Trapping of waves by a submerged elliptical torus

In linear water-wave theory a trapped mode is defined to be a non-zero solution of the homogeneous equations and boundary conditions, which doesn’t radiate waves and has finite energy. The existence of such modes supported by three-dimensional, surface-piercing bodies in the open sea was first established by McIver & McIver[1]. They constructed an axisymmetric potential and defined the shape of the body to be one of the stream surfaces of the flow. In practice, however, it would be more useful to be able to specify the shape of the obstacle and determine whether there are any trapped modes for that geometry. The purpose of this work is to show how this may be done for certain classes of bodies by exploiting the link between three-dimensional trapped modes and zeros of transmission for two-dimensional obstacles. First a plane wave approximation to a trapped mode is derived, then a numerical technique for finding exact trapped mode wave numbers is described. Finally some results which show that modes may be trapped by a submerged elliptical torus are presented.