The analytical determination of dispersion relations of stress waves propagating within an axisymmetric member has been well established during the past four or five decades. For dynamic analysis associated with fundamental modes of vibration, various one-dimensional (1-D) member theories have been proven to be able to reproduce reasonable approximations for the dispersion relationship. These 1-D member theories provide more efficient and simpler solutions than those obtained by the three-dimensional (3-D) elasticity theory since the need for evaluating special functions such as Bessel functions is eliminated. To assess the loss in prestressing force, elastic waves were applied by various researchers to slender members of axisymmetric cross sections, such as seven-wire-strand tendons and underground pipelines. Thus the feasibility of using the 1-D member theories to predict the associated dispersion phenomenon should be addressed. In this work, formulations of most applicable 1-D linear theories were summarized and corresponding dispersion approximations were obtained in terms of dimensionless parameters. These formulations include the three-mode Mindlin-McNiven theory for axial vibrations, the two-mode Timoshenko beam theory for transverse vibrations, and the nonlinear theory for cable dynamics.