Dynamic response of soil deposits to vertical SH waves for different rigidity depth-gradients

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The response of soils to vertically propagating shear waves has hitherto been analytically studied by assuming a rigidity increase with depth according to some power-law. Solutions in terms of Bessel functions are well-known. However, the near-surface stress distribution due to the static foundation load or the presence of a stiff surface layer leads to a notable rigidity value at the surface that may decrease with depth. In the present paper, a three-parameter depth-function with bounded value at large depths, one that is capable of reproducing both positive and negative depth-gradients, is adopted. The analytical solution derived in terms of power series is used to study elastic-rock amplification characteristics and modal shapes. Approximate expressions for the fundamental natural frequency of an elastic layer are given, covering a wide range of the parameters involved. Using the transfer matrix approach for multi-layered ground and the solution for a linear depth-profile for the individual layers, the accuracy of discretization schemes is examined. Finally, the case of a 2D rigidity variation is investigated by means of a numerical code.