unsteady water wave modulations: fully nonlinear solutions ...jwd/articles/99-uwwm.pdf ·...

Wave Motion 29 (1999) 341–361

Unsteady water wave modulations: fully nonlinear solutions andcomparison with the nonlinear Schrodinger equation

K.L. Henderson1,∗, D.H. Peregrine, J.W. Dold2

School of Mathematics, University of Bristol, Bristol BS8 1TW, UK

Received 10 November 1997; received in revised form 20 May 1998; accepted 8 September 1998

Abstract

The time evolution of a uniform wave train with a small modulation which grows is computed with a fully nonlinearirrotational flow solver. Many numerical runs have been performed varying the initial steepness of the wave train and thenumber of waves in the imposed modulation. It is observed that the energy becomes focussed into a short group of steepwaves which either contains a wave which becomes too steep and therefore breaks or otherwise having reached a maximummodulation then recedes until an almost regular wave train is recovered. This latter case typically occurs over a few hundredtime periods. We have also carried out some much longer computations, over several thousands of time periods in whichseveral steep wave events occur. Several features of these modulations are consistent with analytic solutions for modulationsusing weakly nonlinear theory, which leads to the nonlinear Schrodinger equation. The steeper events are shorter in bothspace and time than the lower events. Solutions of the nonlinear Schrodinger equation can be transformed from one steepnessto another by suitable scaling of the length and time variables. We use this scaling on the modulations and find excellentagreement particularly for waves that do not grow too steep. Hence the number of waves in the initial modulation becomesan almost redundant parameter and allows wider use of each computation. A potentially useful property of the nonlinearSchrodinger equation is that there are explicit solutions which correspond to the growth and decay of an isolated steep waveevent. We have also investigated how changing the phase of the initial modulation effects the first steep wave event that occurs.c©1999 Elsevier Science B.V. All rights reserved.

1. Introduction

The stability and evolution of a uniform wave train on deep water has been a subject of much interest over theyears. It is well known that a periodic uniform deep water wave train suffers an instability known as the Benjamin–Feir instability [1,2], which appears in the form of growing modulations. Unlike other instabilities it occurs for wavetrains of steepness well below the maximum steepness of steady waves at which crests approach 120◦ and so may beespecially relevant to the evolution of ocean waves. The wave groups that form are a useful model for the sea surface.

∗ Corresponding author. Fax: +44-117-976-3860; e-mail: [email protected] Present address: Department of Mathematical Sciences, University of the West of England, Bristol BS16 IQY, UK.2 Present address: Department of Mathematics, UMIST, Manchester M60 1QD, UK.

0165-2125/99/$ – see front matterc©1999 Elsevier Science B.V. All rights reserved.PII: S0165-2125(98)00045-6

342 K.L. Henderson et al. / Wave Motion 29 (1999) 341–361

From our deterministic computations full hydrodynamic details of such wave groups are available and hence may beused for further studies such as evaluating the forces on off-shore structures. The main limitation is that this study isrestricted to two-dimensional motions. However, we show that the slowly varying modulation approximation whichleads to the nonlinear Schrodinger equation is surprisingly effective. This effectiveness of the nonlinear Schrodingerequation can be of value for applications such as specifying wave groups in studies of the hydrodynamic forceson marine structures and also suggests that it may also be particularly useful for three-dimensional waves wherecomputation of fully accurate solutions is impractical for more than a few wave periods.

The analysis of Benjamin and Feir was for a linear perturbation to the weakly nonlinear solutions and predictedexponential growth over a finite band of perturbation frequencies. They conclude that the most unstable modescorrespond to a pair of side-band modes which emerge distinctly if disturbances in the initial state have an appropriatebandwidth. When the side-band amplitudes become approximately equal the wave profile corresponds to a uniformmodulation of the primary wave train. The qualitative nature of these theoretical results was confirmed by theirexperiments.

Further experiments have been performed to look at the evolution of nonlinear wave trains on deep water;Lake et al. [3], Melville [4] and Su et al. [5]. They obtained good quantitative agreement for the initial growth ofdisturbance as predicted by Benjamin and Feir [2]. However, rather than a disintegration of the wave train, theyobserved that further evolution is characterised by a spreading out of the energy over many spectral componentsfollowed by demodulation and return of the energy to the initial spectral components. Over a longer timescalethe modulation increases and decreases periodically and the wave train exhibits the Fermi-Pasta-Ulam type ofrecurrence phenomenon. An interesting result of these experiments is that for sufficiently steep initial waves therecurrence to a uniform wave train is accompanied by a decrease in frequency. Recently Trulsen and Dysthe [6] havepublished results on frequency down-shifting in three-dimensional wave trains in a deep basin. We have not observedfrequency down-shifting numerically for inviscid waves except for a restricted time as mentioned in Section 3.3.Further stability analyses have been carried out by Longuet-Higgins [7] and McLean [8,9], which have confirmedthe Benjamin–Feir instability for a range of initial wave train steepnesses.

There is another important instability of regular waves, the Tanaka [10,11] instability which occurs at the crestof steep waves. This is a relatively local response by the wave crest to disturbances, that affects only the steepestwaves, that isak > 0.43, H/L > 0.137. The evolution of this instability in the ideal situation may take one oftwo courses, either breaking or adjustment to a lower wave; but for practical purposes it can be considered to leadto wave breaking since in the ocean environment there are many disturbances available to trigger the instability inthis direction. Recent studies [12,13] clearly show that the Tanaka instability is not related to the wave train but isessentially related to the wave crest and hence is of equal significance for steep waves within wave groups. However,it is perhaps the Benjamin–Feir instability which is more relevant to the evolution of ocean waves as it occurs forwave trains of steepness much less than the maximum steepness of steady waves.

In considering the evolution of nonlinear deep water waves analytically, Lake et al. [3], Stiassnie and Kroszynski[14] and Lo and Mei [15] have used approximate equations. To third order in the initial wave steepness the slowlyvarying modulation amplitude satisfies the nonlinear Schrodinger equation [16–18]. The weakly nonlinear theorypredicts strong modulation and recurrence of uniform waves; however, it cannot describe the wave breaking thatoccurs at high initial wave steepness.

In this paper we use an efficient and accurate numerical code [19–21] to calculate the time evolution of smallamplitude modulations on two-dimensional periodic deep water waves. It solves the fully nonlinear inviscid irrota-tional flow in a spatially periodic domain. Laplace’s equation is solved using boundary integrals with the advantagethat only a point discretisation of the surface is required. Boundary conditions are used in a Lagrangian form andthe wave profile may be followed through to the initial overturning at breaking. We have completed many numericalruns varying the initial steepness and number of waves in the original wave train. The early evolution is characterisedby a growth in modulation which either contains a steep wave which subsequently breaks or provided the initialsteepness is not too great, demodulation occurs and a nearly uniform wave train recurs. We have also carried outsome much longer runs over which several peak growths in the modulation are recorded. Dold and Peregrine [22]

K.L. Henderson et al. / Wave Motion 29 (1999) 341–361 343

showed how modulations grow to fully modulated waves for a range of initial perturbations to a uniform wave train.This paper reports work which focusses on the evolution of such modulations with a brief report extending the Doldand Peregrine results. Banner and Tian [23,24] report results from the same Dold and Peregrine program and studythe details of waves in a group as they approach breaking. Here we are mainly concerned in the wave evolution overlong time periods and hence frequently keep the initial wave conditions such that the waves do not break.

Some of the features of the modulations are consistent with the weakly nonlinear theory, in that for higher initialsteepnesses the modulations are larger and occur more quickly than for lower initial steepnesses. By using appro-priate scalings for the nonlinear Schrodinger equation we compare several computations and find good agreementparticularly at lower steepnesses. This result is powerful in that, provided the initial steepness of the wave train isnot too high, it renders the number of waves in the original wave train an almost redundant parameter, and allowsmore use to be made of each computation.

2. Water wave modulation

In the next two sections we describe the numerical code and results of the evolution of small amplitude modulationson a uniform wave train.

2.1. Numerical code

The numerical code used for the calculations was developed by Dold and Peregrine [19,20] and a detaileddescription of the method can be found in [21]. The code is based on a Cauchy theorem boundary integral for theevaluation of multiple time derivatives of the surface motion, thus the entire motion of an inviscid, incompressible,irrotational fluid may be modelled using a point discretisation of the surface. Cauchy’s integral theorem is usediteratively to solve Laplace’s equation for successive time derivatives of the surface motion and time-steppingis performed using truncated Taylor series. Thus the code is computationally fast and efficient and a detailedinvestigation into the efficiency and accuracy of the code was performed by Dold [21]. Time-stepping in the codeis adjusted to ensure overall conservation of energy. Boundary conditions are chosen to make the spatial domainperiodic. This domain may contain many waves. In order to run the code an initial wave surface together with theappropriate velocity potential on that surface must be supplied. This gives enormous freedom in the choice of initialconditions. In this paper we describe the case which is easy to specify yet appears to lead to ‘natural’ waves: that isa small modulation of a uniform wave train. Given some suitable initial conditions the program can either track thesurface profile of the fluid for many thousands of linear time periods, with a slight smoothing imposed as describedin [21], or up to the initial overturning stage of wave breaking.

For a particular computation the main parameters of the system are the steepness of the initial wave train andthe number of these primary waves in one spatial period of the imposed initial modulation. The steepness of theinitial wave train is measured in the formak wherea andk are the dimensional amplitude and wave number ofthe waves. We adopt length and timescales which set the dimensionless wavelength of each initial wave to 2π andscale gravity to 1. Another way of thinking of initial steepness is the ratio of the overall height of the wave dividedby its wavelength, this quantityH/L is related to initial steepness by the formulaH/L = ak/π . Initial steepnessesconsidered in this work are mostly in the range 0.05 < ak < 0.12, i.e. 0.016 < H/L < 0.038. If a comparisonis made with actively generated wind wave fields the energy per unit areas are equivalent to values of the wavesteepness,ak, in the region 0.11–0.13 where breaking occurs for all but the shortest wave groups.

The initial conditions used throughout this work for the surface elevationη and corresponding velocity potentialφ on the surface are made up in the following three stages:

(i) Once the initial steepnessak is specified we compute an accurate steady wave form [25,26] on deep waterwhich to first order in wave steepness would have the form

η = ak cos(x)


for the surface elevation and

φ = ωak sin(x)

for the velocity potential. We choose to take 16 discretisation points per wave since tests indicate that thisaffords a high level of accuracy whilst keeping running times reasonably low [21].

(ii) The number of waves in the initial modulation is specified and the computational region is then taken to containn waves, where typically 4< n < 16.

(iii) To this wave train perturbations of the form

ηp = εak cos

(n + m

nx + θ

)+ εak cos

(n − m

nx + θ

)= 2εak cos(x + θ) cos

(mx

n

)

and

φp = εak

(n

n + m

)1/2

sin

(n + m

nx + θ

)+ εak

(n

n − m

)1/2

sin

(n − m

nx + θ

)

are added to the surface elevationη and the velocity potentialφ, respectively. This gives a modulation lengthof 2πn/m wherem was taken to be 1 or 2. We will use the quantityl = n/m to denote the number of wavesin the modulation. For the majority of runs the phase shiftθ was taken to be−π/4 as it has been indicatedto give the most rapid growth of the modulation [27]; the effect of changing the phase is discussed later inthe paper. The value ofε was typically taken to be 0.1, though in other studies a value of 0.05 has also beenconsidered [22].

The periodic nature of the computational domain means that the effect of the above initial conditions is to give aweak periodic modulation on the wave train. The number of waves in one modulation must be consistent with theperiodic nature of the spatial domain. Thus with one modulation in the domain up to 25 waves have been included,if two modulations are in that same domain then there are 12(1

2) waves in each modulation. For five modulationsin the same domain the modulation length is five waves. Of course, this latter case is much quicker to calculate byusing a spatial domain of five waves ab initio.

Once the initial conditions have been established the computation is followed in time until a specified end point isreached, in some cases illustrated in this paper this is over thousands of periods. The program stops as the numericalapproximations fail if the surface curvature becomes too high as in the tip of an overturning jet in wave breaking.For waves of high initial steepness breaking typically occurs within 100 periods. For longer runs, over severalhundred linear time periods, a slight level of smoothing is required to eliminate a very slowly developing steep waveinstability as described in [21], while maintaining a high level of accuracy in the simulations.

2.2. Results

For the first part of our results we have continued the work of Dold and Peregrine [22]. They, using the initialconditions described in Section 2.1, numerically followed the evolution of a modulated wave train for up to 200linear time periods. For the number of waves in one modulation in the range 2(1

2) ≤ l ≤ 10 they indicated thevalues of the initial steepness,ak, for which: (i) waves grew to breaking, (ii) the modulation grows and uniformwaves recur, and (iii) there is no growth of modulation. In this paper we have extended the range of the parameterspace in investigating the first evolution of the modulation. We report results from many computations with dif-fering values of the two main parameters of the system, the initial steepnessak and the number of waves in themodulationl.

The computation was continued until either a nearly uniform wave train was recovered or until wave breakingoccurred and the computation broke down. A summary of these results is illustrated in Fig. 1, where open circlesindicate the modulation grows and uniform waves recur, the crosses indicate the waves grew to breaking and thefilled circles indicate no growth of modulation was observed. Some of the smaller modulations of 2(1

2) and three


Fig. 1. Summary of wave modulation computations. Crosses indicate waves grew to breaking, open circles indicate a growth in modulation witha recurrence of uniform waves, filled circles indicate no growth in modulation was observed.

waves are too short to be unstable. Details of an instability of just two waves and its evolution can be found in[7,28].

Dold and Peregrine [22] considered the particular case of an initial five-wave modulation with steepnessak = 0.11and followed its evolution over 200 linear time periods. This is a representative example of cases where the smallinitial modulation evolved into a steep, short wave group and back to a recurrence of a near uniform wave train.They also reported that: (i) for times of a few periods, linear theory gives a fair indication of wave behaviour, evenfor the steepest non-breaking waves, but the wave shape is nonlinear and (ii) for long time intervals, the weaklynonlinear NLS equation gives a qualitatively correct evolution, with quantitative discrepancies which are consistentwith Dysthe’s [17] higher order approximation.

The results of a computation with an initial steepness ofak = 0.092, H/L = 0.0293 and nine waves in themodulation is illustrated in Fig. 2. The surface and modulation profiles are plotted every 10 linear time periodsagainstx − cgt , wherecg = 1

2 is the non-dimensional linear group velocity of waves on deep water. The dottedlines represent the wave surface and the shaded regions enclosed between the solid lines represent the modulationenvelope. In plotting the surface and modulation out at intervals of 2n periods the effects of the linear phase velocityand the linear group velocity lead to a first-order superposition of the wave and modulation positions and one canclearly see the evolution of the modulation. The main features of the evolution of the modulation illustrated in Fig.2 are discussed in the following paragraph.

It can be seen that the modulation grows developing a short group of steep waves of about 112 wavelengths, a steep

wave event (SWE), whilst either side of this group the amplitude of the remaining waves is small relative to initialvalues. In fact the modulation has near-zero minima either side of this short group. It can also be seen that over thetime range of maximum growth the modulation travels slightly to the right indicating that the group is moving fasterthan the linear group velocity. Around the time of maximum modulation a wave crest is lost, so atT = 110 periods


Fig. 2. Surface and modulation profiles plotted againstx − cgt for a nine wave modulation, initial steepnessak = 0.092 at times of0, 10, 20, . . . , 160. The dotted lines represent the wave surface and the shaded regions enclosed between the solid lines represent the modulationenvelope. Vertical exaggeration is five times.

there are eight waves in one spatial period of the computation; however, as the surface evolves the initial number ofnine waves is recovered and thus no permanent frequency down-shifting is observed. A fairly regular wave train isrecovered atT = 160 periods with a change in phase relative to the initial profile. These features can also be seenin Fig. 3 in which the surface height for every second, linear time period is contoured in the(x − cgt, t) plane. Theloss and subsequent recovery of one wave crest can clearly be seen.

The more detailed structure of the steep wave event is displayed in Fig. 4 in which we plot the wave profile againstx from T/2π =108–113 at intervals of1T/2π = 1/8. This picture clearly shows the effects of group velocity,that is that individual waves travel with speedc whilst the wave group, for waves on deep water, travels with speedcg = c/2. Even though during the SWE the waves are highly nonlinear the linear relationship for the velocity is stilla good approximation over several time periods. A consequence of the group velocity being half the phase velocityis that if the surface elevation in the ocean is measured by a wave gauge at a fixed point in space then twice as manywaves will be observed as that occur at a fixed point in time. To illustrate this a time-trace is plotted in Fig. 5 at thespatial positionx = 5.77 over a time period ofT/2π =108–113. It can be seen that the group has an approximatelength of 3, twice as long as the groups shown in the spatial domain, as in Fig. 4. To further illustrate the groupmoving through the field we plot contours of the surface height in space–time in Fig. 6. This figure clearly showsthe enhanced phase velocity of the steeper waves.

The evolution of a modulation that evolves to breaking follows a similar pattern, until a few periods beforebreaking, to the evolution structure of a modulation that grows and then decays. This is illustrated in Fig. 7 whichis a similar picture to Fig. 2 but with an initial steepness ofak = 0.093, i.e.H/L = 0.0296. It can be seen that forthe first 100 time periods the surface and modulation profile is very similar to the case ofak = 0.092. However, in


Fig. 3. Contour plot of the surface height,y, plotted againstx − cgt at every second time period for a nine wave modulation and initial steepnessak = 0.092.

Fig. 4. Surface profiles betweenT = 108 andT = 113 periods at time stepsδT = 1/8. Initial steepnessak = 0.092, H/L = 0.0293,modulation of nine waves. Vertical exaggeration is three times.


Fig. 5. Time trace of surface height,y, atx = 5.77 for a nine wave modulation, initial steepnessak = 0.092.

Fig. 8(a) we plot the wave surface for an initial steepness of 0.093 atT = 109 and it can be seen that an extreme eventhas occurred, the same wave is drawn to scale in Fig. 8(b). This steep wave has a sharp crest and the computationbreaks down shortly after.

Some much longer numerical runs have been computed for a range of initial steepnesses and number of wavesin the modulation. The results of one such computation are illustrated in Figs. 9 and 10. Fig. 9 shows a time series,at frequent time intervals, of the maximum surface height for the case of an initial 14-wave modulation with initialsteepness of 0.06. The graph shows a thick line since there is a relatively rapid modulation, of period approximately2, in the maximum height. The higher edge of the ‘shaded area’ occurs when a wave crest is at the maximum ofthe modulation. The lower part of the ‘shaded area’ shows the case when a wave trough is at the maximum of amodulation, if there is only one modulation. The range between trough and crest lines also indicates the spatiallength of each modulation. In this example there are four particularly steep wave events in a time of 3500 waveperiods, and some lesser events. The ‘thickness’ of the line depicting this time series can be almost completelyreduced by taking time series every two periods. That is just the time it takes for one crest to replace another in themodulation. Now, with much smoother data a better picture of the modulation can be obtained.

The whole wave profile is stored every two periods, then an estimate is made of the modulation that gives thewaves’ envelope. This modulation is then contoured in the modified space–time,(x − 1

2t, t), plane to give Fig. 10.Fig. 10 uses the same computation as Fig. 9, but now we can see the location of each SWE event and see that insome instances two smaller ones are occurring at the same time.

An idea of the sparsity of the maximum SWEs can immediately be obtained from Fig. 10. For example supposethe initial waves have a 10 s period so that their wavelength is approximately 160 m. The spatial domain is then2.2 km and the duration of Fig. 10 is almost 10 h, yet only four very steep SWEs have occurred.


Fig. 6. Contour plot of surface height,y, over the time range of maximum modulation for a nine wave modulation, initial steepnessak = 0.092.

Fig. 7. Surface and modulation profiles plotted againstx − cgt for a nine wave modulation, initial steepnessak = 0.093 at times of0, 10, 20, . . . , 100. The dotted lines represent the wave surface and the shaded regions enclosed between the solid lines represent the modulationenvelope. Vertical exaggeration is five times.


Fig. 8. (a) Wave profile atT = 109 periods of a nine wave modulation, initial steepnessak = 0.093. Vertical exaggeration is five times. (b)Same wave profile as that in (a) drawn to scale.

Fig. 9. Time series of the maximum waveheight within the computational domain, for initial steepnessak = 0.06, H/L = 0.019, initialmodulation= 14 wavelengths.

3. Nonlinear Schrodinger equation

Several features of these modulations are consistent with analytic solutions of the nonlinear Schrodinger equation(NLS) which is the simplest approximate equation for weakly nonlinear modulations, e.g. see [29]. The steeperevents are shorter in both space and time than the lower events.

3.1. Introduction

The velocity potential of a two-dimensional modulated water wave train on deep water can be described, to firstorder by the expression

φ(x, y, t) = A(x, t)ekyei(kx−ωt) + complex conjugate, (1)

whereA(x, t) is the ‘slowly varying’ complex modulation amplitude,ω the frequency andk is the wave number ofthe modulated wave, or ‘carrier wave’ which has an amplitude ofs = 2k|A|/ω. Then to first order in the length of


Fig. 10. Modulation amplitude contours for the same run as Fig. 9 (white indicates> 0.17, black< 0.02).

the modulation, i.e. for very long modulations the modulation amplitudeA satisfies the equation

At + cgAx = 0, (2)

wherecg = ω′(k) is the linear group velocity of the modulated wave. That is long modulations of a uniform wavetrain travel at the group velocity.

The next approximation, which is correct to third order in small wave steepness and long modulation length, takesinto account the weakly nonlinear dispersive effects and the gradient of the modulation. This gives the equation forthe complex modulation amplitude as

2iω(At + cgAx) − c2gAxx = 4k4|A|2A. (3)

This equation forA can be transformed to the self-focussing nonlinear Schrodinger equation

iqT + qXX + 2|q|2q = 0 (4)


Fig. 11. Scaled modulation histories. (a) Initial steepness:ak = 0.07, H/L = 0.0223, modulation of 12 waves; (b) initial steepness:ak = 0.056, H/L = 0.0178, modulation of 15 waves.

by the transformation

T = 1

2ωt, X = kx − 1

2ωt, q =

√2k2A∗

ω, (5)

where∗ denotes complex conjugate [16–18].Solutions of the NLS equation can be transformed from one steepness to another by the scaling

q ′ = q/q0, x′ = q0X, t ′ = q20T . (6)

As a result we make the hypothesis that for waves that do not grow too steep (a criterion to be made specificby considering computational results), the modulation growth should scale as solutions of the NLS equation andthe actual number of waves in the initial modulation becomes an almost redundant parameter. To evaluate thishypothesis several computations have been run with appropriate matching of the number of waves with the initialwave steepness so that if this hypothesis held then the same modulation pattern would emerge. Fig. 11 shows tworesults similar to Fig. 9 since the length, time and steepness scales have been suitably adjusted. Although there are


Fig. 12. As Fig. 11. Initial steepnessak = 0.084, H/L = 0.0267, modulation of 10 waves.

some discrepancies the major features of the wave evolution are closely comparable to a scaled time of almost 1000,i.e. over 2000 periods for Fig. 11(a) and over 3000 periods for Fig. 11(b). The NLS equation is formulated includingterms of orders2 in wave steepness, thus the timescale for the evolution of the modulation could be expected tovary as 1/s2, e.g. 200/π wave periods in the case ofs = 0.1. What we find from our computations is that there isgood qualitative agreement over a much longer time scale than is expected, particularly when the initial waves arenot too steep. In Fig. 12 the length, time and steepness scales have been suitably adjusted to be comparable withthe plots in Fig. 11. In this case the initial steepness wasak = 0.084 and the results are qualitatively similar to ascaled time of 400.

Inspection of these results and those of other similar computations gives an indication that this approach holdsfor waves which grow no steeper thanak = 0.3, H/L = 0.095. Once waves become steeper than this criticalsteepness the modulation regime appears to take on a different character, see the latter portion of Fig. 12 and seethe irregular behaviour of Chereskin and Mollo-Christensen [30]. This more irregular behaviour is partly becausethere are often more than one or two wave groups within the calculations. However, we note that these solutionsare like the ‘homoclinic orbit’ solution of Ablowitz and Herbst [31] (see Section 3.3). They show how although theNLS equation is integrable, and hence does not have chaotic solutions, the perturbations introduced by numericalapproximations may lead to the numerical version of this solution being chaotic. It would therefore not be surprisingif the stronger nonlinear effects that arise for the steeper waves may also bring the possibility of chaotic solutions.Even these very long numerical computations are far too short to give any definite indication about chaos.

3.2. Changing the phase

The numerical results presented so far have all been for an initial phase of the modulation relative to the car-rier wave of−45◦. We now investigate how changing the initial phase effects the growth of the modulation.Many numerical runs have been carried out with an initial modulation of 12 waves, initial steepnessak = 0.07,modulation amplitudeε = 0.1, varying the value of the initial phase,θ , of the modulation. Some of the resultsare presented in Fig. 13 in which we plot a time series of the maximum surface height for values of the phaseshift θ = −300◦, −285◦, −270◦, . . . , 15◦, 30◦, 45◦. If the initial number of waves in the initial modulation iseven then the results for phaseθ and phaseθ + 180◦ will be identical, so computations were only made forθ = 0◦, −15◦, −30◦, . . . , −150◦, −165◦. Fig. 13 shows that changing the phase has a dramatic effect on the timeand size of the SWEs that occur over the first 1000 linear time periods. Note we are considering differences where5◦ corresponds to a shift of the modulation by only1

72 of a wavelength of the underlying carrier wave. The value


Fig. 13. Time series of the maximum surface height for a 12-wave modulation of initial steepnessak = 0.07, for values of the phase shiftθ = −300◦, −285◦, −270◦, . . . , 15◦, 30◦, 45◦.

of the phase for which the most rapid growth occurs is aroundθ = −30◦ with the first SWE occurring at about200 linear time periods and the second SWE occurring at about 550 linear time periods. However, as we moveaway from this initial phase the position in time of both the first and second SWEs is later. At the initial phase ofθ = −120◦ it can be seen that the first SWE occurs close to 350 linear time periods and the second at 900 lineartime periods. In addition we can see that the two SWEs are much smaller than for other values of the phase.

The initial conditions used in the numerical code are equivalent to an initial condition of

q∗ = h√

2[1 + 2εe−iθ cos(X/l)]


Table 1The maximum surface elevation during the first SWE,ymax and the time, measured in linear time periods,Tymax at which it occurs for an initial12-wave modulation with steepnessak = 0.07 and phase shift,θ .

θ(◦) Tymax ymax

−60 193.7 0.2356−45 189.7 0.2378−35 179.6 0.2401−30 179.6 0.2407−27.5 179.6 0.2407−25 179.6 0.2407−22.5 179.6 0.2406−20 179.6 0.2404−15 179.6 0.23960 189.8 0.2369

in the NLS equation, wherel = n/m represents the number of waves in the modulation. We can investigate theearly development of this solution by performing a linear analysis on the solution

q(X, T ) = bei2b2T (1 + ε(X, T )),

whereb = h√

2 andq(X, 0) = q, assumingε � 1. Following the methods of Ablowitz and Herbst [31] we findthat the early development of the modulation is governed by the equation

ε(X, T ) = 2εbl

�{(l−2 + i�) sin(β − θ)e�T + (l−2 − i�) sin(β + θ)e−�T } cos(X/l), (7)

where� =√

(4l2b2 − 1)/ l2 and tan(β) =√

(4l2b2 − 1). Thus there is an initial condition for which there is noinitial growth in the modulation; the expression multiplying the e�t term must be equal to 0 which implies thatθ = β. Recalling thatb = h

√2, whereh = ak is the initial steepness, we have that

β = tan−1{√

8l2(ak)2 − 1}

and for an initial 12-wave modulation of steepnessak = 0.07 this givesβ = −114.9◦.Further time series of the maximum surface elevation are plotted in Fig. 14 for an initial 12-wave modulation with

steepnessak = 0.07 with values of the phase close toβ. We can see that for values ofθ very close toβ the SWEoccurs at around 550 linear time periods which is much later than for other values ofθ , see Fig. 13. In addition theSWE is much larger forθ = −114.9◦, −115◦, −115.1◦, −115.2◦ than forθ = −113◦, −114◦, −116◦, −117◦. Thelargest SWE occurs forθ = −115.1◦ rather than forθ = β, probably due to nonlinear effects becoming important.It can also be seen that the profiles are symmetrical aboutθ = −115◦, in that there is qualitative agreement betweenthe times series forθ = −114.5◦ andθ = −115.5◦, θ = −116◦ andθ = −114◦, θ = −117◦ andθ = −113◦, asis indicated from the linear analysis (7).

Returning to the linear solution (7), it could be argued that in order for the first SWE to occur most rapidly, theexpression multiplying the e�t should be maximised. This gives a value for the phase angle ofθ1 = β + 90◦. Thusfor an initial modulation containing 12 waves of steepnessak = 0.07, θ1 = −24.9◦. This result is confirmed inTable 1. Here we present the maximum surface height attained during the first SWE,ymax and the time at which itis attained, measured in linear time periods,Tymax for an initial 12-wave modulation with steepnessak = 0.07 forphase anglesθ = −60◦, −45◦, −35◦, −30◦, −27.5◦, −25◦, −22.5◦, −20◦, −15◦, 0◦. It can be seen that the SWEoccurs most quickly for values ofθ between−15◦ and−35◦ and that the maximum height attained is largest in themiddle of this range. As the phase moves out of this range the SWE occurs later and is smaller in amplitude.


Fig. 14. Time series of the maximum surface height for a 12-wave modulation of initial steepnessak = 0.07, for values of the phase shiftθ = −113◦, −114◦, −114.5◦, −114.9◦, −115◦, −115.1◦, −115.2◦, −115.5◦, −116◦, −117◦.

3.3. Soliton solutions

One of the potentially useful analogies with the NLS equation is that there are explicit solutions, in particularthe Ma soliton [32], which corresponds to the periodic growth and decay of an isolated SWE. Peregrine [18] tookthe analytic solution for the Ma soliton and performed a double Taylor Series expansion about the amplitude peak(which occurs atx′ = t ′ = 0) to arrive at the soliton

q ′ = e2it ′(

1 − 4(1 + 4it ′)1 + 4x′2 + 16t ′2

). (8)

Formally this is the nonlinear sum of a finite plane wave of unit scaled amplitude and a soliton of zero amplitude.The maximum amplitude, of three times that of the uniform plane wave, is isolated in both space and time. Thesimplicity of the analytical expression in Eq. (8) makes it the most convenient approximation to one of the SWE.This soliton grows to three times the initial uniform wave amplitude. In considering the results of our numerical


Fig. 15. (a)|A| for the isolated Ma soliton in solid line scaled such that the amplitude is the same as|A| estimated from an initial 14-wavemodulation,ak = 0.06 atT = 247.6, shown in dashed line. The velocity potential for the isolated Ma soliton is the dotted line and the velocitypotential from the water wave modulation in long dashes. (b)T = 249.1 (c) T = 252 (d)T = 255 where the isolated Ma soliton is takento travel with the group velocity calculated to third order and the wave itself travels with the phase velocity calculated to third order. Verticalexaggeration is 35 times.

computations in the region for which the NLS scalings gives good agreement, we observe that three times theoriginal wave amplitude appears to be a good measure of the steepest SWEs, see Figs. 9 and 11. Considering inparticular the results in Fig. 9 of a 14-wave modulation of initial steepnessak = 0.06, we find that the largest SWEoccurs atT = 247.6 × 2π (taken to be the time at which the maximum wave elevation occurs). An estimate of themodulation amplitude|A| at this time was made and the isolated Ma soliton fitted around these data, using the onefree parameter to match the maximum height of|A| at the amplitude peak. These results are shown in Fig. 15(a),where the amplitude of the isolated Ma soliton is drawn in a solid line and the water wave modulation amplitude as


a dashed line. Also illustrated in this figure is the velocity potentialφ from the water wave modulation computation,drawn as a long dashed line and the velocity potential of the isolated Ma soliton, drawn as a dotted line, calculatedfrom Eq. (1) and centred to get the best fit in the SWE. It can be seen that there is excellent agreement between theisolated Ma soliton and the water wave modulation particularly close to the amplitude peak; however, the isolatedMa soliton has two zeros either side of the peak whilst the modulation has two non-zero minima. Further from theamplitude peak the isolated Ma soliton has a higher value than the modulation amplitude. Turning to the comparisonbetween the two velocity potentials, there is excellent agreement in the phase and amplitude close to the amplitudepeak and reasonable agreement beyond the modulation amplitude minima. However, there is a difference betweenthe two around the amplitude minima. In this region the isolated Ma soliton has two crests where the water wavemodulation solution has only one, representing a temporary down-shift in frequency.

In Figs. 15(b)–(d) the same quantities as in Fig. 15(a) are plotted at times ofT = 249.1 × 2π, T = 252× 2π

andT = 255× 2π , respectively. The isolated Ma soliton is calculated from Eq. (8) with appropriate values oft ′and is taken to travel at the group velocity taken to be half the phase velocity calculated to third order. The velocitypotential is again calculated from Eq. (1) with the wave itself travelling at the phase velocityc =

√1 + (ak)2.

Again there is good qualitative agreement between the isolated Ma soliton and the modulation at the three times;however, it can be seen that the modulation has moved slightly ahead of the isolated Ma soliton atT = 255× 2π .In comparing the velocity potentials over the three cases although there is good agreement away from the amplitudepeak, it can be seen that the numerical steep water wave travels faster than the wave enclosed by the isolated Masoliton and this effect is most noticeable in Fig. 15(d) at the furthest time from the amplitude peak. However, thisgood agreement over several time periods is encouraging and it has led us to look at other analytical solutions ofthe NLS equation satisfying periodic boundary conditions.

Ablowitz and Herbst [31] have developed an analytical solution of the NLS equation which is periodic in space.It can be written as

u(X, T ) = aoexp(2ia2oT )

1 + 2 cos(X/k)exp(�T + 2iφ + γ ) + A12exp(2�T + 4iφ + 2γ )

1 + 2 cos(X/k)exp(�T + γ ) + A12exp(2�T + 2γ ), (9)

which they call a ‘homoclinic orbit’, wherek−1 = 2ao sinφ, � = ±k−1√

4a20 − k−2, A12 = sec2φ. ChoosingT

such thatε0 = exp(�T + γ ) is small and second order terms inε0 are neglected, the solution (9) can be linearisedto give an initial condition of

u(X, 0) = ao

(1 + εo

a0kcos(X/k)e−i(β+π)

).

Thus comparison with Eq. (7) shows that the homoclinic orbit identified by Ablowitz and Herbst is only appropriateto water waves whenθ = β + π or, if there is an even number of waves in the modulation, whenθ = β. For aninitial modulation of 12 waves, steepnessak = 0.07, phase angleθ = β = −114.9◦, we find that the largest SWEoccurs atT = 560.95× 2π . An estimate of the modulation amplitude|A| at this time was made and the Ablowitzand Herbst solution calculated such that it has maximum amplitude (this occurs when exp(�T +γ ) = cosφ). Theseresults are shown in Fig. 16, where the amplitude of the Ablowitz and Herbst solution is drawn in a solid line andthe water wave modulation amplitude as a dashed line. Also illustrated in this figure is the velocity potentialφ fromthe water wave modulation computation, drawn as a long dashed line and the velocity potential of the Ablowitz andHerbst solution, drawn as a dotted line, calculated from Eq. (1) and centred to get the best fit in the SWE. It canbe seen that the Ablowitz and Herbst solution is noticeably lower than the water wave estimate of the modulationamplitude and consequently there is not such good agreement with the velocity potential profiles. We see that overa long time period corresponding to the growth of a SWE some discrepancies develop. The qualitative picture isstill good, and as illustrated with the isolated Ma soliton if the match is made with the maximum height then formoderate time intervals agreement is usefully close.


Fig. 16.|A| for the Ablowitz and Herbst solution in solid line,|A| estimated from an initial 12-wave modulation,ak = 0.07, θ = −114.9◦ atT = 560.95, shown in dashed line. The velocity potential for the Ablowitz and Herbst solution is the dotted line and the velocity potential fromthe water wave modulation in long dashes. Vertical exaggeration is 35 times.

3.4. Higher-order approximations for modulations

The NLS equation is derived by using approximations for small wave steepnesses and long modulation scales.Further approximations can be made and the next order equations were obtained by Dysthe [17]. A new featurearises at this approximation: the waves are influenced by small currents that are set up by the modulation gradients.They have a velocity potential8 which appears in the Dysthe equation:

2iω(At + cgAx) − ω2

4k2Axx − 4k4|A|2A = ik3(6A2A∗

x − 6AA∗Ax − 2|A|2Ax)

+ iω2

8k3Axxx + 2ωkA(8x − i8z).

The first group of terms on the right-hand side, in brackets, are nonlinear terms. It can be seen, by consideringA = Reiθ , that the first two terms have a steepness dependent effect on phase but no effect on amplitude. Thethird term is easily seen to give an increase of1

2cgk2s2 to the group velocity, recalling that the amplitude of the

modulation is given bys = 2k|A|/ω. This corresponds to half of the Stokes correction to the phase velocity andis clearly seen in the computations. The third-derivative term is the next order improvement in approximation tothe linear dispersion, and the final term gives the interaction with the induced current. This current comes fromsolving Laplace’s equation for8 with the boundary condition8z = 2k2(|A|2)x/ω at the mean free surface. Thereis a corresponding deviation of the mean surface equal to(1/g)8t , but this is too small to be significant at thisapproximation. Note that we are now including terms as small ass3 in wave steepness. These might be expected tobe important only on a timescale of 1/s3, e.g.(2000/π)T in the case ofs = 0.1.

Solutions of the Dysthe equations have been studied by Lo and Mei [15] and show differences from the NLSequation. The strongest is the group velocity increase already noted; however, the group velocity/phase velocityratio appears to be unchanged as is also found from computations of the steepest SWE. Lo and Mei [15] show asteepening of the modulation at the front of wave groups; however, this does not appear in the evolving wave groupsthat are characteristic of our accurate computations. This is probably because these wave groups evolve too rapidly,on the shorter timescale of the NLS equation. This evolution on the faster NLS timescale could be because thesegroups are not sufficiently close to the steadily propagating solutions for envelope soliton solutions, hence givinglittle opportunity for the nonlinear contribution to group velocity to have a significant effect on the modulationshape.

4. Conclusions

We have carried out many computations following the nonlinear evolution of a modulated wave train varyingthe initial wave steepness and the number of waves in the modulation. We have extended the work of Dold and


Peregrine [22] in characterising the growth of the modulation over these two parameter ranges. They fall into twomain categories; either the group contains a sharp-crested wave which breaks or otherwise having formed a shortgroup of steep waves at the peak modulation, demodulation occurs and we find a recurrence to a near uniformwave train results. During the stages of peak modulation the linear group velocity is still a good approximation overseveral periods.

We have also completed some much longer runs, over thousands of time periods. The initial wave train soondevelops a strong modulation and then shows moderately regular variation with times of strong modulation. Incomparing numerical runs of different initial steepnesses there are similarities to analytic solutions of the weaklynonlinear theory, in that modulations of steeper initial wave trains are both shorter in space and time. A scalingin space and time allows the initial steepness to be scaled out of the nonlinear Schrodinger equation. We compareappropriately scaled numerical computations and find excellent agreement over a much longer timescale than isexpected particularly for lower initial steepnesses. This is a significant result and allows greater use to be madeof each computation, reducing the parameter space to include only the steepness of the initial wave train. Anotherimportant result of the favourable comparison with weakly nonlinear theory is that the nonlinear Schrodingerequation has analytic solutions, in particular the isolated Ma soliton which closely corresponds to a short group ofsteep waves.

The longer numerical computations that have been performed provide a realistic model of the ocean surface.Several of the SWEs that form could be used as ‘design’ waves in modelling structures that need to withstand thefull range of wave and current conditions that are expected for its lifetime in that position. Although the computationalmethod uses a boundary integral technique, for any chosen time, a further simple computation gives any subsurfacefluid properties that are required.

As experiments and theory show, the long time evolutions shown here are vulnerable to three-dimensional insta-bilities, especially for the steeper waves. Nonetheless, the results and comparisons with the nonlinear Schrodingersolutions and scaling can be of value for practical application and give encouragement for the use of the nonlinearSchrodinger equation in three-dimensional modelling, particularly for cases where unsteady evolution means thatthe waves are not close to the steadily propagating solutions which give an opportunity for higher-order effects tobe significant. The major higher-order effect showing in our computations is the enhanced group velocity which isdescribed by the Dysthe equation. However, although the groups travel faster than the linear group velocity, theydo not survive long enough for it to have much effect on their shape.

Acknowledgements

Support from the Marine Technology Directorate of EPSRC and from industrial contributors to the programmeon Uncertainties in Loads of Offshore Structures under contract GR/J23624 is gratefully acknowledged.

Dr. K.L. Henderson gratefully acknowledges the support of the Nuffield Foundation.

References

[1] T.B. Benjamin, Instability of periodic wavetrains in nonlinear dispersive systems, Proc. Roy. Soc. A 299 (1967) 59–75.[2] T.B. Benjamin, J.E. Feir, The disintegration of wave trains on deep water. Part 1. Theory, J. Fluid Mech. 27 (1967) 417–430.[3] B.M. Lake, H.C. Yuen, H. Rungaldier, W.E. Ferguson, Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous

wave train, J. Fluid Mech. 83 (1977) 49–74.[4] W.K. Melville, The instability and breaking of deep-water waves, J. Fluid Mech. 115 (1982) 165–185.[5] M.-Y. Su, M. Bergin, P. Marler, R. Myrick, Experiments on nonlinear instabilities and evolution of steep gravity-wave trains, J. Fluid Mech.

124 (1982) 45–72.[6] K. Trulsen, K.B. Dysthe, Frequency downshift in three-dimensional wave trains in a deep basin, J. Fluid Mech. 352 (1997) 359–373.[7] M.S. Longuet-Higgins, The instabilities of gravity waves of finite amplitude in deep water. II Subharmonics, Proc. Roy. Soc. London A

360 (1978) 489–505.[8] J.W. McLean, Instabilities of finite-amplitude water waves, J. Fluid Mech. 114 (1982) 315–330.


[9] J.W. McLean, Instabilities of finite-amplitude waves on water of finite depth, J. Fluid Mech. 114 (1982) 331–341.[10] M. Tanaka, The stability of steep gravity waves, J. Phys. Soc. Japan 52 (1983) 3047–3055.[11] M. Tanaka, The stability of steep gravity waves, II, J. Fluid Mech. 156 (1986) 281–289.[12] M.S. Longuet-Higgins, D.G. Dommermuth, Crest instabilities of gravity waves. Part 3. Nonlinear development and breaking, J. Fluid Mech.

33 (1996) 33–50.[13] M.S. Longuet-Higgins, M. Tanaka, On the crest instabilities of steep surface waves, J. Fluid Mech. 336 (1997) 51–68.[14] M. Stiassnie, U.I. Kroszynski, Long-time evolution of an unstable water-wave train, J. Fluid Mech. 16 (1982) 207–225.[15] E. Lo, C.C. Mei, A numerical study of water-wave modulation based on a higer-order nonlinear Schrodinger equation, J. Fluid Mech. 150

(1985) 395–416.[16] A. Davey, K. Stewartson, On three-dimensional packets of surface waves, Proc. Roy. Soc. London A 338 (1974) 101–110.[17] K.B. Dysthe, Note on a modification to the nonlinear Schrodinger equation for application to deep water waves, Proc. Roy. Soc. London

A 369 (1979) 105–114.[18] D.H. Peregrine, Water waves, Nonlinear Schrodinger equations and their solutions, J. Austral. Math. Soc. Ser. B 25 (1983) 16–43.[19] J.W. Dold, D.H. Peregrine, Steep unsteady waves: an efficient computational scheme, Proceedings of the 19th International Conference on

Coastal Engineering, A.S.C.E., Houston vol. 1 (1984) pp. 955–967.[20] J.W. Dold, D.H. Peregrine, An efficient boundary-integral method for steep unsteady water waves, In: K.W. Morton, M.J. Baines (Eds.),

Numerical Methods for Fluid Dynamics II, Clarendon Press, Oxford, 1985, pp. 671–679.[21] J.W. Dold, An efficient surface-integral algorithm applied to unsteady gravity waves, J. Comp. Phys. 103 (1992) 90–115.[22] J.W. Dold, D.H. Peregrine, Water-wave modulation, Proceedings of the 20th International Conference on Coastal Engineering, Taipei, vol.

1 (1986) 163–175.[23] M.L. Banner, X. Tian, Energy and momentum growth rates in breaking water waves, Phys. Rev. Lett. 77 (1996) 2953–2956.[24] M.L. Banner, X. Tian, On the determination of the onset of breaking for modulating surface gravity water waves, J. Fluid Mech. 367 (1998)

107–137.[25] M. Tanaka, The stability of solitary waves, Phys. of Fluids 29 (1986) 650–655.[26] A.F. Teles Da Silva, D.H. Peregrine, Steep steady surface waves on water of finite-depth with constant vorticity, J. Fluid Mech. 195 (1988)

281–302.[27] M. Stiassnie, Private communication.[28] M.S. Longuet-Higgins, E.D. Cokelet, The deformation of steep surface waves on water. II Growth of normal mode instabilities, Proc. Roy.

Soc. London A 364 (1978) 1–28.[29] C.C. Mei, The Applied Dynamics of Ocean Surface Waves, Wiley, New York, 1983.[30] T.K. Chereskin, E. Mollo-Christensen, Modulational development of nonlinear gravity-wave groups, J. Fluid Mech. 151 (1985) 337–365.[31] M.J. Ablowitz, B.M. Herbst, On homoclinic structure and numerically induced chaos for the nonlinear Schrodinger equation, SIAM J.

Appl. Math. 50 (1990) 339–351.[32] Y.-C. Ma, The perturbed plane-wave solution of the cubic nonlinear Schrodinger equation, Stud. Appl. Math. 60 (1979) 43–58.

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