Formation of three-dimensional surface waves on deep-water using elliptic solutions of nonlinear Schrödinger equation

Formation of three-dimensional surface waves on deep-water using elliptic solutions of nonlinear Schrödinger equation

Shahrdad G. Sajjadi, Stefan C. Mancas and Frederique Drullion Department of Mathematics, ERAU, Florida, U.S.A.
Trinity College, University of Cambridge, U.K.

A review of three-dimensional waves on deep-water is presented. Three forms of three dimensionality, namely oblique, forced and spontaneous type, are identified. An alternative formulation for these three-dimensional waves is given through cubic nonlinear Schrödinger equation. The periodic solutions of the cubic nonlinear Schrödinger equation are found using Weierstrass elliptic functions. It is shown that the classification of solutions depends on the boundary conditions, wavenumber and frequency. For certain parameters, Weierstrass functions are reduced to periodic, hyperbolic or Jacobi elliptic functions. It is demonstrated that some of these solutions do not have any physical significance. An analytical solution of cubic nonlinear Schrödinger equation with wind forcing is also obtained which results in how groups of waves are generated on the surface of deep water in the ocean. In this case the dependency on the energy-transfer parameter, from wind to waves, make either the groups of wave to grow initially and eventually dissipate, or simply decay or grow in time.

02.30.Hq, 02.30.Ik, 02.30.Gp

1 Introduction

Finite amplitude water waves have been studied since the pioneering work of G.G. Stokes in 1847 STK47 (). Over the years very important methods (such as the method of inverse scattering) have been developed for obtaining exact solutions to water wave problems, for instance, analytical solutions to Korteweg-deVries equation for shallow water waves and the cubic nonlinear Schrödinger equation which describes wave envelopes for slow modulation of weakly nonlinear water waves. The mathematical richness in the field of water waves appears surprising at first sight, since the governing differential equation is simply that of Laplace’s equation but, of course, since the boundary conditions are nonlinear this gives rise to wealth of problems. It is also remarkable that in a classical field such as that of water waves, new physical phenomena are still being discovered, both theoretically and experimentally, and many open questions still remain unanswered.

Figure 1: Photographs of progressive wave trains, of fundamental wavelength 7.2 ft, showing disintegration due to Benjamin-Feir instability; (Top) Close to wavemaker; (Bottom) 200 ft away from wavemaker. From Benjamin benj67 ().

For the two-dimensional periodic, irrotational surface waves of permanent form propagating under the influence of gravity on water of infinite or finite depth, such as Stokes waves, much has been discovered over the last few decades. For example, instability of steady finite amplitude waves to long wave two-dimensional disturbances was first postulated by Lighthill lighthill () by the use of Whitham’s variational principle, and an approximate Lagrangian. Zakharov zakharov () used Hamiltonian methods and showed that weakly nonlinear gravity waves are unstable for modulations longer than a critical wavelength depending upon their waveheight both analytically and numerically for two as well as three-dimensional disturbances. Benjamin & Feir bf67a () examined the case of two-dimensional disturbances to weakly nonlinear waves, by adopting standard perturbation methods, found that two-dimensional disturbances, of sufficiently long wavelength, are unstable for which experimental evidence was demonstrated by Benjamin & Feir bf67b () and Feir feir (). In a series of experiments reported by Benjamin & Feir bf67b () deep-water wave trains of relatively large amplitude were generated at one end of a tank, and were seen traveling many wavelengths. They observed that these wave trains developed conspicuous irregularities when traveled far enough, and they completely disintegrated. The severity of the instability can be seen in Fig. 1 where the fundamental wavelength is 7.2 ft at the water depth of 25 ft. Fig. 1 (a) shows the wave train close to the wavemaker, and Fig. 1 (b) shows the same wave train at a distance of 200 ft (28 wavelengths) further along the tank.

Extension to Benjamin-Feir instability for three-dimensional oblique waves on deep water was first studied by Ross & Sajjadi rossaj () using the same perturbation method as that of Benjamin & Feir bf67a (). They discovered a new instability criterion for these waves which reduces to the classical Benjamin-Feir instability when the angle between the waves is zero sajrev (). They also demonstrated that in the limiting case, where the waves are propagating at an angle , the instability represents standing waves such as those studied by Penny & Price penpri ().

Figure 2: (Top) Wave patterns in a basin showing two-dimensional subharmonic instabilities (, Hz). (Bottom) A typical three-dimensional spilling breakers in a basin. From Su et al. suetal82 ().

Longuet-Higgins LH6 (); LH7 () investigated, by combination of numerical analytical techniques, the stability of finite amplitude water waves to superharmonic and subharmonic two-dimensional disturbances. His work was an extension of results obtained by Zakharov and Benjamin & Feir to finite amplitude waves for disturbances of shorter wavelength. Longuet-Higgins analysis LH6 (); LH7 () confirmed Lighthill’s prediction that the long wave instability is no longer present when the waves become very steep (that is when the steepness ) and gave values for growth rates that agree well with the observations of Benjamin & Feir, as well as Benjamin benj67 () and Lake & Yuen lakeyuen (). Longuet-Higgins also discovered that when the wave is sufficiently steep, two-dimensional subharmonic disturbances of twice the wavelength of the undisturbed wave become unstable and have growth rates substantially larger than the type studied by Lighthill, Zakharov and Benjamin & Feir. Fig. 2 (left) shows such wave patterns up to 23 wavelengths. This photograph shows the first few waves contain small subharmonic disturbances which grow in size and height. These perturbations are three-dimensional, with wavelength of of original basic wave. Fig. 3 (right), on the other hand, depicts the three-dimensional spilling breakers due to subharmonic instabilities in which the nature of the crescent-shaped breakers can clearly be seen. As Su et al. suetal82 () commented, the three-dimensionality of the spilling breakers is an intrinsic characteristic of the three-dimensional subharmonic instabilities.

2 Three-dimensional water waves

Following the advances mentioned above for two-dimensional water waves, attention has been focused to three-dimensional propagating water waves, where an even greater richness of phenomena has been encountered SaffYuen80 (). We remark that an important distinction in three-dimensional water waves has to be made, namely between forced and spontaneous three-dimensional waves. The forced three-dimensional water waves is where the dependence upon the second horizontal dimension is forced by boundary or initial conditions. This is the case when one is concerned with studying the effect of nonlinearity on the interaction of two equal but non-parallel wave trains or the reflection of an obliquely incident wave train on a wall rossaj (), see Fig. 3 (bottom). As it has already been remarked, a special case when the waves are at an angle of to each other represents a two-dimensional standing wave rossaj ().

Forced three-dimensional waves, from a mathematical stand point, are essentially superharmonic modifications of the fundamental waves. The basic linear state is given by sajrev ()


with being the linear dispersion relation for waves with wavenumber where two wave trains make an angle with each other. The case is the limit of two-dimensional propagating waves, and the case is a two-dimensional standing wave. The steady propagating finite amplitude forced three-dimensional waves of permanent form is a solution which may be expressed in the form


where is the wave complex speed. These wave profiles are commonly referred to as short crested waves and were first studied by Fuchs fuchs () and Chappelear chapp (). Perhaps the most important property of these waves is that they exist in the infinitesimal limit, and consequently can be calculated formally by expansions in powers of wave height . However, there are serious concerns as to whether such expansions converge and thus the existence of these steady short crested waves is still not fully determined. For standing waves, that is when and with being finite, the first comprehensive study was made by Penney & Price penpri () and later on by Schwartz & Whitney schwhit (). The existence of these waves are also uncertain.

Figure 3: (Top) A perspective plot of a spontaneous three-dimensional wave for and . (Bottom) A perspective plot of a forced three-dimensional wave for .

In contrast, the spontaneous three-dimensional waves are completely different to that of forced three-dimensional waves because these waves originate by instabilities or bifurcation of a uniform two-dimensional wave train and, in general, they cannot have an arbitrary small amplitude. Mathematically speaking, they can be described or interpreted as subharmonic bifurcations LH7 (), where a two-dimensional wave of wavelength


bifurcates at a critical height into a steadily propagating three-dimensional wave of the form


where and are arbitrary real numbers with . However, if is an integer, these represent short crested waves. The critical wave height at which bifurcation occurs depends upon the values of and . The surface elevation, , given by (2.4), is periodic in the transverse direction and having wavelength . The longitudinal variation of these waves can be thought of having wavelength . Note incidentally, these waves are not exactly periodic unless is rational. In the particular case when these waves correspond to those whose wavelength is doubled in the direction of propagation.

The existence of such waves was first demonstrated by Saffman & Yuen SaffYuPRL () for general nonlinear dispersive systems where the nonlinearity is described by four wave interactions and their detailed calculations for water waves can be found in the paper Saffman & Yuen SaffYuen80 () using the Zakharov equation. Saffman & Yuen have pointed out that if the medium is isotropic, the bifurcation is degenerate and the solutions on the new branches can be either skew or symmetric. For skew branches and in this case the wave surface is not symmetric about the direction of propagation. On the other hand, for symmetric branches and the surface is symmetric about the direction of propagation as in short crested waves. In this case, the surface elevation may be described by


with the corresponding velocity potential, satisfying equations (3.1) below, is given by


where , and both and are Fourier coefficients, see Section 3. Solutions of this type have been calculated by Meiron et al. MeieEtAl () using the exact water wave equations (3.1). A typical example is shown in Fig. 3 (top).

Figure 4: Three-dimensional symmetric waves (, Hz) in a wide basin. From Su Su81 ().

A typical experimental example of symmetric waves in a wide basin generated by a wavemaker with frequency Hz, wave steepness and wavelength m is shown in Fig. 5. We remark that the anti-symmetrical modes of skew waves will lead to a branch of bifurcated solutions in which the surface can be expressed as


with . These skew waves have the property that a frame of reference can be chosen in which the surface is stationary.

Figure 5: Skew wave patterns for wave steepness . From Su suetal82 ().

Fig. 6, on the other hand, depicts photographs of four skew patterns that propagate to the right side of the -axis. These are three-dimensional wave-forms of the skew wave groups in which each pattern propagates at an angle with respect to the -direction of the initial Stokes wave train. It is to be noted that skew waves are generally observed in a narrow range of steepness of about . However, the predominantly two-dimensional envelope modulations of Benjamin-Feir instability are uniform over a wider range of steepness McLetal80 (); Su81 (). This observation implies that the bifurcation rate for the skew waves is much lower than the growth rate of Benjamin-Feir instability for and Su81 ().

The approach to the instability and shapes of permanent waves, described above, relies on solving exact formulations, given by equations (3.1). An alternative approach, valid for weak nonlinearity and slow modulations, is solving the cubic nonlinear Schrödinger equation exactly.

The aim of this paper is to classify various possible exact solutions of the elliptic ordinary differential equation that arises from nonlinear Schrödinger equation under the decomposition given by (4.1). To the best of our knowledge as yet no such classification has been reported in the open literature. We anticipate the classification presented in this paper will assist in adopting physical solutions to nonlinear Schrödinger equation, for three-dimensional waves, amongst various possible ones.

3 Formulation of the problem

The governing equations for an irrotational, inviscid, incompressible surface gravity waves on deep water are given by


where is the acceleration due to gravity, is the velocity potential, and describes the free surface elevation.

Following zakharov () the surface elevation , where , of weakly nonlinear deep-water gravity waves may be expressed as


where superscript * denotes complex conjugates and is the wave frequency which satisfies the dispersion relation


In equation (3.2) is the time evolution of spectral components of a weakly nonlinear system for dominating four-wave interactions and its governing equation is given by


where is the linear frequency and the real interaction coefficient , given by originally by Zakharov zakharov () and later, with some minor corrections, by Crawford et al. CrawEtAl (), characterize the properties of the system. For clarity we have listed these coefficients in the Appendix.

Zakharov zakharov () in his pioneering paper showed that equations (3.2)-(3.4) yield the surface elevation for deep-water gravity wave (note, in his original paper he also took into account the surface tension) may be reduced to


where is the complex envelope of the slowly modulated carrier wave propagating in the -direction. The real envelope is given by , and and represent, respectively, the modulation wave vector and frequency. Zakharov showed that satisfies the two-dimensional nonlinear Schrödinger equation


Note incidentally, the complex envelope function is related to the Fourier components through the equation

Zakharov zakharov () also showed that for oblique plane modulations, where , equation (3.6) may be reduced to one-dimensional nonlinear schrödinger equation


As was originally shown by Saffman & Yuen SaffYuPhysFl () for , equation (3.7) has the following soliton solution


whose profile is depicted in Fig. 6 for . We remark that for , there are no steady solutions to equation (3.7) that decay as . Note that Saffman & Yuen SaffYuPhysFl () state that no steady solution exist for and they plot their result, using equation (3.8), for the angle . This is an error because for the term will be negative and thus will no longer be a  sech profile but instead it will yield a periodic solution. Note, in this case , where represents the argument of  sech in equation (3.8). Fig. 6 (bottom) shows the plot of solution for values of arguement slightly below and above the range . This is because the solution of the equation equation (3.8) becomes unbounded for since at these values is infinite.

Figure 6: (Top) Free surface elevation for a plane oblique solitary wave making an angle of with the carrier wave vector. (Middle) A different prospective of three surface elevation as that of the top figure except for an angle . (Bottom) is the same as the middle profile for an angle , where the  sech-profile becomes a profile. Note at values , where is an integer (including zero), the -profile becomes infinite.

Following a common practice, we adopt the dimensionless variables

and obtain the equation (3.6) in non-dimensional form


This equation has been used to study stabilities and bifurcation of three-dimensional water waves. This two-dimensional form of nonlinear Schrödinger equation (3.9) has attracted a great deal of attention because it can be solved exactly by inverse scattering and other techniques, but we emphasize that this equation has a rather physically limited range of validity. In the range of its validity, however, it provides an easy way to produce all kinds of three-dimensional wave patterns. For example, substituting


into equation (3.9) we obtain


Now, depending on the values of the parameters and , and the magnitude of , many bounded solutions of this equation exist (see Roberts & Peregrine RobPer () and Sections 4 and 5 below). For example, there are stationary dislocation-type slip line solutions


which describes a surface in which the propagating wave has a phase jump of across the line (see the case (2a)(ii) below). The problem with the nonlinear Schrödinger equation is that it has an overabundance of solutions, and it is not easy to decide which, if any, are of physical significance. There is also the mathematical problem of determining if the nonlinear Schrödinger solutions are genuine in the sense that there are limits, as the wavenumber of the modulation and the amplitude go to zero, of solutions of the exact equations, and not the leading terms of expansions whose radius of convergence is zero.

4 Analytical solutions

To find analytical solutions to (3.9) we employ the polar form of ansatz


with phase , is the wave number, is the angular frequency, and is the real magnitude of the field. Substituting the ansatz into (3.9), yields


By multiplying by and integrating once we obtain the elliptic equation


where is the integration constant that depends on the boundary conditions used, and


is the dispersion relation. We will make use of this disperion relation in order to classify the solutions and will be written as therein where , see Fig. 7 which gives the graph of the dispersion relation.

It is well known Wei (); Whi () that the solutions of


where is a quartic polynomial in can be expressed in terms of in Weierstrass’ elliptic functions via


where is not necessarily a root of , and are elliptic invariants of . Due to the biquadratic nature of , an analysis of the nature of solutions of (4.6) is made, and we will show that solutions of (4.6) are reduced to solitary waves, periodic or Jacobi type of elliptic functions.

To see this we let in (4.3) which yields the Weierstrass equation


Using a linear transformation , (4.7) can be written in normal form


The germs of the Weierstrass equation (4.8) are


and together with the modular discriminant


are used to classify the solutions of (4.7). Also, are the three solutions to the cubic polynomial equation


and are related to the two periods of the function for , and , seeSteg ().

Since the solution of equation (4.8) is


then the solution of equation (4.3) may be expressed as


5 Results

We shall now proceed to classify the solutions of equation (4.3) case-by-case.

Case (1).

We consider the simpler case with zero boundary conditions, that is when . In this case equations (4.9) become


Also, in this case, (4.10) reduces to


which implies that either has repeated root of multiplicity two () or three ().

Case (1a).

(), we now let then , hence


Also, from equations (4.9) we see that


see Fig. 7 (left), and hence, in this case Steg (), solution of (4.8) is given by


Using (5.2), (5.5) in (4.13), we may express the solution of equation (4.3) as


For , equation (5.6) becomes


see Fig. 8 (blue).

Case (1b).

(), here we let then , hence


Also, from equations (4.9) we have


see Fig. 7 (center), thus in this case Steg () solution of equation (4.8) is given by

Figure 7: Domain of solutions (left), (center), (right), when .

Using (5.9), (5.10) in (4.13), we may write the solution of (4.3) in the form


For then , equation (5.11) becomes


see Fig. 8 (red).

Case (1c).

(): here we obtain , hence then and this yields . For then . Hence




see Fig. 8 (green).

Figure 8: Solutions when , for , (blue); , (red); and , (green).
Case (2).

Now we consider the general case with nonzero boundary conditions, i.e., , then equation (4.3) is factored as




For real solutions of the amplitude , we require .

Case (2a).

For then , and equation (5.15) becomes

Case (2a)(i).

, then


Now, for then gives , and equation (5.18) becomes


see Fig. 9 (blue).

Case (2a)(ii).



For and equation (5.20) becomes


see Fig. 9 (red).

Figure 9: Solutions when , for , , (blue); and , (red), .
Figure 10: Time series -solution of cubic nonlinear Schrödinger equation, for the same parameters as that in Fig 9.
Case (2b).

Let , then there are two distinct roots which are given by


and we need to consider the following sub-cases.

Case (2b)(i).

, with then , . Hence, equation (4.3) becomes


In this case the solution of equation (5.23) is


For then . If we let this gives , and we thus have


see Fig. 11 (blue).

Case (2b)(ii).

, with then , . Hence, equation (4.3) becomes


Here the solution of equation (5.26) is


For then . Letting gives , and we obtain


see Fig. 11 (red).

Figure 11: Solutions when , for , , (blue); , , (red); and , , (green).
Figure 12: Time series -solution of cubic nonlinear Schrödinger equation, for the same parameters as that in Fig 11.
Case (2b)(iii).

, with then , . Hence, equation (4.3) becomes


The solution of equation (5.29) is now given by


For then . Now if gives , and we get


see Fig. 11 (green).

6 Wave groups generated by wind

Yuen & Lake YuLake () proposed that a nonlinear wind-driven wave field may be characterized, to a first approximation, by a single nonlinear wave train. They based this proposal on experimental data obtained earlier for nonlinear deep-water wave trains without wind forcing. Using their laboratory experimental results they argued that the wind-wave interaction transports energy predominantly at a single speed corresponding to the group velocity, which is based on the dominant frequency corrected by wind-induced drift.

Based on their findings, they developed a model for the evolution of a nonlinear wave train in the absence of wind (which they argued is a satisfactory model) through the nonlinear Schrödinger equation


where is the complex wave envelope, and are the carrier-wave frequency and wavenumber, and is the inital steepness of the wave train. Similar to our analysis outlined in this paper, they related the free surface to the wave envelope by the expression

Yuen & Lake remarked that in a frame of reference moving with the group velocity the variation of with and are of order and , respectively, and is therefore slow compared to the oscillations of the dominant wave, characterized by and .

Based on recent studies, e.g. SHD1 (); SHD2 (), and our analysis here, it is very unlikely that Yuen & Lake’s model can adequately (if at all) represent nonlinear surface waves induced by wind, particularly in three dimensions.

A more consistent model, based on relatively a recent study by Leblanc labanc (), for three-dimensional deep water waves induced by wind forcing is to adopt the following two-dimensioanl version of nonlinear Schrödinger equation (cf. Sajhari ())


where are slow variables

and is the group velocity. The coefficients are given by

On the right-hand side of equation (6.2), and are the interfacial impedance, related to normal and tangential stresses at the air-sea interface Miles57 (); is commonly known as the energy-transfer parameter.

Figure 13: (Left) Formation of a surface wave with wind forcing, calculated from by Miles’ Miles57 () formula, obtained from -solution of cubic nonlinear Schrödinger equation given in case (2b)(i) above. (Right) Variation of with for .

A solution of equation (6.2) may be sought in the form

where , and . An exact homogeneous solution of (6.2) maybe obtained by writing labanc () where


where the suffix zero indicates the initial state. The solution corresponding to a spatially uniform plane Stokes wave is then obtained by letting and in this limit . On the other hand for an inhomogeneous solution is of the form provided .

Following Leblanc labanc (), we seek a perturbation to the Stokes solution by writing

and expressing and in the form

where satisfies the following ordinary differential equation


As in the analysis of Leblanc labanc () for two-dimensional waves, we see that in the absence of wind forcing () the solution grows exponentially provided . This represents the three-dimensional Benjamin-Feir instability and is in direct contradiction with Yuen & Lake YuLake (). However, when wind forcing is present (), it is easy to show the differential equation (6.4) admits two linearly independent solutions in terms of modified Bessel function of complex order where . Thus we see that the decaying () solution of equation (6.4) oscillates in time and eventually damps due to viscous dissipation. On the other hand, the nondissipative solution () either decays if or grows superexponentially labanc () when . In the latter case the asymptotic behaviour of equation (6.4) is given by labanc ()

This results seems to suggest the three-dimensional waves for which suffer superharmonic instability initially but they are then suppressed by subharmonic instability. However, as yet no experimental evidence has been reported to support or reject this conjecture.

Figure 14: (Top) Formation of wave groups which gradually grow in time. (Bottom) Formation of wave groups with initial growth and subsequent decay in time.

7 Concluding Remarks

In this paper we have reviewed several types of three-dimensional waves on deep-water. We have identified that three dimensionality of surface waves are essentially of three kinds namely those of oblique, forced and spontaneous type. Although perturbation techiques can be used to describe these waves, we have shown an alternative formulation, through cubic nonlinear Schrödinger (NLS) equation, which in some cases may be superior. It is shown that, when adopting the alternative approach the solution of NLS equation must be carefully classified since this equation has overabundance of solutions and it is often difficult to decide which, if any, have physcial significance. Here, we have obtained various periodic solutions of the NLS equation using Weierstrass elliptic functions. It is shown the classification of solutions depends on the boundary conditions, wavenumber and frequency. We have demonstrated that in certain cases the solutions are of solitary type, or simply periodic, while in other cases it is shown that solutions can be expressed in terms of Jacobi elliptic functions. For the formation of a group of waves, analytical solution of forced (or inhomogeneous) NLS equation, which arises from wind forcing, is found whose that clearly shows group of waves can form on the surface of deep water similar to those that are commonly observed in the ocean. In this case the depencency on the energy-transfer parameter, from wind to waves, is clearly a very influential factor, making either the groups of wave to grow initially and eventually dissipate or simply decay or grow in time.

Appendix A Second and third order interaction coefficients

Adopting the short-hand notation we have