Edge waves on a gently sloping beach: uniform asymptotics

Abstract

Edge waves on a beach of gentle slope ε [Lt ] 1 are considered. For constant slope, Ursell (1952) has obtained a complete set of trapped modes and shown that there exists only a finite number n of such modes, (2n + 1)β < ½π, β = tan−1ε. For non-uniform slope the formulae for the trapped-mode frequencies were heuristically derived by Shen, Meyer & Keller (1968). For small n ∼ O(1) Miles (1989) has obtained formulae which coincide with Shen et al.'s (1968) with accuracy to O(ε) and differ from them by O(ε2). However, Miles’ formulae fail at n ∼ 1/ε. In this paper it is proved that Shen et al.'s (1968) formulae are valid for all n (including n ∼ 1/ε) with accuracy to O(ε) and corrections of any order in ε are given. Uniform asymptotic expansions are obtained for the corresponding eigenfunctions. These expansions give Miles’ (1989) result for small n. The formulae for the frequencies and the eigenfunctions have the same structure for both the full dispersion system and the shallow-water equation. For small n the frequencies for both models coincide with accuracy to O(ε2), but for n ∼ 1/ε they differ by O(1). In the last section the effect of rotation following Evans (1989) is taken into account. All the asymptotics have formal character, i.e. they satisfy the corresponding equations with accuracy to O(εN), N being arbitrarily large. The rigorous justification of these asymptotics is under way.