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The detection of oil reservoirs in the subsurface is an important problem of exploration geophysics. For the solution of this problem seismic crosshole measurements can be used. The anomalous behavior of a wave elastic field in an oil-saturated seam in comparison with this field in a dry seam can be used as the theoretical background for solving this problem. At the beginning of the 1960's it was proposed to regard a liquid layer sandwiched between two elastic half-spaces as a simple model of a fluid-filled collector (see [1]). It was shown that in this model a slow, weakly attenuated wave exists for arbitrary sets of parameters of the media. The properties of this slow wave heavily depend on the rigidity of the half-spaces surrounding the liquid layer. This wave is absent if the layer is solid. Therefore, the existence of a slow wave is an indicator of the saturation of a collector. The absence at that time of any reliable experimental data about wave dynamics in an oil-saturated layer and a strong idealization of the model proposed did not allow one to draw any conclusions concerning the physical realization in nature of the theoretical prediction. However, in the 1980's the interest in this problem arose again in connection with the investigations of a low-frequency oscillation appearing during volcanic eruption [2]. Theoretical studies had found a slow wave arising in a magma flow, and it was shown that the oscillation observed earlier during volcanic activity depends on the motion of this slow wave [3]. Recently, slow waves have been observed in a number of cases during crosshole measurements in the Tyumen oil region. These slow waves are propagating in oil-saturated layers and have rather intensive amplitudes. A typical acoustic signal of a slow wave will be presented below. All these facts show that the extensive study of wave propagation from a point source in a liquid layer surrounded by elastic half-spaces is an urgent problem. Such a consideration will also facilitate further passage to more complicated models of an oil-saturated seam.In a cylindrical coordinate system (r, z, 0) we consider a liquid layer (t) (0 < z < h) sandwiched between two half-spaces (0) (z < 0) and (2) (z > h). We denote by a~ -1 (i = 0, 1, 2), b~ -1 (i = 0, 2) the velocities of the longitudinal and transverse waves in media 0, 1, 2, respectively, and by pi (i = 0, 1, 2) the densities of the media, which are related with the Lain6 constants )~i, #i as follows:A point source of the type "center of dilatation" is located at the point (0, 0, -H) of the medium and is described by the Heaviside step time function. In medium (0)

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