Soliton spectra of random water waves in shallow basins

Abstract

Interpretation of random wave field on a shallow water in terms of Fourier spectra is not adequate, when wave amplitudes are not infinitesimally small. A nonlinearity of wave fields leads to the harmonic interactions and random variation of Fourier spectra. As has been shown by Osborne and his co-authors, a more adequate analysis can be performed in terms of nonlinear modes representing cnoidal waves; a spectrum of such modes remains unchanged even in the process of nonlinear mode interactions. Here we show that there is an alternative and more simple analysis of random wave fields on shallow water, which can be presented in terms of interacting Korteweg-de Vries solitons. The data processing of random wave field is developed on the basis of inverse scattering method. The soliton component obscured in a random wave field is determined and a corresponding distribution function of number of solitons on their amplitudes is constructed.The approach developed is illustrated by means of artificially generated quasi-random wave field and applied to the real data interpretation of wind waves generated in the laboratory wind tank.

Soliton-Like Waves in the Vicinity of the Southern Kuril Islands

“…Strictly speaking, interpreting a random wave field in shallow water in terms of Fourier spectra is unsatisfactory when the wave amplitudes are not small. Nonlinearity of wave fields leads to harmonic interactions and random changes in the Fourier spectra [12]. Taking into account this prerequisite, but also recognizing that the amplitudes of the detected waves are modest and nonlinearity only appears sporadically, the authors decided to accept the use of Fourier spectra for sea level fluctuation analysis.…”

“…To study the spatial evolution of the sea surface level measured at a certain point, the KdV equation can be used in the so-called temporal form (given in [12]), with an initial disturbance at this point. However, the equation in [12] describes only the soliton wave envelope, which coincides with the envelope of the observed packets. In the present case, wave packets which take the form of modulated oscillations are recorded.…”

Untitled

“…Harmonic breathers. The presence of 1.6-hour oscillations within the envelope of single wave packets, i.e., modulation, suggests that the well-known classical solution of the KdV equation [12] is incomplete and that further research is needed. On the other hand, the KdV equation is non-linear and, although researchers have made significant progress in solving it, invoking the inverse scattering transform (IST) as an alternative to a linear Fourier transform remains challenging.…”

“…This equation was introduced by J. V. Boussinesq in 1877, but acquired its modern form in the works of Diederik Korteweg and Gustav de Vries, who reinterpreted it, investigated it more fully [11] and obtained a nonlinear equation for describing long solitary waves on water. To study the spatial evolution of the sea surface level measured at a certain point, the KdV equation can be used in the so-called temporal form (given in [12]), with an initial disturbance at this point. However, the equation in [12] describes only the soliton wave envelope, which coincides with the envelope of the observed packets.…”

Experimental Evidence of a Hydrodynamic Soliton Gas

“…24, 25] while experimental evidence in a hydrodynamic context is scarce. In [26] it has been claimed that the low frequency component of sea surface elevations measured in the Currituck Sound (NC, USA) behave as a dense soliton gas, displaying a power law energy spectrum with exponent equal to -1; another approach is described in [27] where the soliton content in laboratory shallow water wind waves is estimated.In this Letter we describe a unique experiment that is designed to build and monitor a hydrodynamic soliton gas in a laboratory. We focus on shallow water gravity waves where the dynamics is described to the leading order in nonlinearity and dispersion by the KdV equation.…”