Exponential asymptotics for flow past sheets and boundaries

The problem of irrotational inviscid incompressible free-surface flow over a step under a thin elastic sheet is considered. This problem is examined in the limit of a small elastic parameter, corresponding to a sheet with low rigidity. The behaviour of the waves along the free surface is approximated using an asymptotic series. This asymptotic series is divergent due to the singularities located at each corner of the step. Exponential asymptotic techniques are employed to observe the switching on of exponentially small waves in the system across curves known as Stokes lines. This problem is highly nonlinear, and therefore challenging to study. The system is first linearized about small step height, and the amplitude is calculated using both asymptotic and exact methods. The results of the two methods are compared. In order to study the effects of nonlinearity in the system, a reduced model problem is considered. It is found that there are particular geometries that minimise the wave amplitude dramatically due to destructive interference.