Assuming that the earth is a visco-elastic sphere with constant density and visco-elasticity, forced radial oscillations of the earth are studied.
1. Introduction to the problems.
2. We have 3-dimensional stress-strain relations for rheological bodies of two models different in kind, viz. Voigt model and Maxwell model.
3. Fundamental equations are derived for the problems in which the earth is considered as a self-gravitational sphere.
4. Equations in case of a forced radial oscillation are solved and energetics is discussed in connection with dissipation of energy.
5. The above theory is applied to earth's phenomena on the assumption that earth tide corresponds to the forced radial oscillation concerned.

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A computer program was made by which we can calculate sound wave dispersion characteristics in vertically heterogeneous liquid media. By this program, group velocity U is calculated without numerical differentiations. An example given by Pekeris is worked out in detail. Phase and group velocities thus obtained are shown in Table 1. Corresponding vertical particle velocity and pressure perturbation are shown in Fig. 1 and 2.

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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 T<0,
and α 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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Numerial calculations of the analytical expressions obtained in Part I are presented in this paper as Part II.
At first, several cases, where Q is infinite and a/r is finite are considered. Main results in these cases are that the smaller the ratio a/r is, the larger the amplitude ratio A_S/A_P becomes remarkably and the ratio of the apparent period τ_S/τ_P a little, where subscripts S and P refer to S and P waves, respectively.
Next, several cases, when a/r is very small compared with unity and parameter Q and S take various values, are studied. It is found that the amplitude ratio A_S/A_P lies between about 3.3 and 4.5 and the ratio of the apparent period τ_S/τ_P between about 1.5 and 2.3 except when the force given as the source oscillates extremely and then the ratio τ_S/τ_P becomes nearly unity.

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