# A Perturbation Analysis of an Interaction between Long and Short Surface Waves

01 Jun 1994-Studies in Applied Mathematics (Massachusetts Institute of Technology)-Vol. 92, Iss: 2, pp 159-189

About: This article is published in Studies in Applied Mathematics.The article was published on 1994-06-01 and is currently open access. It has received 12 citations till now. The article focuses on the topics: Surface wave & Capillary wave.

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TL;DR: In this article, a simple and effective multiscale approach for nonlinear asymptotic short-wave dynamics in dispersive systems is devised for wave dynamics and perturbative methods, which allows for a general classification of classical model equations as Boussinesq, Benjamin-Bona-Mahony-Peregrine and Camassa-Holm equations.

Abstract: Nonlinear monochromatic short surface waves in ideal fluids are studied and, by the general consideration of wave dynamics and perturbative methods a simple and effective multiscale approach is devised for nonlinear asymptotic short-wave dynamics in dispersive systems. In particular the evolution of a monochromatic surface wave in an ideal fluid is shown to lead to a modified Green-Naghdi system of equations and a Green-Naghdi system with surface tension. Short surface waves exist in these systems and the nonlinear asymptotic analysis produces the nonlinear model equations that govern their dynamics. Particular solutions are shown. Moreover the method allows for a general classification of classical model equations as Boussinesq, Benjamin-Bona-Mahony-Peregrine and Camassa-Holm equations. Their related nonlinear short-wave model limits are then derived. A relation between the short-wave limit of the integrable Camassa-Holm equation and the Harry-Dym hierarchy is finally unveiled.

24 citations

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TL;DR: In this paper, a nonlinear equation governing asymptotic dynamics of monochromatic short surface wind waves is derived by using a short wave perturbative expansion on a generalized version of the Green-Naghdi system.

22 citations

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TL;DR: Parasitic ripple generation on short gravity waves (4 cm to 10 cm wavelengths) was examined using fully nonlinear computations and laboratory experiments in this article, where the maximum ripple steepness (in time) was found to increase monotonically with the underlying (lowfrequency bandpass) wave steepness.

Abstract: Parasitic ripple generation on short gravity waves (4 cm to 10 cm wavelengths) is examined using fully nonlinear computations and laboratory experiments. Timemarching simulations show sensitivity of the ripple steepness to initial conditions, in particular to the crest asymmetry. Signicant crest fore{aft asymmetry and its unsteadiness enhance ripple generation at moderate wave steepness, e.g. ka between 0.15 and 0.20, a mechanism not discussed in previous studies. The maximum ripple steepness (in time) is found to increase monotonically with the underlying (lowfrequency bandpass) wave steepness in our simulations. This is dierent from the subor super-critical ripple generation predicted by Longuet-Higgins (1995). Unsteadiness in the underlying gravity{capillary waves is shown to cause ripple modulation and an interesting ‘crest-shifting’ phenomenon { the gravity{capillary wave crest and the rst ripple on the forward slope merge to form a new crest. Including boundary layer eects in the free-surface conditions extends some of the simulations at large wave amplitudes. However, the essential process of parasitic ripple generation is nonlinear interaction in an inviscid flow. Mechanically generated gravity{capillary waves demonstrate similar characteristic features of ripple generation and a strong correlation between ripple steepness and crest asymmetry.

14 citations

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TL;DR: In this article, a theoretical and experimental study of the propagation of a short gravity wave packet (modulated Stokes wave) over a solitary wave was performed using a WKB-type perturbation method.

Abstract: This paper is a theoretical and experimental study of the propagation of a short gravity wave packet (modulated Stokes wave) over a solitary wave. The theoretical approach used here relies on a nonlinear WKB-type perturbation method. This method yields a theory of gravity waves that can describe both short and long waves simultaneously. We obtain explicit analytical solutions describing the interaction between the soliton and the short wave packet: phase shifts, variations of wavelengths and of frequencies (Doppler effects). In the experimental part of this work the phase shift experienced by the Stokes wave is measured. The theoretical conclusions are confirmed.

13 citations

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TL;DR: In this article, a new nonlinear equation governing asymptotic dynamics of short surface waves is derived by using a short-wave perturbative expansion in an appropriate reduction of the Euler equations.

Abstract: A new nonlinear equation governing asymptotic dynamics of short surface waves is derived by using a short-wave perturbative expansion in an appropriate reduction of the Euler equations. This reduction corresponds to a Green-Naghdi-type equation with a cinematic discontinuity in the surface. The physical system under consideration is an ideal fluid (inviscid, incompressible and without surface tension) in which takes place a steady surface motion. An ideal surface wind on a lake which produces surface flow is a physical environment conducive to the above-mentioned phenomenon. The equation obtained admits peakon solutions with amplitude, velocity and width in interrelation.

12 citations

##### References

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TL;DR: In this article, the stability of steady nonlinear waves on the surface of an infinitely deep fluid with a free surface was studied. And the authors considered the problem of stability of surface waves as part of the more general problem of nonlinear wave in media with dispersion.

Abstract: We study the stability of steady nonlinear waves on the surface of an infinitely deep fluid [1, 2]. In section 1, the equations of hydrodynamics for an ideal fluid with a free surface are transformed to canonical variables: the shape of the surface η(r, t) and the hydrodynamic potential ψ(r, t) at the surface are expressed in terms of these variables. By introducing canonical variables, we can consider the problem of the stability of surface waves as part of the more general problem of nonlinear waves in media with dispersion [3,4]. The resuits of the rest of the paper are also easily applicable to the general case.

2,425 citations

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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.

1,021 citations

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TL;DR: In this article, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated by taking into account the non-linear interactions between the two wave trains, and the results differ in some essentials from previous estimates by Unna.

Abstract: Short gravity waves, when superposed on much longer waves of the same type, have a tendency to become both shorter and steeper at the crests of the longer waves, and correspondingly longer and lower in the troughs. In the present paper, by taking into account the non-linear interactions between the two wave trains, the changes in wavelength and amplitude of the shorter wave train are rigorously calculated. The results differ in some essentials from previous estimates by Unna. The variation in energy of the short waves is shown to correspond to work done by the longer waves against the radiation stress of the short waves, which has previously been overlooked. The concept of the radiation stress is likely to be valuable in other problems.

597 citations

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TL;DR: In this article, the spectral properties of short surface waves superimposed upon, and interacting with, a long, finite-amplitude dominant wave of frequency N were studied, and it was shown that gravity wavelets continue to propagate freely provided the dominant wave does not break.

Abstract: The characteristics are studied of short surface waves superimposed upon, and interacting with, a long, finite-amplitude dominant wave of frequency N . An asymptotic analysis allows the numerical investigation of Longuet-Higgins (1978) to be extended to higher superharmonic perturbations, and it is found that, although they are distorted by the underlying finite-amplitude wave, gravity wavelets continue to propagate freely provided the dominant wave does not break. Capillary waves can, however, be blocked by short, steep, non-breaking gravity waves, so that in a wind-wave tank at short fetch and high wind speed, freely travelling gravity-capillary waves can be erased by the successive dominant wave crests. A train or group of short gravity waves suffers modulations δ k in its local wave-number because of the straining of the long wave, and large modulations C δ k in its apparent frequency measured at a fixed point (where C is the long wave phase speed), largely because of the Doppler shifting produced by the dominant wave orbital velocity. The spectral signatures of a wave train are calculated by stationary phase and are found to have maxima at the upper wavenumber or frequency in the range. If an ensemble of short-wave groups is sampled at a given frequency f at a fixed point, the signal is derived from groups with a range of intrinsic frequencies δ, but is dominated by those at the long-wave crest for which f = δ + k.u 0 , where u 0 is the orbital velocity of the dominant wave. The apparent phase speed measured by a pair of such probes is the sum of the propagation speed c of the wavelet and the orbital velocity u 0 of the long wave. When f / N is large, the apparent phase speed approaches u 0 , independent of f . These results are consistent with measurements by Ramamonjiarisoa & Giovanangeli (1978) and others in which the apparent phase speed at high frequencies is found to be independent of the frequency — the measurements do not therefore imply a lack of dispersion of short gravity waves on the ocean surface.

110 citations

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TL;DR: In this article, it was shown that the linearized theory used previously for the longer waves is generally inadequate and the fully nonlinear theory used here indicates that for longer waves having a steepness parameter AK = 0.4, for example, the short-wave steepness can be increased at the crests of the longer wave by a factor of order 8, compared with its value at the mean level.

Abstract: To understand the imaging of the sea surface by radar, it is useful to know the theoretical variations in the wavelength and steepness of short gravity waves propagated over the surface of a train of longer gravity waves of finite amplitude. Such variations may be calculated once the orbital accelerations and surface velocities in the longer waves have been accurately determined – a non-trivial computational task.The results show that the linearized theory used previously for the longer waves is generally inadequate. The fully nonlinear theory used here indicates that for longer waves having a steepness parameter AK = 0.4, for example, the short-wave steepness can be increased at the crests of the longer waves by a factor of order 8, compared with its value at the mean level. (Linear theory gives a factor less than 2.)The calculations so far reported are for free, irrotational gravity waves travelling in the same or directly opposite sense to the longer waves. However, the method of calculation could be extended without essential difficulty so as to include effects of surface tension, energy dissipation due to short-wave breaking, surface wind-drift currents, and to arbitrary angles of wave propagation.

96 citations