Analytical solution to scattering of SH waves by a circular lined tunnel embedded in a semi-circular alluvial valley in an elastic half-space

In this study, the dynamic interaction between a surficial alluvial valley and an underground tunnel embedded in it under plane shear horizontal (SH) waves is investigated. Within the framework of classical elastodynamics, an analytical solution to the wave propagation and scattering around a circular lined tunnel embedded in a semi-circular alluvial valley in an elastic half space is proposed. Using the wavefunction expansion and the mirror image methods, the wave fields with unknown coefficients in three regions (i.e., the tunnel lining, alluvial valley, and rutted half space) are constructed separately in different polar coordinates to satisfy the traction-free boundary conditions along the horizontal ground surface as well as the inner surface of the tunnel lining. Using a series of transformations of the Graf's addition formula, the coefficients of the wave fields are obtained by matching the three regions to satisfy the stress and displacement continuity conditions at the interfaces. Both the surface displacement amplitude of the alluvial valley and the dynamic stress concentration factor of the tunnel are calculated to determine the dynamic interaction effects between the surficial valley and underground tunnel. A mutual enhancement of the dynamic responses in the alluvial valley and embedded tunnel is found owing to the wave reflection, scattering, interference, and amplification.