In order for a better understanding of the effect of initial stress on flow in elastic tubes, the propagation of time harmonic waves in a prestressed elastic tube filled with an inviscid fluid is studied. Although the blood is known to be a non-Newtonian fluid, for simplicity in the mathematical analysis, it is assumed to be non-viscous while the tube material is considered to be incompressible, isotropic and elastic. Utilizing the theory of small deformations superimposed on large initial static deformation, for a non-symmetrical perturbed motion, the governing differential equations are obtained in the cylindrical polar coordinates. Due to variability of the coefficients of the resulting differential equations of the solid body, the field equations are solved by a truncated power series method. Applying the boundary conditions, the dispersion relation is obtained as a function of inner pressure, axial stretch and the thickness ratio. It is observed that the wave speed of the non-symmetrical wave is large as compared to the symmetrical case. Various special cases as well as some numerical results are also discussed in the paper.