schwartz82 strongly nonlinear waves

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Ann. Rev. Fluid Mech. 1982. 14:39-60Copyright 1982 by Annual Reviews Inc. All rights reserved.

STRONGLY NONLINEARWAVESL. W. SchwartzExxonResearch and Engineering Company, inden, New ersey 07036

J. D. FentonSchool of Mathematics, University of NewSouth Wales, Kensington, N.S.W.,Australia 2033.1. INTRODUCTIONAs steep waves have recently come to be described with increasing accu-racy, a number of unexpected physical and mathematical phenomena avebeen revealed. Until ten years ago it had been assumed that accuratesolutions for high waveswould hold few surprises. Examplesof such sup-positions are that deep-water solutions wouldconverge for all wavesshortof the highest, that important integral quantities such as speed, energy,and momentumwould increase with wave height until the highest isreached, that the solutions for periodic waves would be unique, and thatif one solitary wave overtakes another any change of wave height wouldbe a decrease. It is nowknownhat all these suppositions are false, havingbeen disproved in the last decade. The nonlinearity of the describing equa-tions producesa complexityof solution structure that is only nowbeginningto be appreciated.This review will deal with effectively exact solutions for nonlinear wavesand the phenomena evealed by such solutions. The governing equationwithin the fluid is taken to be Laplaces equation, corresponding to irro-tational flow of an incompressible fluid. Excludedare the physical effectsof viscosity, density gradients, compressibility, and rotation. This modelof the flow is the simplest, but one which s an excellent approximation nmanycases of wavemotion, and is the traditional avenue of approach tomost problems of fluid flow. Throughout his review, however, he problemsand solutions described are those where the complete nonlinear boundaryconditions have been included. It has been the nonlinearity of these con-ditions which has made the accurate solution of water-wave problemsso difficult.

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40 SCHWARTZ & FENTONThe fluid is assumed o be inviscid and the flow irrotational, such thatthe velocity q may be expressed as the gradient of a potential ~b, q =Vq~. If the fluid is assumed o be incompressible, such that ~ .q = 0, the

equation that holds throughout the fluid is Laplaces equationwhere the subscripts denote partial differentiation. The x and z coordinatesare taken to be in a horizontal plane, the y axis vertically upwards. If thefluid is partly boundedby solid boundaries that are free to move uch thaty = h(x, z, t) on the boundary, it can be shown hat the condition thatno fluid pass through the boundary is

h, = ~ -- ~hx -- ~zh~on y = h. In many situations the boundary can be taken as stationary,h~ = 0, and horizontal, h~ = hz = 0, so that the boundary conditionbecomes

~y(x, h, z, t) = (1.2)The free-surface boundary conditions are to be satisfied on y = ~ (x, z,t), which is also unknown. The kinematic requirement that a particle onthe surface remain on it is expressed by

DDt (y ~) --~, --~ ~n~ + ~ (1.3)

on y = 0. The dynamic boundary condition can be written~ + ~/~(~ + ~ + ~) + g~ + p~/p

+ T(1/R, + lIRa) = C(t) (1.4)on y = ~, where g is gravitational acceleration, p~ is the pressure at thesurface, ~ is fluid density, T is surface tension, R~ and R2 are principalradii of surface curvature, and C(t) is a function only of time. In manysituations, and throughout the rest of this review, air motion above thesurface is neglected and the surface pressure taken to be a constant. Theequations maybe simplified in any or all of the cases of (a) steady flow,O/Ot ~ O, (b) two-dimensional flow, O/Oz ~ 0, and (c) surface tensionrelatively unimportant, T set to zero. However, in all these physicalsimplifications, nonlinear terms remain in the free-surface boundaryconditions.2. THE CANONICAL PROBLEM : STEADY WAVESThe problem of a periodic train of waves propagating without change ofform allows a considerable simplication by the addition of a suitable hor-izontal velocity to the reference frame, so that the fluid motion maybemade steady and all time dependence and time derivatives vanish from


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STRONGLY NONLINEAR WAVES 41(1.1-1.4). By considering a coordinate frame in which one axis is parallelto the direction of propagation the problem is made two-dimensional.Surface tension will be neglected in this section, and its inclusion describedin Section 3. The problem now formulated, the two-dimensional steadyperiodic gravity wave, is the simplest of all, and has often been the avenueby which solutions to more difficult and general problems have been ap-proached. Despite its relative simplicity, it contains the full nonlinearityof the surface boundary conditions and has succumbed o accurate solutiononly in the last decade, during which several interesting phenomena avebeen discovered.The existence of solutions to this problem has been studied by severalauthors. Krasovskii (1960) has devised the most significant proof. A recentdiscussion of his work has been given by Keady & Norbury (1978), whoalso established existence of a set of solutions of which his are a subset.Toland (1978) has shown that a solution exists for a wave of greatestheight, and that this wave s the uniform limit of wavesof almost extremeform.A numberof general theorems for the motions due to steady wave rainshave been established, a summaryof which is given in Wehausen& Lai-tone (1960, Section 32). These include theorems on the decrease of motionwith depth, as well as relations between energy and momentumntegrals.Longuet-Higgins (1974, 1975) has established a number of relations be-tween the integral quantities of a wave rain, and these have been gener-alized by Cokelet (1977a) to allow for the wave rain movingat arbitraryspeed relative to the frame of reference.A steady wave train can be uniquely specified by three lengths--thepeak-to-trough wave height H, the wavelength ~,, and the mean depth D.From hese, two independent dimensionless ratios can be formed, so thatthe steady-wave problem has a two-parameter family of solutions, althoughrecently (see Section 2.2.1) it has been shown hat for waves near thehighest there maybe more than one solution. Special limiting cases of thesteady-wave problem are (a) D/~ --~ ~o, ?~ and H finite, called the deep-water wave, and (b) D/~ --~ O, Dand H finite, called the solitary wave.2.1 Solution by Perturbation Expansion Methods2.1A STOI~ES X~ANSIONStokes (1847) first used a systematic perturba-tion technique to solve the steady-wave problem. He assumed that thefree-surface elevation maybe represented by an infinite Fourier series

rt(x) = a~ cos x + 2 cos 2x +. . . (2.1a)and that the velocity potential maybe similarly represented by

4~(x, y) = cx + b~ cosh (y + Y0) sin+ bz cosh 2(y + Y0) sin 2x + . . (2.1b)


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42 SCHWARTZ & FENTONwhere the problem has been made dimensionless by referring lengths toA/2~r and velocities to Co = (gA/2r)~/2, the speed of an infinitesimal wavein deep water. The coefficient c and the an and bn are functions of thewater depth. In some situations Y0 has been assumed to be D, while ininverse formulations of the problem (see Section 2.2.1) it is more conve-nient to let Yo, the height of the origin above the bottom, be d ---- Q/cwhere Q is the area flow rate under the stationary wave. The leadingcoefficient c is the phase speed of the wave rain in another frame throughwhich the waves propagate such that the time-mean fluid velocity at allpoints is zero. The coefficients c, an, and bn are assumed o be power-seriesexpansions in a~, such that the leading orders of an and bn ~ O(aT). Whenthese series are substituted into (2.1a and b), which are then substitutedinto the steady free-surface conditions, an ordered set of equations isobtained from which the coefficients in the power series can be foundsuccessively. The complexity of the manipulations makes a manual high-order calculation impractical. Fifth-order solutions have been obtained byDe (1955), Chappelear (1961), who claimed mistakes in Des solution,and by Skjelbreia & Hendrickson (1961).If the inverse problem s studied, where $ and ~b rather than x and y arethe independent variables, the calculations can be greatly reduced. Thefree surface becomes a knownboundary, ~b = 0.

Stokes (1880) computed the deep-water case to O(a~) and found resultsfor finite depth to O(a{). In the most ambitious manual computations usingthe inverse method, Wilton (1914) carried the infinite depth calculationto O(al) (but has errors at the eighth order) and De (1955) has publisheda fifth-order solution for general depth.In order to reveal details of highly nonlinear wavesby the series method,solutions of muchhigher order must be obtained. Schwartz (1972, 1974)simplified the formalism of the inverse methodby using complex unctions.Each wave cycle in the physical z = x + iy plane is mapped onto theinterior of an annulus in the ~" plane, where ~ = exp(--if/c) wherefis thecomplexpotential f = ~b + i~k. The mapping unction is an infinite poly-nomial in ~" with coefficients an, each being taken to be a powerseries in~, a parameter that is zero for the undisturbed stream and assumed toincrease monotonically with waveheight. By substituting into the dynamicboundary condition, a set of recurrence relations is found, and the coef-ficients in the power series determined successively by computer. Thesystem of equations is closed by defining ~. Choosing ~ = al reproducesthe procedure of Stokes (1880).

Stokes (1847) showed that the highest wave, assumed to be sharp-crested, must have an included angle of 120 at the crest. He conjecturedthat this limiting wavewouldcorrespond to the critical value of a~ in the




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STRONGLY NONLINEAR WAVES 43series expansion. In fact this is not the case, for al is not a monotonicfunction of waveheight but achieves its maximumalue before the highestwave s attained, about 10%before for the deep-water wave. To surmountthis difficulty Schwartz introduced the waveheight as a new parameter,in fact, ~ = HI2. He found that all the coefficients in the expansionsreached maxima efore the highest wave s achieved. Thus when a~ is usedas the independent parameter, the maximum f at as a function of Hbecomes square-root singularity, which limits the convergenceof Stokesexpansion in al. Schwartz subsequently used series-analysis techniques,including Pad6 approximants and Domb-Sykes lots, to estimate the lim-iting waveheight in deep water. He found this to be (H/A)max = 0.1412.From the accurate results of the enhanced series, Schwartz showed thatthe mass of the wave has a maximumn H. This has important implicationsfor other integral quantities of the wave train such as energy andmomentum.Longuet-Higgins (1975) used Schwartzs program for the deep-waterwave, recomputed to 32 decimal places by Cokelet, and re-expressed theseries in terms of the parameter o~ = 1 - (qcqt/cco)2, whereq is the fluidspeed at the crest and qt that at the trough. The parameter ~o has theuseful property that its range is known b initio, undisturbed flow corre-sponding to ~o = 0 and the limiting wave o o~ = 1. Longuet-Higgins oundmaxima n each of the integral quantities : wave speed, momentum, ndpotential and kinetic energy, before the highest wavewas attained, as hadbeen found by Longuet-Higgins & Fenton (1974) for the solitary wave.The variation of the integral quantities with wavesteepness H/Ais shownin Figure 1 for the deep-watercase. It is clear that the highest wave s not

1 .~1~; .10 . 141Figure 1 Dimensionless wave speed, impulse and kinetic and potential energies versus waveheight for deep-water progressive gravity waves.


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44 SCHWARTZ& FENTONthe fastest nor the most impulsive nor the most energetic. The physicalimplications of this are not well understood, but it maybe responsible forthe instability of the crest of high waves (see Section 4.1), for the multi-plicity of solutions (see Section 2.2.1) and be relevant to the observedintermittency of spilling breakers (Longuet-Higgins & Turner 1974).the maxima ccur for high waves, when surface tension and viscosity wouldalso be important, especially on a laboratory scale, experimental confir-mation wouldbe difficult.Cokelet (1977a) used a method very similar to that of Schwartz, andfor a wide range of finite water depths produced a number of accurateresults for integral quantities of the wave train. Each showed a maximumbefore the highest wave was reached. Finite-depth results for wave speedare displayed on Figure 2, using data from Schwartz (1972), Cokelet(1977a), and Longuet-Higgins & Fenton (1974).2.1.2. SHALLOW-WATERXPANSIONSt has been shown by Ursell (1953)that the linear theory of periodic waves the first term in Stokes expansion)is valid only if the shallowness parameter HX2/Das well as the wavesteepness///~, is small. That is, the waves must not be too long relativeto the water depth. This is shownby the results given by Schwartz(1974),where the radii of convergence of the Stokes expansions becomesmallerfor longer waves. Analytical theories show the dependence on wavelengthmore explicitly; it can be shown hat whereas the nominal Stokes expansionparameter is a~k, wherek is the wavenumber7r/~, the ratios of successiveterms in the expansion actually behave.like a~k/sinh3kD.For shallow water

0.2 ~ t~" ~

0.0 ~ I ~0.4 0.6 0.8 1 .0 I .2 1.4 I .6c=/gdFigure 2 Wave speed versus amplitude and depth for progressive gravity waves. Numberson curves are values of d/P,.--sharp-crested waves.

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STRONGLY NONLINEAR WAVES 45kD becomes small so that the effective expansion parameter varies likealk/(kD)3, proportional to Ursells parameter.A theory for waves in shallow water was put forward by Korteweg &de Vries (1895), who howed hat to first order, the free surface of steadywaves s of the form of a Jaeobian elliptic cn function squared, giving riseto the term "cnoidal" waves which show the long flat troughs and shortcrests characteristic of waves n shallow water. Higher-order cnoidal wavetheories have been obtained by Laitone (1960) and Chappelear (1962)second and third order respectively. A systematic method using power-series expansions in terms of shallowness, (D/X)2, was used by Fenton(1979), who used computer manipulation of the long series to produceninth-order solution. It was shown hat the most appropriate depth scaleis h, the water depth under the troughs, and that the natural expansionparameter when the series are recast in terms of wave height is H/mh,where n is the parameterof the elliptic functions. For the long-wavelengthlimit of the solitary wave m --~ 1; however, for shorter waves m becomessmaller and the expansion parameter larger, complementary o the mannerin which the sinhakD denominator in Stokes expansion invalidates theirapplication to shallow water. The cnoidal-wave solutions were found to benot accurate for very high waves, for reasons associated with the maximaof integral quantities as a function of waveheight. While the cnoidal-waveresults should be used instead of Stokess wavesolutions in shallow water,for physical applications where accurate solutions are necessary, both havebeen somewhat superseded by the numerical method of Rienecker & Fen-ton (1981), described in Section 2.2.2.2.1.3 SOLITARYAVEhe solitary wave is a steady wave of infinite wave-length. A recent specialized article on solitary waveshas been written byMiles (1980), so that the treatment here will be brief. The only high-orderseries results are those presented by Fenton (1972) to ninth order, andLonguet-Higgins & Fenton (1974) to fourteenth order. It was found thatthe series in terms of waveheight H/h did not give accurate solutions forvery high waves.. The series was recast in terms of the parameter ~o = l- qE~/gh, where o~ has the known ange (0,1). Using these series, Pad6methods were implemented, and convergent results were obtained for allwave heights. It was found that the integral quantities of the wave allshowed maximaas functions of wave height.The other approach has been through an integral-equation formulation,the history of which is given by Miles (1980). Accuracyof the alternativeequations and numerical methods has usually been measured by theirability to describe the wave of greatest height. Someof the more notableresults for the maximum eight (H/h)max are 0.827 _ 0.008 (Yamada1957b), 0.827 (Lenau 1966), 0.8262 (Yamada t al. 1968), and the result


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46 SCHWARTZ & FENTONof 0.827 from Longuet-Higgins & Fenton (1974). All these results seemto support one another. However, Witting & Bergin, in an unpublishedwork mentioned by Witting (1975), obtained a value of 0.8332, preciselythe result obtained by Fox (1977) in an unpublished dissertation. Thisagreement s rather striking, despite the use of someextrapolation in bothcases. Finally, it should be noted that Witting (1975) has suggested thatthe method developed by Fenton (1972), on which the high-order seriesresults are based, is defective in that the assumed expansions areincomplete.2.2. Solution by Numerical MethodsRecent numerical solutions of the steady-wave problem are as accurate asthe series results, are often easier to implement omputationally, and gen-erally do not need the convergence improvement echniques of the series.They do suffer from the usual disadvantage of numerical solutions in thatthey reveal less about the nature of the problem and its solution. In thecase of high waves, however, so muchnumerical smoothing and extendingof the series solutions is necessary that they too suffer from this disadvan-tage. Numerical methods can be divided into two categories depending onwhether the authors choose to solve the problem in the physical (x,y) planeor the inverse (4~,ff) plane.2.2.1 ~yvERsE I~AYEMErHOOSn this case the problem is to be solved ona region which is known priori. This huge advantage is offset by the factthat as the sharp-crested wave s approached, the singularities near thecrest are stronger in the inverse plane. Also, the calculation of local quan-tities such as pressure and velocity as functions of position becomes aseparate problem to be solved subsequently if the inverse plane is used,Schwartz & Vanden-Broeck (1979) used an algorithm that is typicalof inverse plane methods to solve the capillary-gravity waveproblem oninfinitely-deep fluid and that was generalized to the finite-depth case inVanden-Broeck & Schwartz (1979). One wavelength of the flow wasmapped onto an annulus, the dynamic boundary condition becoming anonlinear differential equation for x and y on the unit circle. To satisfyLaplaces equation within the annulus, a Cauehy integral was written,valid on the unit circle and satisfying the bottom boundary conditionidentically. The equations were approximatedby finite-difference expres-sions to give a system of nonlinear algebraic equations which were solvedby Newtons method. The spacing of computational points at equal inter-vals of velocity potential was found to workwell for capillary-gravity waveswhere the points tended to be clustered around the narrow troughs wherevelocities were greatest. For pure gravity waves, on the other hand, the




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STRONGLY NONLINEAR WAVES 47sparse spacing occurred in the vicinity of the sharp crests and producedpoor results for waves near the highest. This was overcomeby a simpletransformation which could be used to cluster points near the crest. Theresults given for some wave speeds and energies are probably the mostaccurate to date.A remarkable result has recently been found by Chen & Saffman(1980a), providing convincing evidence that solutions for permanent grav-ity waves of finite amplitude are not unique when hey are sufficientlyhigh They formulated the deep-water steady-wave problem as a nonlinearintegro-differential equation, and approximated it by finite-differencemethods to give a system of nonlinear algebraic equations which weresolved by Newtons method. Having noted the existence of the "prema-ture" maxima f the various integral properties of periodic waves, and itspossible analogy, with the analytic structure 6f the capillary-gravity waveproblem, where multiple solutions were known o exist, they carefullymonitored the determinant of the Jacobian matrix used in the Newtonsmethod olution. Zeroes were found, identifying bifurcation points in thesolutions, and the separate branches were followed. The new families ofsolutions, found to occur for very steep waves, H/~, ~ 0.13, correspondedto a doubling and tripling of the fundamental wavelength. In the doublingcase, for example, the solution obtained was a train of steep waves n whichalternate wavesdiffered slightly in waveheight.2.2.2 PHYSICALLANEMETHODShese have an important role to play inpractical applications where, despite an approximate doubling of the num-ber of unknowns ecause the free surface is also unknown,t is considerablyeasier to solve the problem from the beginning in the physical plane. Suchsolutions include the recent work of Rottman& Olfe (1979) and Rienecker& Fenton (1981). In the first of these, a boundary-integral technique wasused to formulate an integro-differential equation, Newton teration beingused to find the vector of surface points. The methodworkedwell for steepgravity waves in that the now well-known speed maximum as found. Itis, however, not well suited to its nominal objective, the computation ofcapillary-gravity waves, since it fails whenn(x) becomesdouble-valued.Rienecker & Fenton (1981) used a method in which the stream functionis represented by a truncated Fourier series similar to that of Stokes in(2.1b). However,for a given wave he coefficients of the expansion arefound by numerical means, obviating the introduction of general powerseries with their finite radius of convergence and breakdown n the inap-propriate depth limit. The numerical method depends for its accuracy onthe ability of a Fourier series to describe the wave rain. This approachwas originated by Chappelear (1961) and Dean (1965), but the method


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48 SCHWARTZ & FENTONof Rienecker & Fenton is substantially different in that, for example, thesolution method s simpler because Newtons method s used directly, theonly approximation is in the truncation of the Fourier series, and themethod recognizes that the waves may propagate at speeds determined byquantities such as mass flux. In comparing esults for fluid velocity withexperimental results, good agreement was found. Very close agreementfor all waves ncluding the highest was found between he results for phasespeed and those reported by Cokelet (1977a) and Vanden-BroeckSchwartz (1979).2.3 The Highest and Almost-Highest Waves2.3.1 THEHIGHESTWAVE number of attempts have been made to solvethe problem of the sharp-crested wave of greatest height, usually incor-porating Stokes discovery that it has an included crest angle of 120. Fora sharp-crested wavewith its apex at z = x + iy -- 0, it follows from theBernoulli condition that the complex potential f = ~ + i~b varies likez3/2 locally. Thus the complex velocity df/dz varies like f~/3 near thecrest. If the computation s restricted to deep water, an expansion can beassumed (Michell 1893):

af _ c(1 - if)l/3 (1 + b, g- + b2 ~-z + ...), (2.2)dzwhere ~" = exp(--if/c). Michell substituted this into the dynamicboundarycondition and determined he first few coefficients of the expansion to givea result for the limiting steepness of (H/X)max = 0.142. The same methodwas used by Havelock (1919) whoobtained a limiting steepness of 0.1418.He also calculated solutions for waves short of the highest by displacingthe cube-root singularity above the crest. This technique is apparentlydefective, however, since Grant (1973) has shown hat only square-rootsingularities are admissible in all cases but the limiting one. He showedthat the singularity structure of the highest wave solution is muchmorecomplicated than had previously been assumed and that the sharp crestis not a regular singular point. The Stokes singularity is merely the firstterm in a local expansion about the crest in which irrational exponentsoccur.Meanwhile Yamada 1957a) had assumed a solution equivalent to thatof Michell, truncated after twelve terms, and obtained a limiting steepnessof 0.1412. In a later paper, Yamada& Shiotani (1968), the method wasextended to finite depth. McCowan1894) and Lenau (1966) usedparable techniques for the solitary wave. Schwartz (1972, 1974) analyzedhis high-order Stokes series and incorporated the inferred singularity struc-ture in a recast form of the series. The highest wave was graphicallyindistinguishable from Yamadasand had the same steepness.


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STRONGLY NONLINEAR WAVES 49Michells expansion (2.2) has been used again, but with a deli,ghtfuldifference, by Olfe & Rottman (1980). They observed that the nonlinearequation resulting from substitution of a one-term expansion into the dy-

namic boundary condition has multiple roots, one corresponding to Mich-elis solution but another real root almost eliminating the fundamentalFourier term so that the series is dominated by a higher harmonic--pre-cisely the behavior found by Chen& Saffman (1980a) for irregular gravitywaves. Subsequently, using (2.2) with up to 120 terms, Olfe & Rottmanexperimented with Newtons method and found other solutions in additionto Michells, corresponding to every second, third, or fourth crest beingsharp, the intermediate ones being lower and more rounded, as found byChen&Saffman for waves lower than the highest.Somesimple and accurate irrational approximations, not part of sys-tematic schemes, have been found by Longuet-Higgins (1973, 1974) forthe highest steady, standing, and solitary waves.

2.3.2 ALMOST-HIGHESTWAVESlocal expansion in the vicinity of the crestfor waves just short of the highest was devised by Longuet-Higgins & Fox(1977). They found a class of self-similar flows with a length scale of= q2~/2g, which have a smooth crest and whose free surface oscillatesabout the Stokes corner flow with a decaying amplitude like (~/r) 1/2, wherer is the distance from the Stokes corner. The oscillations cause the max-imum nclination of the surface to be greater than 30, namely 30.37, aresult confirmed from extrapolated numerical results of Sasaki & Mura-kami (1973) and Byatt-Smith & Longuet-Higgins (1976). In a secondpaper, Longuet-Higgins & Fox (1978) matched the local-crest solutiona form of Michells expansion for deep-water waves, valid far from thecrest. A small parameter proportional to qc was introduced, and a numberof asymptotic expressions found for the height of the waves, the phasespeed and other integral quantities. Unlike almost all other results for non-linear waves, these are valid in the limit of the highest wave. These expres-sions have the unusual feature that they showan infinitude of local maximaand minima as the highest wave is approached. The global maximumndthe first local minimum f speed and energy was found in the numericalsolution of Schwartz & Vanden-Broeck (1979) for deep water and forparticular case of finite depth D/k = 0.110. No doubt all finite-depthwave rains, and perhaps the solitary wave, show such behaviour.

Longuet-Higgins & Fox (1978) analytically continued their solutionacross the free surface in the crest neighborhood.They found a stagnationpoint above the crest corresponding to the square-root singularity foundby Grant (1973) and Schwartz (1972, 1974). The apparent transitionthe 2/3-power form for the sharp-crested wave was explained by Grantas the coalescence of several square-root singularities. Longuet-Higgins


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50 SCHWARTZ& FENTON(1979a) has subsequently developed a simpler approximation to the crestflow found by himself and Fox. This result, and other approximations,have been used to calculate fluid particle paths (Longuet-Higgins 1979b).

3. OTHER PERIODIC WAVESThere are two other important classes of spatially periodic surface wavesthat will be discussed here. The emphasis, as before, will be on stronglynonlinear effects.3.1 The Inclusion of Surface TensionIn this case the waves considered are also two-dimensional, periodic, andthe flow is steady, but the surface-tension term in (1.4) is retained. Fortwo-dimensional waves, l/R2 = 0, and 1/R~ = ~xx/(l + r/x2) 3/2 a highlynonlinear function of 7. The parameter K = 4~r2T/pg~2 is used to measurethe relative importance of surface tension and gravity. WhenK is large;corresponding to short wavelengths, surface tension is the dominant re-storing force. For longer wavesgravity is most significant; however, cap-illarity becomes ncreasingly important as the wavesteepens and the crestbecomes more sharply rounded.

The problem of pure capillary waves in deep water has been resolvedcompletely by Crapper (1957). In this most remarkable work, he obtainedan exact closed-form, solution for waves of arbitrary amplitude. Morerecently, closed-form solutions for finite depth were found by Kinnersley(1976), a result anticipated by Crapper. Unlike pure gravity waves, steepcapillary waves are characterized by deep troughs and broad fiat crests.The limiting wave occurs when he two sides of the trough meet, enclosinga pendant-shaped bubble. Steeper "waves" can be computed but they arephysically impossible since the free surface crosses itself. Recently Vanden-Broeck & Keller (1980) discovered a new family of capillary wavesallowing a nonzero pressure within the enclosed pendant.Wilton (1915) treated the combined capillary-gravity wave problemusing a Stokes expansion which he carried to fifth-order in amplitude,invoking Stokes hypothesis that the nth Fourier coefficient is nth order.When he parameter r = 1In for n = 2,3 ..... this expansion failsbecause certain series coefficients become nfinite. For the particular caseof ~ = 1/2, Wilton was able to find two solutions, each of which he carriedto third order, by revoking Stokes hypothesis and re-ordering the termsin the series.Recently the numerical methods described in Section 2.2.1 have been




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STRONGLY NONLINEAR WAVES 51applied to the problem (Schwartz & Vanden-Broeck 1979, Chen & Saff-man 1980b). Waves of maximum eight can be computed without diffi-culty for most values of K. In all cases for K > 0, the highest waves aretopologically limited, in that the surface encloses one or more bubbles,just as for pure capillary waves, and unlike pure gravity waves wherethe limiting wave is sharp-crested. The critical values of K, where theStokes-series solution fails, are indicative of multiple solutions. When

1 1-- < r < -, for moderate steepness the series yields a family of pro-n+l nfiles with approximately n inflection points or "dimples." The numericalmethods showed that each solution family can be analytically continuedoutside its "natural" domain. Thus there can be manydifferent waveformsfor a given wavelength.Certain of the multiple solutions arise via a finite-amplitude bifurcation from the regular wave train. As ~ -~ 0 the numberof solution families increases. The structure of this highly singular limit,where surface tension should be important only in the neighborhoodof thesharp crest, remains to be explored. A stability analysis for each of theseveral possible wave forms would be a useful contribution and mightresolve the non-uniqueness in the ripple regime. Experimental results arenot completely consistent, but Schooley (1960), for example, has publishedphotographs of several multi-dimpled profiles. Analytical work, dealingwith the genesis of multiple solutions, has been presented by Pierson &Fife (1961), Nayfeh (1970), and Chen & Saffman (1979).Hogan (1979) obtained a number of relations between integral quan-tities of the steady wave rain, which for Crappers pure capillary wavereduced to very simple expressions. He then (Hogan 1980) extended theseries technique of Schwartz (1972, 1974) to include surface tension forthe deep-water case, and obtained membersof the family of solutionscorresponding to the gravity wave. For K sufficiently small, each integralproperty was found to have a maximums a function of wave height, butthe effect of increasing r was to move he maximumloser to the highestwavepossible, showing how urface tension acts so as to makeexperimentalverification of the maximamoredifficult. Beyond certain value of r, themaximumwave height became limited by the wave geometry, and nomaxima n the integral quantities were found before the limiting bubblewas attained.Figure 3 shows wo steep waveprofiles for the case r = 1/2, taken fromSchwartz & Vanden-Broeck 1979). The profile labelled 1 belongs to thefamily of Crappers limiting wave and exhibits one trapped bubble perwavecycle; the other wave s from a different family which, while alsosymmetric, is limited by two bubbles per cycle.


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SCHWARTZ & FENTON3.2 Finite-Amplitude Standing WavesA time-dependent relative of the steady gravity wave is the two-dimen-sional time and space periodic standing wave. Physically the problem isthat of the periodic "sloshing" or "seiching" of water between wo verticalside walls. To first order it corresponds o the reflection of a periodic wavetrain by a vertical barrier or the interaction of two oppositely propagatingwave rains. Unlike the steady wave, no rigorous existence proof for finiteamplitudes has been obtained.In the spirit of Stokes, Rayleigh (1915) solved the problem by a third-order expansion in wave height. Penney & Price (1952) computeda fifth-order solution, and obtained a maximumteepness, for deep water, of 0.22.They also suggested that this highest wave has a 90 included angle atthe crest, but the premises in their argument have been questioned bysubsequent workers. Experiments of Taylor (1953) and Edge & Walters(1964) confirmed that the crest angle is close to .Recently Schwartz & Whitney (1977, 1981) have produced a 25th-order solution by a time-dependent conformal mapping method. Theyfound that Penney & Prices procedure is defective in that it producesnon-periodic secular time dependence f carried to higher order. This "res-onance" may be suppressed by exploiting a degree of arbitrariness incertain of the series coefficients. Very steep waves were found to possessseveral inflection points near the crest, reminiscent of those for steadywaves. The highest wave steepness was found to be about 0.208, with anincluded angle of about 90. The oscillating water surface is never flat andhas no nodes. The wave frequency, which in general decreases with in-creasing amplitude, appears to reach a minimum alue just short of thelimiting steepness.

.5

Figure 3 Two teep gravity-capillary wave profiles for x = 1/2.


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STRONGLY NONLINEAR WAVES 53Unlike the deep-water case, shallow-water standing waves show an in-crease in frequency with amplitude, the critical value of the D/k ratio atwhich the first nonlinear correction to the frequency changes sign being

0.17. This was obtained as part of the third-order series solution forfinite depth obtained independently by Chabert-dHi~res (1960) andTadjbakhsh & Keller (1960), and has been experimentally confirmedFultz (1962). An interesting feature of the series method s that certainvalues of D/k must be excluded so as to suppress secular terms, whichmay ndicate multiple solutions; however, the set of excluded depths be-comes nfinite as the order of the series solution becomes nfinite (Concus1964). Until this can be explained, the existence of standing-wave olutionsfor finite amplitude and depth remains in doubt.A numerical methodfor finite depth, employing truncated Fourier seriesto represent space and time variation has been used by Vanden-Broeck& Schwartz (1981). While inappropriate for very steep waves becausecomputerstorage limitations, the results confirmed the series results formoderate steepness and showed hat, for somedepths, the frequency is nota monotonic function of wave height.Miche (1944) established that deep-water standing waves have a second-order contribution to the pressure field that is unattenuated with depthand varies with twicethe frequency of the surface displacement. Longuet-Higgins (1.9,50) proposed this effect as a likely cause of observed micro-seismic activity and explained the phenomenon ery simply in terms ofvariation of the potential energy of the water mass. The phenomenon asbeen demonstrated experimentally by Cooper & Longuet-Higgins (1951)for low waves, but recent calculations by Schwartz (1980) indicate thatthe leading-order theory used by Longuet-Higgins may overpredict thepressures for very steep waves by as muchas 40%.

4. UNSTEADY WAVESThe standing-wave problem described in the previous section is unsteady,but it is periodic in time and space and all flow quantities can be expandedin Fourier series in both variables. In this section, problems, methodsandresults are described in which the motion is more generally unsteady andwhere the effects of nonlinearities maybring about irreversible changes.Unlike the problems described in Sections 2 and 3, for which someof themost powerful methods were devised last century, almost all of the methodsand results described here have appeared in the last decade.4.1 Stability of Steady WavesThere has been mucheffort expended on the stability of weakly nonlinearwaves, but very little on the stability of steep steady waves. The reason is


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54 SCHWARTZ & FENTONsimple enough--it was not until 1972 that accurate solutions for waves ofarbitrary height became available. Longuet-Higgins (1978a, 1978b) ex-amined he stability of steady waves n deep water for all heights, testingthe linear stability of the nonlinear wave solutions to small harmonicperturbations. For low waves t was found that they were neutrally stable,but above a certain steepness the waves became unstable to subharmonicperturbations (disturbance wavelength greater than the fundamental). Thegrowth rates obtained agreed quite well with experiment. For higher wavesthese modes ecame table again, but for wavesclose to the highest, a veryfast-growing superharmonic (small wavelength) instability was found.This was associated with the crest, and it seemed o owe its existence tothe fact that the wavewas steep enough that the first Fourier coefficientof the unperturbed wave had a maximum.The results of this linear stability analysis were checked by Longuet-Higgins & Cokelet (1978), whofollowed the evolution of the perturbationby numerically solving the full nonlinear equations, and confirmed theaccuracy of the linear analysis. It was found that the local superharmonicinstability could quickly lead to overturning at the crest, providing thebeginning of insight into the causes of that phenomenon.More emarkably,however, they found that the slow subharmonic nstability, if followed forlong enough, could also cause an overturning of the crest. Results fromthe linear theory have had some uccess in supporting a conjectc,"re of Lake& Yuen (1978) that in very steep wind waves the modulation frequencyof the wave amplitude maycorrespond to.the fastest-growing subharmonicinstability (Longuet-Higgins 1980d).4.2 Breaking WavesThe plunging or overturning wave is the most important and dramatic ofall breaker types and probably is the orisin of the spillin 8 breaker. It isone of the most difficult of all wavephenomenao analyze because of therapidity with which it can occur, the large amplitudes and accelerationsinvolved, and the contortions of the free surface. The origin of plungingin deep water remains unexplained; however, as described in Section 4.1,it has been shown hat the wave crest is vulnerable to perturbations andthat overturning of the crest is a common utcome even for waves thatare considerably lower than the highest.Someprogress has been made in describing the overturning. Severalmethods are described in Cokelet (1977b), but the only methodso far ableto describe the overturning wavefor most of its duration is that of Longuet-Higgins & Cokelet (1976, 1978), which will be outlined in Section 4.3.the original 1976 paper they studied an idealized problem of a steady wavein deep water to which an asymmetric pressure distribution was applied


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STRONGLY NONLINEAR WAVES 55

for a finite time, after which he wave urned over. Cokelet (1977b) studiedan initially sinusoidal wave rain with an excess of energy and also foundthat the crest steepened and overturned. Subsequently Longuet-Higgins& Cokelet (1978) followed the motion of steady waves with a small per-turbation and found the overturning as described in Section 4.1. A moredetailed picture of the dynamics and kinematics inside a wave as it ap-proaches breaking has been given by Peregrine, Cokelet & Mclver (1980).The wave crest before breaking has been found to travel faster than themaximum hase speed for a wave of that length, while the whole frontface of the wavehad an acceleration several times greater than gravityLonguet-Higgins (1980e) has studied simple analytical models of thedevelopment of the overturning, and for the evolution of the tip of aplunging wave (Longuet-Higgins 1980b). In a very different study (Lon-guet-Higgins 1980c) he calculated the angular momentumf steady wavesin deep water, and found that the relative persistence of wind-wave restsmaybe explained by the fact that for a force applied to the wave o havea minimumffect, it should act at about the wave-crest level; this is whatactually occurs for both wind forces and drag forces due to whitecapping.4.3 Solution of the Unsteady EquationsAnearly approach to the numerical solution of the full nonlinear equations(1.1-1.4) was through a marker-and-cell technique. The furthest devel-opment of this method is that of Chan & Street (1970) who used itstudy th.e reflection of a solitary waveby a wall. This problemwas alsoconsidered by Byatt-Smith (1971 ), whoobtained a second-order analyticalsolution from a pair of nonlinear integro-differential equations. Othermethods for the full equations include those of Brennen & Whitney 1970)(summarized in Brennen 1970), Whitney (1970), Multer (1973),Chan (1975). None, however, seems to have been adopted and exploitedfurther.In 1976 a method was introduced which seems to have considerablepotential. Longuet-Higgins & Cokelet (1976) studied the evolutionwaveson water of infinite depth. If the initial free surface and $ on it areknown, then a Cauchy-type integral equation may be formulated andapproximated using discrete computational points on the surface, to givesubsequently he velocity of each point normal o the surface as the solutionof a matrix equation. The tangential velocity can be found by numericaldifferentiation. The surface particles are allowed to move finite distancein a small time step, giving a new surface location. The new values of $maybe calculated from the dynamic ree-surface condition and the processrepeated, with a predictor-corrector method, for a large numberof timesteps. Longuet-Higgins& Cokelet found that a slow instability developed,


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56 SCHWARTZ & FENTONbut this could be countered by regular smoothing. This methodhas provedcapable of describing the evolution of high waves, including the overturn-ing of the crest (see Sections 4.1 and 4.2). Fenton & Mills (1976) showedhow the method may be applied to water of finite depth with arbitraryboundary geometry, but were unable to produce solutions.In two recent works, the free-surface boundary conditions have beengiven in different forms. Longuet-Higgins (1978a) expressed them in in-verse form with dependent variables x and y as functions of q~ andAnother technique (Longuet-Higgins 1980a) has been developed in whichthe usual dynamic boundary condition with p = 0 is used, but the kine-matic condition is written in terms of the material derivative of pressure,Dp/Dt = 0. This has been used (Longuet-Higgins 1980b, 1980e) to findexact solutions which mimic the overturning breaker.A Fourier method has been developed by Fenton & Rienecker (1980,but described in greater detail in a manuscript submitted for publication).The method s applicable to irrotational flows over arbitrary bed topog-raphy and makes use of Fourier approximation throughout to representhorizontal variation. The truncation of Fourier series for q5 and ~?, similarto (2.1), is the only approximation. The solution maybe advanced in timeby a leapfrog scheme, although it is necessary to solve a matrix equationat each time step. The method was found to be stable and accurate indescribing solitary wave nteractions, the use of Fourier series automatinga numberof numerical operations as well as facilitating deductions aboutstability or spectral growth. This makes t well suited to studies of finite-wave interactions, although it loses accuracy if at any stage a wavecomes sharp-crested, and it cannot describe overturning wavesbecause itdepends upon the surface elevations being a single-valued function of x.4.4 Solitary Wave InteractionsThe problem of two interacting solitary waves has come to be considereda classical problem of nonlinear waves because it is completely specifiedby 0nly two parameters (the incident wave heights), and because of thefundamental nature of the first-order equations which describe it, theexistence of an exact solution to one of these equations, and the fact thatthe solution shows that the waves emerge unchangedfrom the interaction.4.4.1 OVERTAKINGOLITARYAVESOfirst order in wave height and shal-lowness, the interaction of one solitary wave overtaking another is de-scribed by the Korteweg~le Vries (K~le V) equation, for which an exactsolution has been obtained (see Miles 1980). The only change after inter-action contained in the solution is that the high and fast wavehas receiveda finite phase shift forward and the low wave has been shifted backward.Weidman& Maxworthy (1978) conducted a number of experiments and




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STRONGLY NONLINEAR WAVES 57found generally good agreement between experiment and K~le V theory,but with some consistent differences. Nothing was reported on the wavesafter interaction; this has been studied numerically using the full nonlinearequations by Fenton & Rienecker in the report mentioned in Section 4.3.For the one interaction studied they found that the wavesafter interaction,contrary to expectation, had a larger height difference than the incidentwaves--the high wave had grown slightly higher, at the expense of thelower wave. After interaction the waves propagated almost without furtherchange, and there was no trace of a trailing wave rain.4.4.2. COLLIDINGOLITARYAVEShe other form of interaction, when thewaves ravel in opposite directions, has received muchattention, as de-scribed in Miles (1980). In this case the interaction, although brief,more nonlinear. Nevertheless, to second order in waveheight, theory pre-dicts that the waves emerge rom the interaction unchanged, with a finitebackward phase shift (Oikawa & Yajima 1973, Byatt-Smith 1971).recent work, a third-order calculation has been made by Su & Mirie(1980). The waves after interaction were shown o have the same heightas before, but the profiles are tilted backwards,and each sheds dispersivewaves. Fenton & Rienecker (1980, and the above-mentioned report) stud-ied a numberof collisions with the numerical methodoutlined in Section4.3. They found that both waveswere actually degraded by the interaction,but that this change was slightly greater than that for overtaking. Ac-cordingly, they recommend hat the adjectives "weak" and "strong" notbe used for the colliding and overtaking interactions, as the former hasmore effect on the waves than the latter. In contradiction of the third-order theory, they found strong evidence that the change of wave heightis actually of third order, and that the wavesafter interaction were trav-elling faster than before, also noted experimentally by Maxworthy1976).This change of speed has important implications for the measurementsofphase changes due to the interaction, for it means hat the change dependson the measuring location, which may explain some unusual features ofMaxworthys esults for the phase change. In view of the ambiguity of thespatial phase shift, Fenton & Rienecker recommend se of the temporalshift at the wall, and show that this is considerably underestimated bysecond- and third-order theories.Literature Cited

Brennen, C. 1970. Somenumerical solutionsof unsteady free surface wave problemsusing the Lagrangian description of theflow. Proc. 2nd Int. Conf. Numer. Meth.in Fluid Dynam., Berkeley, Sept., 1970,pp. 403-9

Brennen, C., Whitney, A. K. 1970. Un-steady, free surface flows; solutions em-ploying the Lagrangian description of themotion. 8th Syrup. Naval Hydrodynam.,Pasadena, Aug., 1970, pp. 117-45Byatt-Smith, J. G. B. 1971. An integralequation for unsteady surface waves anda comment on the Boussinesq equation.J. Fluid Mech. 49:625-33


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58 SCHWARTZ& FENTONByatt-Smith, J. G. B., Longuet-Higgins, M.S. 1976. On he speed and profile of steepsolitary waves. Proc. R. Soc. LondonSet.A 350:175-89Chabert-dHi~res, G. 1960. Etude du cla-potis. Houille BI. 15:153-63Chart, R. K. C. 1975. A generalized arbi-trary Lagrangian-Eulerian methodfor in-compressible flows with sharp interfaces.J. Comput. Phys. 17:311-31Chan, R. K. C., Street, R. L. 1970. A com-puter study of finite-amplitude waterwaves. J. Comput. Phys. 6:68-94Chappelear, J. E. 1961. Direct numericalcalculation of wave properties. J. Geo-phys. Res. 66:501-8Chappelear, J. E. 1962. Shallow-waterwaves. J. Geophys. Res. 67:4693-4704Chen, B., Saffman, P. G. 1979. Steady grav-ity-capillary waves on deep water I.Weaklynonlinear waves. Studies in Appl.Math. 60:183-210Chen, B., Saffman, P. G. 1980a. Numericalevidence for the existence of new types ofgravity waves of permanent form on deepwater. Studies in Appl. Math. 62:1-21Chen, B., Saffman, P. G. 1980b. Steadygravity-capillary waves on deep water 11.Numerical results for finite amplitude.Studies in Appl. Math. 62:95-111Cokelet, E. D. 1977a. Steep gravity waves

in water of arbitrary uniform depth.Philos. Trans. R. Soc. London Ser. d286:183-230Cokelet, E. D. 1977b. Breaking waves.Nature 267:769-74Concus, P. 1964. Standing capillary-gravitywaves of finite amplitude: Corrigendum.J. Fluid Mech. 19:264-66Cooper, R. I. B., Longuet-Higgins, M. S.1951. An experimental study of the pres-sure variations in standing water waves.Proc. R. Soc. London Set. A 206:424--35Crapper, G. D. 1957. An exact solution forprogressive capillary waves of arbitraryamplitude. J. Fluid Mech. 2:532-40De, S. C. 1955. Contributions to the theoryof Stokes waves. Proc. CambridgePhilos.Soc. 51:713-36Dean, R. G. 1965. Stream function repre-sentation of non-linear ocean waves. J.Geophys. Res. 70:4561-72Edge, R. D., Waiters, G. 1964. The periodof standing gravity waves of largestamplitude on water. J. Geophys. Res.69:1674-75Fenton, J. D. 1972. A ninth-order soluti6nfor the solitary wave. J. Fluid Mech.53:257-71Fenton, J. D. 1979. A high-order cnoidalwave theory. J. Fluid Mech. 94:129-61Fenton, J. D., Mills, D. A. 1976. Shoalingwaves: numerical solution of exact equa-tions. Lecture Notes in Physics 64: Waves

on water of variable depth, pp. 94-101.Berlin: SpringerFenton, J. D., Rienecker, M. M. 1980. Ac-curate numerical solutions for nonlinearwaves. Proc. 17th Int. Conf. CoastalEngrg., Sydney, March, 1980Fox, M. J. H. 1977. Nonlinear effects in sur-face gravity waves on water. PhD hesis.Cambridge Univ.Fultz, D. 1962. An experimental note on fi-nite-amplitude standing gravity waves.J. Fluid Mech. 13:193-212 (2 plates)

Grant, M. A. 1973. The singularity at thecrest of a finite amplitude progressiveStokes wave. J. Fluid Mech. 59:257-62Havelock, T. H. 1919. Periodic irrotationalwavesof finite height. Proc. R. Soc. Lon-don Ser. A 95:38-51Hogan, S. J. 1979. Someeffects of surfacetension on steep water waves. J. FluidMech. 91:167-80Hogan, S. J. 1980. Some ffects of surfacetension on steep water waves. Part 2.J. Fluid Mech. 96:417-45Keady, G., Norbury, J. 1978. On the exis-tence theory for irrotational water waves.Math. Proc. Cambridge Philos. Soc.83:137-57Kinnersley, W. 1976. Exact large amplitudecapillary waveson sheets of fluid. J. FluidMech. 77:229-41 ~Korteweg, D, J., de Vries, G. 1895. On hechange of form of long waves advancingin a rectangular canal and on a new typeof long stationary wave. Philos. Mag.(5)39:422-43Krasovskii, Yu. P. 1960. The theory ofsteady-state waves of finite amplitude.Dokl. Akad. Nauk SSSR 130:1237-40;also Zh. Vychisl. Mat. i Mat. Fiz. 1:836-55(1961); transl, in USSR Comp. Math.Math. Phys. 1:996-1018Laitone, E. V. 1960. The second approxi-mation to cnoidal and solitary waves.

J. Fluid Mech. 9:430-44Lake, B. M., Yucn, H. C. 1978. A new modelfor nonlinear wind waves. Part 1. Physicalmodeland experimental evidence. J. FluidMech. 88:33-62Lenau, C. W. 1966. The solitary wave ofmaximum amplitude. J. Fluid Mech.26:309-20Longuet-Higgins, M. S. 1950. A theory ofthe origin of microseisms. Philos. Trans.R. Soc. London Ser. A 243:1-35Longuet-Higgins, M. S. 1973. On the formof the highest progressive and standingwaves n deep water. Proc. R. Soc. LondonSer. A 331:445-56Longuet-Higgins, M. S. 1974. On the mass,momentum,energy and circulation of asolitary wave. Proc. R. Soc. LondonSer..4 337:1-13




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STRONGLY NONLINEAR WAVES 59Longuet-Higgins, M. S. 1975. Integral prop-erties of periodic gravity wavesof finiteamplitude. Proc. R. Soc. London Set. A342:157-74Longuet-Higgins, M. S. 1978a. The insta-bilities of gravity wavesof finite amplitudein deep water. I. Superharmonics. Proc.R. Soc. London Set. A 360:471-88Longuet-Higgins, M. S. t978b. The insta-bilities of gravity wavesof finite amplitudein deep water. II. Subharmonics.Proc. R.Soc. London Ser. A 360:489-505Longuet-Higgins, M. S. 1979a. The almost-highest wave: a simple approximation.J. Fluid Mech. 94:269-73Longuet-Higgins, M. S. 1979b, The trajec-tories of particles in steep symmetric rav-

ity waves. J. Fluid Mech. 94:497 517Longuet-Higgins, M. S. 1980a. A techniquefor time-dependent free surface flows.Proc. R. Soc. London Ser. ,4 371:441-51Longuet-Higgins, M. S. 1980b. On he form-ing of sharp corners at a free surface.Proc. R. Soc. London Ser. A 371:453-78Longuet-Higgins, M. S. 1980c. Spin and an-gular momentum n gravity waves. J.Fluid Mech. 97:1-25Longuet-Higgins, M. S. 1980d. Modulationof the amplitude of steep wind waves. J.Fluid Mech. 99:705-13Longuet-Higgins, M. S. 1980e. The unsolvedproblemof breaking waves. Proc. 17th Int.Conf. Coastal Engrg., Sydney, March,1980Longuet-Higgins, M. S., Cokelet. E. D.1976. The deformation of steep surfacewaves on water. I. A numerical method ofcomputation. Proc. g. Soc. London Set.A 350:1-26Longuet-Higgins, M. S., Cokelet, E. D.1978. The deformation of steep surfacewaves on water. II. Growth of normal-mode nstabilities, Proc. R. Soc. LondonSer. A 364:1-28Longuet-Higgins, M. S., Fenton, J. D. 1974.On the mass, momentum, nergy and cir-culation of a solitary wave. II. Proc. R.Soc. London Set. A 340:471-93Longuet-Higgins, M. S., Fox, M. J. H. 1977.Theoryof the almost-highest wave: the in-ner solution. J. Fluid Mech. 80:721-41Longuet-Higgins, M. S., Fox, M. J. H. 1978.Theory of the almost-highest wave. II.Matchingand analytic extension. J. FluidMech. 85:769-86Longuet-Higgins, M. S., Turner, J. S. 1974.An "entraining plume" modelof a spillingbreaker. J. Fluid Mech. 63:1-20Maxworthy, T. 1976. Experiments on colli-sions between solitary waves. J. FluidMech. 76:177-85McCowan, J. 1894. On the highest waveof permanent type. Philos. Mag. (5)38:351-58

Miche, R. 1944. Mouvementsondulatoiresde lamer en profondeur constante oud6croissante. Ann. Ponts Chausskes114:25 6t

Miehell, J. H. 1893. The highest waves inwater. Philos. Mag.(5) 36:430-37Miles, J. W. 1980. Solitary waves.Ann. Rev.Fluid Mech. 12:11-43Multer, R. H. 1973. Exact nonlinear modelof wave generator. J. Hydraul. Div. ASCE99:31-46Nayfeh, A. H. 1970. Triple and quintuple-dimpledwave profiles in deep water. Phys.Fluids 13:545-50Oikawa, M., Yajima, N. 1973. Interactionsof solitary waves--a perturbation ap-proach to nonlinear systems. J. Phys. Soc.Jpn. 34:1093 99Olfe, D. B., Rottman, J. W. 1980. Somenewhighest-wave solutions for deep-waterwaves of permanent form. J. Fluid Mech.100:801-10Penney, W. G., Price, A. T. 1952. Finiteperiodic stationary gravity waves n a per-fect fluid. Philos. Trans. R. Soc. LondonSet. ,4 244:254-84Peregrine, D. H., Cokelet, E. D., Mclver, P.1980. The fluid mechanics of waves ap-proaching breaking. Proc. 17th Conf.Coastal Engrg., Sydney, 1980

Pierson, W. J., Fife, P. 1961. Some onlinearproperties of long-crested periodic waveswith lengths near 2.44 centimeters. J.Geophys. Res. 66:163-79Rayleigh, Lord 19t5. Deep water waves,progressive or stationary, to the third or-der of approximation. Proc. R. Soc. Lon-don Ser. A 91:345-53Rienecker, M. M., Fenton, J. D. 1981. AFourier approximation method for steadywater waves. J. Fluid Mech. 104:119-37Rottman, J. W., Olfe, D. B. 1979. Numer-ical calculation of steady gravity-capillarywaves using an integro-differential for-mulation. J. Fluid Mech. 94:777-93Sasaki, T. K., Murakami, T. 1973. Irrota-tional, progressive surface gravity wavesnear the limiting height. J. Oceanogr. oc.Jpn. 29:94-105Schooley, A. H. 1960. Double, triple, andhigher-order dimples in the profiles ofwind-generated water waves in the capil-lary-gravity transition region. J. Geophys.Res. 65:4075-19Schwartz, L. W. 1972. ,4nalytic continua-tion of Stokes expansion for gravitywaves. PhD hesis. Stanford Univ.Schwartz, L. W. 1974. Computer extensionand analytic continuation of Stokes ex-pansion for gravity waves. J. Fluid Mech.62:553-78Schwartz, L. W. 1980. On he pressure fieldof nonlinear standing water waves. Rep.JIAA TR-33. Joint Inst. Aeronautics &


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60 SCHWARTZ& FENTONAcoustics, Stanford Univ.Schwartz, L. W., Vanden-Broeck, J. M.1979. Numerical solution of the exactequations for capillary-gravity waves. J.Fluid Mech. 95:119-39Schwartz, L. W., Whitney, A. K. 1977. Ahigh-order series solution for standingwater waves. Proc. 6th Australasian Hy-draul. & Fluid Mech. Con f, Adelaide,pp. 356-59Schwartz, L. W., Whitney, A. K. 1981. Asemi-analytic solution for nonlinear stand-ing waves in deep water. J. Fluid Mech.107:147-71Skjelbreia, L., Hendrickson, J. 1961. Fifthorder gravity wave heory. Proc. 7th Conf.Coastal Engrg. pp. 184-96Stokes, G. G. 1847. On he theory of oscil-latory waves. Trans. Cambridge Philos.Soc. 8:441-55, and in Mathematical &Physical Papers 1:197-229. Cambridge,Univ. PressStokes, G. G. 1880. Supplement to a paperon the theory of oscillatory waves. Math-ematical & Physical Papers 1:314-26.Cambridge, EnglandSu, C. H., Mirie, R. M. 1980. On headroncollisions between two solitary waves, d.Fluid Mech. 98:509-25Tadjbakhsh, I., Keller, J. B. 1960. Standingsurface wavesof finite amplitude. J. FluidMech. 8:442-51Taylor, G. L 1953. An experimental studyof standing waves. Proc. R. Soc. LondonSet. A 218:44-59Toland, J. F. 1978. On the existence of awave of greatest height and Stokes con-jecture. Proc. R. Soc. London Set. A363:469-85Ursell, F. 1953. The long-wave paradox inthe theory of gravity waves. Proc. Cam-bridge Philos. Soc, 49:685-94Vanden-Broeck, . M., Keller, J. B. 1980. Anew family of capillary waves. J. Fluid

Mech, 98:161-69Vanden-Broeck, J. M., Schwartz, L. W.1979. Numerical computation of steepgravity waves in shallow water, Phys.Fluids 22:1868-71Vanden-Broeck, J. M., Schwartz, L. W.1981. Numerical calculation of standingwaves n water of arbitrary uniform depth.Phys. Fluids. 24:812-15Wehausen, J. V., Laitone, E. V. 1960. Sur-face waves. In Encyclopaedia of Physics9:446-778. ed, S. Fltigge, Berlin: SpringerWeidman, P. D., Maxworthy, T. 1978. Ex-periments on strong interactions betweensolitary waves. J. Fluid Mech. 85:417-31Whitney, A. K. 1970. The numerical solu-tion of unsteady free surface flows by con-formal mapping. Proc. 2nd Int. Conf.Num. Meth. Fluid Dynam., Berkeley,Sept. 1970, pp. 458-62Wilton, J. R. 1914. On deep water waves.Philos. Mag. (6) 27:385-94Wilton, J. R. 1915. On ripples. Philos. Mag.(6) 29:688-700Witting, J. 1975. On the highest and othersolitary waves. SIAM J. Appl. Math.28:700-19Yamada, H. 1957a. Highest waves of per-manent type on the surface of deep water.Rep. Res. Inst. Appl. Mech. KyushuUniv.5, no. 18:37-52

Yamada, H. 1957b. On the highest solitarywave. Rep. Res. Inst. Appl. Mech. KyushuUniv. 5, no. 18:53-67Yamada,H., Kimura, G., Okabe, J. I. 1968.Precise determination of the solitary waveof extreme height on water of a uniformdepth. Rep. Res. lnst. Appl. Mech. Kyu-shu Univ. 16:15-32Yamada,H., Shiotani, T. 1968. On he high-est water waves of permanent type. Bull.Disaster Prev. Res. Inst. Kyoto Univ.18:1-22


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