Edge Waves and Beach Cusps

VOL. 80, NO. 21 JOURNALOF GEOPHYSICALRESEARCH JULY 20, 1975

EdgeWaves and BeachCusps

ROBERTT. GUZA AND DOUGLASL. INMAN ScrippsInstitution ofOceanography, Unioersity ofCalifornia, LaJolla, California 92037

Genetically,beach cus. psare of at least two types: those linked with incident waves which are surging andmostly reflected (reflective systems) andthose generated onbeaches where wave breaking and nearshorecirculation cells are important (dissipative systems). The spacings ofsome cusps formed under reflectivewave conditions both in the laboratory and in certain selected natural situations areshown to be consistentwithmodels hypothesizing formation byeither (1) subharmonic edgewaves (period twice that ofthe incident waves) ofzero mode number or(2) synchronous (period equal to'that of incident waves) edgewaves of lowmode. Experiments showthat visible subharmonic edgewave generation occurs on nonerodableplanelaboratory beaches only when the incident waves are strongly reflected atthe beach, andthis observation isquantified. Edge wave resonance theory and experiments suggest that synchronous potentialedge wave generation canalso occur on reflective beaches andis a higher-order,weaker resonancethan the subharmonic type. In dissipativesystems, modes of longshoreperiodic motion other thanpotential edge waves may be important incontrolling thelongshore scaleof circulation cellsand beachmorphologies. Onreflective plane laboratory beaches, initially large subharmonic edgewaves rear- ragesand tracers into shapes which resemble natural beach cusps, butthe edge wave amplitudes decrease asthe cusps grow. Cusp growth isthus limited by negative feedback from the cusps tothe edge wave ex- citationprocess. Small edge waves can form longshore periodic morphologies byproviding alestabilizing perturbationsona berm properly located inthe swash zone. In this case the retreating incident wave surge ischannelized intobreeches inthe berm caused by the edge waves, and there is an initially positive feed- backfrom the topography to longshoreperiodic perturbations.

INTRODUCTION modelsare feasibleon erodablebeaches, they cannotexplain later laboratoryexperiments [Gah)in, 1965; Harris, 1967; Rhythmicallongshore patterns have often been observed in Bowen,1967; Bowen and Inntan, 1969, 1971] on nonerodable shorelineand nearshøre morphologies, both in thefield and in planebeaches which showed the existence of longshore wavetanks [e.g., Køntar, 1973; Bowen and Inntan, 1971; Dolan, periodicwave motions (edge waves), some of whichdevelop 1971'Harris, 1967; Johnson, 1919]. Cuspate patterns, which circulationcells. Either the edgewaves or the associatedcir- will be consideredhere, are concaveseaward, are usually culationcells can imposetheir longshoreperiodicity on an formedat theshoreline, and have longshore wavelengths vary- erodablebed. The present experiments indicate that although ingfrom less than 1 m onlakeshores [Komar, 1973] to thescale the inducedtopographic changes eventually have a feedback of capes(105 m) or larger[Dolan and Fernt, 1968]. Dolan and to the waves and currentsand thus alter the further rearrange- Ferm enumerate some traditional names given cuspate mentof sediment,the primary longshore periodic generative featureson ocean ShorelineS: (1) 'typical beach cusps,' 8-25 m; mechanismof beach cusps issometimes present in theewaves (2)'storm cusps,' 70'120 m; and (3) 'giant cusps,' also known andcurrents occurring on nonerodablebeds, Inclusion of the as,shOreline rhythms,' 700-1500 m.A Classificationaccording feedback of changingtopography is fOUndto be necessaryto to sizeis, hOweVer ' genetically unconvincing. Kuenen [1948] determinethe equilibriumamplitude and the permanenceof statedthat 'tYpical' and 'storm ' cusps are 'practically the same' morphologicchanges. exceptfor size.The presentstudy centers on a single It hasalso been hypothesized thatrhythmic topograPhie son generatingmechanism, edge waves, which have wavelengths an unboundedcoast are 'sand wave' trains which result from rangingfrom centimeters onlakeshores to hundreds of meters an instabilityof thesurf zone bed to perturbationsby (or more)near oceanic coastlines. Edge waves capable of longshorecurrents [e.g., Hom-nta and Sonu, 1963; Sonu., I968, producingcuspate morphologies are generatedin the 1973;Schwartz, 1972]. The analogy here is to thedunes and laboratorywith Wavelengths varying from tens of centimetersantidUnes of fluvialsystems; initially small bottom ir- to l0 m, thewave basin size precluding longer waves. regularitiesgrow because of a reinforcingcoupling to the culationsCuspsareaSsociated noted for withtheir them.regularity JOhnson and [1919]the distinctive summarizes cir- longshore current.Their proponents hypothesize thatsand earlyattempts tø explain theserhythmic morphologies and wavesinfluence theincident wavefield insuch away asto citesLane[1888] ashYPothesiZing thatrandom irregularities producetheobserved circulation Cells.Like the earlier onthe beach become evenlyspaced through someprocess of hypotheses, thismechanism cannotexplain thelongshore adjustmenttoequilibrium notClearly understood andthat this periodicmore, many motionsauthors observed[e.g., JohnSOn, onnonerodable 1919; Lønguet-Higgins beaches.Further-and equilibriumdistance between Cusps isrelated to theheight of Parkin, 1962;Russell and MClntire, 1965] claim that natural the waves.Johnson basically agreed with this hypothesis,as cuspsform only with normally incident waves, Which generally did laterauthors [e.g., Kuenen, 1948; Otvos, 1964], although do not produceunidirectional, coherent flows like those someauthors stressed the importanceof alepositionalover responsiblefor classicaldune systems. Others have observed erosionalprocesses, and vice versa. This model stresses the im- cuspformation with bothnormal and nonnormal incidence portanceof theinteraction of wave motions and topographic [Worrall, 1969; Otoos, 1964; Mii, 1959].Clearly, any theory changesindetermining cusp spacing. While bottom interactive which requires a longshorecurrent for cuspformation cannot Copyright¸ 1975by theAmerican Geophysical Union. explaina largeportion of observednatural cusps. It should 2997 2998 , GUZAAND INMAN: EDGE WAVES AND BEACH CUSPS

notbe assumed, however, that cusps of largephysical dimen- EDGEWAVES ON CONSTANTSLOPE BEACHES sionformed at locationswith large process scales (i.e., incidentwave wavelengths) arenecessarily genetically different Coriolisforces being neglected, the dynamics of inviscid from muchsmaller cusps formed elsewhere. edgewaves on a nonerodablebeach of constantslope ex- Edgewaves are the normaltrapped modes of longshoretending into deep water have been discussed byEckart [1951], periodicwave motion that occuralong the edgeof water Ursell[1952], Bowen and Inman [1969, 1971], and others. The bodies,and they may be either standing orprogressive. They edgewave wavelength L in the longshore direction isgiven by existequally well along straight or gentlycurving shorelines Ursell [1952], with no assumptionsabout the shallownessof the water, as andfor uniformlysloping, concavely or convexlycurving offshoredepth profiles. Numerical methods show the structure L = (g/2;r)Te:sin (2n + 1)/•= 2;r/ky (1) of edgewaves on morerealistic topographies [Munk et al., 1964].Edge waves are noteworthy because they are trapped; where x andy arethe offshore and longshore coordinates, theycannot radiate energy out into deep water. Unlike or- respectively;Teis thePeriod; /• is thebeach slope; g is the dinaryswell waves, they cannot propagate offshore and away gravitationalacceleration; n is aninteger known as the mode froma shallowwater energy source. Their energy can only be number,and ky is the longshorewave number. A further dissipatedthrough friction and through interaction with cur- restrictionon n is(2n + 1)/•< ;r/2,where/• is in radians. Thus rentsand other waves. The possible importance of edge waves for a givenedge wave period there are a seriesof possible in thenearshore region was first stressed by lnman'•[Bowen, wavelengths corresponding to the possible values of n. Thex 1967].It hasbeen demonstrated theoretically that edge waves behaviorofthe Ursell solutions isalgebraically quite complex. canbe excited by w•aves incident from deep water [Gallagher, One of us(R.T.G.) has found that Tait [1970] made errors in 1971;Guza and Davis, 1974] or by traveling atmospheric pres- computingthese functions, thereby invalidating his results. It surefronts [Donn and Ewing, 1956]. It is thecontention of the is usefulto considerthe simplershallow water solutions in presentwork that edge waves, both directly and via theirin- termsof thevelocity potential 4•; the velocity is V4•,and the teractionswith other water motions, are responsible formany surfaceelevation • is (-I/g) Ock/Ot.Eckart [1951] gives the casesof cuspatetopography. Related ideas are discussed by shallowwater velocity potential solution for edge waves on a Bowen[1967, 1969, 1972], Bowen and Inman [1969, 1971, beachsloping into deepwater as 1972],Komar [1971, 1973], and Inman [1971]. Thedynamics of water motion on the beach face change ck= (a•g/co)cos k•y exp(-k•x) L,,(2k•x)cos cot (2) drasticallyaccording towhether the incident wave is strongly where ae is the wave amplitude, L,,is the Laguerre polynomial reflectedor breakscleanly and propagates onshore as a dis- of ordern, co= 2;r/Teis theradian frequency, and sipativebore. A distinctionis thereforemade between 'reflective' and 'dissipative'systems. The presentwork deals L = (g/2;r)Te•'(1 - 2s)tan /ff = 2;r/k• (3) primarilywith the generation ofedge waves and beach cusps in where-s = n = 0, 1,2, ßß .. Thecondition (2n + 1)tan/ff << 1 reflectivesystems, because the appropriateequations are is requiredeither by comparison of (3) withthe exact linear analyticallytractable and because beach cusps are frequently relation (1) or by the requirementthat theseshallow water observedwith stronglyreflected incident waves. Reflective solutionsbecome asymptotically small while still in shallow systemscorrespond to surgingor collapsingincident waves ac- water[Guza and Davis, 1974]. The profilesof thesea surface cordingto theclassification scheme of Galvin[1972]. Cusps for n = 0, 1, 2, 3 aregiven in Figure1 as functionsof the alsooccur on clearly dissipative beaches, but the edge wave dimensionlessspace variable X, where X = co:x/gtan • andX resonancestheoretically predicted and observedfor reflective isproportional to the number of zero mode wavelengths off- systemsare not visible on the beach face when the incident shore.The profile of a wavenormally incident from deep wavescleanly plunge. waterand totally reflected at the coastis alsoshown. Because

1.0-

0.8 Edge waves Reflected normally 0.6 incidentwave

0.4 I• Lf• n:2

oL I x,,j //I,,7 I x. % • , •7•-..t..•, I "1,

-o.2 t -06• Fig. 1. Offshoredependence ofprofiles ofEckart-type edgewaves ofmodes n = O,1, 2, 3 andof normally incident reflectedwaves, in termsof the dimensionless variable X = w•'x/gtan •. GUZA AND INMAN: EDGE WAVES AND BEACH CusPs 2999 of the similarities,data consistingonly of onshore-offshore TABLE I. Minimum Beach Width Xmlnfor Existence of Mode n and Maximum Beach Width Xc for Any Effect of Slope amplitudeprofiles close to shorecannot differentiate between Truncation edgewaves (particularly of high mode number)and waves incidentat any angle from deep water and reflectedat the Mode Xmln Xc (2n + 1)•' coast. The laboratory experimentswhich have been conducted 0 2 1 i 6.5 13 9 were on a beachwith slope/3 when x < Xoand then constant 2 21.0 30 25 depth ho for x > x0. It can be shown by using a simple 3 43.5 53 49 matching of sea surface and onshore-offshorevelocities betweensloping and flat bottom shallowwater solutionsat x .= Xothat edgewaves can still existand are easilyunderstood in pletely exposed.Table 2 gives the maximum periods (in termsof the Eckart-typesolutions. In termsof the nondimen- seconds)for which mode n = 1, 2, 3 can exist for various sional beachwidth Xo,suggested by Bowenand lnman [1969], depths.Periods given for the zero mode correspondto the edgewave period for whichthe distanceoffshore needed for Xo = co•xo/gtan/• the edgewave to decayto e-• of its valueat the shoreis more the dispersionrelation of edge waveswill be basicallyun- than 2 times the length for the Eckart wave. The ability of changed(less than 1% changein the parameterss in (3)) this topographyto trap edge wavesof a given period and from the Eckart form if Xo > xc, where xc is a critical beach modeis clearly(theoretically) strongly dependent on thetidal width and dependson the modenumber, as givenin Table 1, level. for the first four modes.As would intuitively be expected,x• THEORY OF EDGE WAVE EXCITATION ON occurs after the last maximum of the Eckart edge wave (com- NONERODABLE BEACHES pare valuesin Table I with those in Figure l): if the wave amplitudeis somesmall value at theslope break and is already The excitationof low-modestanding edge waves by totally exponentiallydecaying, the effect of the break should be reflectednormally incident waves of periodTt on a beachof minimal. constantslope extending far offshorehas been theoretically In order for inviscidwaves to be trapped and not to radiate shownto occurvia a resonantenergy transfer resulting from energy away they must satisfy an instabilityof the incidentwave to edgewave perturbations [Guzaand Davis, 1974]. The classical shallow water equations ky > kf (4) wereused to showthat initiallysmall standing edge waves of wherek r is thewave number of ordinarygravity waves in shal- variousperiods and low modenumbers can grow with.time, low water of constantdepth ho, extractingtheir energy from the incident wave field. The edge co•'ho/g= krhotanh krho (5) wavesresonantly excited by this mechanismalways have periodslonger than thoseof the primaryincident waves. Equation (4) and the shallowwater limit of (5) lead to a Laminar viscouseffects were included to show that there is a minimum beach width Xmin(a function of mode number), critical amplitudeof the incidentstanding wave at the below which waves of mode n cannot be trapped. These are shoreline, below which a particular resonant incident also shownin Table 1 and from Figure 1 correspondapprox- wave-edgewave interaction does not lead to edgewave growth imately to the location of the last maximumof the Eckart becausethe viscousdamping of edgewaves exceeds the rate of wave. Note that for x0 < 6.5, only the zero modeis possible. nonlinearexcitation. For edgewave growth it wasshown that For very 'short' sloping sectionsthe simple matching a? > vK(N•,No.)/cot (6)

procedureß is not sensible,and sono valueof Xmlnfor the zero mode is given. For values of X, whereat is approximately theincident wave amplitude atx = 0; v isthe kinematic viscosity; N• andN •' are the mode numbe rs Xmln(rt)< X < x•(n) of excitedwaves; cot is the incidentwave frequency; and K(N•, edgewaves of moden still exist,but the parameters in (2) N0 is a constant.The most easily excited edge wave (requiring changes.The waveProfiles (except for the zero mode)are not the smallestincident wave) is the zeromode subharmonic; coe changedsignificantly on the slopingsection, but as x ap- = 0.5cot,N• = No.= 0, and K(0, 0) = 5.25 X 10•, where proachesXmln, the long exponential tail will decreasethe wave arithmeticalerrors of Guzaand Davis [1974, Table 1] are cor- amplitudefor a givenenergy and will probablymake that rectedby dividingK(N•, No.)by 4.0. wave lesslikely to be of importance.Figure 2 showsmode 2 The minimumincident wave amplitudewhich can overcome edgewave profiles for variousx0. Only for valuesof Xo> (2n viscosityand excite the zero mode subharmonic is not a func- + 1)•' (as a rule of thumb seeTable 1) can the moden wave tion of beachslope and is shown on Figure3 asthe solid curve. effectivelybe trapp ed . Unfortunately,it has not been possible For incidentwave amplitudes theoretically large enough to ex- to verifyexperimentally the effectsof slopetruncation in the cite otherresonant edge waves the initial growthrate of the la_boratorybecause edge waves with significant energy on the zero modesubharmonic is alwaysat leasttwice that of other flat sectionare evidentlyaffected by couplingto the paddle; possibleresonances, and experimentsconfirm that the zero this effect has not been considered here. modesubharmonic ispreferentially excited. The other possible The possibleimportance of slope truncation in natural resonances were never observed. situationsis illustratedby El Moreno Beachon the northwest The resonancetheory assumes a totally reflectedincident coast of the Gulf of California [Bowenand Innnan,1969]. A wave,and experiments [Galvin, 1965; Harris, 1967] show that surveyconducted on May 22, 1967,showed a steep-faced(fl = whenthe incidentwave is largeand breaksby plunging,edge 6ø) beach rising above a nearly horizontal low tide terrace wavesare no longervisible in the run-up.This may be ex- (slopeless than 1/100). At hightide the depthat the toe of the plainedby oneor moreof severalpossible reasons: beach was about 3 m, but at low tide the terrace was com- 1. With the onsetof wavebreaking there is a changein the 3000 GuzA AND INMAN' EDGE WAVES AND BEACH CUSPS

1.0 EFFECT OF SLOPE TRUNCATION ON WAVE PROFILE

0.8 MODE2 Xmin=2l

0.6

o.4 - ...... 0.2 '..•...•.-.._ • ß.-w.ß ,-rr..-.•-.(-• :-- :_-t .... •-' -' -r'- , -i øll ....,. so -0.2 • X=o•-x/g tanl• ....Xo22 -1.834 -o.4•- x,/ .... 2s • ---- 28 -}.968 -0.6 ...... 30•4-2-1.995 Fig. 2. Changesin mode2 wave profile and dispersionparameter s (seeequation (3)) for variousdimensionless widths Xo of the sloping section. With Xo < Xml-, no mode 2 edge waves exist.

essential nature of the nearshore from a weakly nonlinear, important in determiningbreaker type [Galvin, 1972]. It will weakly dissipative,potential regime to a strongly nonlinear, be assumed that the maximum size incident wave which can dissipative, turbule9t one. Potential edge waves with excite potential-type edge waves through the present wavelengthsof the order of those of the turbulent surf zone mechanismdepends upon the parameterr. Neither the theory are now irrelevant to the problem. nor the experiments,however, are refinedenough to determine 2. Elementsof the potential theory are still valid, but the whether the amount of edge wave excitationdepends on the effectiveviscosity of the systemhas becomeso high that edge amount of reflection in a continuousway or changesmore waves can no longer grow. dramatically for a specificrange of r related to the onset of 3. Only standingincident wavescan effectivelydrive edge plunging breakers. Experiments,discussed later, do indicate waves.As the percentageof the total incidentenergy reflected that subharmonic edge wave excitation was always weak or decreases,so doesthe overall importanceof standingincident nonexistentfor r > 2. For simplicity,the maximum incident wave related instabilities. wave amplitude leading to subharmonicexcitation is written Regardlessof the exact reasonfor the cessationof subhar- as monic excitation an approximate range of incident wave a,w?/g tan2/• = 2 (8) amplitudes capable of exciting subharmonicpotential edge wavescan be obtainedby quantifyingthe observationthat the which correspondsto the transition region between surging incident wave must be strongly reflected. Carrier and and plungingbreakers [Galvin, 1972]. We will usethe valueof Greenspan[1958] usedthe full nonlinear,sloping bottom, shal- r (lessor greater than 2) as indicatingwhether the systemis low water equations to show that for a standing wave reflectiveor dissipative;reflective systems may exhibit the in- theoretically to be possibleon a sloping beach, stabilitiespredicted by potential standingwave theory. The theoreticalmaximum incident amplitudes are indicatedby the r = atw?/g tan•'/5 < I (7) dashedcurve on Figure 3 for/• = 6ø and variousperiods and A matched asymptotic expansion between the Carrier and on Figure 6 for variousbeach slopes and Tt = 2.7 s. Of course, Greenspan solution and slowly varying Stokes-typesolutions a well-defineddividing line betweenthe two wave regimesdoes valid in deeperwater may be usedto show that the maximum not exist even for the simplestincident wave conditions, and at in (7) is approximatelyequivalent to the value given by the classificationsare meantto be qualitative.The existenceof Miche [1951] for the maximum standing wave amplitude in maximum and minimum incidentwave amplitudeswhich can deep water which will not break on a slope. Existing experi- excitesubharmonic edge waves is confirmedby laboratory ex- ments [e.g., Ursell et al., 1960] have shownthat the deepwater perimentsdiscussed in the next section, and the agreement [Miche,1951 ] equivalentof r isindeed an importantparameter with theory is qualitatively good. in determiningthe amount of energyreflected at a coast.If the Present and previous experiments indicate that a syn- incident waves break some distance offshore and at is taken as chronousedge wave (same period as the incident wave) is the breakeramplitude, then the parameterr (equation(7)) is sometimesexcited by stronglyreflected incident waves. These synchronousedge waves have beenproposed as beinga critical TABLE 2. Maximum Edge Wave Periods(Seconds) for Each Mode elementin the formation of rip currentsand as beingrelated to and Various Offshore Depths, With Beach Face Slope of 6ø cuspformation [e.g., Bowen, 1967, 1972;Harris, 1967; Bowen and Inntan, 1969]. It is now shown that the same systemof ho(m) equations used to explain subharmonicedge wave growth

Mode 1 2 3 [Guza and Davis, 1974] suggestsa higher-order excitation mechanismfor synchronousexcitation. The nonlinearshallow 0 44.9 63.4 77.8 water horizontal momentum equationsmay be vertically in- I 7.5 I0.6 13.0 tegrated and, by using the principle of mass conservation, 2 4.2 5.9 7.2 combinedto form a singleequation for the velocitypotential 3 2.9 4.1 5.0 4•. In nondimensionalvariables (indicated by circumflex ac- GUZA AND INMAN: EDGE WAVES AND BEACH CUSPS 3001

cent) with e as a (small) expansionparameter of nonlinearity, wave is the most easily excited higher-order resonance,and this conclusionis verified by experiments. L• = eQ(•, •) + e2C(•,•, •) + O(e8) (9) It is emphasizedthat the resonanceformalism requires, for whereL is the usualliftear long-wave differential operator and strict validity of the ordering procedure,that the nonlinear Q and C are quadratic and cubic differential operators, re- parameter e << 1. However, spectively[Guza and Davis, 1974]. The linear descriptionof • = a•o•a/g tana • = r edgewaves (equations (2) and (3)) and the forms for incident wavesare obtained by neglectingterms of O(e) and higher in where r is the reflectivityparameter discussedpreviously and (9). Keeping terms of O(•) (i.e., the first correctionof non- does become O(1) in the experiments.This means that the linearity) showsthat the primary incidentwave of frequency assumptionof weak nonlinearityand validity of the ordering has a first harmonic of frequency2•o•, that the mean sea sur- proceduresis breaking down. Therefore the formally higher- face is changed,and that certain edge waves can extract en- order synchronousedge wave resonancemay be almost as ergy from the primaryincident wave. The initial growthrate of strong as the lower-ordersubharmonic. Interestingly, the as- edge wave amplitudesis O(e). These are the resonanttriads sumptionof weak nonlinearityis beginningto lose validity at already discussed:two initially small edgewaves and the inci- about the samevalue of e (or r) for which plungingbreakers dent wave interacting via the Q term to exchangeenergy begin and the visuallyobservable edge wave resonancescease. resonantly.Generally, the lowest-orderresonances (here O(e)) Gallagher [1971] considerededge wave excitation by in- will be strongerthan higher-orderones (O(e2)). However, if the cident wave spectrawith many frequenciesand anglesof inci- primary wave amplitude (or other parameterssuch as tank dence and showedthat an edge wave can be forced by the geometry)does not allow growth of the lowest-orderinstabil- interaction of two totally progressiveincident wave compo- ities, then higher-orderinstabilities may be evident. The form nents with different frequenciesand anglesof incidence.Gal- of (9) leadsdirectly to the conclusionthat O(e•) resonancesare lagher's [1971] work concernsonly dissipativebeaches, but possibleand that synchronousedge waves might be excitedin further study may show that interactionsamong various inci- this way. The calculationsnecessary to find growth rates and dent wave componentscan be important in exciting edge critical amplitudesand to insurethat synchronousgrowth is waveson reflectivebeaches. In the presentlaboratory studies, theoretically predictedrather than merely possiblehave not however,and for someselected field casesthe modelingof the yet beendone. However, by analogyto the lowest-orderinsta- incident wave field as a unidirectional, monochromatic wave bility it might be assumedthat the zero mode synchronous train appearsto be consistentwith the observededge waves.

,•=6 ø

MAXIMUM,MINIMUMr= 2.}- a/.., THEORY RESONANCE OBSERVED NO RESONANCE

/ /

/ o/ / /7/• o ø xc,,0,,oT

2 3 4 5 (sec) Fig. 3. Observedexcitation of zero modesubharmonic edge waves as a functionof incidentwave period and amplitude. The dashedcurve indicates the incidentwave amplitude for transitionfrom reflectiveto dissipativesystems (equation (8)), and the solidcurve indicatesthe amplitudesrequired to overcomeviscous damping (equation (6)). 3002 GUZA AND INMAN: EDGE WAVES AND BEACH CUSPS

EXPERIMENTS ON NONERODABLE BEACHES for a free wave, so that various possibleresonances may be The experiments were conducted in the 15.2 m X 18.2 m studied by varying the parametersof (11). Galvin [1965] ob- wave basin at the Hydraulics Laboratory at ScrippsInstitu- servedthe resonantlyexcited zero mode subharmonic(n = 0, tion of Oceanography.The beachwas a concretevariable slop- Te = 2T0 edge wave on eight plane beachesof slopevarying ing sectionextending from the shorelineend of the basin for from 6 ø to 20.5 ø. The tank width satisfied(11)with m = 1, but 8.7 m, the depth being constantfor the 5.1 m betweenthe toe no incident wave periods were reported. Bowen and lnman of the beachand the plunger-typewave maker. The sidesof the [1969] producedthe zero mode subharmonicedge wave, with basin were lined with wave absorbers, about 1 m wide and 40 low-amplitude normally incident waves, for variousvalues of cm thick, made of aluminum shavingsin wire meshcontainers. m and Te. Theseedge waves, of periodsup to 7.9 s and wave- These absorbers were used to reduce cross waves, tank seiche, lengths to 7.3 m, grew so large that they occasionallyover- and other motionsnot relevant to the study. The actual work- topped the experimentalbeach of slope4 ø. Harris [ 1967]more ing area was of width b, separatedfrom the basinproper by closelyapproximated an unboundednatural beach in that his solid barriers which extended offshore from above still water wave tank was six or more longshorewavelengths wide, the ex- level to within 1 m of the wave maker, althoughsome experi- act satisfactionof (11) beingless important. He observedsub- ments were conducted with barriers only to the toe of the harmonic excitation,with a longshorewavelength close to that beach. given by (3) with n = 0 and Te = 2T• on 6 ø and 22ø slopes.This resonance was observed for about 20 different combinations of Edge wave excitationwas studiedby sendingnormally incident low-amplitudewaves onto the beachand visuallyobserv- edge wave period and beach slope;the maximum edge wave ing the resultant edge wave growth, if any. When sub- period was 3 s. Bowen [1967], Harris [1967], and Bowenand harmonic edge waves occurred,they were obviousbecause of lnman [1969] also reported exciting synchronousedge waves, the distinctiverun-up pattern that they produce. Each sub- sometimesaccompanied by small rip currentsand circulation harmonic edge wave antinode was alternately the location of cells. The effects of finite offshore depth and basin length, run-up maxima and minima as the edge wave alternated be- variations in beach slope, and importance of incident wave tween constructive and destructive interference with the inci- parametersto subharmonicexcitation were not consideredin dent wave uprush. At any edge wave antinode the difference any detail by the aboveworkers. The lack of edgewave ampli- betweenrun-up maxima and run-up minima along the sloping tude data, in particular, left the actual importance of edge beach face, Re, was measured with a meter stick and a crude waves to the nearshorezone open to question. For example, estimate of the edge wave amplitude ae obtained from Sonu [1972] inferred that edge waveswould not have amplitudes larger than one quarter of the incident wave amplitude, ae = (Re/2) tan/• although the experimentshere commonly showededge wave amplitudes2 times larger than the incident wave amplitude. This is actually a lower limit of the actual edge wave ampli- Sets of measurementswere conducted with/• = 4.17 ø, 5.08 ø, tude becausethe edge wave may not have its maximum dis- 6ø, and 6.83ø with l•/b constant.If slopetruncation is not im- placement at the same time as the incoming wave reachesits portant, s = -n in (11), and the same edge wave periodswill maximum run-up. The simple measurementstaken, however, then be possible resonanceswith the same value of m, on certainly give a reasonableestimate of the edge wave excita- different slopes.The barrier spacingwas 10.8 m when/• = tion at differentperiods and slopesrelative to eachother. The 6.83ø and was proportionallysmaller when/• was smaller.Ex- incident wave amplitudesat the shorelinewere estimatedby cept for m = 1 and possiblysome caseswith m = 2 the zero measuringthe swashlength before the edge wavesgrew or at mode subharmonicsare unaffectedby slopetruncation and/or an edgewave node. The incidentwave amplitudeoccasionally the finite onshore-offshorelength of the wave basin.The inci- was not uniform acrossthe tank, varying by as much as 30%. dent wave periods Tt were chosen so that (11) was satisfied An average value was used in thesecases. When the incident with Te = 2Tt and s - 0. The maximum and minimum inci- wave surgedon the beach face, there was good agreementbe- dent wave heights which produced visually observablezero tween an amplitude determinedthrough the run-up and a di- rect measurement of the surge height. Near the maximum mode subharmonicedge waves and the edge wave 'ampli- valuesof incidentwave amplitudestudied the wavesbroke and tudes'were measured.Figure 3 showssome results for/• = 6 ø, the solid and dashedlines representingthe theoreticalmini- plungedsome small distanceoffshore, and the breakeramplitude was as much as 50% larger than the value calculatedfrom mum (equation (6)) and maximum (equation (8)) incident the swashlength. An averagewas usedin thesecases. Because waves, respectively,for edge wave excitation. The circles in squares and the circles in Figure 3 representincident wave of the crudenessof the measurementswe regard the ex- amplitudes and periods for which edge wave excitation was perimental resultsas being qualitative but as agreeingwith the and was not observed,respectively. The solidlines connecting trends suggestedby theory. If standingedge waves are going to be excitedon a beachof the circlesin squaresindicate that excitationwas alwaysob- served for an amplitude in this range. There is obviously finite width b, then in order to satisfythe boundarycondition qualitative agreement between theory and experiment. The of no flow through the sidewalls, increasingincident wave amplitudes,compared to theory, b = m(L/2) rn = 1, 2, ... (10) neededto excite the longer periodsmay be due to the paddle directly affecting the edge wave, becausethe longer-period is an integer indicatingthe number of half wavelengthsin the incidentwaves do not have the profile assumedby theory or longshoredirection. By meansof (3) it follows that an equiv- becausethe simple model of viscousdissipation used is not alent relation for freestandingedge waves is quantitativelyadequate. Additional viscousdamping from the b = (g/4•')Te•'m(1 - 2s) tan/• (11) sidewallswould be most important for smallerm, but a cal- culationshows that the bottomdamping already considered is Typically, resonantenergy transfers occur most readily when always much larger. The longshorewavelengths of the excited the resonantlyexcited wave satisfiesthe boundary conditions edge waves shown in Figure 3 vary between 2 and 10 m. GUZA AND INMAN: EDGE WAVES AND BEACHCUSPS 3003

Experimentson the other slopes,with the sameincident wave plitudesfor edgewave excitation have been given without con- periods, gave minimum amplitudes very similar to those siderationof edgewave amplitudes.No theorypresently ex- shown in Figure 3. This is as is expected,since the critical istswhich predicts edge wave amplitudes for prescribedbeach amplitudes(equation (6)) do not dependon •. Maximum inci- and incidentwave conditions,but sometrends are indicatedby dent wave amplitudeswith subharmonicexcitation are shown the data. The subharmonicedge wave responseto incident in Figure 4 for all slopesand clearly demonstratethe impor- wavesof fixed period (2.7 s) and constantrn was measuredon tance of the parameter r (equation (8)). four slopes.The incidentwave amplitude varied over the range The minimum subharmonicedge wave periodswhich could which excitededge waves, and the edgewave inducedrun-up visibly be excitedvaried from 2.18 s when • -- 6.83ø to 4.8 s variation R, was determined.On Figure 6 the theoreticalmini- when • = 4.17ø. The minimum-periodincident wave which mumincident amplitude (not a functionofslope) isindicated can resonantly excite edge waves is theoretically predicted by a dotted line, and the theoreticalmaximum for the four through the amplitude limits of (6) and (8). As is shown in slopesby dashedlines. Solid lines, showing trends in the data, Figure 3 for short-periodwaves, increasing the incidentwave indicate that the excited edge wave amplitudesgrow with amplitude cannot overcomeviscous damping before wave increasingincident wave amplitudeuntil the incidentwave plungingtakes place. The minimumexcitable period is shown amplitudeis near the theoreticalmaximum. The edgewaves by the intersectionof the maximum(a functionof •) and mini- visiblein the run-up then decreasein amplitudeas the inci- mum amplitude lines. The theoreticaland observedminimum dent wavesbegin to plungecleanly rather than surge.Other incident wave periodsfor zero mode subharmonicexcitation experiments,not shownhere, indicate that the decreasein edge are shown in Figure 5. An aluminum beach, made of thin, wave amplitudewith increasingbreaker size can be rather smooth aluminum sheets,was installed on the 6 ø slope, and sharpin somecases, a smallincrease in incidentwave ampli- the minimum period dropped from 1.40 to 1.12 s. Harris tudecausing a we!l-developededge wave system almost to dis- [1967] observeda minimum incidentperiod of about 0.95 s on appear. a 6ø slope of concretewhich was speciallycoated to insure With incident wave amplitudesnear the upper limit for smoothness.As is expected,a reductionof viscousdamping al- excitationthe edgewave responseoccasionally was not stable. lows higher-frequencywaves to be excited.The minimumex- Largeedge waves would sometimes grow and thenrapidly dis- citable period might vary considerablyaccording to the appear,only to reappearagain. Generally, however, once the roughness(and perhapsthe porosity)of the bed. The assump- edgewaves grew to their equilibriumamplitude, no significant tion that swashlength can be simply related to wave ampli- changesoccurred. When the incidentamplitudes are chosento tude also breaks down on porous beds. developmaximum edge wave amplitudes,the edgewave in- Thus far, only minimum and maximum incident wave am- ducedswash variations are generallyat leastequal to the inci-

v,v f•: 4.17 ø I,rl • -. 5.08 o I,O •=•o i,l •; 6.83ø v - v o THEORY

0.4 0.6 0.8 1.0 oiz/!lto. (½rn'l Fig.4. Maximumincident wave amplitudes at theshoreline for zeromode subharmonic edge wave excitation (theory from (8)). 3004 GvzAAND INMAN: EDGE WAVES AND BEACH CUSPS

5!•!:i:!:i:i:!:i;!;!;i:':•"::':.![ • EXCITATION] Iii!!ii!!!iii!!i!!!i!i!i;;•••it".:::..::•NOEXCITATION ITHEORY' r=2

:::::::::::::::::::::::::::::::::::::::::::::::::::::::

0 0.02 0.04 0.06 0,08 0.10 0.12 0.14 0.16 0.18 0.20t0n/• 0 1.15 2.29 3.43 4.57 5.7 6,8 7.96 9.09 10.2 !1.3 /•ø FiB.5. Observedminimum incident wave periods (solid symbols) for. = 0 subharmonicexcitation onbeaches oœvarious slopesand smoothness (theory from (6) and(8)). dentwave swash length, frequently more than twice as large. ingbreaker amplitude varied along the shore from about 5 cm The maximumamplitude edge wave, of a givenperiod, is wherethe edgewave was out of phasewith the incidentwave muchsmaller on gentler slopes than on steeperones. Viewed to 10 cm whereit wasin phase.The maximumshoreward in- simplistically,the gentleslopes cannot reflect sufficient inci- cursionof water was more than 1 m greaterwith sub- dentwave energy to producelarge incident standing waves harmonics than without. whichcan drive large edge waves. Similarly, on a fixedslope, Experiments(/• = 6ø, b = 10.8m) weremade to determine long-periodincident waves can generate larger edge waves theimportance of exactlysatisfying the condition (equation thanshorter-period incident waves because larger long-period (11) with s = 0 and Te = 2Tt) that the excitedsubharmonics incidentwaves are reflected. With Tt = 3.82s,/• = 6ø,and the havethe longshorewave number-frequency relation of free incidentwave swash length 135 cm (at = 6.75cm) a sub- edgewaves (equation (3)). If Tt is chosenso that (with Te = harmonicedge wave developed which had a 220-cmdifference 2Tt) equation(11) is not satisfied,then the longshorewave in run-upbetween adjacent antinodes (ae = 11cm). The surg- numbersof excitedsubharmonics do not satisfy (3). The inci-

T./= 2.7 sec. 160 _ ß /9= 4.170 ß /9= 5.08 ø _ ß j9=6 ø ß •9:6.850

120 _ß ...... MAXIMUMMINIMUM} øi,THEORY

o 1 2 3 4 5 6 oi (cm) Fig.6. Edgewave run-up amplitudes versus incident wave amplitudes forvarious beach slopes. The dashed lines arc maximumincident wave amplitudes given by equation (8)and also correspond tothe onset of plungingbreakers. GUZA ASD INMAN' EDGE WAVES AND BEACH CUSPS 3005 dent wave periods coveredthe range 1-4 s. As was expected, dent wave periods which allowed subharmonicexcitation but edgewave excitationwas greatestwhen (11) was satisfied.For with amplitudestoo smallto overcomeviscous damping of all periodsexcept rn = 1 the effectsof slope truncationand subharmonics,small synchronouswaves were usually ob- finite onshore-offshoretank lengthwere negligible.For rn = 1, served.With larger incidentwaves of the sameperiod the syn- Te = l l.5 s, and the change in the dispersionrelation (3) is chronouswave appearedalmost as soon as the incident waves small (s = 0.02), but the edge wave is so long that it would impinged on the beach, but the subharmonicsoon grew to theoreticallyextend to the paddlestill having 10%of its ampli- large amplitudes and clearly dominated the system. Syn- tude at the shoreline. This wave could not be excited, indicat- chronousedge waveswere also observedat periodstoo short ingthat in thiscase the coupling to thepaddle and the effect of (F,igure 5) for subharmonicexcitation. energy leakage to the wave absorbersprevented resonant Incidentwave periodswere studiedfor which both mode0 growth. The shortestedge wave visiblyexcited was at rn = 16 and mode ! synchronouswaves were possiblebut the mode 0 and Te = 2.8 s. Figure 7 showsthe edge wave responseob- subharmonicwas not. With • = 6.83ø, b = 10.8 m, and T• = served,as a function of incidentwave period. At eachrn the 4.4 s the zero mode subharmonicresponse was in the 'dead incidentwave amplitude was approximately constant and cor- zone' betweenrn = 1 and rn = 2 (analogousto Figure7). Very respondedto surgingincident waves. The edgewave ampli- gentleswash clearly excited only the zero modesynchronous tudesgenerally decreased with increasingwave frequency and wave,but with increasingincident amplitudes, mode 1 became with rn > l0 weretoo smallto measureaccurately. When the progressivelymore important and eventuallydominated t[ae tank width is of the order of a subharmonicedge wave wave- longshoreperiodicity. Bowen and lnman [1969]also found that length(m = 2, 3, 4), the possiblefree edge wave periods are far largerincident waves excited higher-mode synchronous waves, apart,and incident wave periods may be chosen that will not resultingin largerrip currentspacings, and they experimental- excite subharmonicedge waves.When the tank is wide in ly studiedthe nondimensionalsurf zonewidth for transition comparisonwith a wavelength(rn > 4), the possibleresonant from mode 0 to mode 1. They further reportedthat syn- periodsare so closetogether that anyincident wave period chronousedge waves were not visible in thesurf with cleanly between1.4 and 3.1 s of properamplitude will excitea sub- plungingbreakers. We reacheda similarconclusion, finding harmonicedge wave. The fluctuationsof edgewave amplitude that bothsynchronous excitation and subharmonicexcitation within a givenrn also becomesmall. Thus the narrowband becamemuch less apparent when r (equation(7)) wasgreater of incidentwave periods for subharmonicexcitation observed than about 2. Thusaccording to thepresent classification the by Galoin[1965] and the continuousresonant response oh- experimentsof Bowenand lnman [1969],with well-defined servedby Harris [1967] are functionsof the valuesof rn longshoreperiodic circulation cells and visiblesynchronous appropriateto thoseexperiments. With rn > 5 •thenumber of edgewave excitation, were reflective systems. edgewave antinodes sometimes alternated between adjacent On reflectivelaboratory beaches, synchronous edge waves m, but the edge.wave amplitude remainedapproximately had smaller amplitudesthan subharmonics,occurred best constant.No caseswere observed with visiblesimultaneous when the tank geometryexcluded the subharmonic,and were excitation of different values of m. observed even with very small nonbreaking incident waves. Whenthe incident wave period was chosen so that the tank Theseobservations support the presenthypothesis that the'-' geometry excluded subharmonicresonances, synchronous synchronous edge wave resonanceon nonerodablereflective edgewaves were sometimesclearly visible in the run-up. For beachesis weakerthan the subharmonicand can be explained example,with the tank geometryof Figure7 and T• = 3.64 s, a without referenceto rip currentsand other phenomenaas- synchronousedge wave with rn = l0 ands = 0 satisfiedthe sociatedwith wavebreaking. boundaryconditions for freeedge waves (equation (l l)) and A few experimentson dissipativebeaches showed less well wasobserved. Synchronous edge wave amplitddes were gen- definedlongshore periodicities than those on reflective beaches erally muchsmaller than subharmonicamplitudes observed anda generaltrend toward increasing rip currentspacing with with similar incidentwave amplitudesand periods.With inci- increasingamplitudes. Harris' [1967]work and the presentex- ß

SUBHARMONIC,n-O, EDGEWAVE RESPONSE TO SURGINGINCIDENT WAVES

ß ttt

•'0.• •=•o b=•O.•m

0.2 0 I I I I I I I I I I I I I I I I1__. L I • I I _ 1.6 1.8 Z.0 Z.2 Z.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0 T.i'(sec) Fig. 7. Normalized edge wave responseto surgingincident wavesof various periods. The vertical axis indicatesthe edgewave amplitudenormalized by the maximum amplitudeobserved at a particularm. Synchronousedge waves were observedin the stippledregion. 3006 GUZA AND INMAN: EDGE WAVES AND BEACH CusPs periments show that the dominant longshorespacing of rip cretion, not merely an absenceof erosion by the swash.The currentson nonerodabledissipative beaches exceeds the max- 'offshoredeltas' near the surgeline appear to be analogousto imum possible longshore wavelength of synchronousedge those associated with natural cusps [e.g., Johnson, 1919; waves(equations (4) and (5)). This indicatesthat the rip cur- Kuenen, 1948]. With more sand added 'to the system, ac- rents are related to a longshoreperiodic instability other than cumulationswould occasionallyoccur at an edge wave an- the classicalsynchronous edge wave mechanismoperative with tinode,forming a very distinctridge down the centerof the bay collapsingincident waves [Bowenand Inman, 1969]. LeBlond and resultingin two smaller embayments.A seriesof cusps and Tang [1974] haveproposed a criterionfor rip currentspac- would then have intermittent spacingshalf as large as the ing based on the minimization of relative energy dissipation dominantspacing. Such 'compound cusps' have been observed rates, but the resultsdo not comparewell with observations. in the field [Johnson,1919, p. 484; Russelland Mclntire, 1965]. Hino [1972, 1973]has suggestedthat the set-upsea surface as- The cusphorns and deltasbecame increasingly well developed sociatedwith breakingwaves is unstableand that the in- and usually connected,forming crescenticbars with the same stabilities are rip currents. This model does not require the wavelengthsas the cusps.With the continuedaddition of sand presenceof edgewaves or other time periodic intermediaries. the morphologiesinterfered with continuededge wave excita- Bowenand Inrnan [1969] suggestthat, in the field, long-period tion.The edge waves now died away, the incident wave fieM edgewaves may interactwith a two-dimensional1ong-perioO revertedto a two-dimensionalstate, and the •usps began to 'beat wave' to give circulationswith very long wavelengths. lose their definition. Edge wave amplitudescould be signifi- They also speculate that edge wave-like motions might cantly reducedby placinga long metal bar about 5 cm high on primarily exist seawardof the surf zone, sinceconsiderable the beachface at the edgewave node. Naturally, becausethere reflectionand/or scatteringof the incident wave occursat the are maximum longshorevelocities at nodes,the partial barrier break point. There is also the possibilitythat circulationson inhibits edgewave growth. Perhapswell-developed cusp horns the smallestscale are related to edgewave instabilitiesand that act in a similarway. Becauseof the problemswith establishing the largest-scalerip currents are driven by the smaller ones, simpleequilibrium beach profiles it was not possibleto deter- energycascading to a final longshorewavelength determined mine if the edge waves could reform after the cuspswere by the surf zone width. Observationson dissipativebeaches reworkedinto shapesmore conduciveto edgewave excitation. oftenshow small'scale circulation cells within a lar•er,Fieranting [1964] conductedexperiments on an initially very stronger'circulationcell [e.g., Harris, 1967]. steep beach (/• = 30ø) with small incident waves. Cusps generally appeared but were continuously destroyed and OBSERVATIONS ON ERODABLE REFLECTIVE BEACHES reformed. This may demonstratea cycle of edgewave excita- The gjY½•ctof edgewaves on nearshoremorphologies was tion, cusp growth, cessationof excitation, cusp destruction, modeledwith sandtracers. A thin veneer(2 cm) of fine sand etc., but may also reflect the overall instability of the entire was spreadon a 6ø concreteslope, and low-amplitudewaves beach profile. were sentnormally incidentonto the beach.After edgewaves The swashcirculations with small amountsof sand (Figure developed,more and more sandwas randomlyscattered "into 8) were basicallysimilar to the circulationswith edgewaves on the swashzone, and the resultingaccumulations were noted. the nonerodable beach. With the retreat of each incident wave It was not possibleto beginwith a large quantity of sand on surge,water would flow from an in-phaseantinode (a bay with the slopebecause the incidentwaves very quickly reworked the maximum run-up) around the nearest horn to neighboring sand into an entirely different topographythan the simple bays,where it would rushup the beachface with the nextinci- plane beach configurationunder consideration.Edge waves dent wave. There wasvery little longshoretransport across an- can developon complexequilibrium beachprofiles, but the tinodes,as would be expected,since edge wave theory (equa- longshorewavelengths and incident wave conditions needed tion (2)) predictsonly onshore-offshoremotion there. With for excitationare not known. Even thoughit was not practical cuspsand deltasdeveloped to the point that edgewave excita- to producetrue cuspson initially fully erodableequilibrium tion was somewhatinhibited but was not stoppedcompletely beaches,the featuresproduced have much in common with the flow of water with each retreatingsurge tended to become naturally observedcusps. channelizedbetween the horns and the deltas, concentrating Well-develop.edsubharmonic edge waves resulted in a cusp- flow nearer the horns than the bay centers. like configuration,as shownin Figure8; here Te = 4.8 s, and Longshore periodic morphologies of two different the cuspspacing is about 1.8 m. Initially, the sandveneer (dyed wavelengthscould be generatedwith synchronousedge waves, black for contrast)was eroded by the swashand depositedin but thesewere not nearlyas well developedas cuspsdue to the accumulatiisnsseaward of cusp bays and at the cusphorns. larger subharmonics.With gently surgingwaves no rip cur- Sincethe run-up is maximum at antinodes,cusp horns corres- rentsdeveloped, but synchronousedge waves were clearly visi- pond to edgewave nodesand are spacedone-half wavelength ble in the run-up. Bowen [1972] discussestheoretically how apar't.Erosion at antinodesmay occur because the fine sand is synchronousedge waves (with no rip currentspresent)can not in equilibriumwith the relativelysteep 6 ø slope,and once producecusps every half edgewavelength, possibly with alter- it is suspendedby wave action, it movesoffshore. The steady nate cusps of differing size. The present observationswith drift velocitiesassociated with edgewaves are offshoreon the gently surging incident waves often showedhalf-wavelength beachface [Bowenand Inntan, 1971], and this would comple- cuspsof equal size which, following Bowen [1972], suggestsa ment erosionalprocesses. On an accretingbeach, depositional phase differenceof ;r/2 between incident and synchronous processeswould be important, and location of maximum run- edge waves. Minimum run-up occurred at edge wave nodes, up (antinodes)might develop into cusp horns, as has been where very small horns developed. observedin the laboratory by Galoin [1965] and in the field by As the incidentwave amplitudeincreased, synchronous edge Russelland Mclntire [ 1965]. waves were less apparent in the run-up, and rip currents The horns were observedto progradeseaward several cen- developedat alternateantinodes, as was predictedby Bowen timetersand to grow severalcentimeters thicker, indicatingac- and Inntan[1969]. Cusps were not well defined,and horns Guz^ ^NO INM^N: EO6E W^VES ^NO BE^Cn CusPs 3007

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Fig. 8. Cuspatemorphologies produced by subharmonicedge waves; water has been drained below still level (marked 'A') to increasevisibility. Black is sand, white is exposedconcrete, and a meter stick is shown for scale. sometimesoccurred onshore of rips, and at other times, in citation. With T½= 2.7 s and a½= 2.5 cm, synchronouswaves between rips. As was true on the nonerodable slopes, syn- appearedfirst and formed small cuspsspaced about I m apart. chronous edge waves generally would not dominate the With the onset of subharmonic edge waves a few moments longshoreperiodicity if conditions allowed subharmonic ex- later the surge greatly increasedin strength, rapidly washed 3008 GUZA AND INMAN: EDGE WAVES AND BEACH CUSPS out the synchronouscusps, and began forming cuspsspaced and cusp slopes,7 of the 9 reportedspacings are reasonably 2.25 cm apart. However, once in an apparentlyidentical ex- closeto synchronousedge wave wavelengths(as suggestedby perimentthe synchronouswaves formed cusps sufficiently well Komar), or half a zero mode subharmonicwavelength (Figure developedto prevent growth of the subharmonic!Whether 9). Six of these7 caseshave averagebeach slopes greater than this is an isolatedcase and whethersimilar topographicfeed- 5ø and incidentwave periodsclose to the theoreticalminimum back is important in the field are not known. (Figure 5) for subharmonicexcitation. One point (• = 3.25ø, Incidentwaves of period 1.02 s did not producevisible sub- Tt = 1 s) is distinctlybelow the minimumincident period line, harmonic excitation on the 6ø concreteslope. If, however,a and this is the only case which givesbetter agreementwith a small berm (',•3 cm high) wasconstructed just seawardof the synchronousthan with a subharmonicspacing. No longshore maximum uprush so that water would pond on the flatter variation in run-up was evident, and so the presenceof syn- slopebehind the berm, the return flow was in continuousnar- chronousor subharmonicedge waves could not be confirmed row streamswhich cut evenlyspaced channels in the berm and visually. Longuet'Higgins and Parkin [1962] and Williams depositederoded material offshore. These flows were very nar- [1973] found that cusp spacing on dissipativebeaches in- row and were evenly spacedat 30 cm, compared with a half creasedwith increasingswash length, but no such trend ap- subharmonicwavelength of 31 cm, but no edge waves were pears in Komar's data. The incident wave conditionsfor the visiblein the run-up. The channelsgradually widened slightly, two anomalouscusp spacingsshould excite subharmonicsby and so the berm was alternatelyhigh and low, but the high the criterionused for all the otherpoints, but the cuspspacings points did not taper seawardinto not,iceably cuspate forms. are very short, about half a zero mode synchronous Onshoretransport of water by incident surgesoccurred over wavelength. King [1965] reported cusps formed by 'low' the broad high points,converged, and returnedseaward in the regular swell waves of period 9.5 s normally incident on narrower bays. These circulationsresemble the flows most Trahane Beachin Ireland 05 = 2.6ø). The averageof 36 cusp often observedon natural cusps[e.g., Bagnold,1940; Russell spacingswas 14.7 m, comparedwith the 12.7 m predictedfor and Mclntire, 1965]. Note that eventhough subharmonicedge cuspsformed by zero mode subharmonics.On nearby Cruet waves are presumably responsible for the longshore Island Beach,which is approximatelytwice as steep,the cusp periodicities,the circulationis much differentfrom that with spacingwas also doubled, in accordwith theory. large, well-developedsubharmonics. With artificial bermsof Kuenen[1948] describedin detail somecusps and associated proper size and location, high-frequencysurging incident circulationson the Isle of Wight. Figure l0 [from Kuenen, wavesof other periodswhich excitedno visiblesubharmonics 1948] shows cusps formed under surging wave conditions. on the concreteslope produced similar narrow flows at half Notice in particularthe longshoreperiodic highs (H) and lows subharmonicwavelength spacing, with occasionallya whole (L) of the swashline. Kuenennoted that a bay surgeappeared wavelengthor irregular intervals. to alternate(in time) betweenbeing stronger and beingweaker Evans' [1938, 1945] observationsof 'ideal cusps'on various than a horn surge.This descriptionand this photographare lakes are consistentwith a model hypothesisthat small edge completelyconsistent with the presenceof a subharmonicedge wavesprovide an initial periodiclongshore perturbation which wavewith baysat eachantinode. The surgeat out-of-phasean- is rapidly accentuatedby ridge breechingand channelingof tinodes (correspondingto points of minimum run-up on a the return flow. Evans describes formation of a small beach nonerodablebeach) is lower than that at in-phaseantinodes berm by a small,very regularseries of waveslocally generated and appearsto be retardedin relation to a horn. For the next by a gentle steady breeze. With a slight increase in wind incidentsurge the locationsof in- and out-of-phaseantinodes velocityand wave height the beachberm was simultaneously are reversed,and the 'alternating action' of the surge at a breechedat many placesby a wave train with some regular givenbay and horn is explained.The retreatingsurge was con- longshoreamplitude variation. In less than 100 s a perfect centratedalong the horns, as was true in the laboratory with systemof cuspswas formed. With a further increasein wind well-developedsubharmonic edge waves and in the field with velocity and wave height the cusps were completely 'gentle swash' [Williams, 1973]. obliterated,leaving the beachsmooth. Repeated observations OBSERVATIONS ON ERODABLE DISSIPATIVE BEACHES showedthat for cusp formation, closeadjustment is required betweenamplitude and uniformity of incident wavesand size Komar [1971] produced cuspate features with waves of and shape of the berm. The experimentson nonerodable period 1.46 s breaking normally incident on a granular coal beachesshow that with high-frequencyincident waves, as would occur in lakes, only a very narrow range of incident wave amplitudeswill excite edge waves (Figure 3) and that thesewill be of small amplitude.Without the exact tuning of the incident waves, edge waves will not occur; without the 50 • ßT e: 2T i ,n=O positive feedbackof berm breechingand channelingthe edge o wavesmight not effectthe topography;and Evans[1938, 1945] n..as 0 _ • •'Te =Ti,n=l actually concludedthat breechingof sand or seaweedridges was required for cusp formation. Subsequentobservations -50 have shown that breechingof bermsor ridgesis not essential for cuspformation, evenwith high-frequencywaves, but some positive correlation has been noted [Mii, 1959; Otvos, 1964; -I00 I I I I I I I Williams, 1973]. SPACING (crns) Komar [1973] observed small cusps at Mono Lake, best Fig. 9. Seven of nine cusp spacingsobserved by Komar [1973], formedwith surgingincident waves and disappearingwhen the comparedwith modelshypothesizing formation by synchronousand incident wave energy increasedso as to produceappreciable subharmonicedge waves producing cusps with spacingsequal to and wavebreaking. If thebeach slope is takbn as the average of bay half of the edge wave wavelength,respectively. GUZA AND INMAN: EDGE WAVES AND BEACH CUSPS 3009 beach.The observedwavelength was about 6 m, althoughthe .bars,reform as smallerpotential waves, and eventuallyreflect maximum potential synchronousedge wave wavelength for off the beach face. Thus on the largestscale, with a wide surf Komar's experiment is 228 cm (equations (4) and (5)). zone, the nearshoresystem is dissipativebut may behaveas a Contrary to Komar's assertionthe cusp spacing is therefore reflectivesystem on the much smaller scaleof the beach face. not due to rip currents associatedwith synchronousedge Reflectivebeach cusps with spacingsof tensof metersmay ap- wavesof the usualpotential type. The cuspspacing may be the pear on very high energy dissipativebeaches with surf zone resultof large-scalecirculations of unknown genesis,like those widths and rip spacingsof hundredsof meters [lnman, 1971]. observedby Harris [1967]on nonerodabledissipative beaches. Different frequency components of an incident wave It is also conceivablethat the cusp spacingis somehow spectrummay underg,o quite different amounts of reflection related to cross waves. Cross waves are standing waves with [Suhayda,1972]. It is not known if a stronglyreflected long crestsat angles(usually 90ø ) to the wave maker (their energy wave can exhibit reflective edge wave instabilities in the source),just as edgewaves have crests at right anglesto nor- presenceof a dissipativewave breakingat higher frequencies. mally incident waves. Longshore periodicities observed on Further work is also needed to investigatethe possibilityof laboratory beachesmay be due to crosswaves driven at the many incident frequencybands, each satisfyingthe reflective wave maker and extendingdown the entire tank but are not resonanceconditions, acting independentlyof each other, and due to edge waves. Garrett [1970] has theoretically exciting edge waves of •many frequenciesand wave numbers. demonstratedthat the lowest-ordercross-wave instability is a Incident wave frequency and directional spectra also allow subharmonicof the paddle frequencyand that a higher-order edge wave excitation through the interaction of different inci- resonancemay be responsiblefor driving synchronouscross dent wave components[Gallagher, 1971 ] as well as through the waves, exactly as we have suggestedis the case with edge instability of a single incident wave component. Gallagher's waves.McGoldrick [1968] usedprogressive primary wavesto [1971] work suggeststhat 'surf beat,' the low-frequencyoscil- generate synchronousand subharmoniccross waves having lations characteristicof dissipativebeaches, is, in fact, low- the longshorewave number of free gravity waveswith crests mode edgewaves. The frequenciesof theseedge waves have no perpendicularto the wavemaker, kt in (5). Subharmoniccross simple relationshipto the frequencyof the principal incident waveswere easierto exciteand of larger amplitudethan syn- wave component;they may be progressiveor standingand, for chronouswaves. Garrett [1970] theoretically showed that with .typicalincident wave conditions, may have approximately standing primary waves, additional cross-waveresonances equal energy in the three lowest modes. These edge waves may occurwith crestsat anglesother than 90ø to the paddle would typically (Te = 25 s, • = 6ø , n = 2) have longshore and longshorewave numbersky < kr. Theseinstabilities have wavelengthsof severalhundred (325) meters.Innnan and Bowen beenobserved experimentally but havenot beenstudied in any [1967] have measuredspectra which suggestthe presenceof detail [McGoldrick, 1968]. Regardlessof the exact cross-wave such waves. Much more theoreticaland experimentalwork is instabilitythey will alwayssatisfy ky < kr, while inviscidedge needed to determine the relative importance of the different wavesobey k• > kt (equation (4)). possibleedge wave excitation mechanisms for variousincident Escher[1937] produced cusps with crosswaves, although he wave conditions.Furthermore, Kenyon [1970] has shown understoodthese oscillationsonly as standing waves with theoreticallythat edge waves can transfer energy to otheredge crestsat right,angles to the primaryprogressive waves, not dis- waves via nonlinear resonances.This means that significant tinguishing between cross-wavesand edge waves. The edge wave energy might occur at frequenciesand wave observedtransverse oscillations were synchronous, and the numbersnot directly driven by the incident wave field. What- numberof antinodesreported can be shownto be generally everthe edgewave energy source, if manyedge wave frequen- • consistentwith ky.= kt (equation(5)). In someof theexperi- cies and wave numbers occur simultaneously, no single ments[Escher, 1937, p. 86]a boardwas placed between the fiat periodicitymay be obviousin the run-up and/or morphologies and the sloping basin sections,and the primary waveswere [Bowen,1972]. The presentwork hasstressed the edgewave preventedfrom reachingthe beachuntil good transverseoscil- and the edgewave induced topographic response to single- lations had developed.The board was then removed,and the frequency, normally incident, strongly reflected, incident waveswere allowed to reworkthe beach. This mode of genera- waves.Only selected,special field casessatisfy this assumP- tion rulesout thepossibility that the transverseoscillations tion, and so directapplication to cuspformation on all were high-modeedge waves. Kornar [1973] believedEscher's beachesis not appropriate. [1937] oscillationsto be edge waves, but both observations Some confusion'has •arisenconcerning the presenceof may representthe effectsof crosswaves. The frequenciesand standingedge waves on beachesnot boundedin the longshore paddlestrokes for WhichCross waves will beexcited are a com- directionby headlandsor groins.Sonu [1973] stated that 'On a plicatedfunction of paddleconfigurations and tank geometry long straightbeach an edge wave is progressive,'and LeBlond [Garrett, 1970]; thus no simple rule exists for determining and Tang[1974] agreed. Huntley and Bowen [1973], however, when crosswaves will occur. Low-steepnessprimary waves, measuredvelocities on a reflective (r • l) beach consistent however,generally cannot initiate the cross-waveinstability, with thepresence of thestanding zero mode subharmonic edge so that cross-wave COntamination will be most serious in ex- waves.The beachranin a gentlecurve for more than 5 km, perimentson dissipativebeaches. and the edgewave had a periodof l0 s anda theoretical wavelengthof 32 m, sø that end effectswere indeed negligible. DISCUSSION Standingedge waves (of longerperiod) have been observed in Under the very simplifiedconditions of the laboratory the the openwaters of theCalifornia shelf [Munk et al., 1964]. distinction between reflective and dissipative systems is It is alsotrue that beachcusps, taken as indicatingthe qualitative but conceptuallyclear. On real beaches,however, presenceof standing edge waves, seem to occur more fre- the complicationsof irregular topography and incident wave quently near jetties, sharp curves in the beach, etc., than on spectramake the application of this classificationsomewhat very long straightbeaches. The standingedge wave excitation ambiguous.Waves on real beachesmay break on a seriesof theory of Guza and Dat)is [1974] assumeda beach which was 3010 GUZA AND INMAN' EDGE WAVES AND BEACH CUSPS

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GUZAAND INMAN: EDGE WAVES AND BEACH CUSPS 3011 unboundedin the longshoredirection and an incidentstand- REFERENCES ing wavewhich was completelycoherent and of uniform Baghold,R. A., Beachformation by waves:Some model experi- amplitudealongshore. Edge wave growth occurs because the ments in a wave tank, J. Inst. Civil Eng., 15, 27-52, 1940. Bowen,A. J., Rip currents,Ph.D. dissertation, 115 pp., Univ. of Calif.- inputof edgewave energy via nonlinearforcing exceeds the at San Diego, La Jolla, 1967. energyloss of viscousdamping, with no requirementfor Bowen,A. J., Rip currents,1, Theoreticalinvestigations, J. Geephys. onshore-offshorebarriers. On real beaches,however, the inci- Res., 74(23), 5467-5478, 1969. dentwaves may not be phase coherent for largelongshore dis- Bowen,A. J., Edgewaves and the littoral environment, in Proceedings tances,and this must reducethe strengthof the resonance. of the ThirteenthCoastal Engineering Conference, pp. 1313-1319, Council on Wave Research, London, 1972. Furthermore, if the edge wave resonance stops at some Bowen,A. J., andD. L. Inman,Rip currents,2, Laboratoryand field longshorelocation, the resonant part of thesystem will radiate observations,J. Geephys.Res., 74(23), 5479-5490,1969. edgewave energy along the shoreinto the nonresonantzone. Bowen,A. J., and D. L. Inman, Edgewaves and crescenticbars, J. Theseradiative losses could greatly decrease the amplitudes of Geephys.Res., 76(36),8662-8671, 1971. excitededge waves. A headland,a groin, or evena gentlycurv- Bowen,A. J., and D. L. Inman, Reply, J. Geephys.Res., 77(33), 6632-6633, 1972. ingshoreline may provide end effects which limit the spatial Carrier,G. F., and H. P. Greenspan,Water waves of finiteamplitude extentover whichresonance must occur, resulting in localized on a slopingbeach, J. Fluid Mech.,4, 97-109, 1958. edgewave excitation and/or cusp formation. When the inci- Dolan, R., Coastal landforms:Crescentic and rhythmic,Geol. Sec. dentwave field is verylong crested and of uniformamplitude Amer. Bull., 82, 177-180, 1971. Dolan, R., and J. C. Ferm, Crescenticlandforms along the mid- alongshore,there is no needfor suchend effects because any Atlantic coast, Science,159, 627-629, 1968. radiativeenergy losses out the 'ends'of thesystem are only a Donn, W. L., and M. Ewing,Stokes' edge waves in Lake Michigan, small fraction of the total nonlinear energy input. Many Science,124(3234), 1238-1242, 1956. authors[e.g., Longuet-Higgins and Parkin, 1962;Worrall, Eckart,C., Surfacewaves on waterof variabledepth, WaveRep. 100, Ref 51-12,99 pp., Univ. of Calif.,Scripps Inst. of Oceanogr.,La 1969]have noted a correlationbetween regularity of theinci- Jolla, 1951. dent wave field and cusp formation. Escher,B. G., Experimentson the formationof beachcusps, Leidse No theoreticalwork on edgewave generationhas con- Geol. Meded., 9, 79-104, 1937. sideredthe importanceof topographicfeedback. Hino [1972, Evans,O. F., Classificationand origin of beachcusps, J. Geol.,46, 1973]and Noda[1972] presented bottom interactive theories 615-627. 1938. Evans,O. F., Further observationson the origin of beachcusps, J. for dissipativesystems, but theaveraging procedures used do Geol., .53, 403-404, 1945. not allowany edge wave resonances. Resonantly excited sub- Fiemining,N. C., Tankexperiments on thesorting of beachmaterial harmonicedge waves with nonerodable beds and reflective in- duringcusp formation, J. Sediment.Petrology, 34(1), 112-122, 1964. cidentwaves often had amplitudeslarger than those of the in- Gallagher,B., Generationof surfbeat by nonlinear wave interactions, cidentwaves, but the experimentswith sandtracers on plane J. Fluid Mech., 49, 1-20, 1971. Galvin, C. J., Jr., Resonantedge waveson laboratorybeaches beachesindicate that edgewave induced topographic changes (abstract),Eos Trans.AGU, 46, 112, 1965. providenegative feedback to theedge wave excitation process. Galvin, C. J., Jr., Wavesbreaking in shallowwater, in Waveson On the otherhand, ridges properly located and of appropriate Beaches,edited by R. E. Meyer,pp. 413-456, Academic, New York, sizeprovide strong positive feedback to edgewave induced 1972. Garrett,C. J. R., On crosswaves, J. FluidMech., 41,837-849, 1970. longshoreperturbations in the topography.Thus edge waves Guza, R. T., and R. E. Davis,Excitation of edgewaves by wavesinci- which are too smallto be readilyapparent in the run-upcan dent on a beach,J. Geephys.Res., 79(9), 1285-1291,1974. exert someinfluence in formingperiodic morphologies. The Harris,T. F. W., Fieldand model studies of thenearshere circulation, fact that the alternatingsurge strengths, which characterize Ph.D.dissertation, 188 pp., Dep. of Phys.,Univ. of Natal,South subharmonicedge waves, have rarely been observedon Africa, 1967. Hino, M., Theory on formationof shore current systemand naturalcusps emphasizes the importanceof consideringthe systematicdeformation of coastaltopography, Tech. Rep. 13, pp. coupled(fluid-bed) system• The synchronousedge wave 99-113,Dep. of Civil Eng.,Tokyo Inst. of Technol.,Tokyo, 1972. resonance,although it is weaker than the subharmonicHino, M., Hydrodynamicinstability theory on the formationof resonanceon nonerodablebeds, may be more importantwith systemof shorecurrent and coastal topography, 2, Tech.Rep. 14, movablebeds. It is alsoconceivable that topographicfeedback pp. 27-41,Dep. of Civil Eng.,Tokyo Inst. of Technol.,Tokyo, 1973. to edgewave excitation is sostrongly negative that there is no Hom-ma, M., and C. Sonu, Rhythmicpatterns of longshorebars importanttopographic response. We believe this to behighly related to sediment characteristics,in Proceedingsof the Eighth unlikely.Edge wavesmost probablyprovide an initial CoastalEngineering Conference, pp. 248-278,Council on Wave longshoreperiodic perturbation in thetopography, the further Research, London, 1963. Huntley,D. A., andA. J. Bowen,Field observations of edgewaves, evolutionof cuspsbeing due to positivecoupling between the Nature, 243(5403), !60-162, 1973. primaryincident wave train and the perturbed bed form, berm Inman,D. L., Nearshereprocesses, in McGraw-Hill Encyclopaedia of breechingbeing an extremeexample. This contention is sup- Scienceand Technology,vol. 9, pp. 26-33, McGraw-Hill, New portedby observations [e.g., Russell and Mclntire, !965; Wor- York, 1971. rail, 1969;Williams, 1973] that regular longshore variations in Inman, D. L., and A. J. Bowen,Spectra of breakingwaves (abstract), Eos Trans. ,4GU, 48(1), 146, 1967. swash,topography, and/or circulation are necessaryfor cusp Johnson,D. W., Shore Processesand ShorelineDevelopment, pp. development. 473-476, John Wiley, New York, 1919. Kenyon,K. E., A noteon conservativeedge wave interactions, Deep Acknowledgments.This researchwas supported by the Officeof Sea Res., 17, 197-201, 1970. Naval Researchunder contract with the Scripps Institution of King,C. A.M., Someobservations onthe beaches of thewest coast of Oceanography,University of California,La Jolla,California, and by CountyDonegal, Irish Geegr.,$(2), 46-50, 1965. the NOAA Officeof Sea Grant, Departmentof Commerce,under Komar, P. D., Nearsherecell circulationand distributionof giant grantUSDC 2-35208.Anthony J. Bowenprovided invaluable sugges- cusps,Geol. Sec. Amer. Bull., 82, 2643-2650,1971. tionsconcerning many aspects of the work. ChristopherJ. Garrett Komar,P. D., Observationsof beach cusps at MonoLake, California, helpedclarify points regarding cross waves. Andrea Lindberg drafted Geol. Sec. Amer. Bull., 84, 3593-3600, 1973. the figuresand aided substantially in their design. The staffof the Kuenen,P. H., The formationof beachcusps, J. Geol.,56, 34-40, ScrippsHydraulics Laboratory helped set up the experiments. 1948. 3012 GUZA AND INMAN' EDGE WAVES AND BEACH CusPs

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