Rotational Multiplets in the Spectrum of the Earth

Abstract

The doublets in the spectrum of the free oscillations of the earth which have been observed on the gravimetric (UCLA) and strainmeter (Pasadena) records of the great Chilean earthquake of May 22, 1960, are interpreted as multiplets arising from the rotation of the earth. The phenomenon is similar to the Zeeman effect, and is indeed a realization of the mechanical analog from which Larmor deduced the "Larmor precession" in his interpretation of the Zeeman effect. A first-order perturbation calculation yields the result that the degenerate frequency ${\sigma }_0\mathrm{}\left(n\right)\mathrm{}$ in the absence of rotation is resolved by a slow rotation into ($2n+1$) lines, ${{\sigma }_n}^m$ given by ${{\sigma }_n}^m={\sigma }_0\mathrm{}\left(n\right)+m\tau \mathrm{}\left(n\right)\mathrm{}\omega , -n<~m<~n,$ where $\omega$ denotes the angular velocity of rotation of the earth, and $m$ is the azimuthal number of the wave function. $\tau \mathrm{}\left(n\right)\mathrm{}$ is determinable from the zero-order solution in the case of spheroidal oscillations, and is equal to ${\mathrm{}\left[n\right(n+1\mathrm{}\left)\right]}^{-1\mathrm{}}$ in the case of purely torsional oscillations. The relative intensities within the quintet $n=2\mathrm{}$ and the septet $n=3\mathrm{}$ have been determined for an observing station at Los Angeles, on the assumption of an explosive pointsource at the earthquake focus in Chile. The strongest lines should be the pair $m=\pm 1$ for $n=2\mathrm{}$, and the pair $m=\pm 2$ for $n=3\mathrm{}$. These agree in separation with the pairs observed on the strain-meter and with the gravimetric pair at $n=3\mathrm{}$, but less so with the gravimetric pair at $n=2\mathrm{}$. There are indications in the strain-meter spectrum for $n=3\mathrm{}$ of a weaker line at $m=0\mathrm{}$, while the other lines are theoretically of an intensity not exceeding the background noise. The separation in the observed gravimetric doublet for the first over-tone of $n=3\mathrm{}$ agrees with the interval of the strongest pair $m=\pm 2$. The intensities of the lines in the rotational multiplets of the components of displacement for an observing station at Palisades, New York, have also been determined.