In order to explore the spatial distribution and temporal variation of ground motions near a semicircular canyon with a dipping edge, a simplified mathematical model is constructed. Based on the region-matching technique, a Fourier-Bessel series solution for the plane SH-wave excitation is derived and then applied to theoretically simulate the seismic response of the canyon. The use of the adequate wavefunctions and a newly derived Graf’s addition formula can solve the unknown expansion coefficients. Parametric analyses with respect to the frequency of input motion, angle of incidence, and canyon geometry are illustrated. Both frequency- and time-domain computations are presented. The canonical case, a completed semicircular canyon, which has the exact analytical solution, and the horizontally truncated case analyzed in previous works are considered as particular cases of the proposed general model. Comparisons with boundary-element solutions show good agreement. Steady-state results show that the phenomenon of wave focusing tends to be significant when the incident angle bends toward the horizontal ground surface. Propagation and attenuation of scattered waves that originated from the surficial anomaly are exhibited in transient-state simulations.

Ground motions around a semicircular canyon with a dipping edge under SH plane wave incidence

The purpose of this study is to theoretically probe how the joint effects of near-field source and surface topography impact the seismic wavefields. A simplified canyon model, which couples a semicircular model with a model for minor segments, is used to predict characteristic changes in ground motions due to line-source excitation. Based on a semi-analytical procedure using the region-matching technique, a Fourier–Bessel series solution is derived. Distinct coordinate systems are unified with the aid of Graf’s addition formula. Comparisons with boundary-element solutions verify the modelling framework proposed. Steady-state and transient simulations are both conducted.

Antiplane response of a flat-bottomed semicircular canyon to cylindrical elastic waves

Steady state responses of a circular cavity and a semi-circular canyon subjected by plane SH wave in an elastic quarter are presented by using Fourier-Hankel wave function expansion method and image method with Fourier series expansion on the boundary conditions to determine linear algebraic equations of unknown wave function coefficients. Especially, displacement and stress component expressions are formulated for incident, reflected, scattering waves, respectively. This method can provide an analytical ideas and methods for further studies of elastodynamic interface problems.

Dynamic Stress Concentration of a Circular Cavity Subjected by Steady Plane SH Waves in an Elastic Quarter with a Semi-Circular Canyon

In this study, the region-point-matching technique (RPMT) is applied to examine the scattering problem of truncated semi-elliptic canyons under plane SH-wave excitation. The partition of the entire analyzed region into two subregions is carried out via an introduction of the elliptic-arc auxiliary boundary. Taking advantage of appropriate wavefunctions in elliptic coordinates, the expression of antiplane motions for each subregion can be obtained. To accomplish the indispensable coordinate shift, the coordinate-transformed relation, intended as a substitute for the addition theorem involving Mathieu functions, is well utilized. Integration of the coordinate-transformed relation into the RPMT brings about the rapid construction of simultaneous equations. Effects of pertinent parameters on steady-state and transient surface motions are demonstrated. Computed results show that, for horizontal incidence, the potential high level of ground shaking may occur near the illuminated upper corner of the canyon. In such a small localized region, due to the occurrence of constructive interference between the reflected waves from the horizontal ground surface and the scattered waves from the corners of the canyon, the peak amplifaction may be at least two times that of free-field response.

M.-S. Hsu

Papers

Scattering of SH Waves by a Truncated Semi-Elliptic Canyon