= Pressure-area law of a collapsible tube
{title2=$P-P_e=\mathcal P(A),\quad c^2=(A_0/\rho)\mathcal P'(A_0)$}
The <pressure-area law of a collapsible tube> specifies transverse <mechanical equilibrium> of a deformable tube wall. With physical internal and external <pressures> $P,P_e$ and <cross-sectional area> $A$, write $P-P_e=\mathcal P(A)$. For constant <fluid> <mass density> $\rho$,
$$
c^2=\frac{A_0}{\rho}\mathcal P'(A_0)
$$
is the squared <tube wave speed> about an undisturbed <area> $A_0$.
The <kinematic pressure> convention used in the parent model instead writes $p=P/\rho$, $p_e=P_e/\rho$ and $p-p_e=\widetilde P(\alpha)$. It gives the equivalent expression $c^2=\alpha_0\widetilde P'(\alpha_0)$, whether $\alpha$ denotes <area> or <area> divided by a fixed reference <area>. Linearized <area> and <momentum conservation> yield this coefficient of the resulting <wave equation>. A stable local wall branch has positive derivative; prescribed external <pressure> is held fixed when taking it. This distinguishes a wall stiffness from a change in applied external loading.
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