One-dimension nonlinear and dispersive seismic wave modeling in solid media

Abstract

The nonlinear theory in Earth Science is very important for solving the problems of the earth. When considering some of the nonlinear properties of the medium, solitary wave (a special wave with a finite amplitude and a single peak or trough) may appear. Previous studies showed that it may be related to the rupture in the earthquake process. Therefore, it would be very helpful to explain some special phenomena in actual observation data if we fully understand the characteristics of nonlinear waves.#br#In this paper, based on the nonlinear acoustic wave equation, we first perform 1-D nonlinear acoustic wave modeling in solid media using a staggered grid finite difference method. To get the stable and accurate results, a flux-corrected transport method is used. Then we analyze several different types of nonlinear acoustic waves by setting different parameters to investigate their nonlinear characteristics in the solid media. Compared with the linear wave propagation, our results show that the nonlinear coefficients have important influences on the propagation of the acoustic waves. When the equations contain only a third-order nonlinear term (consider the case β 1 ≠ 0, β 2=0, α =0), the main lobe of the wave is tilted backward and its amplitude gradually attenuates with the wave spreading, and the amplitude of its front side-lobe attenuates slowly while the back side-lobe attenuates quickly. The whole shape and amplitude of the wave remain unchanged after propagating a certain distance. When the equations contain only a fourth-order nonlinear term (consider the case β 2 ≠ 0, β 1=0, α =0), the main lobe and the two side-lobes of the wave are all slowly damped, but the shape of the whole wave is unchanged with the wave spreading.#br#In addition, for some combinations of nonlinear and dispersive parameters (consider the case β 1 ≠ 0, α ≠ 0, β 2=0), the wave acts like the linear wave, and the nonlinear acoustic wave is equal to solitary wave which is usually obtained by Kortewegde de Vries (KdV) equation. We validate our modeling method by comparing our results with the analytic solitary solutions. Solitary wave propagates with a fixed velocity slightly less than that of the linear compressional wave, which is probably due to the balance between nonlinear and dispersion effects, making the stress-strain constitutive relations show the nature of linear wave.