Exponential asymptotics for elastic and elastic-gravity waves on flow past submerged obstacles

Abstract

Linearized flow past a submerged obstacle with an elastic sheet resting on the flow surface are studied in the limit that the bending length is small compared to the obstacle depth, in two and three dimensions. Gravitational effects are included in the two-dimensional geometry, but absent in the three-dimensional geometry; the Froude number is chosen so that gravitational and elastic restoring forces are comparable in size. In each of these problems, the waves are exponentially small in the asymptotic limit, and can be computed using exponential asymptotic methods. In the two-dimensional problem, flow past a submerged step is considered. It is found that the relative strengths of the gravitational and elastic restoring forces produce two distinct classes of elastic sheet behaviour. In one parameter regime, constant-amplitude elastic waves and gravity waves extend indefinitely upstream and downstream from the obstacle. In the other parameter regime, all waves decay exponentially away from the obstacle. The equivalent nonlinear two-dimensional geometry is then studied; this asymptotic analysis predicts the existence of a third intermediate regime in which waves persist indefinitely in only one direction, depending on whether the submerged step rises or falls. In the three-dimensional geometry, it is predicted that the elastic waves extend ahead of the submerged source, decaying algebraically in space. The form of these elastic waves is computed, and validated by comparison with numerical computations of the elastic sheet behaviour.