Theoretical seismograms, for both simplified and realistic models of the earth, can be calculated and analyzed to learn the relations between free oscillations and traveling waves. These studies can help to clarify the influence of the physical properties of the earth upon the generation, propagation, attenuation, and dispersion of elastic waves. Principal results include the equivalence of mantle and crustal Rayleigh waves to the fundamental radial modes of spheroidal free oscillations a•d the equivalence of G waves and ordinary Love waves to the fundamental radial modes of to,rsional free oscillations. In contrast, the higher radial modes produce all the direct, reflected, refracted, and diffracted body phases, as well as the higher-mode surface waves. Stoneley-wave propagation along the core-mantle boundary introduces a large number of interesting phenomena into the propagation of the spheroidal overtones. The confinement of the Stoneley modes to the vicinity of this boundary is responsible for the discontinuities in the spectra and the relative difficulty in the excitation of long-period P and SV diffracted arrivals, as compared with the corresponding SH phases. The trapping of body-wave energy for sources within a velocity inversion is demonstrated. The particle motion ratio at the surface may be as useful as phase velocity in the understanding of the physical properties beneath a recording site; realistic curves of this parameter are presented for a wide range of periods and radial orders. 1. IN'TRODUC.Y[ 0 N The calculation and study of theoretical seismograms for elastic, gravitating, spherical models of the earth have seen remarkable progress in just the past few years. These studies stem directly from SatS's fundamental observation that, since the sphere is a finite, bounded body, the entire disturbance on the surface of the sphere can be found by an appropriate summation of the contributions from the simple poles corresponding to the free oscillations [Sate, Usami, and Ewing, 1962]. The lack of a half-space in this problem removes the necessity of considering the branch points normally associated with the body waves in most problems of elasticRy [Sat•, Usami, and Ewing,'•1962]. These studies have systematically explored many basic questions in seismology, including the fascinating problem of the relation of modes to waves and •Contribution 152 of the Geosciences Division, University of Texas at Dallas, Texas 75230. 2University of Texas at Dallas, Texas 75230. rays. They have also permitted an improvement in the understanding of the generation, propagation, attenuation, and dispersion of seismic waves as related to the properties of the source, the physical properties of the terrestrial elastic transmission medium, and the recording system. Even though there remain many outstanding problems that are still under study, the authors hope that a synthesis of some of the principal results of this program may serve as an appropriate contribution in honor of l•orman Haskell, one of America's great theoretical ...