The theory of Rayleigh's principle as applied to continuous systems

Abstract The object of this paper is to examine and extend a method frequently applied by Lord Rayleigh to the calculation of the frequency of the gravest mode of a vibrating system. In this method no attempt is made to obtain an accurate solution of the differential equations of vibration in any normal mode, but any fairly simple function satisfying the boundary conditions is adopted as an approximate solution. This solution will involve the time only through a harmonic factor such as sin 2π vt, so that the mean value of the kinetic and potential energies of the system may be calculated—always on the supposition that frictionless constraints are applied so that the displacements of the system follow the law of the approximate solution adopted. By equating these mean values of the kinetic and potential energies there is obtained an expression for v, the frequency of the constrained motion. According to Rayleigh’s principle, this value of the frequency is always in excess of the natural frequency of the gravest mode of vibration. Moreover, it is usually found that almost any differentiable function satisfying the boundary conditions of the problem may be made the basis of a calculation yielding a close approximation to the fundamental frequency. Although these results follow easily enough from the Lagrangian method of treating the oscillations of a system possessing only a finite number of degrees of freedom, they do not appear to have been investigated in the far more important case of a continuous system. In this paper a closer examination of Rayleigh’s principle is made by considering a series of successive approximations to the accurate solutions of the problems proposed, and a method is devised for obtaining an upper bound to the error involved in the approximate values of the frequency. Rayleigh’s method is also extended to the calculation of the frequency of the first overtone, and analogous methods are given for more general problems of the computation of “characteristic numbers.”