Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 81 / 1 / ii

Once $C$ is removed, the remaining connected segments form the chain $K\to L\to N\to M$, of total length $3\ell$. For $RY\gg 1$, each hydraulic resistance draws a small flow, so to leading order the far end at $M$ is reflecting, with zero segment flow there. Successive wave transfer matrices of an arterial segment giveThe incident-plus-reflected inlet relation remains $q_K=2Y(2P_A-p_K)$. Therefore, defining $D=3e^{3i\beta }+e^{-3i\beta }$,The modulus of $D$ equals $\gamma =|3e^{4i\beta }+e^{-2i\beta }|$, because the expressions differ by a unit-modulus factor. Comparison with the normal flows gives the amplitude ratiosFor small phases the cosines are positive and the absolute values can be removed. The original PDF prints an additional $\cos (\beta /2)$ in the middle numerator. That factor does not follow from the displayed equal-length network: the transfer relation from $M$ to $K$ above fixes $p_M=8P_A/D$, with no such factor. The corrected middle ratio is shown here.

Since ${\gamma }^2=10+6\cos 6\beta$, one has $\gamma =4[1-27{\beta }^2/8+O({\beta }^4)]$. Thus the three corrected ratios, in the order $N,M,L$, areAll three exceed one for sufficiently small nonzero $\beta$; they equal one at zero frequency. This is an oscillatory arterial wave reflection effect, not an increase in steady conductance. Removing a branch changes the standing wave pattern and the total inlet pressure associated with a fixed incident wave. Reflected waves can enhance junction pressures and hence peripheral flow amplitudes. Neither total inlet pressure nor total inlet flow was held fixed, and no extra energy source is implied.