Amplitude of Body Waves in a Heterogeneous Sphere Comparison of Wave Theory and Ray Theory

Abstract

According to ray theory, the path of propagation of body waves is curved due to hetogeneity of the medium, and the amplitude is obtained by calculating the geometrical spreading effect. The ray theory is valid when the variation of physical constants such as velocity and density is slight within a wavelength. It is frequently required, however, to know the behaviour of waves in rather long period range where it is uncertain whether the ray theory is applicable or not.In this paper, the expression for the disturbance due to a point SHsource when the medium consists of many heterogeneous spherical shells with second order discontinuities is obtained in an exact form. The solution is computed numerically and compared with that expected from the ray theory.When there is no singular point on the travel-time curve, the ray theory seems to give a fairly good approximation in a wide frequency range, although the existence of the second order discontinuity effects observed amplitudes with long period if the observing point is close to the boundary.At the caustic or the cusp point of the travel-time. curve, where the amplitude cannot be calculated from simple ray theory, the amplitude is nearly proportional to (period)-) and is not extremely large. The phase shift is nearly equal to n/4.

On the Correspondence Relation Between Normal Modes and Rays

“…The ray theory is primarily based on the assumption of the short wave approximation and is, therefore, invalid in some cases. SATO (1968SATO ( , 1969a and SATO and ESPINOSA (1966) have discussed the validity of the ray and wave theories in deriving the reflection coefficient at the core-mantle boundary, the amplitude of body waves near the shadow boundary, and the amplitude in a heterogeneous sphere, all concerning SH waves.…”