This paper studies the propagation of disturbances in a pre-stressed circular rod composed of a general compressible hyperelastic material. Two coupled four-parameter dependent equations are derived as the governing equations for small axial-radial deformations superimposed on a pre-stressed rod. It is found that one parameter plays a crucial role. Depending on its value, the shear-wave velocity could be larger or smaller than or equal to the bar-wave velocity. In the case that these two velocities are equal, there exist travelling waves of arbitrary form. An initial-value problem with an initial discontinuity in the longitudinal stress is also studied. The technique of Fourier transform is used to express the solutions in terms of integrals. Then, by combining a general technique developed by us previously and the method of stationary phase, we managed to derive the asymptotic expansions for the longitudinal stress, which are uniformly valid in the whole spatial domain. The novelty is that we are able to provide completely analytical descriptions for the transient waves. The general characteristics of the disturbances is then determined. It is found that pre-stresses affect the motion greatly. In particular, the presence of pre-stresses can change the wave front from a receding type to an advancing type, or vice versa. Quantitative results for two concrete examples are also provided. [References: 24]
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