General equations with reflection and friction
For the linearized, frictionless case obtained previously in (72): $$ \begin{equation*} p=p_0\; f(z-ct) + p_0^*\; g(z-ct)=p_f+p_b \end{equation*} $$ and for a frictionless vessel the forward component was conveniently expressed as: $$ \begin{equation*} p_f = p_0\;f(z-ct) = P_0 \; e^{j\omega (t-x/c)} \end{equation*} $$ Viscous friction may conveniently be incorporated by the introduction of a complex \textit{propagation coefficient}: $$ \begin{equation} \tag{98} \gamma=\frac{j\omega}{\hat{c}} = a + j b \end{equation} $$ where \( a \) is the \textit{attenuation constant}, \( b \) the \textit{phase constant}, and \( \hat{c} \) a complex pulse wave velocity. An expression \( a \) may be obtained from the equation for the shear stress (13), however this is discarded here for brevity. The phase constant \( b \) is related to the pulse wave velocity by: \( c = \omega / b \). Having adopted this convention, the forward propagating waves incorporating viscous friction may be expressed: $$ \begin{equation*} p_f = p_0\; e^{j\omega t}\;e^{ - \gamma z}, \quad Q_f=Q_0\; e^{j\omega t}\;e^{ - \gamma z} \end{equation*} $$
And further, by assuming that the waves propagates in a vessel depicted in Figure 7, one may obtain expressions for the backward propagating components also: $$ \begin{align*} p_f(L) &= p_f(0) e^{-\gamma\, L}, \qquad p_f(0) = p_0 e^{j\omega t} \\ p_b(L) &= \Gamma \, p_f(L) = \Gamma \, p_f(0)\, e^{-\gamma L} \\ p_b(0) &= p_b(L)\,e^{-\gamma\,L} = \Gamma \, p_f(0)\, e^{-2 \gamma L} \end{align*} $$
Figure 8: Spatial variation in pressure and velocity in the arterial tree (McDonald, 1974)
The forward and backward components at the inlet of the vessel may subsequently be superimposed to form the resulting pressure: $$ \begin{equation} \tag{99} p(0) = p_f(0) + p_b(0) = p_f(0) \; (1 + \Gamma e^{-2\gamma \, L}) \end{equation} $$ This solution incorporates both viscous friction and reflections.
By taking into account that the reflected flow wave opposes the forward wave, an expression for the total flow at the inlet may be obtained in the same manner: $$ \begin{align*} Q_f(L) &= Q_f(0) e^{-\gamma\, L}, \qquad Q_f(0) = Q_0 e^{j\omega t} \\ Q_b(L) &= -\Gamma \, Q_f(L) = -\Gamma \, Q_f(0)\, e^{-\gamma L} \\ Q_b(0) &= Q_b(L)\,e^{-\gamma\,L} = -\Gamma \, Q_f(0)\, e^{-2 \gamma L} \end{align*} $$ which yields: $$ \begin{equation} \tag{100} Q(0) = Q_f(0) + Q_b(0) = Q_f(0) \; (1 - \Gamma e^{-2\gamma \, L}) \end{equation} $$
Further, from (99) and (100) an expression for the input impedance may also be found: $$ \begin{equation} Z_{in} = Z_c \; \frac{1 + \Gamma\;e^{-2\,\gamma \,L}}{1 - \Gamma\;e^{-2\,\gamma \,L}}, \quad Z_c = \frac{p_f(0)}{Q_f(0)} \end{equation} $$ or alternatively: $$ \begin{equation} Z_{in} = Z_c \; \frac{Z_T + Z_c \; \tanh( \gamma \,L)}{Z_c + Z_T\; \tanh( \gamma \,L)} \end{equation} $$ where: $$ \begin{equation*} \Gamma = \frac{ Z_T -Z_c}{ Z_T + Z_c}, \qquad Z_T = \frac{p(L)}{Q(L)} \end{equation*} $$ From (99) and (100) it is clearly seen that a positive reflection factor \( \Gamma \) will cause an \textit{amplification in pressure}, whereas the flow will be reduced. Thus, the presence of reflections will cause the pressure and flow waves to have different forms. Such reflections are believed to explain the streamwise increase in pressure amplitude in the arterial tree (Figure 8).