Using the shear-coupled PL wave hypothesis of Oliver as a basis, a method is developed for computing synthetic long-period seismograms between the onset of the initial S-type body phase and the beginning of surface waves. Comparison of observed and synthetic siesmograms shows that this hypothesis can explain, in considerable detail, most of the waves with periods greater than about 20 sec recorded during this interval. The synthetic seismograms are computed easily on a small digital computer; they resemble the observed seismograms much more closely than the synthetic seismograms obtained through the superposition of normal modes of the Earth that have been reported in the literature. The synthesis of shear-coupled PL waves depends on a precise knowledge of the phase-velocity curve of the PL wave and travel-time curves of shear waves. Hence, in principle, if one of these quantities is well-known the other can be determined by this method. Phase-velocity curves of the PL wave are determined for the Baltic shield, the Russian platform, the Canadian shield, the United States, and the western North-Atlantic ocean, on the assumption that J-B travel-time curves of shear waves apply to these areas. These dispersion curves show the type of variations to be expected on the basis of the current knowledge of the crustal structures in these areas. Examples are presented to show that J-B travel-times of shear waves along paths between Kenai Peninsula, Alaska and Palisades, equatorial mid-Atlantic ridge and Palisades, and Kurile Islands and Uppsala need to be revised. Shear-wave travel-time curves that are not unique for reasons explained in the study but that give synthetic seismograms in agreement with the observed seismograms were obtained. The new S curves are compared with the J-B travel-time curves for S; and they all predict S waves to arrive later than the time given by J-B tables for epicentral distances smaller than about 30°. The new S curve for the Alaska to Palisades path appears to agree with one of the branches of a multi-branched S curve proposed recently by Ibrahim and Nuttli for the ‘average United States’ insofar as travel-times are concerned, but there are some differences in the slopes of the two curves.