Two water waves in subsurface interaction on a shear flow

Abstract

Two plane-traveling water waves are superposed, with a background shear flow. The horizontal shear flow varies exponentially, with infinite depth. The present idea is to satisfy the two-dimensional vorticity equation for the second-order subsurface interaction between the two waves. The free-surface conditions are linearized, while the remaining (subsurface) nonlinearity has its amplitude set by the product of the two wave amplitudes. The first traveling wave is assumed steady, satisfying the same Helmholtz equation as the shear flow. Thereby, one single frequency enters the model, dictated by the second traveling wave. There are in total four wave components generated by subsurface interactions. Two of these are rotational waves arising from the exact quadratic interactions between the two first-order waves. The two other waves are compensational irrotational waves setup by linearized free-surface conditions, since the rotational waves alone are unable to satisfy these conditions.