风化半圆形河谷对柱面SH波的散射解析解
An Analytical Solution to the Scattering of Cylindrical SH Waves by a Weathered Semi-circular Canyon

风化河谷地震效应对建在此类场地上工程结构（如大坝、桥梁）的动力稳定性具有潜在的威胁，为揭示河谷风化层对地面运动的影响，利用波函数展开法推导风化半圆形河谷对线源柱面SH波散射问题的解析解。计算不同震源位置条件下风化半圆形河谷的地表位移反应和地面运动放大因子，分析地面运动的幅值和形态，发现随着震源距离的增大，在河谷附近地震动的放大和衰减的交替更频繁。
Site conditions can cause scattering and diffraction of seismic waves and therefore, the amplification and de-amplification of earthquake ground motions may occur. This study focuses on the influence of a weathered canyon on the ground motion that is key input to the seismic design of critical structures. In fact, the site effect of a weathered canyon is crucial to the seismic stability of engineering structures like dams and bridges. To reveal the influence of the weathered layers on the earthquake ground motions, an analytical model of a locally layered semi-circular canyon subjected to a cylindrical SH wave emanating from a line source is established. With the aid of the wavefunction expansion method and the region-matching strategy, a rigorous analytical solution to this problem is derived. The entire region of the half-space with a weathered canyon is firstly divided into three sub-regions. Then the wavefields for the sub-regions, in terms of the series of Bessel and Hankel functions with unknown complex coefficients, are constructed to satisfy the Helmholtz equation by means of the separation of variables method. The solution is finally obtained as the unknown coefficients are sought using the continuity conditions on the interfaces of the three sub-regions. It is worth noting that the wavefield coefficients are in an explicit form which can guarantee high efficiency of the solution. To clearly identify the amplification or de-amplification of the ground motion, the near-source amplification factor is calculated by normalizing the seismic response to the free field of the line source. Based on both results of the surface displacement amplitudes and the ground motion amplification factors for the weathered semi-circular canyons, it is found that the location of the near source is an important factor influencing the site effects. There will be more amplifications and reductions of ground motions within a certain area near the canyon if the source is located farther from the canyon site.