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This paper considers the propagation of anti-plane shear waves in a half-space whose shear modulus and mass density have an arbitrary dependence on the distance from the free surface. An appropriate reformulation of the anti-plane displacement produces a governing equation for the reformulated displacement that is amenable to a solution in the high-frequency range. The boundary condition on the free surface subsequently yields an equation which relates the speed of surface waves to the wavenumber and to the functions that define the depth dependence of the shear modulus and the mass density. Restrictions for the existence of surface waves are discussed, and numerical results are presented.