Third-Order Interactions in Kelvin Ship-Wave Systems

Abstract

This paper considers the possible existence of third-order resonant interactions in Kelvin ship-wave systems. Such interactions will generally occur when two discrete plane wave systems are present, but Phillips's criterion for third-order interactions is applied to the Kelvin pattern to show that there is no resonant interaction, except possibly along the cusp line. A more accurate analysis is then performed in the vicinity of the cusp line, using triple-integral solutions of the third-order free-surface condition, and it is shown that the resulting solutions are divergent on the cusp line, for large distances behind the disturbance. It is concluded that significant nonlinear interactions can occur on the cusp line, involving energy transfer among adjacent wave numbers.