Geometric focusing of internal waves

Abstract

The spatial structure of the streamfunction field of free, linear internal waves in a two-dimensional basin is governed by the canonical, second-order, hyperbolic equation on a closed domain. Its solution can be determined explicitly for some simple shapes of the basin. It consists of an algorithm by which ‘webs’ of uniquely related characteristics can be constructed and the prescription of one (independent) value of a field variable, related to the streamfunction, on each of these webs. The geometric construction of the webs can be viewed as an alternative version of a billiard game in which the angle of reflection equals that of incidence with respect to the vertical (rather than to the normal). Typically, internal waves are observed to be globally attracted (‘focused’) to a limiting set of characteristics. This attracting set can be classified by the number of reflections it has with the surface (its period in the terminology of dynamical systems). This period of the attractor is a fractal function of the normalized period of the internal waves: large regions of smooth, low-period attractors are seeded with regions with high-period attractors. Occasionally, all internal wave rays fold exactly back upon themselves, a ‘resonance’: focusing is absent and a smooth pattern, familiar from the cellular pattern in a rectangular domain, is obtained. These correspond to the well-known seiching modes of a basin. An analytic set of seiching modes has also been found for a semi-elliptic basin. A necessary condition for seiching to occur is formulated.