On a special case of forced capillary–gravity waves in a circular basin under Hocking’s edge condition

Abstract

In [M.C. Shen, N.S. Yeh, Exact solution for forced capillary–gravity waves in a circular basin under Hocking’s edge condition, Wave Motion 26 (1997) 117], an exact solution was constructed for the linear problem of forced capillary–gravity waves in a water-filled circular basin generated by a prescribed horizontal vibration of its side wall under Hocking’s edge condition [L.M. Hocking, The damping of capillary–gravity waves at a rigid boundary, J. Fluid Mech. 179 (1987) 253] at the contact line. The solution becomes not valid in the special case that the forcing frequency ω is equal to one of the resonance frequencies Ω mn , m= non-negative integer, n=1,2,…, under the classical edge condition for the circular basin. In this paper we show that the limiting solution as ω→Ω mk for a fixed k satisfies the governing equations for ω=Ω mk . A uniqueness result is established by showing that Ω mn , n=1,2,…, are not eigenvalues of the linear problem.