Trapped modes about multiple cylinders in a channel

The trapped modes which can occur in a long narrow wave channel containing any number of different-sized bottom-mounted circular cylinders arbitrarily spaced along the centreline of the channel are considered. The modes, all of which are antisymmetric with respect to the centreplane of the channel are of two types: Neumann modes, in which the fluid has normal velocity zero on the channel walls corresponding to a localized sloshing near the cylinders, and Dirichlet modes, in which the dynamic pressure vanishes on the channel walls. These latter modes have no physical meaning in the water-wave context but have been observed in a related acoustic context where the same governing equations and boundary conditions apply. It is shown that in general there are less than or equal to N trapped modes for any configuration of N cylinders, the precise number depending critically on the geometry of the configuration. Both types are of importance in predicting the exciting forces on individual cylinders within a large but finite periodic arrangement of cylinders.