Stable and Accurate Computation of Dispersion Relations for Layered Waveguides, Semi-Infinite Spaces and Infinite Spaces

This paper studies the dispersion characteristics of guided waves in layered finite media, surface waves in layered semi-infinite spaces, and Stoneley waves in layered infinite spaces. Using the precise integration method (PIM) and the Wittrick–Williams (W-W) algorithm, three methods that are based on the dynamic stiffness matrix, symplectic transfer matrix, and mixed energy matrix are developed to compute the dispersion relations. The dispersion relations in layered media can be reduced to a standard eigenvalue problem of ordinary differential equations (ODEs) in the frequency-wavenumber domain. The PIM is used to accurately solve the ODEs with two-point boundary conditions, and all of the eigenvalues are determined by using the eigenvalue counting method. The proposed methods overcome the difficulty of seeking roots from nonlinear transcendental equations. In theory, the three proposed methods are interconnected and can be transformed into each other, but a numerical example indicates that the three methods have different levels of numerical stability and that the method based on the mixed energy matrix is more stable than the other two methods. Numerical examples show that the method based on the mixed energy matrix is accurate and effective for cases of waves in layered finite media, layered semi-infinite spaces, and layered infinite spaces.