Influence of irregular geologies and inhomogeneity on SH-wave propagation

Abstract

In the present paper, a study is performed in an irregular earth crust, layered over a semi-infinite half-space under the effect of gravity. The irregularities at the interface are possible combinations of geometric shapes such as rectangular, paraboic and triangular notches. The aim of the study is to come up with the influence of these irregularities on the phase velocity of shear horizontal waves. The current work also explores how inhomogeneities affect SH-wave propagation. The medium is assumed to exhibit inhomogeneities as a function of depth. These functions are the product of a linear algebraic function and an exponential function of depth. By means of separation of variables and the substitution method, the equation of motion is reduced to the hypergeometric equation. Suitable boundary conditions are employed to derive a closed form of the dispersion equation. Numerical computations are performed to visualize the impact of irregularity and inhomogeneity. It is observed that the irregular interfaces and the inhomogeneity involved in the medium have a significant effect on SH-wave propagation.

Appendix

$$\begin{aligned} q= & {} i \omega \frac{v_0}{\mu _0} \quad \hbox { where } \omega =c k \\ a_1= & {} 1+q \\ a_2= & {} a+q b \\ a_3= & {} \sqrt{d^2+4k^2} \\ a_4= & {} \frac{d+a_3}{2} \\ a_5= & {} \frac{a_2(d _3)+2k^2c^2/c_0^2}{2 a_2 a_3} \\ a_6= & {} \frac{-a_1 a_3}{a_2} \\ l_1= & {} l+\frac{g \rho _1}{2 \mu _1} \\ l_2= & {} \sqrt{4k^2l_1^2+l^2m^2} \\ l_3= & {} \frac{-lm+l_2}{2l_1} \\ l_4= & {} (ll_1^2m+l^2m^2-ll_1m^2+l_1^2l_2-l m l_2+l_1m l_2+2c_{01}k^2l_1^2(c/c_0)^2)/2l_1^2l_2 \\ l_5= & {} 1+\frac{(l-l_1)m}{l_1^2} \\ l_6= & {} \frac{-l_2}{l_1^2} \\ l_7= & {} \frac{l_2}{l_1} \\ a_{11}= & {} -a_4 U(a_5,1,a_6-a_3H)-a_3 a_5 U(1+a_5,2,a_6-a_3H) \\ a_{12}= & {} -a_4 L(-a_5,a_6-a_3H)-a_3L(-1-a_5,1,a_6-a_3H) \\ a_{21}= & {} e^{-a_4\epsilon h} U(a_5,1,a_6+a_3\epsilon h) \\ a_{22}= & {} e^{-a_4\epsilon h} L(-a_5,a_6+a_3\epsilon h) \\ a_{23}= & {} -e^{-l_3\epsilon h}U(l_4,l_5,l_6+l_7\epsilon h) \\ a_{31}= & {} (1+q)e^{-a_4\epsilon h}\left[ (-a_4+i \epsilon k h')U(a_5,1,a_6+a_3 \epsilon h)-a_3a_5U(1+a_5,2,a_6+a_3 \epsilon h)\right] \\ a_{32}= & {} (1+q)e^{-a_4\epsilon h}\left[ (-a_4+i \epsilon k h')L(-a_5,a_6+a_3 \epsilon h)-a_3L(-1-a_5,1,a_6+a_3 \epsilon h)\right] \\ a_{33}= & {} -\frac{\mu _1}{\mu _0}e^{-l_3\epsilon h}\left[ (-l_3+i \epsilon k h')U(l_4,l_5,l_6+l_7 \epsilon h)-l_4l_7U(1+l_4, 1+l_5,l_6+l_7 \epsilon h)\right] \end{aligned}$$