Wave diffusion in the composite soil layer with the lined tunnel structure is often encountered in the field of seismic engineering. The wave function expansion method is an effective method for solving the wave diffusion problem. In this paper, the wave function expansion method is used to present a semi-analytical solution to the shear horizontal (SH) wave scattering problem of a circular lined tunnel under the covering soil layer. Considering the existence of the covering soil layer, the great arc assumption (that is, the curved boundary instead of the straight-line boundary) is used to construct the wavefield in the composite soil layer. Based on the wave field and boundary conditions, an infinite linear equation system is established by adding the application of complex variable functions. The finite term is intercepted and solved, and the accuracy of the solution is analyzed. Although truncation is inevitable, due to the Bessel function has better convergence, a smaller truncation coefficient can achieve mechanical accuracy. Based on numerical examples, the influence of SH wave incident frequency, soil parameters, and lining thickness on the dynamic stress concentration factor of lining is analyzed. Compared with the SH wave scattering problem by lining in a single medium half-space, due to the existence of the cover layer and the influence of its stiffness, the dynamic stress of the lining can be increased or inhibited. In addition, the lining thickness has obvious different effects on the dynamic stress concentration coefficient of the inner and outer walls of different materials.

An analysis of the scattering of steady-state SH waves by a shallow buried unlined elliptical tunnel is presented. For the horizontal boundary, a mirror method in elliptic coordinates is proposed, which is implemented using the Mathieu function addition theorem. Taking into account the stress-free condition of the tunnel, an infinite equation system was created and solved after truncating it. The orientation of the tunnel is arbitrary, and typical calculations are performed for tunnels arranged horizontally and vertically in the long axis. The effects of depth, the ellipse’s axis ratio, and wavenumber on the dynamic stresses around the tunnel are investigated.

The aim of this article is to investigate the scattering of anti-plane shear waves by elliptic notches in a quarter-space piezoelectric material. The wave field and electric field expressions in this region are derived using the wave function expansion method. An infinite linear system of equations is established by applying the boundary conditions of no traction and no potential. The results obtained are accurate due to the good convergence of the wave function. The displacement amplitude and dynamic stress in the vicinity of the elliptic arc notch under different characteristic parameters and different wave numbers are computed numerically.