A weakly inhomogeneous vibrating membrane and the solotone effect in two dimensions

Abstract

The phrase solotone effect refers to a persistent irregular pattern within a set of eigenvalues. This effect arises when the underlying differential equations contain discontinuities. Models of the Earth contain such discontinuities and research into solotones was originally motivated by vibration problems in geophysics. More recently, solotone effects have been discovered within heat conduction, and within quantum wells. However, at present, only one-dimensional arrangements have been shown to give rise to a solotone effect. In this paper, we compute the eigenvalues of an inhomogeneous vibrating rectangular membrane. The resulting eigenspectra, when taken as a whole, fail to display a solotone effect. Nevertheless, we show that irregular patterns do exist within selected subsets of eigenvalues. Our analysis therefore provides the first example of a solotone effect in a spatial dimension greater than one. Furthermore, this discovery could be used to extend the solotone inverse method already implemented in one dimension.