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RB-04Room & Building Acoustics

Helmholtz Resonator

Resonance of a cavity and neck - bass traps, bottles, ports and silencers.

Geometry

L
750 cm³
cm
cm
End correction
°C
Resonance f₀
109.6Hz
Neck area
2.84cm²
Effective length
9.39cm
Wavelength / size
34.5×
lumped model valid
Resonator schematicnot to scale
■ air plug = mass (ρ·A·L_eff)▨ cavity air = spring (ρc²A²/V)V = 0.75 LL = 8 cm, Ø = 1.9 cmf₀ = 109.6 Hz
Resonance vs cavity volumesame neck
110 Hz0.010.11101001k1101001kCavity volume (L)f₀ (Hz)
§ Theory of operation

A mass on an air spring

A Helmholtz resonator is a rigid cavity connected to the outside through a narrow neck. The plug of air in the neck acts as a mass, the air trapped in the cavity acts as a spring. Like any mass–spring system it has a single natural frequency:

\begin{aligned} f_0 &= \dfrac{c}{2\pi}\,\sqrt{\dfrac{A}{V \cdot L_\mathrm{eff}}} \\ A &= \pi r^2 \quad \text{(neck area)} \qquad V = \text{cavity volume} \\ L_\mathrm{eff} &= L + \Delta L \qquad \text{(neck length + end corrections)} \end{aligned}

End corrections

The air just outside each end of the neck also moves with the plug, so the effective length is longer than the physical one. Typical values: 0.85\,r for a flanged end, 0.61\,r for an unflanged end. Hence \Delta L \approx 1.7\,r (both flanged), 1.46\,r (one of each - e.g. a bottle mouth flanged inside the body), or 1.22\,r (both unflanged).

Applications

  • Bass traps and slotted/perforated panels, arrays of small resonators absorbing a band at low frequency.
  • Loudspeaker bass-reflex ports, tuned to extend the bass response.
  • Side-branch silencers in exhausts and HVAC ducts, removing a specific tonal component.
  • The hum you get blowing across a bottle, the boom of a car with one window open, and the body resonance of a guitar.

Validity

The lumped model holds when all dimensions are much smaller than the wavelength (typically below λ/8). The bandwidth (Q) depends on radiation and viscous losses; adding porous material in the neck broadens the absorption but lowers its peak.