Steep waves in free-surface flow past narrow topography

Abstract

In this work, we compute steep forced solitary wave solutions for the problem of free-surface flow over a localised topographic disturbance in an otherwise flat horizontal channel bottom. A single forced solitary wave and a double-crested forced solitary wave solution are shown to exist, both of which approach the Stokes limiting configuration of an included angle of 120∘ and a stagnation point at the wave crests. The solution space for the topographically forced problem is compared to that found in Wade et al. [“On the free-surface flow of very steep forced solitary waves,” J. Fluid Mech. 739, 1–21 (2014)], who considered forcing due to a localised distribution of pressure applied to the free surface. The main feature that differentiates the two types of forcing is an additional solution that exists in the pressure-forced problem, a steep wave with a cusp at a single wave crest. Our numerical results suggest that this cusped-wave solution does not exist in the topographically forced problem.