The study presented in this dissertation aims at the ground motion assessment of various two-dimensional hill models subjected to incident plane SH waves in an elastic half-space. The exact, analytic solutions to the boundary-valued problem, based on method of wave function expansion in elliptical coordinate are presented. These require the technique of angular half-range expansion in elliptical coordinate to deal with mixed-boundary condition arisen during mathematical implementation of all hill models; the traction-free boundary condition at the surface of the elliptical hill and the continuity of displacement and stress at the semi-elliptical interface.

In the study we found out that the existence of a hill results in complex pattern of surface displacement. Generally for nearly grazing angle, a hill shields the propagating waves, resulting in a standing wave pattern in the coming direction and the focusing of the amplitude at the far edge of the hill. In addition the presence of full-elliptical tunnels resulted in more prominent standing waves, and amplitudes of ground motion in the neighborhood of unity or less on the far end may be monitored without the abrupt jump at that edge of the. We discovered that the presence of a full-elliptical tunnel amplifies maximum displacement by 30%–70% from corresponding value of reference (inexistence of a tunnel) for the horizontal incidence and de-amplifies maximum displacement in the range of 1%–15% for the vertical incidence; note that these amplifying and de-amplifying effects are dependent on the size of tunnels. The bigger the tunnel is, the stronger the effects are.

We also found out that the effect of the semi-elliptical tunnels on ground surface motion is dissimilar to the effect of the full-elliptical ones. Since horizontal incidences are able to slip underneath semi-tunnels, the weaker standing waves on the left side and weaker shadow zones associated with high jumps of the displacement amplitude at the far edge of the hill are observed. However, when the incident waves are nearly vertical, the stronger shielding effect due to semi-elliptical tunnels than to full-elliptical ones are detected. It could be said that the hinder efficiency of an elliptical tunnel depends on the direction of wave passage; the tunnel becomes better obstruction when the incidence is propagating normal to its major axis.