  In this paper, we present an analytic solution for pulse wave propagation in
a flexible arterial model with tapering, physiological boundary conditions and
variable wall properties (wall elasticity and thickness). The change of wall
properties follows a profile that is proportional to $r^\alpha$, where $r$
represents the lumen radius and $\alpha$ is a material coefficient. The
cross-sectionally averaged velocity and pressure are obtained by solving a
hyperbolic system derived from the mass and momentum conservations, and they
are expressed in Bessel functions of order $(4-\alpha)/(3-\alpha)$ and
$1/(3-\alpha)$, respectively. The solution is successfully validated by
comparing it with numerical results from 3D fluid-structure interaction
simulations. Subsequently, the solution is employed to study pulse wave
propagation in an arterial model, revealing that the wall properties and the
physiological outlet boundary conditions, such as the RCR model, play a crucial
role in characterizing the input impedance and reflection coefficient. At
low-frequency range, the input impedance is found to be insensitive to the wall
properties and is primarily determined by the RCR parameters. At high-frequency
range, the input impedance oscillates around the local characteristic
impedance, and the oscillation amplitude varies non-monotonically with
$\alpha$. Expressions for the input impedance at both low-frequency and
high-frequency limits are presented. This analytic solution is also
successfully applied to model flow inside a patient-specific arterial tree,
with the maximum relative errors in pressure and flow rate never exceeding
$1.6\%$ and $9.0\%$ when compared to results from 3D numerical simulation.
