${\mathrm{Mg}=\mathrm{P}_{0} \mathrm{A}}{\ldots(1)}$          $  {P_{0} A x_{0}^{\gamma}=P A\left(x_{0}-x\right)^{\gamma}}$

                                          $P=\frac{P_{0} x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}$

Let piston is displaced by distance $x$

$M g-\left(\frac{P_{0} x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}\right) A=F_{\text {restoring }}$

$P_{0} A\left(1-\frac{x_{0}^{\gamma}}{\left(x_{0}-x\right)^{\gamma}}\right)=F_{\text {restoring }} \quad\left[x_{0}-x \approx x_{0}\right]$

$F=-\frac{\gamma P_{0} A x}{x_{0}}$

Frequency with which piston executes $SHM.$

$f=\frac{1}{2 \pi} \sqrt{\frac{\gamma P_{0} A}{x_{0} M}}=\frac{1}{2 \pi} \sqrt{\frac{\gamma P_{0} A^{2}}{M V_{0}}}$