Edge waves on a gently sloping beach

Abstract

Edge waves of frequency ω and longshore wavenumber k in water of depth h(y) = h1H(σy/h1), 0 [les ] y < ∞, are calculated through an asymptotic expansion in σ/kh1 on the assumptions that σ [Lt ] 1 and kh1 = O(1). Approximations to the free-surface displacement in an inner domain that includes the singular point at h = 0 and the turning point near gh ≈ ω2/K2 and to the eigenvalue λ ≡ ω2/σgh are obtained for the complete set of modes on the assumption that h(y) is analytic. A uniformly valid approximation for the free-surface displacement and a variational approximation to Λ are obtained for the dominant mode. The results are compared with the shallow-water approximations of Ball (1967) for a slope that decays exponentially from σ to 0 as h increases from 0 to h1 and of Minzoni (1976) for a uniform slope that joins h = 0 to a flat bottom at h = h1 and with the geometrical-optics approximation of Shen, Meyer & Keller (1968).