Wave reflection from a gently sloping beach

The linear reflection of an obliquely incident gravity wave of frequency ω from a gently sloping beach of shoreline slope σ and characteristic length l is determined for σ [Lt ] 1 [Lt ] ω 2 l/g . An asymptotic (σ↓0), inviscid approximation that is uniformly valid in the shallow-water domain is matched to Keller's (1958) geometrical-optics approximation for non-shallow water. An exact solution is obtained for the profile h = σ l [1−exp (− x/l )] in the shallow-water domain and used to test the asymptotic approximation. The absence of viscosity implies perfect reflection. A model that incorporates both small viscosity and small capillarity predicts a fixed contact line and the reflection coefficient | R | = exp [−πσ −2 g −1 (2νω 3 ) ½ ], where ν is the kinematic viscosity. These predictions are in qualitative agreement with the experimental results of Mahony & Pritchard (1980).