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The propagation of SH waves in a heterogeneous viscoelastic layer lying over a heterogeneous viscoelastic half space from a point source is examined analytically. The significance of the heterogeneity of hyperbolic and exponential variations associated with rigidity, viscosity, and density is investigated mathematically. A dispersion equation and displacement components are computed in a compact form, considering the case that displacement and stress are continuous at the interface and stress vanishes on a free surface. The acquired dispersion equation signifies the relation between phase velocity and dimensionless wavenumber. The dispersion equation is highly influenced by heterogeneous parameters of both the layer and the half space. The dispersion equation is reduced to the classical equation of the Love wave after eliminating all the heterogeneous parameters, in agreement with pre-established results. The analysis uses the technique of Fourier transformation and Green’s function. The wavenumber is supposed to be complex, as the frequency is fixed in the viscoelastic model. Graphs are presented to demonstrate the effect of heterogeneous parameters on phase and damping velocity with respect to wavenumber.

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The present article aims to explore three dimensional Rayleigh-type waves in a layered structure composed of transversely isotropic material. The technique of potential function is utilized to solve the problem. With the aid of appropriate boundary conditions, a dispersion relation (illustrating variation of velocity with frequency) is established for the three dimensional Rayleigh-type waves which depends on the elastic constants. It is worth mentioning that the deduced results are found to be in well agreement with the pre-established and the classical results. Moreover, the obtained results are computed numerically and are illustrated graphically for distinct models. The study further computes the phase velocity of Rayleigh-type waves for the distinct models. The outcomes of the present study provide a better insight into numerous engineering and mechanics problems for various purposes. More specifically, understanding the dispersive nature of Rayleigh waves in three dimensions further help in non-destructive testing, structural health monitoring, exploration of oil, fuels and minerals, and delineating soil properties for geotechnical studies. Hence provides sufficient motivation for exploring the same in similar kinds of structures.