Towards eliminating the nonlinear Kelvin wake

The nonlinear disturbance caused by either a localised pressure distribution moving at constant speed on the free surface of a liquid of finite depth or a flow over a topographic obstacle, is investigated using (i) the weakly nonlinear forced Kadomtsev–Petviashvili equation which is valid for depth-based Froude numbers near unity and (ii) the fully nonlinear free-surface Euler system. The presence of a steady v-shaped Kelvin wave pattern downstream of the forcing is established for this model equation, and the wedge angle is characterised as a function of the depth-based Froude number. Inspired by this analysis, it is shown that the wake can be eliminated via a careful choice of the forcing distribution and that, significantly, the corresponding nonlinear wave-free solution is stable so that it could potentially be seen in a physical experiment. The stability is demonstrated via the numerical solution of an initial value problem for both the model equation and the fully nonlinear Euler system in which the steady wave-free state is attained in the long-time limit.