We assume that the air in the neck of Helmholtz resonator acts as a piston alternately compressing and rarefying the air within the cavity of the resonator.
Let x = Displacement towards the cavity of the piston of sectional area * at any instant t and δP be the increase in the pressure in the cavity. Total force acting on the piston is
where ρ = Density of air and Slρ = Mass of air in the neck. Since the pressure change in the cavity is adiabatic
The velocity of propagation of sound in a gas is given by
The frequency of vibration is thus given by
Since the damping is small, a Helmholtz resonator is highly selective and the response is very sharp.
