Theory of water waves derived from a Lagrangian. Part 1. Standing waves

Abstract

A new system of equations for calculating time-dependent motions of deep-water gravity waves (Balk 1996) is here developed analytically and set in a form suitable for practical applications. The method is fully nonlinear, and has the advantage of essential simplicity. Both the potential and the kinetic energy involve polynomial expressions of low degree in the Fourier coefficients Y_n(t). This leads to equations of motion of correspondingly low degree. Moreover the constants in the equations are very simple. In this paper the equations of motion are specialized to standing waves, where the coefficients Y_n are all real. Truncation of the series at low values of [mid ]n[mid ], say n < N, leads to ‘partial waves’ with solutions apparently periodic in the time t. For physical applications N must however be large. The method will be applied to the breaking of standing waves by the forming of sharp corners at the crests, and the generation of vertical jets rising from the wave troughs.