<p>In this paper, the analytical solution of the SH wave propagation problems in shear modulus and density continuous nonhomogeneous medium is given based on the complex variable function method, and the scattering problem of SH waves by arbitrarily shaped cavities is studied. The shear modulus is expressed as an arbitrary continuous function, and the displacement field is firstly constructed by introducing the auxiliary function and the shear modulus variation function, and then the density variation function is constructed in a specific form by introducing the auxiliary function and the shear modulus variation function again with the help of the functional gradient material design theory. Through the special variability of medium parameters, the range of SH wave frequencies that can be propagated in the medium is limited, while the displacement fields and the inhomogeneous parameter variation function of the medium are interconnected, and further combined with the Commonality Mapping method to equivalently transform the fluctuation equation of variable coefficients into the Helmholtz equation of standard form. Assuming that the medium shear modulus varies as a natural logarithmic function and the density is composed of an exponential function compounded with a function related to the modulus, the dynamic stress distribution at the boundary of the elliptical hole is calculated and analyzed, taking elliptical hole scattering as an example. The results show that wave number, inhomogeneous parameters, hole boundary size and shape have significant effects on the dynamic stress concentration coefficient.</p>