Scattering of a scalene trapezoidal hill with a shallow cavity to SH waves

Both surface ground motion and cavity stress concentration have always been considered in the designs of earthquake engineering. In this paper, a theoretical approach is used to study the scattering problem of circular holes under a scalene trapezoidal hill. The wave displacement function was obtained by solving the Helmholtz equation that meets the zero-stress boundary conditions by the variable separation method and the image method. Based on the complex function, the multipolar coordinate method and the region-matching technique, algebraic equations were established at auxiliary boundaries and free boundary conditions in the complex domain. Auxiliary circles were used to solve the singularity of the reflex angle at the trapezoidal corner. Then, according to the sample statistics, instead of the Fourier expansion method, the least-squares method was used to solve the undetermined coefficient of the algebraic equations by discrete boundaries. Frequency responses for some parameters were calculated and discussed. The numerical results demonstrate that the continuity of the auxiliary boundaries and the accuracy of the zero-stress boundary are good; the displacement of the free surface and the stress of the circular hole are related to the shape of the trapezoid, the position of the circular hole, the direction of the incident wave and the frequency content of the excitation. Finally, time-domain responses were calculated by inverse fast Fourier transform based on the frequency domain theory, and the results have revealed the wave propagation mechanism in the complicated structure.