Oscillations of the earth

Abstract

A study is made of the free and forced oscillations of the earth. The natural periods are determined for radial, torsional and spheroidal types of oscillation. Several models of the earth are used: a homogeneous model, such as was assumed originally by Love, a model consisting of a homogeneous solid mantle enclosing a homogeneous liquid core, Bullen's model B, and Bullard's models I and II. It is found that the spheroidal oscillation of order 2 has a period of about 53$\cdot $5 min in all models, except the homogeneous one, for which this period is only 44$\cdot $3 min. The common period of 53$\cdot $5 min agrees to within the observational error with the period of 57 min observed by Benioff on the seismograms of the Kamchatka earthquake of 1952. In contrast to all the other models, Bullen's model B possesses an additional spheroidal oscillation of order 2 of a period of 101 min. The latter oscillation is confined mainly to the core, its amplitude in the mantle being relatively very small. Benioff's observation of a second oscillation of a 100 min period in the Kamchatka earthquake record might be considered as evidence favouring Bullen's model B. The latter differs from Bullard's models mainly by having a central density of around 18 instead of about 12 g/cm$^{3}$. However, a theoretical investigation of the relative excitation of the various free modes by an impulsive compressional point-source situated at a shallow focal depth, shows that the amplitude of the 100 min oscillation should be more than 1000 times weaker than that of the 53$\cdot $5 min oscillation. It is thus not clear how a near-surface earthquake could have excited the core-oscillation. The spectrum of the free modes of oscillation has also been determined for n = 3 and 4, including the fundamental and the first two overtones for each case. The computed free periods of spheroidal oscillation range from 53$\cdot $5 min down to a period of 8 min for the fourth overtone in the case n = 2. We have also treated the bodily tides for Bullen's and Bullard's models. Love's numbers were determined in the case n = 2 for tidal periods of 6, 6 $\surd $2, 12 h and $\infty $. The dependence of the Love numbers on the period is small, a maximum range of variation of 13% occurring in the k-values between the periods of 6 h and $\infty $.