The symmetrical vibrations of a thin elastic plate

Abstract

In recent years, the theory of integral transforms—often in the guise of ‘operator calculus'—has been employed to obtain the solutions of a wide variety of problems in mathematical physics, particularly in the theory of vibrations. The most commonly employed transforms are those of Fourier, Laplace and Mellin; in problems where the range of one of the variables is (0, ∞) or (− ∞, ∞) these transforms have often been used with success to reduce a partial differential equation in n independent variables to one in n − 1 variables and hence to simplify the boundary value problem involved. In this paper we consider some simple problems in the theory of elastic vibrations in which the Bessel, or Hankel, transform is more useful than those mentioned above and is consequently employed throughout. In the first instance we consider problems in which the variable to be eliminated from the differential equation ranges from 0 to ∞ so that the theory of Hankel transforms is directly applicable; the application of finite intervals is then extended to the Hankel transform by a method similar to that used by Doetsch in his treatment of the Fourier transform.