Calculation of normal and leaky modes in planar waveguides based on a semi-analytical spectral element method

Abstract

A semi-analytical spectral element method (SASEM) is proposed to solve for the normal and leaky modes of elastic waves propagating in a planar waveguide with a half-space substrate. For the SH-wave modes, the transparent boundary condition is used to model the SH wavefields in the half-space substrate. To solve for the PSV-wave normal modes on the (+, +) Riemann sheet and leaky modes on the (+, -) Riemann sheet, the elastic wavefields in the finite-thickness layers are modeled using the displacements, whereas the wavefields in the half-space are modeled using the P-and S-wave potentials. In the substrate, the transparent boundary condition is used for the shear wavefields, whereas semi-infinite elements are introduced to treat the radiative boundary condition of the P wavefields. Then, a polynomial eigenvalue problem is derived, which can be transformed into a standard linear eigenvalue problem. Solving the eigenvalue problem, we can obtain the solutions of the normal and leaky modes. Several numerical tests were performed to verify the effectiveness of SASEM, as well as to demonstrate its high accuracy. Modal analyses of the oscillations of the solved modes demonstrate that the leaky modes differ from the normal modes because of the increasing wavefields in the half-space. Moreover, the guided-P modes are confirmed to be more dependent on the P-waves, whereas the normal and organ-pipe modes are primarily determined by the S-waves. Besides the crustal model composed of several homogeneous layers, SASEM is applied to a vertically inhomogeneous offshore model to demonstrate its applicability. The good agreement between the theoretical guided-P modes and the dispersion spectra not only shows the correctness of SASEM when analyzing waveguides composed of gradient layers but also indicates the potential for constraining the P-wave velocity using the guided-P modes.

Multiple Leaking Mode Dispersion Observations and Applications From Ambient Noise Cross‐Correlation in Oklahoma

“…In this study, we inverted the dispersion curves using both normal and leaking modes (guided waves) by matching the theoretical and measured dispersion curves (the black points in Figures 1b and 2e). The normal and leaking modes were calculated by a semi‐analytical spectral element method (Shi et al., 2021), which can avoid root searching in the complex frequency‐wavenumber domain. The Levenberg‐Marquardt method (Xia et al., 1999) was implemented to retrieve both Vp and Vs.…”