On hearing the shape of a drum

Abstract

AbstractThe problem of determining the shape of a vibrating membrane given all its characteristic frequencies is discussed using a regular array of interacting atoms as a discrete model of a membrane in contrast to the continuum model analyzed recently by Kac. Using elementary methods it is shown how the size, boundary length, and connectivity may be found from the frequencies. The problem is related to the enumeration of closed random walks on a lattice, in analogy with Kac's treatment, and thence to the adjacency matrices of abstract linear graphs