On the relation between two numerical methods for the computation of random surface gravity waves

Recently two numerical spectral methods, based on the use of the Fast Fourier Transform algorithm, have been found to be useful for studying the statistical properties of a large number of interacting random waves: the first one is known as the Higher Order Spectral (HOS) method and the second is based on the computation of the dynamical equation arising from the Hamiltonian description of surface gravity waves. Here we show analytically the relation between these two methods; more in particular, starting from the HOS approach and writing the corresponding evolution equations in spectral space, after a proper symmetrization of the coupling coefficients in the resulting integral terms, the Hamiltonian dynamical equations are recovered.