Application of the reverberation-ray matrix to the propagation of elastic waves in a layered solid

This paper extends the newly developed method of reverberation-ray matrix (J. Sound Vibrat. 230(4) (2000) 743) to the propagation of elastic waves in a layered solid. The steady state waves generated by point source (axisymmetric problem) or a line source (plane strain problem) are expressed by the Sommerfield-weyl integrals of wave numbers. The waves radiated from the source are reflected or refracted at the interface of two adjacent layers, and the process of transmission and reflection is represented by a local scattering matrix; and the process of wave transmitting from one interface to the neighbouring one is represented by a local phase matrix. The local matrices of all layers are then stacked to form the global scattering matrix and global phase matrix of the layered medium separately. The product of these two matrices together with a global permutation matrix gives rise to the reverberation-ray matrix R, which represents the multi-reflected and transmitted steady state waves within the entire medium. The transient waves are then determined by another integration over the frequency, and the integrand of the double integral in frequency and wave number, known as the ray-integrals, contains a power series of R. The ray-integrals so formulated are particularly suitable for evaluating the transient waves involving a large number of generalized-rays by calculating the double integrals numerically as illustrated by the example of a laminated plate in this paper.