= Solution
Take positive <volumetric flow rate> in the direction of increasing $x$. For an <arterial pressure wave>, the <characteristic admittance of an arterial wave> gives
$$
\boxed{q_B=Y(P_BE^--P_{BR}E^+).}
$$
The backward wave contributes the opposite flow sign. Write $p_j$ for each junction's complex <pressure> amplitude, suppressing the common $e^{i\omega t}$ factor, and put $r=RY$, $C_\beta=\cos\beta$, $S_\beta=\sin\beta$. The topology is the two paths $K\to L\to N$ and $K\to M\to N$; their intermediate and common distal junctions each have a <hydraulic resistance>. The coordinate convention in the source puts the upstream end at $-\ell$ and the downstream end at zero in each segment. Consequently the <wave transfer matrix of an arterial segment> is
$$
\begin{pmatrix}p_u\\q_u\end{pmatrix}
=\begin{pmatrix}C_\beta&iS_\beta/Y\\iYS_\beta&C_\beta\end{pmatrix}
\begin{pmatrix}p_d\\q_d\end{pmatrix}.
$$
Equivalently $q_u=Y(C_\beta p_u-p_d)/(iS_\beta)$ and $q_d=Y(p_u-C_\beta p_d)/(iS_\beta)$, with their continuous limits understood when $S_\beta=0$.
Symmetry gives $p_L=p_M=p_s$. Distal <mass conservation> and the resistive relation $Q_N=p_N/R$ give
$$
2\frac{Y}{iS_\beta}(p_s-C_\beta p_N)=\frac{p_N}{R},\qquad
p_s=\left(C_\beta+\frac{iS_\beta}{2r}\right)p_N.
$$
At $L$, the incoming flow supplies both its bed and segment $D$, so
$$
p_K=\left(2C_\beta+\frac{iS_\beta}{r}\right)p_s-p_N.
$$
Choose the incident amplitude $P_A$ at $K$. If $P_{AR}$ is the reflected amplitude in vessel $A$, <pressure> <continuity equation> gives $p_K=P_A+P_{AR}$, while inlet flow is $2Y(P_A-P_{AR})=2Y(2P_A-p_K)$. Equating this to the two outgoing segment flows yields
$$
p_s=e^{i\beta}p_K-2iS_\beta P_A.
$$
Eliminating $p_s,p_K$ from these three relations gives the <wave transmission through a resistively loaded arterial loop> result
$$
\boxed{Q_N=P_A RY^2\frac{8e^{-i\beta}}{e^{i\beta}(2RY+1)^2+e^{-i\beta}(2RY-1)}.}
$$
For large $r$ at fixed phase, $p_N=2P_Ae^{-2i\beta}+O(r^{-1})$ and $p_M=C_\beta p_N+O(r^{-1})$. Hence
$$
\boxed{Q_N\sim\frac{2P_A}{R}e^{-2i\beta},\qquad Q_M=Q_L\sim\frac{2P_A}{R}e^{-2i\beta}\cos\beta.}
$$
These leading terms must not be used as nonzero denominators near a phase at which the normal intermediate flow vanishes.
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