A spectral method for Faraday waves in rectangular tanks

A theoretical study of Faraday waves in an ideal fluid is presented. A novel spectral technique is used to solve the nonlinear boundary conditions, reducing the system to a set of nonlinear ordinary differential equations for a set of Fourier coefficients. A simple weakly nonlinear theory is derived from this solution and found to capture adequately the behaviour of the system. Results for resonance in the full nonlinear system are explored in various depth regimes. Time-periodic solutions about the main (subharmonic) resonance are also studied in both the full and weakly nonlinear theories, and their stability calculated using Floquet theory. These are found to undergo several bifurcations which give rise to chaos for appropriate parameter values. The system is also considered with an additional damping term in order to emulate some effects of viscosity. This is found to combine the two branches of the periodic solutions of a particular mode.