Abstract: We consider a semi-open elastic waveguide structure formed by a transversely isotropic layer which on the one hand is firmly fixed and on the other hand is linked with an isotropic half-space. A general solution of differential equation system is obtained describing the propagation of elastic waves in a transversely isotropic medium. Using the boundary conditions and conjugation conditions at the junction of a strip and a half-space as well as the explicit representations of the fields in each of the media, a characteristic equation for the eigenvalues (longitudinal permanents) is obtained concerning our waveguide structure. We considered separately the intervals of eigenvalues. The range in which the values of the longitudinal permanents form a discrete spectrum is specified. The dependence of the longitudinal permanents real values from oscillation frequencies is studied. It is noted that the waveguide modes may exist only if the substrate (half-space) is acoustically more rigid material than the layer. It is concluded that the eigenvalues are bounded above and below by the values corresponding to wave numbers of the attached media. Also, the range is specified in which the modes are originated. It is noted that the characteristic curves are not intersected anywhere. The claculation results are presented for a transversely isotropic layer, filled with sandstone and coupled with rather solid material close to the foundation.