On Variational Principles and Matrix Methods in Elastodynamics

The equations of elastodynamics are conveniently derived from a varia-tional principle and by considering an averaged Lagrangian for a harmonic field in a stratified elastic medium it is possible to obtain a set of coupled first order differential equations as the Hamilton’s canonical equations. These equations are precisely those which form the basis of the matrix methods commonly used for handling elastic wave propagation in layered media. The variational results are extended to deal with the more complicated ‘h4inor Matrix ’ system of Gilbert & Backus which improves the numerical accuracy of the computations. The equivalent of Rayleigh’s principle is derived for this system and used to consider the effects of perturbations on surface wave dispersion