A membrane is stretched over one end of an airtight vessel so that both the tension $T$ in the membrane and the excess pressure $p$ of the air' in the vessel act upon the membrane. If $\psi$ gives the displacement of the membrane from equilibrium, show that
$$
p=-\left(\rho c^2 / V\right) \int \psi d A
$$
where $\rho, V$, and $c$ are the equilibrium values of the density, volume, and velocity of sound of the air in the vessel. Show that the equation of motion for the membrane is therefore
$$
\left(1 / v^2\right)\left(\partial^2 \psi, \partial t^2\right)=\nabla^2 \psi-\left(\rho c^2 / F T\right) \int \psi d A
$$
where $v^2=T / \sigma$, where $T$ is the teusion and $\sigma$ is the mass per unit area of the membrane. What assumptions have been made in obtaining this equation?