An Interacting Wave Profile of Three Trains of Gravity Waves on Finite Depth by Contracion Method

Abstract

Superposition of three wave trains on finite depth is investigated. This paper is focused on how to improve the linear superposition of three waves. This was realized by introducing the scheme. The idea of the scheme is based on a fixed point approach. Application of the scheme to the superposition makes it possible to obtain a wave profile of wave-wave interaction. With the help of FFT, it was possible to analyze high-order nonlinear frequencies for three interacting Stokes' waves on finite depth.1. IntroductionThe wave interaction theory has been studied by other researchers. Longuett-Higgins (1963) began research on deepwater problems. Dalzell (1999) employed the symbolic computation to neaten the complicated looking second-order wave-wave interaction coefficients. Oscillatory third-order perturbation solutions of sums of interacting Stokes wave on deep water was shown by Pierson (1993). The treatment of this nonlinear wave-wave interaction problem is based on the perturbation theory. Because the perturbation theory has a solid mathematical foundation, it can be a powerful tool for solving problems. However, the derivation of solutions becomes very lengthy. A many of the applications of nonlinear mechanics in the field of engineering and science may be based directly on fixed-point methods (Zeidler, 1986; Deimling, 1985). One example of the application of the fixed point theorem to the wave problem is given in Bona and Bose (Bona et al., 1974). They examined the question of the existence of solitary wave solutions to simple one-dimensional models for long waves in nonlinear dispersive systems. Recently, Jang and Kwon (2005) have proposed about a scheme, which utilizes fixed-point theory to calculate the nonlinear wave profiles (Jang and Kwon, 2005; Jang, 2005b; Jang, 2005c). In this paper, based on the scheme, wave profiles for more than