A note on an alternative derivation of the Benney equations for short wave–long wave interactions

Abstract

Starting from the Zakharov formulation of surface gravity waves, we derive the short wave–long wave interaction equations. The procedure involves writing the original water wave problem in Fourier space in Hamiltonian form and expanding it in powers of wave steepness. The decomposition of long and short waves is then introduced in the evolution equation and a near identity transformation is used in order to remove the non-resonant terms. Some algebra is then needed to calculate the coefficients in the system of equations. The shallow water limit of such a system is also reported.