The generation and propagation of anti-plane surface waves on an inhomogeneous half-space of depth dependent shear modulus and mass density, is discussed in this paper. The radiation of surface waves an anti-plane line load is analyzed by an application of the reciprocity theorem. Next the governing equation for free surface waves is reformulated in a form that is amenable to a surface wave solution in the high frequency range. The boundary condition on the free surface yields an equation for the velocity of surface waves, in terms of the wave number and derivatives of the functions defining the depth dependence of the shear modules and the mass density. This equation does not always have a realvalued solution, and when it does the amplitude of the corresponding wave motion does not always display the decrease with depth that would define a surface wave. Numerical examples are presented to illustrate these observations.