Rayleigh's Principle in Finite Element Calculations of Seismic Wave Response

Abstract

Summary Rayleigh's principle in the context of finite element modelling is shown to provide a powerful and convenient method for estimating the seismic eigenfrequencies of irregular Earth structures. It is not necessary to solve the eigenvalue problem completely, but instead to construct the elastic moduli and density matrices for the irregular structure, multiply them by an approximate eigenfunction vector, and form the Rayleigh quotient. The resulting error in frequency is of second order in the error of the eigenfunction. In order to conserve computer storage for large models the matrices need not be constructed and stored in their entirety, but multiplications can be accumulated one element at a time. Calculations for an inhomogeneous vibrating string and Rayleigh waves in a layered Earth model illustrate the technique. 1. Value of the variational method Rayleigh's Principle is commonly invoked in the determination of periods of free oscillation. It states that when the mean potential and kinetic energies of the system in free oscillation are equated, the solution for the angular frequency takes a stationary value (see Moiseiwitsch 1966). Its use in seismology is not new. Meissner (1926) observed that it may be used to determine group velocity for Love waves without resorting to numerical differentiation. Jeffreys (1961) extended this work to Rayleigh waves, and further noted that perturbations in the eigenfrequency due to changes in the structure can be determined without repeating the entire calculation. The original form of the mode shape is retained, on the assumption that it approximates that for the modified structure. Because of the stationarity, the new eigenfrequency is obtained with an error of only second order. This observation led to the publication of Universal Dispersion Tables for Love and Rayleigh waves (Anderson 1964, and later papers) and for terrestrial eigenvibrations (Derr 1969) from which the effects of changing the structure on the period of surface waves or torsional and spheroidal oscillations with given wave number could be determined.