dX
t,
X•
X2
at,a•
I1+tScøs!•,t,•e)] (33)
Ntl •
I (D2
cos•t,
+Dlsin•X--•)cpnXt,dXt,
X•
whereXt, is themean X e value corresponding
to x= •t,. We
D1 =
can express(33) in terms of breakpointpositionsusing (16)
(34)
whereZt, is the meanbreakpointposition. The upperlimit of
integrationin (32) is determinedfroln the argumentof (34)
sina•Xb2 sinz
6(2n+1) z
( 3•)sinz'
z' Xt,2sintzl(COSZ
6(2n+1) z'' sinz')
•-2
D2= 1+
.
whereX1= Xt, - 8Xt, andX2 = Xt, + 8Xt,.
8670
LIPPMANN
ET AL.: GENERATION
OF EDGE WAVES IN SHALLOW WATER
Results
surf zone mechanismsis shown in Figure 3. The magnitude
The resultsare presentedin two parts. The first focuseson
andphasefor the caseof (a 1)o= (ø:2)o= 10ø are plottedas a
therelative
magnitudes
of FpqrPn
whichcomprise
thesecond- function of nondimensional35b(the legend defining individual
orderforcingfunctions(29) and (37) from theoffshoreandsurf forcing componentsis given in the figure caption). For small
zone regions,respectively. The secondpart comparesinitial 35b,Fxx dominatesthe forcing in each region. At higher35b,
growthrate amplitudesandphasesin the two regions,(30) and longshorefluxes become relatively more important. Because
(38), and for the total combinedforcing. Sensitivityto mode the forcing decays rapidly as 35b increases (for mode 0),
number,n, and wave angle, (o•i)o, are examinedin eachpart. consideringonly Fxx gives a good first-orderapproximationto
Model parameters used in the following analysis are the overall forcing. The forcing at higher modes,n- 1, 2, and
tanfl-0.01,
7-0.42,
6-0.1,
andxb =100 m, chosenas
3, is shownin Figure4 for (al) o =(o:2)o =10ø. Only the
For convenience, incident
forcingmagnitudes
areshownsincethephases
of Fpqarethe
reasonable for field situations.
wave angles were chosento be identical at the breakpoint. stoneas for mode0 (Figure3). The behaviorof the forcingis
similar in each region, with Fxx dominatingall modesdue to
Resultsare plotted againstthe nondimensional
variable
thedependence
of Fxy(and
Fyx)on(2n+l)
-1 andFyyon
35l,= •
g tanfl
(39)
The range of values plotted, 350<10, covers the range of
typical infragravityfrequenciescommonlyobservedin nature.
Forcing
The
Components
relative
strength
of
the
components
of
F(al,a2,x)cPn(x) for mode0 edgewavesby the offshoreand
10-'
(2n+l)-2. Since
Fxxis insensitive
to modenumber,
the
forcing at higher modeswill be nearly the same as for the low
modes, suggestingthat all modes should be about equally
forced.
In general, the amplitudeof the forcing function in the surf
zone is an order of magnitude larger than in the offshore
region. Thus it is expectedthat this part of the forcing will
dominate the growth rate. Since the forcing function in the
surf zone is nearly independentof Zb (to first order), the
Surf Zone
Offshore
104
10'4
10-5
10-5
10-•
10-6
, 10-7
0
I
2
3
4
5
6
7
0
1
2
3
4
5
6
7
10
0
1
2
3
4
5
6
7
8
9
10
10
0
1
2
3
4
5
6
7
8
9
10
180
135
90
-45
-90
-135
-180
8
9
Zb
Figure 3. Contributionsto the total edge wave forcing function, Fen , for a mode0 edgewave, (left) from
the offshoreregion and (right) within the surf zone. The componentsof the forcing arisingfrom Fxx (dashed
lines),Fxy(dash-dotted
lines),Fyy(dotted
lines),andthevector
sums
(solidlines)areplotted
asa function
of
35b
=rYe2Xb/gtanfl
(intheoffshore
region,
Fyx=Fxysothat2Fxyisplotted,
whereas
inthesurfzone,
Fyx
andFyyarezero).Results
areshown
for(al)o=(a2)o=10ø. Forcing
amplitudes
(m/s
2)ona logscale
are
shownin the top panels;forcing phase(degrees)on a linear scaleis shownin the bottompanels. Resultsare
computed
for tanfl= 0.01, y = 0.42, 6- 0.1, and xb = 100 m.
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
8671
10-3Offshore
k Surf
Zone
1i'1-5
•
- •
Mode
!
Mode
2
Mode
3
10
-• ,,//'•"•-'
-• f
10-7I I '• I I I I I I I I
10-3
10•
]
10-s
10•
.
,
10-7 I I I I I I I I I I
I
I
I
10-•
104
10-•
•. ....
•.
/'
10
-*
x
/';'
/'
.I
,.:.
.....
[..'
0
1
2
3
4
5
6
7
8
9
10
3
4
5
6
7
8
9
10
Zb
Figure4. Forcing
components
arisingfrom Fxx(dashed
lines),Fxy(dash-dotted
lines),F
(dottedlines),
andthevector
sum,
F•Pn
(solid
lines),
foredge
wave
modes
1-3,with(Crl)
o=(or2)
o=•o, plotted
asa
function
of ZI,= Cre2Xb/gtanfi
ß Contributions
fromtheoffshore
andsurfzoneregions
areshown
in theleft
and right panels, respectively. Only the amplitudesare plotted since the componentphasesare identical to
mode 0 (Figure 3; the total phase does vary slightly as a function of mode). Results are computed for
tanfi =0.01, ?'=0.42, 6=0.1, and xb =100 m.
spectraldependenceof the forcing term will be determinedby
the form of the edge wave waveform, having nodes at
particular Zb. If the integrationof the couplingoccursover a
narrow range of Zb, then the spectralshapeof the Onwill be
stronglyreflected in the growth rate.
The effect of (oci)o is examined in Figure 5 with
function of Zb. As expected,the growth rates within the surf
zone region are much larger (by a factor of 2-10) than in the
offshoreregion. The growth rates are rapid, with a magnitude
of about10-I across
mostof therangeof Zb plotted.Because
of the choice of normalization, the inverse of G gives the
numberof edge wave periodsfor the edge wave amplitudeto
(Crl)o=(cr2)o=1 ø, 10ø, and30ø. Resultsfor a mode1 edge grow to the size of the incident waves. Thus, assumingno
wave are shownas a representative
example. At small angles, dissipation,resonantlyexcited edge waves could grow to the
Fxx dominatesthe forcing, being an order of magnitudegreater sameamplitudeas the incidentmodulationin as fast as 10 edge
than the other terms. However, the forcing for low modes wave periods(and even fasterfor lower modesand smallZb).
should
beenhanced
for steeper
angles
of incidence
because
Fxy The model predicts only initial, undamped growth rates
andFyyhavea strong
modalandangular
dependence,
although (essentially the rate at which energy is transferred from
on shallow beaches, incident wave angles are not expected to incident to edge waves), and any reasonable damping
be large owing to refraction effects.
mechanismwill reducetheserates. The effect of dampingis
discussed in the next section.
Growth
Rates
Normalized growth rate magnitudes,G, and phases,0, for
the offshore and surf zone regions (equations(30) and (38))
and the total (vector sum) are shownin Figure 6 for edge wave
modes0-3 with (or1)o=(cr2)o =10ø. The amplitudesare
plotted on a log scale and the phaseson a linear scale, as a
The resultsfor the surf zoneforcingmechanismshowsharp
valleys correspondingto nodes (zero crossings)in •Pn. The
shapeof the edge wave profiles is similar to the growthrates
for the surf zone mechanism. The corresponding
valley in the
offshoregrowthrate doesnot occurat the first edgewave node
(Zb =1.5), nor is there any indication of strong nodal
structure at higher Zb. This result is due to integrating the
8672
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
10-3
Offshore
Surf Zone
10-4
10-5
.J
10-6
10-7
I
I
I
I
I
I
I
I
I
I
10-3
0-41
./
10-s
c•-10
10
-6
10-7
I
I
I
I
I
I
10-3
10-41
10-5
at- 30
10-6
10-7
I
012345678910
01234567891
)•b
Figure 5. Effectof incidentwaveangle( (a I )o = (a2)o = 1ø, 10ø,and30ø) on the forcingcomponents
for a
mode 1 edgewave. Formatis the sameas in Figure4.
E
•0
ø
ß
• Mode
1
Mode
2 ] Mode
3
/'
10-2
1
I
'i,i
/ l'\/!' '•'
!'
180 135 90-
450-45
-
-90
-
-135
-
I
I
-180
I
012345678910
012345678910
I
I
I
I
I
I
I
012345678910
I
I
I
I
I
I
I
I
I
012345678910
Zo
Figure 6. Normalizedinviscid growth rate (top) magnitudeand (bottom)phasefor edge wave modes0-3.
Growthratesfor the offshoreregion(dashedlines;(30)), the surf zoneregion(dash-dottedlines;(38)), and the
total(solidlines),areshown
asa function
of Zt,=Cre2Xb/gtan•
ß Theinverse
of thegrowth
rateis the
numberof edgewave periodsnecessaryfor the edgewavesto grow to the size of the incidentwaves. Results
areshownfor (a 1)o=(a2)o = 10ø, tan•=0.01, ?'=0.42, 6=0.1, andxt,= 100m.
I
!
I
LIPPMANN ET AL.: GENERATION
ø
OF EDGE WAVES IN SHALLOW WATER
Offshore
Surf Zone
10
-•
Totals..,
'"'"'
10-2
10-3
8673
_
I I I I I I I
180 -
135
90
45
0
-90
-
-135
-
-180
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
0
1
2
3
4
5
6
7
8
9
10
Zb
Figure 7. Effect of incidentwave angle on inviscid growth rates for a mode 1 edge wave. Resultsare shown
for (left) the surfzone,(middle)offshore,and(right)totalfor (al) o =(a2) o =1 ø (solid lines), 10ø (dashed
lines), 30 ø (dottedlines), and 45ø (dash-dottedlines). Format is the sameas in Figure 6.
predicted growth rates are higher than can be expected in
nature and places an upper bound on edge wave growth.
We have considered the forcing due to an interacting
incident wave field consisting of only two shallow water
waves, yet in natural situations, incident wave fields are
distinctly not bichromatic but rather consist of a spectrum of
energy. If we have a continuous directional spectrum in
n= 1 with (a I )o: (ø•2)o: 1ø,10ø,30ø,and45ø. The effectof
o• is small for the directional range investigated, particularly
nature, some (small) componentsof the forcing will satisfy
for theoffshore
regionwhere2FxyandFyyarerr outof phase. the resonancecondition for a number of different edge waves
The angulardependenceis more pronouncedin the surf zone
of variousfrequenciesand wavenumbers.
In the offshore region where we have referencedamplitudes
regionfor Zb >'" 3.0, increasingthe growthrate abouta factor
of 2 from 1ø to 45 ø. Interestingly, in the offshore region,
to the breakpoint, the spectral problem seems inherently
increases in wave angle tend to reduce the growth rate (a
linear where the superposition of many wave components
consequenceof the rr phase relationships in the longshore leads to the possibility of resonant triad interactions
forcing in the offshoreregion over a larger length of the edge
wave profile. In the surf zone, the length over which the
forcing occurs is very much smaller, determined by the
incident modulation (•=0.1),
and thus nodal points are
strongly reflected in the growth rates.
The effect of o• on growth rates is shown in Figure 7 for
components
of Fpq),whereas
theopposite
occursin thesurf
occurringfor any numberof different (cyi-cyj, ki-kj)
zone where the growth rate becomes larger as wave angle
incident wave pairs. The forcing of a particular edge wave
mode is just a linear sum of all possible interactions
satisfyingthe edge wave dispersionrelation. Bowen and Guza
[1978] discuss the implications of this resonant restriction
increases
(owingto increased
contributions
fromFxy).
Discussion
Growth
Rates
The predicted growth rates for the case of phase-locked
forcing by deterministicwave trains are not unrealisticif we
expect this mechanism to provide reasonable forcing of
progressive edge waves under the stochastic forcing
conditionsfound in nature [Holman, 1981]. The same type of
result arose in the study of energy transfer into internal
gravity waves from surfacewave packets. An initial study of
Watson et al. [1976] showed a very strong forcing for the
phase-lockedcase. Olbers and Hefterich [1979] redid the
problem for a stochasticsurface wave field and found the
strength of the forcing to be several orders of magnitude
weaker thanpredictedby Watsonet al. This suggeststhat our
and show that for
narrow-beam
incident
swells,
some
frequency selection, with a strong modal dependence,may be
expected.
In the surf zone, the situation is more complicatedbecause
amplitudesare functionsof local depth (and hencedistancefor
monotonic beaches). However, for the bichromatic case, the
forcing is dominated by F xx arising from the self-self
interactionof the primary wave. If we consideronly this term,
then the growth rate equationtakesthe form
1 c-)a
ne-t•e.
=
-2rci
xb
(y,t)
(ai)of
ec-')t (ai)ogtanfi(2n+l)
f•(xb)•n(x)dx
0
(40)
8674
LiPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
whereF(xt, ) is the meanforcingarisingfrom Fxx(•l•l).
where G is found from (30) and (38) for the two forcing
The operandof the couplingintegral is simply a functionof regions. Taking the magnitudeof the real part of (42) results
the edge wave waveform with amplitude equal to F. The in a quadratic
equationin 3an/3t which has solutionof the
possibilityof edge wave forcing arisesif the temporal and fo rill
spatial scalesof the oscillatingbreakerline match the edge
•a n
wave dispersionrelation. For the spectralcase, the breaker
•=c, an[-l+f(an)]
(43)
line is describedby a summationof perturbations
abouta mean
breakpoint. Edge wave forcing results from a linear where
combinationof various incident wave pairs contributingto
the alongshoreperturbation,where the contributionfrom any
2•Cre
2
particularpair (with mnplitudesala2) canbe shown(Appendix
c•- •[2+4ere
2
B) to be
Axt,
=Ho?,tanfl
4 ala
2cos(•l
- •2)
(41)
where H o is the mean breaker height calculated over all
incident wave spectralcomponents.
Since zXx
b is a functionof y and t, the Fourier transformof
(41) resultsin a spectralform for the perturbation,now given
in terms of longshorewavenumberand frequency. This leads
to the attractive result that time and space dependent
observations of the width of the surf zone can be related to the
edge wave forcing by the lYequency-wavenumber
spectrumof
the wave breaking distribution.
As indicatedby the growthrate equations(30) and (38), the
behaviorof the forcing as a functionof the variousparameters
turns out to be quite simple. Growth rate contributionsfrom
the surf zone mechanis•nvary with Xl, in a way which largely
follows the shapeof On. In the offshoreregion, the influence
of mode numberon actualrates is largely throughthe variance
distributionof Onin Xb space. That is, the rate of transferring
energy from the incident field appearsto be about the samefor
all modes,where, for highermodes,the energymust be spread
over a larger cross-shoredistance.
The only known measurements of growth rates for
progressiveedge waves are from the laboratoryinvestigation
of Bowen and Guza [1978]. Strict comparisonwith their
results
is difficult
since
their
discussion
was
limited
to
Zb = 0.25, and furthermore,they have 6= 1.0, thus violating
our assumption of small-amplitude modulation. Their
observedgrowthrate is over an order of magnitudeslowerthan
predicted by the model. The effect of viscous damping is
likely to be important in the laboratory case, and since the
scalesof the lab study are much different from thosetypically
found in nature, no si•nple comparisonis readily made. In
addition, any reasonabledamping mechanismis likely to be
different in the two forcing regions and may well have
characteristicswhich lead to preferential damping of high
f (an)=•--•
e
+ 1+;[2 Ir2an2
The caseof interestis the positiveroot, corresponding
to
energytransferfrom theincidentwavesto theedgewaves. We
havealreadyassumed
thatthe frictionaldissipation
hasonly a
negligible effect on the wave solutionsand dispersion
relation;
thus;t2<<4ere
2. Integrating
(43)andconsidering
only edge wave growth give a transcendentalform for the
amplitude
an=Coe-Clteq
If(an)dr
(44)
where co is an integrationconstantand for growthto occur
f(an) •nustbe real and positive. This equationcannotbe
solvedanalytically;however,we can examinethe amplitude
decay(dissipation)
whentheforcingis turned
off (IGI=0) and
theexpected
amplitude
atequilibrium
( 3an/3t= 0 ).
For the casewherethe forcingis turnedoff, (44) becomes
an
_
-e
_l__At
i4•2
o'e
2
e
ISl-0
(45)
where (an)ois the initial edge wave amplitude at t=0.
Takingthe magnitudeof (45), the decaytimescalenormalized
by the edgewaveperiodis found
t__=
-2'n[a•l(a•)o](46)
It still remainsto parameterize
the damping.We takea very
simpleformfor •, followingLonguet-Higgins
[1970]
;t=Caluø•l
h0
(47)
whereho is the depthat the breakpoint
, Cotis a bottomdrag
frequenciesand low modes [Bowen and Guza, 1978' Holman,
19811.
coefficient
of order10-2-10
-3,anduo is themagnitude
of the
Damped
breakpointgiven by
linearorbitalvelocityof theincidentwaveof heightHo at the
Growth
Rates
Up to this pointwe haveconsidered
only the undampededge
wave growth rate. Here we briefly examine the effect of
frictional damping. If we retain the frictional term in (2) and
(3), the normalizedgrowthequation(11) takesthe form
uø=2[
2 o
(48)
Using H o = 7Xotanfi gives
t -4'n[an/(a•)o](tan
(2or
e+i;t)--•-+/•CYea
n(a1)o
O'e2
TT
= 77r3/2
(• Xt,'/2 (49)
(a•)oXtanfl(2n+l)
i2rc FCndx-G
(42)
A plot of tit e as a functionof Zb for various values of
tanfi/Cct
with an/(an)o =0.5 is shownin Figure8. Thehalf-
LIPPMANNET AL.:GENERATION
OFEDGEWAVESIN SHALLOWWATER
Sensitivity
to Model
8675
Parameters
The dependence
of growthrateson Zb, tan/t, and creis
essentially combined into the nondimensionalscaling
[•/Ca
10
parameter
,•b(equation
(39)). Thusmeasured
spectra
obtained
30-
in the field can be interpretedin termsof samplingposition
andlowest-order
profilecharacteristics.
The effectof varying
20-
:
any particularcombination
of parameters
is easilydeduced.
Additionally,
because
theforcingis dominated
by Fxx,incident
wave angleshave little influenceon the growthrate, except
•0
'2
for low modesand large anglesof incidence.
t/T•
In themodel,y = 0.42 is assumed
constant,
consistent
with
1
0
1'0
field data [Thorntonand Guza, 1982;Sallengerand Holman,
1985], and entersthe growth rate equations(30) and (38)
linearly. Sinceall reportedvaluesof yare O(1). varyingyis
notexpected
to significantly
influencethefinal results.Thus
Figure8. Normalized
edgewavedecaytimescales
(equation the formulationof the growthrate (in both the offshoreand
(49))asa function
of Zt,= O'e2Xl•/gtan,
B, forvarious
values
of surf zone regions)is dependenton only one free parameter:
tan[¾C
a . Thevertical
axisisin number
of edgewaveperiods the incident modulation, & Allowing the modulationto get
for the edgewaveamplitude
to decay50% (an/(an)o =0.5)
aftertheforcingis turnedoff. Results
areshownfor ?'= 0.42.
muchlargerthanabout0.1 is notaccounted
for exactlyby the
model, where we have assumedsmall 8, so that incidentwave
travel
times
areshort
compared
totimescales
associated
with
life decayscaleis moderately
insensitive
to Zt,. The ratio
the modulation[Symondset al., 1982].
Since terms in the radiation stresscontainingA (equation
tan/•/C
a is a roughindication
of theQ (resonance)
of the (28)) are small comparedto other terms, the forcing is
of incidentfrequencies.Hencethe
system,
with highervaluescorresponding
to weaklydamped effective!yindependent
conditions
with smalldragcoefficients.Steepbeaches
tendto
havesmallerdissipation
ratesthanflat beaches,
intuitivelyin
incident frequencies serve only to provide a necessary
(Gf, kf) interaction
which
matches
theedge
wavedispersion
in incident
wavevelocitypotentials,
{I)i{I)
j
agreement
withconceptual
damping
mechanisms
[Bowenand relation.Products
Guza, 1978; Komar, 1979; Holman, 1981]. For resonant
(equation
(23)),havean (XiXj)-1/2dependence,
which
products
of incident
frequencies
o'io'
j. However,
we
systems,
half-lives
canbeashighas10-30edgewaveperiods. contains
At equilibrium,we can estimatethe size of edgewave
amplitudes
relativeto incidentwavesfrom(42)
haveremovedthe dependence
on incidentfrequencies
in the
formulationof S((I)i(I)
j) by choosingincident wave
amplitudes
relativeto the breakpoint,
whereproductsaiaj
have
an(XiXj)•/2dependence.
Substituting
(27)into(21)
a•=r4[G•l
fi);(t,l/2c-)an
=0(50)
3/2ltan
eliminates
dependencies
on (J'io'j.
Still, the resonantresponseassumedin the modelrestricts
This form can also be obtainedfrom the simple argumentthat
incidentwavepairsto differencefrequencies
andwavenumbers
whichsatisfythe edgewavedispersion
relation. Bowenand
Guza [1978] discusstheseresonantrestrictionsin terms of
incidentwave anglesandfrequencies.They showthat only a
f(a,•) in (43) mustbe realfor growthto occur. A plotof
a,•/(al)ois shown
in Figure9 asa function
of Zbforvarious
valuesof tan•/Ca andwith Isl estimated
conservatively
fromFigures6 and7 to be about0.05. Guza and Thornton
[ 1982]foundedgewavermsamplitudes
in therun-upof a near
planar beachas high as 75% of the incidentwave rms
amplitude.
HolmanandSallenger
[1985]alsoobserved
edge
3
13/ca
2.75
lO
2.5
9
2.25
wavesof thismagnitude
on a barredbeachundera widerange
2
of conditions. The modelsuggestsamplituderatiosof this
1.75
magnitude
for tan•/Ca values ranging from 3 to 10.
Comparison
with field resultsis not strictlyquantitative an/(aOo•.5
becausewavesin natureare nevertruly bichromatic,although
it is notedthatKomar [1979] suggested
that tan/•/Ca ratios
of thismagnitude
arereasonable
for meancurrents
onbeaches
often found in nature.
The dampingparameterizationused in (47) is only
8
7
6
1.25
4
1
•.•_•_-•--
0.75
3
=_-------2
0.5
1
0.25
applicable
to wavesin thesurfzone.It is notknownwhatthe
damping
mechanism
is in theoffshore
region.Herbetset al.
Zb
[1995a] show that the ratio of shorewardto seaward
propagating
freeinfragravity
wavesdecreases
asseaandswell Figure 9 Normalizededge wave equilibriumamplitude
energyincreases,
suggesting
thatthedampingof edgewaves (equation
i50))asafunction
ofZb=C•e2Xb/gtanl
• forvarious
on the shelfdepends
on the incidentwaveenergylevel, in valuesof tan/J/C
a . Edge wave amplitudeshave been
qualitative
agreement
withthesurfzonedamping
mechanismnormalizedby the incidentwave amplitudeat the breakpoint.
Results
areshown
for 7-0.42 andIGI-0.05.
examined.
0
0
1
8676
LIPPMANN
ET AL.: GENERATION
OF EDGE WAVES IN SHALLOW WATER
finite number of incident frequencypairs satisfy the resonant
condition for a narrow beam incident swell. This places a
restrictionon the possible(o.1, o'2) combinationswhich could
theoretically excite edge waves.
Shallow
Water
Assumption
In our formulation of the interaction radiation stress, the
incident wave surface elevation is approximated by the
shallow water Bessel function solutionfor progressivegravity
waves over a sloping bottom. Inside the surf zone this
formulation seems very reasonablesince the water depth is
alwaysshallowwith respectto incidentwave wavelengthsand
amplitudesare given as a functionof the local depth. Outside
the variation in breaker position due to changing wave
conditionsis ignored. In effect,Foda and Mei assumethat T is
always larger when the incident waves are large. Field
evidence [Guza and Thornton, 1982; Sallenger and Holman,
1985] suggeststhat T is reasonablyconstantfor any particular
beach and the position of the breaker is the property that
varies as a function of incident wave height. This is a central
point in our calculation.
Foda and Mei [1981] also compute growth rates for the
interaction which vary over 2 orders of magnitude as a
functionof the parameter
the surf zone, however, the use of the shallow water solution
(which gives resonanttriad interactions)is only valid for a
limited distanceoffshore (determinedby the beachslope) and
This parameterarisesfrom a particularscalingwhich is not the
natural scaling for the shallow water equationson a sloping
depends
on the incidentwavefrequency.In the offshore beachwe have considered(equation(39)). As a consequence,
region,(24) givesan x'1/4 dependence
on waveamplitude. Foda and Mei have computedvalues of E2 which are very
Thus productsof incidentwave amplitudesdecayin this region small, ranging from 1 to 3. If we consider a 60s beat, 10s
asx'1/2,whereasthey shouldbecomeconstant
in deepwater incident wave, and slope of 1'100, then E2=16. As growth
where the wave profile we have assumedis only approximate. ratesincreasevery rapidly with E2 in Foda and Mei's results,
As there are no resonanttriad interactionsin deep water (only we might expect very large growth rates for values typical of
forcedwaves are generated),the magnitudeof the forcingdies open coast beaches. However, precise values cannot be
away quickly as incidentwavesapproachdeepwater [Okihiro compareddue to their complex (fourth order) representationof
et al., 1992].
the forcing inside the surf zone.
The growthrate equationalso includesconsideration
of OnSince Ondecaysexponentiallyoffshoreas a functionof mode
number, the offshore integral for low modes is not biased Conelusions
significantlyby reducedcontributionsin intermediatewater.
A theoreticalmechanismfor driving resonantedge waves in
Therefore, when evaluating the model, the upper limit of
the nearshore
is derived
from the forced shallow
water
integrationcan be reducedfor low slopingbeacheswithout
equations. Forcing integrals are separatedinto an offshore
substantially underestimating the growth rate.
This
approximationis good for shallow beacheswhere edge wave region outside the breakpoint and within the region of
length scalesare small; for steep beachesthe applicationis fluctuating surf zone width. Contributions to the surf zone
region occur from primary self-self interactions for both
questionable.The approximationis also more accuratefor the
case of low modes which have a relatively rapid offshore breaking and nonbreaking waves, whereas in the offshore
region, forcing is from crossinteractionsof a pair of waves.
decay' for higher modes with slower decay scales,
The strength of the forcing in each region is based on
contributionsare more severely biased.
Comparison
With
Foda and Mei
[1981]
Foda and Mei [1981] consider the problem of long waves
generated by a normally incident swell which has a small
alongshorevariation which is fixed in space but has a slow
modulation in time. This variation in wave height could be
thought of as two incident waves of the same frequency
approachingthe beach from equal, but opposite,anglesto the
normal. The modulation is then the beat frequencybetween
these waves and the normally incident wave. This is therefore
a rather particular case of the general problem which we are
considering. Their discussionwas further limited in that the
dominant (largest) wave is normally incident. However, Foda
and Mei have carried through a very sophisticatedanalysis
which includesresults for the case where the long waves grow
to magnitudesof the same order as the incident modulation.
They can thereforediscussthe processesthat eventuallylimit
edge wave growth. However, the complexity of the
calculationrather precludesany simple interpretationof these
results.
A further complication is that, for the case of breaking
waves, Foda and Mei [1981] have useda representationof the
breaker condition in which the breakpointis constantand the
perturbationamplitude (modulation) extendsto the shoreline;
amplitude modulations which arise from an interacting
bichromaticwave field. Surf zone forcing is derived from
momentumfluxes inducedby temporaland spatialvariations
in initial breakpointamplitudes,expressedfor the planebeach
caseas three-dimensional
modulationsin surf zone width (first
suggestedby Symondset al. [1982] while focusingon the
two-dimensional
problem). The nonlinearforcingis provided
by theunbalanced
gradient
in radiation
stress,
Spq.Following
Phillips[1977],a formforSpqdueto thenonlinear
difference
interactionof two incidentwavesapproaching
the beachat an
angle is derived.
The model indicatesthat the forcingarisingfrom the crossshorecomponentof onshoredirectedmomentumflux provides
the major contributionto the edgewave forcing,particularly
for small anglesof incidence,higher edge wave modes,and
lower frequencies.Increasingincidentwave angletendsto
reduce the contributionin the offshoreregion for all modes
becauseof the relative phase relationshipsof the forcing
components,whereasthe growth rate inside the surf zone is
enhancedtending to favor the breakpointmechanismin the
overall growth rate.
We find that the strengthof the surf zone generation
mechanismis 2-10 times greaterthan in the offshoreregion
for parameterrangesof particularinterest. Thus considering
only the surf zone componentmay provide a reasonablefirst-
LIPPMANN ET AL.: GENERATION
OF EDGE WAVES IN SHALLOW WATER
order estimate of the growth rate. The strength of the growth
rate is inversely proportionalto edge wave mode number and
directly proportional to the incident wave modulation, with a
linear dependencein each region. Initial inviscid growth rates
are found to be quite rapid, with edge waves amplitudes
growing to incident wave mnpntut•esof the ---•-- of •n •.•_•.
wave periods.
Including a simple parameterizationfor bottom frictional
damping and using inviscid growth rates to describe the
energy transferfrom incident to edge waves allow predictions
of edge wave equilibrium amplitudes consistent with field
The first integral on the right-hand side of (A3) is zero since
radiation stress gradients are constant inside the mean
breakpoint. Thus, after evaluatingthe secondintegralin (A3)
•b(1+•cos•e )
f Fsz,Ondx
=Q(•b
+•b cos
•e)-Q(•b)
where Q(x) is the unspecifiedresult of t•e integration.
Expanding(A4) in a Taylor'sSeriesabout•t, p:•,tluces
•b(I+'•COS
•)
•neasurements of Guza and Thornton [1982] and Holman and
3Q(.•b
)
I Fsz
On
dx
=•bcos
I//e•)-••
0
,
for tanfi/Cd valuesof about3-10 (esti•nated
from equilibrium
for the surf zone, where the combination
linear
sum of all resonant
interactions.
Appendix A: Expansion of First-Order Coupling
Integrals
Edge wave growth rates are given as a function of the
coupling integral between the wave forcing, F, and the edge
wave waveform, On
2
3x2
+..
'
(AS)
whichcanbe writtenin termsof the integrandof (first term on
right-handside of (A2)) evaluatedat -•t, plus higher order
harmonics
7b(1+,•cos• )
f Fsz
CPndx
=•b COS
I//eFsz
('•b)On
(•b)
o
of various incident
wave pairs contributesto the alongshoreperturbation of the
oscillatory breaker line. The contribution to spectral forcing
by any particular pair is shown in principal to be due to the
(A4)
0
Sallenger [1985], who found shorelinemnplitudesas high as
75% of the incident waves. Edge wave dissipationrates in the
surf zone are found when the forcing is turned off and suggest
half-life decay scalesof the order of 10-30 edge wave periods
conditions). Dissipationrates outside the surf zone are not
estimated since the appropriate damping mechanism in this
region is not known.
The extensionto spectral forcing is shown to be possible.
The development in the offshore region can be based on a
linear sum of all possible co•nbinations of wave triads
satisfyingthe edge wave dispersionrelation. The same is true
8677
+O(•
2coS2•e)+... (A6)
Thus for the first integral on right-hand side of (A2) we need
onlyconsider
theforcing
at the•nean
breakpoint,
Fsz(jb),
whichhastermsarisingfrom self andcrossinteractions
of •1
and •2. The ••
and •2•2 interactionsproducecoupling
termswhich containprimary•nodulations
(cos•e)of order8
and•3 respectively,
whereas
•2
interactions
produce
additional
harmonic
reruns
(cos2•e)oforder•2. Considering
only first-order, primary modulations (arising from ••
interactions),the surf zone coupling integral is approximated
by
• (l+•co• % )
0
•FszOnd••bCOS•eFsz(OlOl
, •b)On(•b)
(A7)
where (I)1 and (I)2 are velocity potentialsof the incidentwaves
which have mnplitudesa• and a2 -aa•, respectively. The
coupling
integral
canbeseparated
at thebreakpoint,
xb(y, t),
into surf zone (first term on right-hand side of (A2)) and
offshore (second term on right-hand side of (A2)) integrals
with the forcingdefinedin eachregionby F sz andFoff,
respectively,
•,
()
As in the surf zone region, the offshore coupling integral
(secondterm on right-hand side of (A2)) can be separatedat
Xb,
•
Xb
•
•FoffOndx
• FoffOn
• +• FoffOndx
(h8)
• (•+aoo•v,•)
f FOndx
I Fsz•)ndx
q- I Foff
CP
ndX
o
o
(A2)
7b(1+?;
oos% )
The second integral on the right-hand side of (A8) has
modulationswhich arisefrom only the •2
interactionsand
is numericallyevaluatedin the main text (••
and•2(I) 2
interactions
produce
only
nonvarying
components
through
xb(y,t)-•b(l+r•coSgte)andthe functional
dependencies
where
the
breakpoint
has
been
defined
by
have been droppedfor brevity.
The surf zone integral (first term on right-handside) in (A2)
can be further separatedby the mean breakpoint, Y'b, into a
nonvarying component from the shoreline to -•b and a
component arising from modulations in the breakpoint
position
this integral). The first integral describesthe contributionto
edge wave growth by nonbreakingwaves in the fluctuating
regionof the surf zone. Evaluatingthis integralproduces
I FoffOndx
--g(.•
b)- g(.•
bq-(•bcos
life)
(+aco
f FszrPndxIFsjCPndx+
IFsjrPn
dx
0
0
(A3)
(A9)
8678
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
where R(x) is the unspecifiedresult of the integration. and the secondterm representsthe perturbationof the wave
Expandingin a Taylor's seriesabout Sb and keeping only
first-order,primary modulationsgive
height about its •neanvalue.
The problemcan be simplifiedif the perturbationof the
waveheightaboutthe meanvalueH o is not too large(it is, of
course,necessarilylessthanone). So expanding(B5),
f FofflPndx--•
bcos
lff
eFoff
(0101
, -•b
)(Pn(.•b
)
• (l+aoos•,,)
(AlO)
xb
The total couplingintegral(A2) canthusbe approxi•nated
ai
i
IF•ndX-•bcos
lff
e•n(•b
)[Fsz(0101,
•b)- Foff(0101
, '•b)]
o
+leor(OlO2,
x)O(x)a
_-Ho(x)
1+i j •iai2
(All)
whichshowsthe first-ordercontributions
to edgewave growth
occur through self-self interactionsfrom both breakingand
nonbreakingwaveswithin the fluctuatingregion of surf zone
width, and through cross interactionsof a pair of waves
seawardof the averagebreakpointposition. Since the signs
(B6)
Thereis a potentialforcingtermfor edgewavesfrom each
of the secondtermsin the expansion
cos(%•j)- cos[(//-//)x+(k
i- kj)y-(cr
i- cr
i )t+cpi
- cpj
]
of FszandFof
f areopposite
in thefluctuating
regionof surf
zonewidth,the •1•1 forcingcontributions
from breakingand
nonbreakingwaves reinforce.
(B7)
with resonantforcingpossibleif
ki - kj = ke andcri - crj= cre
Appendix B: Breakpoint Forcing in a Wave
Spectrum
where
(cre,ke) satisfy
theedgewavedispersion
relation.
Within the spectra,the interactionbetween several different
sets of incident waves can produce the same values of
If the seasurfaceelevationrI is describedby
1l - • ai(x)cos
I//i
(BS)
(B1)
(or
e, ke), andthenetforcing
isthesumofallthese
potential
inputs.
i
Now the wavebreakswhen H = Th, so the meanbreaker
where V/i=lix+kiy-Git+ cpi, then the significant wave
height is 4 ti•nes the standarddeviationof rI, and the mean
position hb is given by
hb-Ho(xo)/7
wave height H o is given by
(B9)
and for a plane beach, the mean breakpoint is
xo- Ho(xo)/(ytan/•).Theperturbation
in breaker
height
then leads to a variable breakpointposition in both the
longshoredirection and in time, where
The variationin wave height,givenby the envelopeof the
1
xb(y, t)-•H(x,
7tanfi
wave groups,can be defined and is often calculatedin terms of
the low-passedvalues of the squareof the surfaceelevation,
where
"(X,t)--2'•'[(
T•2
>]
1/2
(B3)
where
{ } isthelow-passed
value;
then
from
(B1),
.
Ho
•1+
T'tanfl
t)
Z Zaiajcøs(lffi-Iffj)
i
J
Z ai
(BlO)
i
which is a linear co•nbinationof a large number of
contributions
to the longshoreperturbationfrom pairs of
[1+o42
)]
incidentwaves.The contribution
fromanyparticular
pair is
then
Axe,
=Ho7tanfi
4 ala2
cos(•l
- g2)
(Bll)
the low-pass filter re•noves the high harmonics and sum
interactions, so that
ai2
i
j
cos(
-
Acknowledgments. This research
was sponsored
by the National
Science
Foundation
undercontract
OCE-8007972-01
andby theOffice
Naval Researchunder grant N00014-90-Jll18. This work was
(B5)ofcompleted
while T.C.L was supported
by a NationalResearchCouncil
Postdoctoral
Fellowship. We thankBob Guza and the reviewersfor
LIPPMANN ET AL.: GENERATION OF EDGE WAVES IN SHALLOW WATER
8679
theirinsightfulcomments
whichled to a correction
to the surfzone Longuet-Higgins,M. S., and R. W. Stewart,Radiation stressesin water
waves: A physicaldiscussion,with applications,Deep Sea Res.,11,
forcingmechanism
andin generalgreatlyimprovedthe paper. Joan
529-562, 1964.
Semlerprepared
thetypeset
manuscript.
Mei, C. C., and C. Benmoussa,Long waves induced by short-wave
groupsoveran unevenbottom,J. Fluid Mech.,139, 219-235, 1984.
Munk, W. H., Surf beats,Eos Trans. AGU, 30(6), 849-854, 1949.
Okihiro, M., 1<. t. tauza, and R. J. Seymour,Boundinfragravity waves,
References
J. Gcophys.Res.,97(C7), 11,453-11,469,1992.
Biesel,F., Equations
generales
au secondordrede la hotfieirreguliere, Olbers, D. J., and K. Herterich, The spectral energy transfer from
surface waves to intemal waves, J. Fluid Mech., 92, 349-379, 1979.
Houille Blanche, 7, 372-376, 1952.
Bowen,A. J., and R. T. Guza, Edge wavesand surf beat,J. Geophys. Oltman-Shay, J. M., and R. T. Guza, Infragravity edge wave
observationson two California beaches,J. Phys. Oceanogr.,17(5),
Res., 83C4), 1913-1920, 1978.
644-663, 1987.
Eckart,C., Surfacewaveson water of variabledepth,WaveRep. 100,
Phillips, O. M., The Dynamicsof the Upper Ocean, 2nd ed., 336 pp.,
Scripps
Inst.of Oceanogr.,
Univ.of Calif.,La Jolla,1951.
Foda, M. A., and C. C. Mei, Nonlinearexcitationof long-trappedwaves
CambridgeUniv. Press.,New York, 1977.
Sallenger, A. H., and R. A. Holman, Wave energy saturation on a
by a groupof shortswells,J. FluM Mech.,111,319-345,1981.
Gallagher,B., Generationof surfbeat by nonlinearwave interactions,
natural beachof variable slope,J. Geophys.Res., 90(C6), 11,93911,944, 1985.
J. FluM Mech., 49, 1-20, 1971.
Guza, R. T., and A. J. Bowen, The resonantinstabilitiesof long waves Sallenger,A. H., and R. A. Hobnan, Infragravity waves over a natural
barredprofile,J. Geo?hys.Res.,92(C9), 9531-9540, 1987.
obliquely
incidenton a beach,J. Geophys.
Res.,80(33),4529-4534,
1975.
Schaffer,H. A., Infragravity waves inducedby short-wavegroups,J.
Fluid Mech.,247, 551-588, 1993.
Guza, R. T., and E. B. Thornton,Swashoscillationson a naturalbeach,
J. Geophys.Res.,87(C1), 483-491, 1982.
Schaffer, H. A., Edge waves forced by short-wavegroups,J. Fluid
Mech.,259, 125-148, 1994.
Guza, R. T., and E. B. Thomton,Observations
of surf beat,J. Geophys.
Res.,90(C2), 3161-3172, 1985.
Schaffer,H. A., and I. Svendsen,Surf beat generationon a mild-slope
Hassehnann,K., œ)nthe non-linear energy transfer in a gravity-wave
beach, in Proceedings of the 21st International Conference on
spectrum,
1, Generaltheory,J. FluidMech.,12,481-500,1962.
CoastalEngineering,pp. 1058-1072,Am. Soc.Civ. Eng.,New York,
1988.
Herbets,T. H. C., S. Elgar, andR. T. Guza, Generationandpropagation
of infragravitywaves,J. Geophys.Res.,100(C12), 24,863-24,872, Stoker,J. J., Surfacewavesin water of variable depth,Q. App. Math.,
1995a.
5(1), 1-54, 1947.
Herbets, T. H. C., S. Elgar, R. T. Guza, and W. C. O'Reilly,
Symonds,G., and A. J. Bowen, Interactionsof nearshorebars with
Infragravity-frequency
(0.005-0.05 Hz) motionson the shelf, II,
Freewaves,J. Phys.Oceanogr,25(6), 1063-1079,1995b.
Hohnan,R. A., Infragravityenergyin the surf zone,J. Geophys.Res.,
80(C7), 6442-6450, 1981.
Hohnan, R. A., and A. H. Sallenger,Setup and swashon a natural
beach,J. Gcophys.Res.,90(C1), 945-953, 1985.
Howd, P. A., J. M. Oltman-Shay,and R. A. Holman, Wave variance
partitioningin the troughof a barredbeach,J. Geophys.Res.,
O0(C7), 12,781-12,795, 1991.
incomingwavegroups,J. Geophys.
Res.,89(C2), 1953-1959,1984.
Symonds,G., D. A. Huntley, and A. J. Bowen, Two-dimensionalsurf
beat: Long wave generation by a time-varying breakpoint, J.
Geophys.Res., 87(C1), 492-498, 1982.
Thornton, E. B., and R. T. Guza, Energy saturationand phase speeds
measuredon a naturalbeach,J. Geophys.Res.,87(C12), 9499-9508,
1982.
Tucker, M. J., Surf beats: Sea wavesof 1 to 5 minute period, Proc. R.
Soc. London A, 202,565-573, 1950.
Huntley, D. A., Long-periodwaveson a naturalbeach,J. Gcophys. Watson, K. M., B. J. West, and B. I. Cohen, Coupling of surfaceand
Res., 81(36), 6441-6449, 1976.
internalgravity waves' A mode couplingmodel,J. Fluid Mech., 77,
105-208, 1976.
Huntley,D. A., R. T. Guza,and E. B. Thornton,Field observations
of
surf beat, 1, Progressiveedge waves, J. Geophys.Res., 80(C7),.
6451-6466, 1981.
A. J. Bowen, Depamnentof Oceanography,
DalhousieUniversity,
Halifax, Nova Scotia, Canada B3H 461.
Komar, P. D., Beach-slopedependenceof longshorecurrents,J.
Waterw. Port Coastal Ocean Div. Am. Soc. Civ. Eng., 105(WW4),
R. A. Hohnan,Collegeof Oceanand Am•osphericSciences,Oregon
460-464, 1979.
StateUniversity,()C AdministrationBuilding 104, Corvallis,OR 97331-
List, J. H., A model for the generationof two-dimensionalsurf beat,J.
55O3.
Geophys.
Res.,97(C7),5623-5635,1992.
Longuet-Higgins,M. S., Longshorecurrentsgeneratedby obliquely
T. C. Lippmm•n,Centerfor CoastalStudies,ScrippsInstitutionof
Oceanography,University of California, San Diego, 9500 Gihnan
[)rive, La Jolla,CA 92093-0209,(e-mail:tlippmann@ucsd.edu.
incidentseawaves,2, J. Geophys.Res.,75(33), 6790-6801, 1970.
Longuet-Higgins,
M. S., andR. W. Stewart,Radiationstresses
andmass
transport
in gravitywaves,with application
to 'surfbeats,'J. Fluid
Mech., 13, 481-504, 1962.
(ReceivedJanuary5, 1994; revisedJune24, 1996;
acceptedSeptember27, 1996.)