Some mathematics for splashes: sea-wave impact on coastal structures

Structures built on the sea shore, such as harbour walls and breakwaters, are prone to damage by breaking waves. Such structures often need costly repairs especially after winter storms. The consulting company H.R. Wallingford gives ad- vice to clients who design, build and repair seawalls. H.R. continually seek theories, models and simulations to predict the wave loads on coastal structures. Mathematics helps account for the surprisingly large forces exerted by sea waves hitting seawalls. A case is made for solving Laplace’s equation, with mixed boundary conditions, to treat wave impact. Based on Euler’s equations of fluid dynamics, the theory accounts for the high accelerations and pressures during the brief time of impact. We predict a sudden change in the water-velocity field in the impacting wave. Also there is an impulsive pressure field: the pressure-impulse is a useful concept and variable for an engineer to understand the loads on a structure when hit by a breaking wave. Solv- ing mathematical problems can unveil the mystery and drama of breaking waves and splashes