A Semi-Analytical Solution for Topographic Amplification and Broadband Scattering of SH Waves by Step-like Rock Slopes

Abstract

Local seismic topographic amplification of rock slopes is a primary cause of earthquake-induced damage in mountainous engineering. To resolve the scattering of plane SH waves by individual rock slopes, this study introduces an analytical model derived from the Uniform Theory of Diffraction. Conventional wave function expansion methods (WFEM) often struggle with non-closed free boundary conditions across varying elevations and tend to suffer from series divergence under high-frequency incidence. This approach precisely partitions the slope into two semi-infinite ideal wedge domains sharing a common face, reconstructing the generalized geometric wave field through rigorous global ray tracing. Second-order coupled diffraction terms with a slope-diffraction correction are introduced to describe the two-way interaction between the slope crest and toe, while the UTD transition functions ensure continuity of the wavefield across shadow boundaries. Systematic numerical evaluations in both frequency and time domains reveal a frequency-selective interference mechanism with distinct spatial variability. Under horizontal grazing incidence, energy is strongly focused on the leeward face, while the back-slope forms a high-frequency seismic shadow zone due to intense destructive interference between direct and secondary diffracted waves. At certain oblique incidence angles, repeated scattering of broadband waves between the slope crest and toe leads to strong local resonance in the slope region. Findings indicate that the location of maximum dynamic response shifts sensitively with the incident wavefront inclination. The results show that the maximum amplification does not always occur at the slope crest, highlighting the importance of incidence angle in seismic microzonation and site planning for mountainous areas.