Based on complex function methods and a multipolar coordinate system, the scattering induced by a cylindrical cavity in a radially inhomogeneous half-space is investigated. Mass density of the half-space varies depending on the distance from the centre of the cavity while the shear modulus is always constant. The wave velocity is expressed as a function of radius vector and the Helmholtz equation is a partial differential equation with a variable coefficient. By means of a conformal mapping technique, the Helmholtz equation with a variable coefficient is transferred into its normal form. Then, displacement fields and corresponding stress components are deduced. Applying the boundary conditions, dynamic stress concentration factors around the cavity are obtained and analyzed. Typical numerical results are presented to demonstrate the distribution of dynamic stress concentration factors when influencing parameters are assumed.

Based on the complex function method and the conformal mapping technique, the scattering of SH waves caused by a semi-cylindrical canyon in a radially inhomogeneous density half-space is studied analytically. During the solution, model of a semi-cylindrical canyon is established in a semi-infinite inhomogeneous media. And the Helmholtz equation with a variable coefficient is transformed into a standard Helmholtz equation by conformal mapping technique. Then the infinite algebraic equations are constructed to satisfy the stress boundary conditions on the surface of semi-cylindrical canyon. Finally, the incident waves, reflected waves and scattering waves induced by canyon topography in half space are solved. According to the numerical results, influences of the concave topography and the inhomogeneous medium on ground motion are analyzed.