Adiabatic gas-spring piston period ($\tau =2\pi \sqrt{m{V}_0/(\gamma p_0{A}^2)}$)

A frictionless piston of area $A$ and mass $m$ supported by gas has $p_0=p_{\mathrm{ext}}+mg/A$. Small displacement $x$ changes its volume by $Ax$. The local isentropic compressibility gives $\delta p=-\gamma p_{0}Ax/V_0$, so Newton's second law becomes $m\ddot{x}=-\gamma p_0{A}^{2}x/V_0$. The period follows. Gas inertia and heat leakage are neglected, and internal equilibration must be faster than the oscillation.