A note on the uniqueness of certain water-wave problems involving two vertical cylindrical shells

A simple geometric condition is derived under which certain problems relating to two partially or totally immersed vertical cylindrical shells in water of finite depth are unique. The method, using classical linear water wave theory, is based on an idea of F. John (1950), utilized by N. Kuznetsov and V. Maz'ya (2001) who proved uniqueness for all geometries and frequencies of a single cylindrical shell. The problem is particularly relevent to the possible existence of trapped modes that have been predicted numerically in the case of two circular cylindrical shells. The method is also applied to two pairs of symmetric vertical barriers and extends the results of Kuznetsov et al. (2001). A further example with a geometry defined by confocal ellipses is considered.