Given an air chamber of volume $V$ with a long neck of uniform area of cross-section $A$, and a frictionless ball of mass $m$ fitted smoothly in the neck at position $C$. Fig. The pressure of air below the ball inside the chamber is equal to the atmospheric pressure. Increasing the pressure on the ball by a little amount $p$, so that the ball is depressed to position $D$, where $C D=y$. Then there will be a decrease in volume and hence increase in pressure of air inside the chamber. The decrease in volume of the air inside the chamber $=\Delta V=A y$
Volumetric Strain $=\frac{\text { change in volume }}{\text { original volume }}$
$$
=\frac{\Delta \mathrm{V}}{\mathrm{V}}=\frac{A y}{V}
$$
$\therefore \quad$ Bulk Modulus of elasticity $E$, will be
$$
\begin{aligned}
E &=\frac{\text { stress }(\text { or increas }}{\text { volumetric }} \\
&=\frac{-p}{A y / V}=\frac{-p V}{A y}
\end{aligned}
$$
Here, negative sign shows that the increase in pressure will decrease the volume of air in the chamber.
Now, $\quad p=\frac{-E A y}{V}$
Due to this excess pressure, the restoring force acting on the ball is
$$
F=p \times A=\frac{-F A y}{V} \cdot A=\frac{-F A^2}{V} y \quad \text {...(i) }
$$
Since $F \propto y$ and negative sign shows that the force is directed towards equilibrium position. If the applied increased pressure is removed from the ball, the ball will
start executing linear SHM in the neck of chamber with $C$ as mean position.
In SHM, the restoring force,
Comparing (i) and (ii), we have
Spring factor, $k=E A^2 / V$
Here, inertial factor $=$ mass of ball $=m$.
Time Period, $T=2 \pi \sqrt{\frac{\text { inertial factor }}{\text { spring factor }}}$
$$
=2 \pi \sqrt{\frac{m}{E A^2 / V}}=\frac{2 \pi}{A} \sqrt{\frac{m V}{E}}
$$
$\therefore$ Frequency, $v=\frac{1}{T}=\frac{A}{2 \pi} \sqrt{\frac{E}{m V}}$
If the ball oscillates in the neck of chamber under isothermal conditions, then $E=P=$ pressure of air inside the chamber, when ball is at equilibrium position. If the ball oscillates in the neck of chamber under adiabatic conditions, then $E=\mathrm{g} P$, where $\mathrm{g}=C_p / C_v$