Scattering of SH waves by a scalene triangular hill with a shallow cavity in half-space

In this study, a theoretical approach is used to investigate the scattering problem of circular holes under a scalene triangle on the surface. The wave displacement function is obtained by solving the Helmholtz equation that meets the zero-stress boundary conditions by adopting the method of separation of variables. Based on the complex function, multi-polar coordinate method, and region-matching technique, algebraic equations are established at auxiliary boundaries and free boundaries conditions in a complex domain. The auxiliary circle is used to solve the singularity of the reflex angle at the triangle corner. Then, according to sample statistics, the least squares method is used instead of the Fourier expansion method to solve the undetermined coefficient of the algebraic equations by discrete boundary. Numerical results show that the continuity of the auxiliary boundaries and the accuracy of the zero-stress boundaries are adequate, and the displacement of the free surface and the stress of the circular hole are related to the shape of the triangle, the position of the circular hole, the direction of the incident wave, and the frequency content of the excitation. Finally, time-domain responses are calculated by FFT based on the frequency domain theory, and the results reveal the wave propagation mechanism in a complicated structure.

Automated summary

This study uses a theoretical approach to investigate the scattering problem of circular holes under a scalene triangle on the surface. By solving the Helmholtz equation that meets the zero-stress boundary conditions, the wave displacement function is obtained. Algebraic equations are established based on the complex function, multi-polar coordinate method, and region-matching technique. Numerical results show that the continuity of the auxiliary boundaries and the accuracy of the zero-stress boundaries are adequate, and the displacement of the free surface and the stress of the circular hole are related to various factors. Time-domain responses are calculated to reveal the wave propagation mechanism in a complicated structure.