Ray asymptotic propagation of the spectra of elastic surface waves in a waveguide with laterally smooth inhomogeneities

Abstract

Summary. A technique based on ray asymptotics has been developed to propagate complex spectra of elastic normal mode surface waves in a waveguide with material and geometrical properties varying smoothly in the lateral directions. In the technique, the original problem defined in the unstretched coordinates has been transformed into an eiconal equation as well as into a certain number of transport equations defined in stretched coordinates. The solution of the eiconal equation is equal to the solution of the eigenproblem of the eiconal operator A0. Due to the self-adjointness of A0, in each of the relevant local inner product spaces, LIPS, the solution of the eigenproblem, A0ψ=vψ results in the set {vt} of real local eigenvalues and in the orthonormal system {ψt} of local eigenvectors. As the Hamiltonian function of an initial value problem, each eigenvalues gives birth to a bicharacteristic curve as well as to the related ray. The introduction of the rays induces connections between the vertical cross-sections of the waveguide. Finally, for each asymptotic order j, the LIPS-valued transport equations are reduced to a set of matricial propagation equations in the local spectral amplitude vectors, LSAVs. Consequently, a knowledge of the initial conditions at a vertical cross-section makes it possible to propagate the LSAVs along the rays of the relevant modes. However, to complete the propagation one needs, in addition to the initial values, information about certain additional quantities, non-diagonal terms of order j, diagonal terms of orders lower than j and the auxiliary boundary terms of orders from 1 to j. The treatment has been completed by the propagation of the modal phases along the relevant rays.