Compressional-shear wave coupling induced by velocity gradient in elastic medium

Abstract

In the real ocean environment, the compressional and shear wave velocities in an elastic sediment layer vary with depth, leading to the coupling between compressional and shear waves. As the coupling will affect the underwater sound field, in this paper, a typical sound velocity distribution (where the compression wave velocity has an <i>n</i><sup>2</sup> linear distribution and the square of shear wave velocity has a linear distribution) is analyzed. Based on the wave equation in inhomogeneous elastic medium, coupled equations of wavenumber kernels of scalar and vector potential functions are established. Based on the perturbation method, approximate analytical solutions of integration kernels are acquired by successive differentiation. The comparison between theoretical prediction and experimental data, which are from the pressure sensor of ocean-bottom seismometer (OBS) consisting of three orthogonal hydrophones and one hydrophone, located at the bottom of the sea near Qingdao City, shows that the coupling between shear wave and compression wave has little effect on near-field sound propagation, while the prediction of long-range sound propagation needs to consider the influence of eigenvalue change caused by coupling. Theoretic analysis shows that there will be coupling between the two waves only if the gradient, <i>σ</i>, of the square of the shear wave velocity is nonzero. When <i>α</i>, the gradient of the reciprocal of the square of the compression wave velocity, becomes larger, and <i>σ</i> remains unchanged, the simulation results show that the change of the eigenvalue is very small when considering the coupling effect. Thus, transmission loss curves calculated by the coupled and uncoupled algorithm are almost the same. When <i>σ</i> becomes larger while <i>α</i> remains unchanged, the simulation results show that eigenvalues are changed to some extent if considering the coupling effect, and that the difference between transmission loss calculated by the coupled and uncoupled algorithms increases. That means the effect of <i>σ</i> value on coupling is greater than that of <i>α</i> value. In addition, the coupling between the compression wave and shear wave can lead the eigenfunctions and derivative eigenfunctions in the sediment to change. The horizontal displacement and vertical displacement are the Fourier-Bessel integral functions of eigenfunctions and derivative eigenfunctions. So the displacement field of particle in the sediment layer is different in the coupled case from that in the uncoupled cases. By comparing the transmission loss of sound pressure simulated by COMSOL software and that obtained from our proposed method, the correctness of the proposed method is verified. And the calculation time is much shorter than the calculation time by using COMSOL software.