New analytic solutions for wave propagation in flexible, tapered vessels with reference to mammalian arteries

Novel, closed-form, analytic solutions for the pressure and velocity fields are derived for the linear problem of wave propagation inside a tapered flexible vessel of conical shape. It is shown that pressure and velocity can be written in terms of Bessel functions of orders $1/ 3$ and $4/ 3$ respectively. An expression is also derived that quantifies the effect of the cone angle on the wave propagation velocity. The analytic solutions are general and valid for tube variations at any length scale in relation to the wavelength of the wave. In other words, the requirement that the changes in vessel properties with distance should take place over a length scale large compared to the wavelength of the wave, is not employed or needed. This is the basic condition for the application of WKB theory to tapered vessels. However, this condition is not satisfied in pressure pulses propagating in mammalian arteries. The general expressions derived in this paper are directly applicable to the cardiovascular system of mammals. It is further shown that the presented solution naturally tends to the asymptotic WKB solution when the assumptions of the theory are applied to the general expressions. An explicit formula is provided for the time-averaged energy flux of the wave that shows clearly the effect of the continuous reflection of the wave from the vessel wall. Viscous effects are incorporated by coupling the derived analytic solution with the radial velocity profile of Womersley. The results are compared with full nonlinear fluid–structure interaction simulations and very good agreement is found (maximum differences are ${\ensuremath{\sim} }1\hspace{0.167em} \% $ and 1.6 % for area-averaged pressure and velocity respectively, and 4–6 % for local velocity values).