The force exerted on a column of air with cross sectional area A by a pressure change dP is given by
where V is the volume of air and x is the vertical offset. For an ideal gas, the adiabatic bulk modulus is
where is the heat capacity ratio and is the undisturbed pressure. Plugging in gives
where a is the acceleration, is the density of gas, and L is the vertical offset. Rearranging gives
so the angular frequency of the oscillator is given by
where v is the velocity of sound in the medium.
