The reverberation-ray matrix and transfer matrix analyses of unidirectional wave motion

The unidirectional wave motions of many physical systems may all be represented by a set of state equations describing the dynamic state of certain physical variables in acoustics, mechanics, optics, and geophysics, varying in time and in one-spatial coordinate. Through the application of Fourier transforms in time variable, the state equations are reduced to a linear system of differential equations with variable coefficients, which may be analyzed by the traditional method of transfer matrix (the propagator of state variables) or the recently developed method of reverberation-ray matrix. The mathematical formulations of both matrices and applications to seemingly two unrelated physical systems, the propagation of axial and flexural waves in a multi-branched framed structure, and that of seismic waves in a layered medium, are reviewed in this article. By detailed comparisons with the method of transfer matrix and others, we conclude that the reverberation-ray analysis is a viable alternative to the solutions of initial value and two-point boundary value problems of unidirectional wave motions.