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

Abstract

The interaction of finite-amplitude long gravity waves with a small-amplitude packet of short capillary waves is studied by a multiple-scale method based on the invariance of the perturbation expansion under certain translations. The result of the analysis is a set of equations coupling the complex amplitude of the packet of short waves with.the long-wave velocity potential and surface elevation. The short wave is described by a Ginzburg-Landau equation with coefficients that depend on properties of the long wave. The long-wave potential and surface elevation satisfy the usual free-surface conditions augmented by forcing terms representing effects of the short waves. The derivation removes some of the restrictions imposed in earlier studies.

Renormalization-theoretic analysis of non-equilibrium phase transitions: I. The Becker-Döring equations with power law rate coefficients

“…Of central interest in the present paper is the asymptotic late time macroscopic behaviour of complex dynamical systems. Previously, Bricmont & Kupiainen have combined asymptotic methods with ideas from renormalisation theory to study problems in diffusive processes and special limiting solutions of nonlinear parabolic equations [3,4,5] while Woodruff has cast multiple timescale problems in the form of renormalisation theory [39,42,40,41]. This allowed the equations governing larger-scale phenomena to be derived from a more general theory and to be separated out from the small scale dynamics.…”

Unsteady ripple generation on steep gravity–capillary waves

“…This conclusion is based on the blocking effect of large ka through an effective gravitational acceleration. It is similar to the trapping of free gravity waves between two 'caustics' on a long wave (Phillips 1981;Shyu & Phillips 1990) and the blockage of a free capillary wave packet by a gravity wave (Woodruff & Messiter 1994). In parasitic ripple experiments, however, Yermakov et al (1986) observed a monotonic increase of the maximum ripple steepness θ r with ka.…”

Short-wave dynamics in the Euler equations

“…This interaction happens when the linear dispersion relation is such that the group velocity of a wavetrain of short waves coincides with the phase velocity of the long-wave component (Benney's theory). The earliest studies along this line were made by Nishikawa et al [10], Benney [11], Kawahara and Jeffrey [12] and Woodruff and Messiter [13].…”