Abstract This paper presents a study of incremental wave motion in elastic media with initial stress. Alternative formulations of the governing equations are examined as independent developments as well as special cases of a general theory in which the strain energy function is written in terms of deformation and initial stress. An application to a straight rod of circular cross is considered by means of a semidiscrete finite-element approximation that enables us to explore the effects of initial axial tension on the torsional, axial and flexural spectra and, furthermore, suggest paths toward estimation of the coefficients arising in the general theory.