Ground motions around a deep semielliptic canyon with a horizontal edge subjected to incident plane SH waves

Abstract

We study scattering of antiplane shear waves induced by a deep semielliptic canyon with a horizontal edge. We employ the region-point-matching technique to cope with the problem considered. Through an auxiliary boundary, a part of the circumference of a semiellipse, the whole analyzed region is divided into two subregions. We express the displacement fields in terms of Mathieu functions. We unify two distinct elliptic coordinates via a simple coordinate transformation relation. Integration of the coordinate transformation relation into the region-point-matching technique simplifies the procedure for constructing simultaneous equations. Imposing the continuity conditions and traction-free ones, we obtain the expansion coefficients. Frequency-domain results demonstrate ground motion variability based on several key factors. Ground surface responses under seismic shaking are also simulated in the time domain.

Appendix. Expressions of associated functions in Eqs. (20)–(23)

In Eqs. (20)–(23), the self-defined functions are listed as follows:

$$ {G}_1\left(\eta \right)={ce}_{2n}\left(\eta, q\right), $$

(27)

$$ {G}_2\left(\eta \right)={se}_{2n+1}\left(\eta, q\right), $$

(28)

$$ {\displaystyle \begin{array}{l}R\left(\eta \right)=-4\kern0.2em \sum \limits_{n=0}^{N_0-1}\kern0.2em {\left(-1\right)}^{n+1}{Mc_{2n}^{(1)}}^{\prime}\left({\xi}_d,q\right){ce}_{2n}\left(\alpha, q\right){ce}_{2n}\left(\eta, q\right)\\ {}\kern2em -4\kern0.2em i\kern0.3em \sum \limits_{n=0}^{N_0-1}\kern0.2em {\left(-1\right)}^{n+1}{Ms}_{2n+1}^{(1)\prime}\left({\xi}_d,q\right){se}_{2n+1}\left(\alpha, q\right){se}_{2n+1}\left(\eta, q\right)\\ {}\kern2em -2\kern0.2em i\kern0.1em k\kern0.1em b\exp \left(-2\kern0.2em i\kern0.1em k\kern0.1em b\sinh {\xi}_d\sin \eta \sin \alpha \right)\cosh {\xi}_d\sin \eta \sin \alpha, \end{array}} $$

(29)

where the first two terms and the last one at the right-hand side of Eq. (29) come from the differentiation of Eqs. (10) and (7), respectively.

$$ {\tilde{G}}_1\left(\eta \right)=\frac{Mc_{2n}^{(3)}\left({\xi}_d,q\right)}{{Mc_{2n}^{(3)}}^{\prime}\left({\xi}_d,q\right)}{ce}_{2n}\left(\eta, q\right), $$

(30)

$$ {\tilde{G}}_2\left(\eta \right)=\frac{Ms_{2n+1}^{(3)}\left({\xi}_d,q\right)}{Ms_{2n+1}^{(3)\prime}\left({\xi}_d,q\right)}{se}_{2n+1}\left(\eta, q\right), $$

(31)

$$ {\tilde{L}}_1\left(\eta \right)=\frac{-{Mc}_{2n}^{(1)}\left[{T}_R\left({\xi}_d,\eta \right),q\right]{ce}_{2n}\left[{T}_I\left({\xi}_d,\eta \right),q\right]}{{Mc_{2n}^{(1)}}^{\prime}\left({\xi}_d,q\right)}, $$

(32)

$$ {\tilde{L}}_2\left(\eta \right)=\frac{-{Ms}_{2n+1}^{(1)}\left[{T}_R\left({\xi}_d,\eta \right),q\right]{se}_{2n+1}\left[{T}_I\left({\xi}_d,\eta \right),q\right]}{Ms_{2n+1}^{(1)\prime}\left({\xi}_d,q\right)}, $$

(33)

$$ {\displaystyle \begin{array}{l}\tilde{R}\left(\eta \right)=-4\kern0.2em \sum \limits_{n=0}^{\infty}\kern0.2em {\left(-1\right)}^{n+1}\frac{{Mc_{2n}^{(1)}}^{\prime}\left({\xi}_d,q\right)}{{Mc_{2n}^{(3)}}^{\prime}\left({\xi}_d,q\right)}{ce}_{2n}\left(\alpha, q\right){Mc}_{2n}^{(3)}\left({\xi}_d,q\right){ce}_{2n}\left(\eta, q\right)\\ {}\kern2em -4\kern0.2em i\kern0.3em \sum \limits_{n=0}^{\infty}\kern0.2em {\left(-1\right)}^{n+1}\frac{Ms_{2n+1}^{(1)\prime}\left({\xi}_d,q\right)}{Ms_{2n+1}^{(3)\prime}\left({\xi}_d,q\right)}{se}_{2n+1}\left(\alpha, q\right){Ms}_{2n+1}^{(3)}\left({\xi}_d,q\right){se}_{2n+1}\left(\eta, q\right)\\ {}\kern2em -2\exp \kern0.2em \left(-i\kern0.2em k\kern0.1em b\sinh \kern0.2em {\xi}_d\kern0.2em \sin \kern0.2em \eta \sin \kern0.2em \alpha \right)\cos \kern0.2em \left(k\kern0.1em b\cosh \kern0.2em {\xi}_d\cos \kern0.2em \eta \cos \kern0.2em \alpha \right),\end{array}} $$

(34)

$$ {\displaystyle \begin{array}{l}{L}_1\left(\eta \right)=\frac{-1}{{Mc_{2n}^{(1)}}^{\prime}\left({\xi}_d,q\right)}\cdot \\ {}\kern2em \left\{{Mc_{2n}^{(1)}}^{\prime}\left[{T}_R\left({\xi}_d,\eta \right),q\right]{ce}_{2n}\left[{T}_I\left({\xi}_d,\eta \right),q\right]\right.\operatorname{Re}\kern0.2em \left[\frac{\partial T\left({\xi}_d,\eta \right)}{\partial \xi}\right]\\ {}\kern1.75em +\left.{Mc}_{2n}^{(1)}\left[{T}_R\left({\xi}_d,\eta \right),q\right]{ce}_{2n}^{\prime}\left[{T}_I\left({\xi}_d,\eta \right),q\right]\operatorname{Im}\kern0.2em \left[\frac{\partial T\left({\xi}_d,\eta \right)}{\partial \xi}\right]\right\},\end{array}} $$

(35)

$$ {\displaystyle \begin{array}{l}{L}_2\left(\eta \right)=\frac{-1}{Ms_{2n+1}^{(1)\prime}\left({\xi}_d,q\right)}\cdot \\ {}\kern2em \left\{{Ms}_{2n+1}^{(1)\prime}\left[{T}_R\left({\xi}_d,\eta \right),q\right]{se}_{2n+1}\left[{T}_I\left({\xi}_d,\eta \right),q\right]\operatorname{Re}\kern0.2em \left[\frac{\partial T\left({\xi}_d,\eta \right)}{\partial \xi}\right]\right.\\ {}\kern1.75em +\left.{Ms}_{2n+1}^{(1)}\left[{T}_R\left({\xi}_d,\eta \right),q\right]{se}_{2n+1}^{\prime}\left[{T}_I\left({\xi}_d,\eta \right),q\right]\operatorname{Im}\kern0.2em \left[\frac{\partial T\left({\xi}_d,\eta \right)}{\partial \xi}\right]\right\},\end{array}} $$

(36)

in which the primes stand for differentiation with respect to the argument ξ of pertinent functions.