This paper employs an approximate form of analysis based on the assumption of plane stress to find the transport equation and corresponding evolution law governing the intensity of acceleration wave propagation in an elastic rod of slowly varying area of cross-section. The result is then extended to include the case of slightly bent rods. In each of these cases it is shown that for a medium in which the stran energy function \(\Sigma\) (p) is such that \(d^ 3\Sigma /dp^ 3\neq 0\), with p the displacement gradient, the acceleration wave intensity is governed by a Bernoulli equation. The work is concluded by showing that the analysis may also be applied to the case of a composite rod comprising an arbitrary number of homogeneous isotropic plane layers normal to the direction of acceleration wave propagation.