Diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane

The diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane is investigated in this paper. The scattered wave potentials with unknown coefficients are solved by Fredlund's theory and the Helmholtz's decomposition. The general expressions of displacements, stresses, pore pressures, and seepage are obtained using complex variable method. The half-surface and the canyon surface are mapped onto semi-circles in the image plane with M & ouml;bius transformation and Schwarz-Christoffel mapping. The boundary-valued problems yield an infinite series of linear algebraic equations with unknown coefficients. The truncating numbers are selected to assure the convergence and accuracy of the solution. A parametric study is performed to investigate the site responses of the canyons by incident P1 waves in an unsaturated half-plane. Numerical results show that the aspect ratio, saturation degree, circular frequency, and angle of incident wave significantly affect the site response of the canyon. The effects of saturation degree on the ground motions are significant owing to the complex tri-interaction of the three phases in the unsaturated medium. The dynamic concentrations occur at the corner region of the canyon. The shielding effect of the canyon to obliquely incident P1 waves is obvious.