scispace - formally typeset
Search or ask a question
Journal ArticleDOI

On three-dimensional packets of surface waves

TL;DR: In this article, the authors used the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wavepacket of wavenumber k on water of finite depth.
Abstract: In this note we use the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wave-packet of wavenumber k on water of finite depth. The equations are used to study the stability of the uniform Stokes wavetrain to small disturbances whose length scale is large compared with 2π/ k . The stability criterion obtained is identical with that derived by Hayes under the more restrictive requirement that the disturbances are oblique plane waves in which the amplitude variation is much smaller than the phase variation.
Citations
More filters
Journal ArticleDOI
TL;DR: In this article, a number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated, and several analytical solutions of NLS equations are presented, with discussion of their implications for describing the propagation of water waves.
Abstract: Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

1,318 citations

MonographDOI
24 Jun 2002
TL;DR: Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory have been explored in this article, where the authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged.
Abstract: This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

835 citations

Journal ArticleDOI
TL;DR: In this paper, a significant improvement can be achieved by taking the perturbation analysis one step further O (∊ 4 ) by introducing the mean flow response to non-uniformities in the radiation stress caused by modulation of a finite amplitude wave.
Abstract: The ordinary nonlinear Schrodinger equation for deep water waves, found by perturbation analysis to O (∊ 3 ) in the wave-steepness ∊ ═ ka , is shown to compare rather unfavourably with the exact calculations of Longuet-Higgins (1978 b ) for ∊ > 0.15, say. We show that a significant improvement can be achieved by taking the perturbation analysis one step further O (∊ 4 ). The dominant new effect introduced to order ∊ 4 is the mean flow response to non-uniformities in the radiation stress caused by modulation of a finite amplitude wave.

641 citations


Cites background from "On three-dimensional packets of sur..."

  • ...Later, Davey & Stewartson (1974) extended this to waves on finite depth....

    [...]

Journal ArticleDOI
TL;DR: When the wind blows at modest speeds over natural bodies of water, numerous streaks or slicks nearly parallel to the wind direction may appear on the surface as mentioned in this paper, and under favorable conditions it is readily apparent to the casual observer.
Abstract: When the wind blows at modest speeds over natural bodies of water, numerous streaks or slicks nearly parallel to the wind direction may appear on the surface. This form of surface streakiness is commonplace, and under favorable conditions it is readily apparent to the casual observer. The streaks result from the collection of floating sub­ stances-seaweed, foam from breaking waves, marine organisms, or organic films-into long narrow bands. Flotsam makes the bands visible directly, and compressed films make them visible by the damping of capillary waves, thereby giving the bands a smoother appearance. Naturalists and seafarers often note color variations of the sea due to minute marine organisms. Bainbridge (1957) cited many old descriptions of long narrow "bands," "streaks," or "lanes" including several by Darwin in 1839 during the voyage of the Beagle. James Thomson (1862) described observations of streaks made jointly with his brother, Lord Kelvin, in a paper that also indicated increased abundances of marine life below the streaks. The first connection between the wind and streak directions, among the authors cited by Bainbridge, was made by Collingwood (1868): "if a moderate breeze were blowing and the sea

495 citations


Cites background from "On three-dimensional packets of sur..."

  • ...Patterns of the sort contemplated do arise in wind-generated seas (Kinsman 1965, p. 543), and a possible theoretical basis for bimodal directional spectra has been advanced by Longuet-Higgins (1976) and Fox (976), based on the resonant interaction theory of Davey & Stewartson (1974)....

    [...]

Journal ArticleDOI
TL;DR: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form as mentioned in this paper.
Abstract: A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

474 citations

References
More filters
Journal ArticleDOI
TL;DR: In this paper, a theoretical analysis of the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes is presented, where the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.
Abstract: The phenomenon in question arises when a periodic progressive wave train with fundamental frequency ω is formed on deep water—say by radiation from an oscillating paddle—and there are also present residual wave motions at adjacent side-band frequencies ω(1 ± δ), such as would be generated if the movement of the paddle suffered a slight modulation at low frequency. In consequence of coupling through the non-linear boundary conditions at the free surface, energy is then transferred from the primary motion to the side bands at a rate that, as will be shown herein, can increase exponentially as the interaction proceeds. The result is that the wave train becomes highly irregular far from its origin, even when the departures from periodicity are scarcely detectable at the start.In this paper a theoretical investigation is made into the stability of periodic wave trains to small disturbances in the form of a pair of side-band modes, and Part 2 which will follow is an account of some experimental observations in accord with the present predictions. The main conclusion of the theory is that infinitesimal disturbances of the type considered will undergo unbounded magnification if \[ 0 < \delta \leqslant (\sqrt{2})ka, \] where k and a are the fundamental wave-number and amplitude of the perturbed wave train. The asymptotic rate of growth is a maximum for δ = ka.

2,109 citations


"On three-dimensional packets of sur..." refers background in this paper

  • ...Hasimoto & Ono (1972) established th a t this solution is stable to relatively small disturbances only if Xv > 0 , a condition which leads to kh 1.363 and is the same as tha t found by Benjamin & Feir (1967)....

    [...]

  • ...I f the depth is h and if the wavelength of this train is 2tz/1c, Benjamin & Feir (1967) proved theoretically and Feir (1967) demonstrated experimentally that it is unstable if kh > 1.363 approximately....

    [...]

Journal ArticleDOI
TL;DR: In this paper, a nonlinear plane wave solution to this equation is found to correspond to the so-called Stokes wave, and the linear stability of this plane wave is essentially determined by the sign of the product of two coefficients in this equation, yielding Benjamin and Whitham's criterion.
Abstract: Slow modulation of gravity waves on water layer with uniform depth is investigated by using singular perturbation methods. It is found, to the lowest order of perturbation, that the complicated system of equations governing such modulation can be reduced to a simple nonlinear Schrodinger equation. A nonlinear plane wave solution to this equation is found to correspond to the so-called Stokes wave. The linear stability of this plane wave solution is essentially determined by the sign of the product of two coefficients in this equation, yielding Benjamin and Whitham's criterion. The same equation is found to give a weak cnoidal wave derived from the Korteweg-de Vries equation in the shallow-water limit.

510 citations


"On three-dimensional packets of sur..." refers background or methods in this paper

  • ...Hasimoto & Ono (1972) established th a t this solution is stable to relatively small disturbances only if Xv > 0 , a condition which leads to kh 1.363 and is the same as tha t found by Benjamin & Feir (1967)....

    [...]

  • ...…= kg-lhAj*, A = 0O1 = krtyh fy Moreover if A and ^ 01 are independent of 7] then (2.19), (2.20) reduce to b w ^ A - ~ W \A\*A (2 21)gihtdr kh~ 2 ( Hasimoto & Ono (1972) have pointed out th a t the nonlinear plane wave solution of (2.21) corresponds to the weak cnoidal wave solution of the…...

    [...]

  • ...…° ’ \ 4wcr2 2cr2 9 - 10cr> + 9<r* - {4c * + 4c, c«( 1 - <t») + (rj( 1 - <r*)«} 1>1 = —{2cp + cg( l - c r 2)}, A, = > 0, /t, °g > (2.25) and ghcg 2Cp + cg( 1 — o'2) g h -c 2 The principal equation (4.5) of Hasimoto & Ono (1972) is now recovered on assum ing A to be independent of tj and putting = 0....

    [...]

  • ...Our aim in this paper is to develop a theory for threedimensional wave-packets parallel to tha t of Hasimoto & Ono (1972)....

    [...]

  • ...At first sight the form of the equations for A and 0O1 look rather different from those obtained by Hasimoto & Ono (1972) for the two-dimensional problem but it is possible to write (2.14), (2.15) in an alternative way which makes the connexion obvious....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the initial value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given for values of the Reynolds number slightly greater than the critical value, above which perturbation may grow.
Abstract: The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are et and e½(x+a1rt), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.

440 citations


"On three-dimensional packets of sur..." refers methods in this paper

  • ...We observe also tha t Stewartson & Stuart (1971), using the method of multiple scales, have developed a theory for the evolution of small two-dimensional dis turbances in marginally unstable plane Poiseuille flow, where dissipative effects are important....

    [...]

  • ...I t is the same scaling as used by Stewartson & Stuart (1971) in the first stage of their theory which led to a first order equation for A ....

    [...]

Journal ArticleDOI
TL;DR: In this article, the author's theory of slowly varying wave trains was derived as the first term in a formal perturbation expansion, which was applied directly to the governing variational principle.
Abstract: In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.

141 citations


"On three-dimensional packets of sur..." refers methods in this paper

  • ...Whitham (1970) has discussed the equivalence of his variational method with a particular multiple scale theory for dissipationless systems....

    [...]

01 Jan 1974
TL;DR: In this article, the author's theory of slowly varying wave trains was derived as the first term in a formal perturbation expansion, which was applied directly to the governing variational principle.
Abstract: In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.

131 citations