Fifth-order nonlinear spectral model for surface gravity waves: From pseudo-spectral to spectral formulations (Workshop on Nonlinear Water Waves)

Abstract

We present a fifth-order nonlinear spectral model describing the spectral evolution of nonlinear surface gravity waves in water of finite depth. Using the equivalence between pseudo-spectral and spectral formulations, it is shown that the spectral model can be easily obtained using a truncated Hamiltonian from the pseudo-spectral formulation. The fifthorder model is written explicitly in terms of two canonical variables (the Fourier transforms of the surface elevation and the free surface velocity potential) and preserves the Hamiltonian structure of the original water wave problem. Under discrete approximation, the timeperiodic solutions of the spectral model for progressive and standing waves are shown to be consistent with the classical solutions of Stokes and Rayleigh, respectively, when truncated at the third order.