Variational modelling of extreme waves through oblique interaction
of solitary waves
Abstract. In this work, we model extreme waves that occur due to Mach reflection through the intersection of two obliquely incident solitary waves. For a given range of incident angles and amplitudes, the Mach stem wave grows linearly in length and amplitude, reaching up to four times the amplitude of the incident waves. A variational approach is used to derive the bidirectional Benney-Luke equations, an asymptotic equivalent of the three-dimensional potential-flow equations modelling water waves. This nonlinear and dispersive model has the advantage of allowing wave propagation in two horizontal directions, which is not the case with the unidirectional Kadomtsev-Petviashvili (KP) equation used in most previous studies. A variational Galerkin finite element method is applied to solve the system numerically in Firedrake with a second-order Stormer-Verlet temporal integration scheme in order to obtain stable simulations that conserve the overall mass and energy of the system. Using this approach, we are able to get close to the fourfold amplitude amplification predicted by Miles.