ON THE CORRESPONDENCE RELATION BETWEEN NORMAL MODES AND RAYS

Abstract

This paper deals with the torsional vibration of radial higher modes of a spherical earth consisting of a homogeneous elastic mantle and a liquid core with special reference to the correspondence relation between normal modes and body waves.The normal modes and the rays assigned by the mode-ray correspondence relation originally derived by Ben-Menahem are related very closely in several aspects even for the normal modes of the low radial modes having small colatitudinal order numbers. This relation connects modes and rays through the identity of the phase velocity of normal modes and the apparent velocity of body waves. Group velocities of the normal modes associated with a certain ray show nearly constant values independent of radial mode numbers, constants being a function of ray parameter. The travel time of surface waves specified by this constant group velocity gives almost the same value as that of the corresponding body wave. In addition, there are close correlations between the two radii, one to the lowermost maximum or zero point of the radial distribution of azimuthal displacement of the normal modes and the other to the deepest point of the ray.The criterion is established which assigns normal modes of minimum numbers required for the construction of respective body wave phases. Direct, reflected and diffracted SH pulses produced by a point source at zero depth are synthesized by summing up contributions from a limited number of normal modes assigned by this criterion, which reveal almost identical patterns to those obtained by the summation of contributions from a large number of modes.It is found that the separation of body wave phases is possible even from the standpoint of the normal mode theory except for pulses having mutually close ray parameters and that normal modes with very large order numbers are needed for the precise construction of pulses traveling the shallow region and that pulses penetrating to the inner region require normal modes up to much higher radial modes with relatively small order numbers.