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

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 omega is equal to one of the resonance frequencies Omega(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 omega --> Omega(mk) for a fixed k satisfies the governing equations for omega = Omega(mk). A uniqueness result is established by showing that Omega(mn), n = 1, 2, ..., are not eigenvalues of the Linear problem. (C) 1999 Elsevier Science B.V. All rights reserved.