Seismic wave propagation in a viscoelastic layer with exponentially varying bounded shear modulus

The one-dimensional shear-wave-propagation problem is analytically investigated for an inhomogeneous viscoelastic layer characterized by shear modulus varying exponentially with depth. Unlike commonly employed power-law models which exhibit unbounded stiffness at infinity, the present formulation maintains bounded stiffness in the whole domain, thereby enabling physically meaningful representations of surface layers of arbitrary thickness with either depth increasing or decreasing stiffness profiles. Exact closed-form expressions in terms of Gaussian hypergeometric functions are derived for the first time, by solving the governing wave equation in the frequency domain considering vertical SH wave excitation. The solutions are obtained for the base-to-surface transfer function, natural frequencies, mode shapes, and the depth-wise distributions of displacement, shear strain, and curvature. The formulation is further interrogated to furnish Rayleigh-type approximations for the fundamental frequency of the layer, and asymptotic expressions for the dynamic amplification factor in both low- and high-frequency regimes. Moreover, the proposed analytical treatment is extended to accommodate both rigid- and elastic-base conditions (rock-outcrop excitation) and investigate the limiting case of vanishing surface stiffness, demonstrating that the layer response remains finite, exhibiting no singularities such as those encountered in power-law models. It is further shown that the asymptotic analysis for vanishing surface stiffness is error prone, as the limits of zero stiffness and zero depth are not interchangeable. The solution is validated against conventional numerical multilayer analyses pertaining to two different real-life soil profiles subjected to recorded earthquake excitations in both frequency and time domain. The results demonstrate that the proposed model properly captures the key characteristics of site response, while providing physical insight and a consistent analytical reference for the assessment of other relevant solutions.