It is both a pleasure and privilege to present a paper on the uniqueness of linearised water waves at this mini-symposium in honour of Professor John Wehausen whose classic review article Surface Waves (with E. V. Laitone) has done so much to influence workers in the field in the forty-two years since its publication. The question of the uniqueness of solutions to the linearised water wave equations was settled once and for all in a paper in the Journal of Fluid Mechanics by M. McIver (1996). She constructed a solution for the motion between a pair of fixed rigid surface-piercing cylinders in two dimensions which decayed at large distances from the cylinders. Soon after she was joined by P. McIver (1997) in producing an axisymmetric example in the form of a fixed rigid surface-piercing toroid of a special shape which supported an oscillatory motion in its interior fluid region whilst the motion in the exterior region decayed to zero. This wave trapping effect or non-uniqueness occurred for a particular relation between the wave frequency and the toroid geometry. In the present paper we show that such a phenomenon can occur for simple geometries also. In particular we show that wave trapping can occur in the annular region between two partially immersed vertical concentric circular cylindrical shells for particular values of radii and frequencies.

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