Formation of threedimensional surface waves on deepwater using elliptic solutions of nonlinear Schrödinger equation
Abstract
A review of threedimensional waves on deepwater is presented. Three forms of three dimensionality, namely oblique, forced and spontaneous type, are identified. An alternative formulation for these threedimensional 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 energytransfer parameter, from wind to waves, make either the groups of wave to grow initially and eventually dissipate, or simply decay or grow in time.
pacs:
02.30.Hq, 02.30.Ik, 02.30.Gp1 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 KortewegdeVries 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.
For the twodimensional 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 twodimensional 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 threedimensional disturbances. Benjamin & Feir bf67a () examined the case of twodimensional disturbances to weakly nonlinear waves, by adopting standard perturbation methods, found that twodimensional 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 () deepwater 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 BenjaminFeir instability for threedimensional 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 BenjaminFeir 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 ().
LonguetHiggins LH6 (); LH7 () investigated, by combination of numerical analytical techniques, the stability of finite amplitude water waves to superharmonic and subharmonic twodimensional disturbances. His work was an extension of results obtained by Zakharov and Benjamin & Feir to finite amplitude waves for disturbances of shorter wavelength. LonguetHiggins 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 (). LonguetHiggins also discovered that when the wave is sufficiently steep, twodimensional 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 threedimensional, with wavelength of of original basic wave. Fig. 3 (right), on the other hand, depicts the threedimensional spilling breakers due to subharmonic instabilities in which the nature of the crescentshaped breakers can clearly be seen. As Su et al. suetal82 () commented, the threedimensionality of the spilling breakers is an intrinsic characteristic of the threedimensional subharmonic instabilities.
2 Threedimensional water waves
Following the advances mentioned above for twodimensional water waves, attention has been focused to threedimensional propagating water waves, where an even greater richness of phenomena has been encountered SaffYuen80 (). We remark that an important distinction in threedimensional water waves has to be made, namely between forced and spontaneous threedimensional waves. The forced threedimensional 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 nonparallel 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 twodimensional standing wave rossaj ().
Forced threedimensional waves, from a mathematical stand point, are essentially superharmonic modifications of the fundamental waves. The basic linear state is given by sajrev ()
(2.1) 
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 twodimensional propagating waves, and the case is a twodimensional standing wave. The steady propagating finite amplitude forced threedimensional waves of permanent form is a solution which may be expressed in the form
(2.2) 
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.
In contrast, the spontaneous threedimensional waves are completely different to that of forced threedimensional waves because these waves originate by instabilities or bifurcation of a uniform twodimensional 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 twodimensional wave of wavelength
(2.3) 
bifurcates at a critical height into a steadily propagating threedimensional wave of the form
(2.4) 
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
(2.5) 
with the corresponding velocity potential, satisfying equations (3.1) below, is given by
(2.6) 
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).
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 antisymmetrical modes of skew waves will lead to a branch of bifurcated solutions in which the surface can be expressed as
(2.7) 
with . These skew waves have the property that a frame of reference can be chosen in which the surface is stationary.
Fig. 6, on the other hand, depicts photographs of four skew patterns that propagate to the right side of the axis. These are threedimensional waveforms 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 twodimensional envelope modulations of BenjaminFeir 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 BenjaminFeir 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 threedimensional 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
(3.1) 
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 deepwater gravity waves may be expressed as
(3.2) 
where superscript * denotes complex conjugates and is the wave frequency which satisfies the dispersion relation
(3.3) 
In equation (3.2) is the time evolution of spectral components of a weakly nonlinear system for dominating fourwave interactions and its governing equation is given by
(3.4) 
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 deepwater gravity wave (note, in his original paper he also took into account the surface tension) may be reduced to
(3.5) 
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 twodimensional nonlinear Schrödinger equation
(3.6) 
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 onedimensional nonlinear schrödinger equation
(3.7) 
As was originally shown by Saffman & Yuen SaffYuPhysFl () for , equation (3.7) has the following soliton solution
(3.8) 
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.
Following a common practice, we adopt the dimensionless variables
and obtain the equation (3.6) in nondimensional form
(3.9) 
This equation has been used to study stabilities and bifurcation of threedimensional water waves. This twodimensional 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 threedimensional wave patterns. For example, substituting
(3.10) 
into equation (3.9) we obtain
(3.11) 
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 dislocationtype slip line solutions
(3.12) 
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
(4.1) 
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
(4.2) 
By multiplying by and integrating once we obtain the elliptic equation
(4.3) 
where is the integration constant that depends on the boundary conditions used, and
(4.4) 
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
(4.5) 
where is a quartic polynomial in can be expressed in terms of in Weierstrassâ elliptic functions via
(4.6) 
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
(4.7) 
Using a linear transformation , (4.7) can be written in normal form
(4.8) 
The germs of the Weierstrass equation (4.8) are
(4.9) 
and together with the modular discriminant
(4.10) 
are used to classify the solutions of (4.7). Also, are the three solutions to the cubic polynomial equation
(4.11) 
and are related to the two periods of the function for , and , seeSteg ().
Since the solution of equation (4.8) is
(4.12) 
then the solution of equation (4.3) may be expressed as
(4.13) 
5 Results
We shall now proceed to classify the solutions of equation (4.3) casebycase.
 Case (1).

We consider the simpler case with zero boundary conditions, that is when . In this case equations (4.9) become
(5.1) Also, in this case, (4.10) reduces to
(5.2) which implies that either has repeated root of multiplicity two () or three ().
 Case (1a).

(), we now let then , hence
(5.3) Also, from equations (4.9) we see that
(5.4) see Fig. 7 (left), and hence, in this case Steg (), solution of (4.8) is given by
(5.5)  Case (1b).

(), here we let then , hence
(5.8) Also, from equations (4.9) we have
(5.9) 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 .  Case (1c).

(): here we obtain , hence then and this yields . For then . Hence
(5.13) and
(5.14) 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
(5.15) where
(5.16) For real solutions of the amplitude , we require .
 Case (2a).

For then , and equation (5.15) becomes
(5.17)  Case (2a)(i).
 Case (2a)(ii).
Figure 9: Solutions when , for , , (blue); and , (red), .  Case (2b).

Let , then there are two distinct roots which are given by
(5.22) and we need to consider the following subcases.
 Case (2b)(i).
 Case (2b)(ii).

, with then , . Hence, equation (4.3) becomes
(5.26) Here the solution of equation (5.26) is
(5.27) Figure 11: Solutions when , for , , (blue); , , (red); and , , (green).  Case (2b)(iii).

, with then , . Hence, equation (4.3) becomes
(5.29) The solution of equation (5.29) is now given by
(5.30)
6 Wave groups generated by wind
Yuen & Lake YuLake () proposed that a nonlinear winddriven 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 deepwater wave trains without wind forcing. Using their laboratory experimental results they argued that the windwave interaction transports energy predominantly at a single speed corresponding to the group velocity, which is based on the dominant frequency corrected by windinduced 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
(6.1) 
where is the complex wave envelope, and are the carrierwave 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 threedimensional deep water waves induced by wind forcing is to adopt the following twodimensioanl version of nonlinear Schrödinger equation (cf. Sajhari ())
(6.2) 
where are slow variables
and is the group velocity. The coefficients are given by
On the righthand side of equation (6.2), and are the interfacial impedance, related to normal and tangential stresses at the airsea interface Miles57 (); is commonly known as the energytransfer parameter.
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
(6.3) 
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
(6.4) 
As in the analysis of Leblanc labanc () for twodimensional waves, we see that in the absence of wind forcing () the solution grows exponentially provided . This represents the threedimensional BenjaminFeir 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 threedimensional 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.
7 Concluding Remarks
In this paper we have reviewed several types of threedimensional waves on deepwater. 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 energytransfer 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 shorthand notation we have