Longitudinal wave propagation in a nonhomogeneous semi-infinite elastic rod is considered. The non-homogeneity may be due to the axial variation of Young's modulus or of density or of both. An arbitrary time-dependent stress is applied at the end, x = 0, of the rod. The normal stress as a function of position and time is obtained using the theory of the propagating surfaces of discontinuity by expanding the stress as a Taylor series about the time of arrival of the wavefront. The coefficients of the Taylor expansion are obtained as solutions of linear ordinary differential equations with variable coefficients. For the case where the Young's modulus E(x) varies as (1 + ax)2a, it is shown that the stress at the wavefront varies as (1 + ax)α, where a and α are constants.