Pressure-area law of a collapsible tube ($p-p_e=\tilde{P}(\alpha ),c^2=\alpha {\tilde{P}}^{\prime }(\alpha )$)

The pressure-area law of a collapsible tube describes transverse mechanical equilibrium of a deformable tube wall. On a stable branch ${\tilde{P}}^{\prime }(\alpha )>0$. Linearizing mass and momentum conservation about a stationary uniform tube gives the displayed wave speed. External pressure is prescribed separately and is not part of the elastic wall stiffness.

Linearizing area conservation and the axial momentum equation of a collapsible tube flow gives waves travelling relative to the fluid at speeds $\pm c$, where the displayed relation uses the area derivative at fixed external pressure. For $p=P_0+\rho c_0^2(A/A_0{)}^2/2$, it gives $c=c_{0}A/A_0$.