The method of wave-function expansion in elliptical coordinates, elliptical cosine half-range expansion and Mathieu function were applied to obtain an exact analytical solution of the dynamic stress concentration factor (DSCF) around an elliptical cavity in a shallow, semi-elliptical hill. An infinite system of simultaneous linear equations for solving this problem was established by substituting the wave expression obtained by the Mathieu function including the standing wave expression of elliptical lining given herein into the boundary condition obtained by the region-matching method. The finite equations system with unknown coefficients obtained by truncation were solved numerically, and the results in the case of an ellipse degenerating into a circle were compared with previous results to verify the accuracy of the method. The effects of different aspect ratios, incident wave angles and aperture ratios on the dynamic stress concentration around the elliptical cavity were described. Some numerical results, when the elliptical hill was changed into a circular one, were analyzed and compared in detail. In engineering, this model can be regarded as a semi-cylindrical hill with an elliptical cylindrical unlined tunnel under the action of SH waves, and the results are significant in aseismic design.

运用椭圆坐标中的波函数展开，椭圆余弦半范围展开和Mathieu函数的方法，获得了浅半椭圆山中椭圆腔周围动应力集中因子（DSCF）的精确解析解。通过将包括此处给出的椭圆衬砌的驻波表达式在内的Mathieu函数获得的波表达式代入通过区域匹配方法获得的边界条件，从而建立了求解该问题的联立线性方程组的无限系统。数值求解了通过截断获得的未知系数的有限方程组，并将椭圆退化为圆的情况下的结果与以前的结果进行了比较，以验证该方法的准确性。描述了不同长宽比，入射波角和孔径比对椭圆腔周围动应力集中的影响。分析并比较了当椭圆山变为圆形时的一些数值结果。在工程中，该模型可以看作是在SH波作用下具有椭圆形圆柱无衬砌隧道的半圆柱丘陵，其结果在抗震设计中具有重要意义。