Resonant amplification of gravity waves over a circular sill

The trapping of gravity waves over a circular sill, originally studied by Longuet-Higgins (1967), is re-examined in the limit δ ≡ dl/d ↓ 0 with λ ≡ σ2d/g = O(1), where d1 is the depth over the sill, d the outer depth, and a the angular frequency of the incident wave. Explicit results are obtained for the resonance curves. These results are in qualitative agreement with the corresponding shallow-water approximations (Λ [Lt ] 1) of Longuet-Higgins, which have been questioned by Renardy (1983). A remarkably simple result is obtained for the mean-square response to a broadband, randomly phased incident wave.