The central aim of this paper is to contribute to the theoretical analysis and understanding of the effect of vessel tapering on the propagation of pressure and velocity wave forms. To this end, it presents new analytic expressions for the temporal and spatial variation of these two variables that account for weak fluid compressibility. It extends previous work in which only the effect of wall deformation (i.e., vessel distensibility) was taken into account. The solutions are derived in the frequency domain and can account for the steady solution component (d.c. component) obtained by taking the asymptotic limit for very low frequencies. It is shown that the effect of compressibility makes the equations more complex but it is still possible to derive closed form analytic solutions in terms of Bessel functions of orders 1/3 and 4/3. The analytical solutions are compared with full 3D fluid structure interaction (FSI) simulations for the case of propagation of a step pressure variation at the inlet of a tapered vessel. Good agreement is observed between the 1D analytical and 3D numerical solutions.