Wave Propagation and a Secular Theory of Calculation Based on Hamilton-Jacobi Theory

A Hamilton-Jacobi method, often used in Astronomy, is outlined. It allows one to suppress secular terms in perturbation calculations. Noting the correspondence between classical mechanics and wave propagation, we use the method to treat the effect of small lateral variations in structure on the propagation of elastic surface waves. The basic result is that the horizontal wave vector is conserved, but the change in position, at which a particular spectral density is found, depends linearly on time. The purpose of this note is to outline a Hamilton-Jacobi method for the solution of wave propagation with perturbations acting (Osgood 1965; Smart 1953). This method has been used in astronomical calculations but has not yet received notice from the geophysical community. The basic result is the following. For a physical system with Hamiltonian H o- H, , it is possible to obtain a set of variables (a, B) which satisfy Hamiltonian equations in which the new Hamiltonian is HI. When HI is zero, these variables become constants of the motion for the system with Hamilton-ian H,. Several examples using this method will be shown (Bogoliubov & Mitropolsky 1961; Backus 1962a, b). For a physical system with Hamiltonian H, , the Hamiltonian equations of motion are with q and p the co-ordinates and momenta respectively i = 1,..., n Assume that equations (1) have been solved by a solution S(q, a, t) of the Hamilton-Jacobi equatio