Localised Oscillations Near Submerged Obstacles

Abstract

In previous work Mclver [1] showed that a trapped mode could be supported by two specially shaped obstacles which are held a prescribed distance apart on the surface of water of infinite depth. Physically a trapped mode is a localised oscillation of the fluid which does not radiate any waves to infinity and which has finite energy. Mathematically it is an eigenfunction associated with a certain operator and the trapped mode found in [1] has the property that its corresponding eigenvalue is embedded in the continuous spectrum of that operator. In general embedded eigenvalues are unstable to small perturbations in the body geometry and it is difficult to prove their existence or absence for a given body using standard variational techniques. Despite this, further examples of embedded trapped modes were found in both two and three dimensions [2, 3, 4] and in each case the mode was found to be supported by one or more bodies which isolate a portion of the free surface. However Evans (private communication) gave an argument based on a wide spacing approximation, which suggests that trapped modes at a certain frequency are possible for two suitably spaced, identical bodies provided that the transmission coefficient for the single body is zero at that frequency. There are submerged bodies which have zeros of transmission and so this argument suggests that trapped modes will exist for pairs of such bodies. The purpose of this work is to explicitly construct a pair of submerged bodies which support trapped modes, with the use of the approach described by Mclver [1]. Further details of this method and its use for constructing submerged bodies is available in [5].