The influence of the finite initial strains on the axisymmetric wave dispersion in a circular cylinder embedded in a compressible elastic medium is investigated within the scope of a piecewise homogeneous body model utilizing three-dimensional linearized theory of wave propagation in an initially stressed body. The material of the cylinder and the surrounding elastic medium are assumed to be compressible and the corresponding elasticity relations are described by the harmonic potential. The numerical results are presented and discussed. It is established that the dispersion curves are divided into four parts by the characteristic nondispersive wave velocities regarding the cylinder and the surrounding materials. As a result of the existence of the initial strains the lengths of these parts change and they move wholly up (down) under initial stretching (compressing) strain along the cylinder, i.e. along the wave propagation direction. (C) 2009 Elsevier Ltd. All rights reserved.