Semi-analytical solutions and numerical simulations of 2D SH wave equation

In this paper, we show the analytical solutions of the SH-wave equation based on wave propagation theories. The main results include the analytical solutions of 2D SH-wave equations in the homogeneous medium, the analytical solutions of reflected SH-wave in a two-layer medium derived in detail by using the Cagniard-de Hoop method; we also obtain the analytical solutions for the transmitted SH-wave. Meanwhile, we present some waveforms of various waves including direct wave, reflected wave, head wave, and transmitted wave computed by the analytical solutions of the SH-wave. We present some numerical results by using numerical integral algorithms for the complex integral solutions, and compare the analytical solutions with the numerical results computed by the numerical methods including the optimal nearly analytical discrete method (ONADM) and the fourth-order Lax-Wendroff correction (LWC) scheme to verify the correctness of the numerical methods. The analytical solutions obtained in this paper have great potentials in the applications of testing the new methods for solving the wave equations and the theoretical analysis of wave propagation.