# Helmholtz resonator

Figure 1. The Helmholtz resonator.

This mechanical system consists of a cavity containing gas connected to the outside by a neck (Figure 1). We set up a simplified model by assuming that the only inertia we have to consider is that of the gas in the neck, which moves backwards and forwards like a
piston of mass *ρAl*, *ρ* being the density of the gas. When this piston moves as a whole through distance *x* from its equilibrium position, the change of volume of the air in its cavity is *Ax*, and the pressure changes may be calculated from the equation of state. Using the adiabatic relation *pV ^{ γ}* = constant, we obtain by taking logarithms and differentiating

so that the pressure changes from its equilibrium value by –*γpAx*/*V* for a small displacement *x* of the piston.

Writing the equation of motion for the plug of air

we see that the motion is harmonic with angular frequency