An analysis is presented for the dynamic stress concentration problem of an infinite inhomogeneous rod having a discontinuity of the curved portion in which the radius of cross sections varies continuously. The rod treated in this study consists of two infinitely straight portions and one finite curved portion of arbitrary curvature. The curved portion lies between the two straight portions. The twisting wave propagates from one of the infinite straight portions to the other, passing through the curved portion. The exact solution for the equilibrium equations for a curved rod has been obtained. The transfer matrix has been derived based on the exact solution and upon considering the internal forces and moments. At discontinuous sections, solutions of curved and straight rods have been connected by adjusting the boundary conditions. As examples, stress concentration factors in circular, elliptical and parabolic arc rods have been obtained.