An Exposition of the Classical Problem of Vibrations of Membranes and Plates

Abstract

Several boundary value problems which occur in the study of vibrations of membranes and plates are discussed briefly in the present exposition. The differential equations are formulated by the use of Hamilton's variational principle applicable to general physical systems and the various boundary problems are noted. These are followed by theorems dealing with the nature, properties and distribution of the frequencies of vibration and an enumeration of known exact solutions of the problems. The last part of the exposition is concerned with recent developments for estimating the fundamental and higher tones of a vibrating membrane. These are in the nature of isoperimetric inequalities.