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in Physics by (79.8k points)

Show that the natural frequency of vibration of Helmholtz resonator is given by

where v = Velocity of propagation of sound in air 

l = Length of the neck of the resonator 

V = Volume of the resonator 

S = Area of cross-section of the neck.

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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 S 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.

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