Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

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 4 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 weakly 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 Störmer–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 4-fold amplitude amplification predicted by Miles.