Propagation of Elastic Waves from a Spherical Origin: Part I

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Abstract
Theoretical studies on the propagation of elastic waves generated by given forces acting on a spherical cavity in an infinite medium have been made by many authors, mainly in order to explain the mechanism of deep earthquakes. In this paper, the present authors investigate the problem of the same kind in more detail, when an observing point is not only distant from the surface of the cavity but also close to it. The stress conditions at the surface of the spherical cavity are taken as (γγ)γ=α=-F sin 2θ cos φ f (T), (γθ)γ=α=-F cos 2θ cos φf (T), (γφ)γ=α=+F cos θ sin φ f(T), where f (T), the time variation of the stress, is given by f(T)=1-exp (-QT) cos (ST); for T≥0, T=t/(αc2), 0: for Tα is the radius of the spherical cavity, F is an arbitrary constant, Q and S are dimensionless parameters, t is the time and c2 is reciprocal of the velocity of shear waves. These stresses correspond to the so-called ‘Type II’ source or the force system of double couple in the theories of focal mechanism. In Part I, we will derive the analytical solutions of the problem and discuss the general behaviours of free waves, forced waves of the first and second kinds, especially in the case when S=0 and Q is very small compared with unity. Numerical results will be presented in the next paper.

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