Nonlinear Periodic Waves in Shallow Water

Abstract

Two classical types of periodic wave motion in polar coordinates has been studied using a computer algebra system. In the case of polar coordinates, the usual perturbation techniques for the nonlinear shallow water equation leads to overdetermined systems of linear algebraic equations for unknown coefficients. The compatibility of the systems is the key point of the investigation. The found coefficients allow to construct solutions to the shallow water equation which are periodic in time. The accuracy of the solutions is the same as of the shallow water equation. Expanding the potential and surface elevation in Fourier series, we express explicitly the coefficients of the first two harmonics as polynomials of Bessel functions. One may speculate that the obtained expressions are the first two terms of an expanded exact three-dimensional solution to the surface wave equations, which describe the axisymmetrical and the simplest unaxisymmetrical surface waves in shallow water.