Can you hear the shape of a cavity ?

Mansfield College MMath in Mathematics Can you hear the shape of a cavity? In this paper we analyse the question " Can one hear the shape of a drum? " [1] presented by Mark Kac in 1966 [1]. We start in Chapter 1 by formalising the question in the form of a differential equation and introducing a simple finite difference scheme to analyse the question. We show how this relates to a matrix eigenvalue problem and then in Chapter 2 present the QR Algorithm which we will use to find the eigenvalues of this problem. In Chapter 3 we present an error analysis of the numerical scheme. That is we show that the difference between the eigenvalues of the continuous problem and that of the analogous discrete problem is of order h 2 where h is the size of the mesh used in the finite difference scheme. We show how this bound can be improved by using Richardson Extrapolation and then in Chapter 4 go on to apply the numerical scheme to various two dimensional and three dimensional regions. We show that a pair of regions given by Gordon, Webb and Wolpert [2] that have the same eigenvalues in the continuous context have the same eigenvalues in the discrete context. We continue to prove that if two regions in two dimensional space have the same eigenvalues in the continuous context then their three dimensional extensions have the same eigenvalues in the continuous context. We also prove the same result in the discrete context, and show this numerically. Finally we present 45 shapes that do not have the same eigenvalues in the discrete setting and hence we conjecture that those same shapes do not have the same eigenvalues in the continuous setting. Acknowledgements I would like to thank my supervisor, Dr Andy Wathen, for presenting this problem to me and for guiding me through it with insightful discussions. ii Contents Abstract i Acknowledgements ii 1 Introduction 1 1.