Zero-stress, cylindrical wave functions around a circular underground tunnel in a flat, elastic half-space: Incident P-waves

The zero-stress boundary conditions at the surface of the half-space in the presence of surface and sub-surface cavities for in-plane, incident cylindrical P- and SV-waves have always posed challenging problems. The outgoing cylindrical P- and SV-waves can be represented by Hankel functions of radial distance coupled with the sine and cosine functions of angle. Together, at the half-space surface the P- and SV-wave functions are not orthogonal over the semi-infinite radial distance from 0 to infinity. Thus, to simultaneously satisfy the zero in-plane, normal, and shear stresses, an approximation of the geometry is often made. This paper presents an analytical formulation of the boundary-valued problem, where the Hankel wave functions are expressed in integral form, changing the representation from cylindrical to rectangular coordinates, so that the zero-stress boundary conditions at the half-space surface can be applied in a more straightforward way.It is sometimes desirable to use alternate, simpler, or approximate solutions to the boundary-valued problems, without going to great lengths to have all of the boundary conditions satisfied. The free-stress boundary conditions at the half-space surface to be solved here are among the most complicated boundary conditions to be satisfied in the present class of problems. It is thus of interest to illustrate what the solution would be like if the half-space boundary conditions are not imposed—in other words, if they are “relaxed.” A section of this paper is devoted to this approach, and the results of the solutions with the half-spaced, stress-free boundary conditions “imposed” and “relaxed” are presented and compared.The method introduced here may also be applied to more complicated wave-propagation problems, like the diffraction problems involving surface and sub-surface inhomogeneities in a poroelastic half-space medium.