A Fourier approximation method for steady water waves

A method for the numerical solution of steadily progressing periodic waves on irrotational flow over a horizontal bed is presented. No analytical approximations are made. A finite Fourier series, similar to Dean's stream function series, is used to give a set of nonlinear equations which can be solved using Newton's method. Application to laboratory and field situations is emphasized throughout. When compared with known results for wave speed, results from the method agree closely. Results for fluid velocities are compared with experiment and agreement found to be good, unlike results from analytical theories for high waves. The problem of shoaling waves can conveniently be studied using the present method because of its validity for all wavelengths except the solitary wave limit, using the conventional first-order approximation that on a sloping bottom the waves at any depth act as if the bed were horizontal. Wave period, energy flux and mass flux are conserved. Comparisons with experimental results show good agreement.