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Helmholtz resonator

The resonance frequency of a Helmholtz resonator is computed as:

\[ f = \frac{1}{2\pi} \sqrt{\gamma\frac{P_0}{\rho} \frac{A_n}{V_c L_n}} \]

$\gamma$ is the ratio of specific heats (1.4 for air),
$P_0$ the atmospheric pressure in N.m$^{-2}$ (or $Pa$),
$\rho$ the mass density of air at rest (1.2 kg.m$^{3}$,
$A_n$ the cross section area of the resonator neck,
     for a circular cross-section of radius $r_n$, $A_n = \pi r_n^2$
$L_n$ the length of the resonator neck,
$V_c$ the volume of the resonator cavity
     for a spherical cavity of radius $r_c$, $V_c = 4/3 \times \pi r_c^3$.

Numerical application:
$L_n$ = mm,
$r_n$ = mm,
$r_c$ = mm,
→ $f$ = Hz.

Helmholtz resonator can be considered as a mass-spring system with a resonance frequency computed as:

\[ f = \frac{1}{2\pi} \sqrt{\displaystyle\frac{k}{m}} \]

where $m$ is a mass and $k$ a stiffness.
The mass is composed of the particles inside the neck while the much larger number of particles inside the cavity constitutes the spring stiffness.

Hypothesis: all dimensions are smaller than the acoustic wavelength.
For more accurate results, the radiation effects of the air particles in the neck should be accounted for (usually by adding a length correction to the the geometrical length of the neck).