The SH wave equation in a vertically inhomogeneous elastic medium has been considered. It is possible to determine all vertical inhomogeneities for which the SH wave equation can be solved in terms of standard transcendental functions. The inhomogeneities where either shear-wave velocity, modulus of rigidity, or density is constant have also been considered. A few simple inhomogeneities and their solutions in terms of hypergeometric, Whittaker, Bessel and exponential functions have been given.

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