Helmholtz resonance

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A brass, spherical Helmholtz resonator based on his original design, from around 1890-1900.

Helmholtz resonance is the phenomenon of air resonance in a cavity. The name comes from a device created in the 1860s by Hermann von Helmholtz to show the height of the various tones. An example of Helmholtz resonance is the sound created when one blows across the top of an empty bottle.

1 Qualitative explanation
2 Quantitative explanation
3 Applications
4 References

[edit] Qualitative explanation

When air is forced into a cavity, the pressure inside increases. Once the external force that forces the air into the cavity disappears, the higher-pressure air inside will flow out. However, this surge of air flowing out will tend to over-compensate, due to the inertia of the air in the neck, and the cavity will be left at a pressure slightly lower than the outside, causing air to be drawn back in. This process repeats with the magnitude of the pressure changes decreasing each time.

This effect is akin to that of a bungee-jumper bouncing on the end of a bungee rope, or a mass attached to a spring. Air trapped in the chamber acts as a spring. Air, being compressible, has a definite spring constant. Changes in the dimensions of the chamber adjust the properties of the spring: a larger chamber would make for a weaker spring, and vice-versa.

The air in the port (the neck of the chamber) is the mass. Since it is in motion, it possesses some momentum. A longer port would make for a larger mass, and vice-versa. The diameter of the port is related to the mass of air and the volume of the chamber. A port that is too small in area for the chamber volume will "choke" the flow while one that is too large in area for the chamber volume tends to reduce the momentum of the air in the port.

[edit] Quantitative explanation

It can be shown^[1] that the resonant frequency is: $\omega_{H} = \sqrt{\gamma\frac{A^2}{m} \frac{P_0}{V_0}}$ (rad/s) ,

where:

$γ$ (gamma) is the adiabatic index or ratio of specific heats. This value is usually 1.4 for air and diatomic gases.
A is the cross-sectional area of the neck
$m$ is the mass in the cavity
P₀ is the static pressure in the cavity
V₀ is the static volume of the cavity

For cylindrical or rectangular necks $A = \frac{V_n}{L}$ ,

where:

L is the length of the neck
$V n$ is the volume of air in the neck

thus: $\omega_{H} = \sqrt{\gamma\frac{A}{m} \frac{V_n}{L} \frac{P_0}{V_0}}$

By the definition of density: $\frac{V_n}{m} = \frac{1}{\rho}$ , thus:

$\omega_{H} = \sqrt{\gamma\frac{P_0}{\rho} \frac{A}{V_0 L}}$

$f_H = \frac{\omega_H}{2\pi}$ ,

where:

f_H is the resonant frequency (Hz)

The speed of sound in a gas is given by:

$v = \sqrt{\gamma\frac{P_0}{\rho}}$ ,

thus, the frequency of the resonance is:

$f_{H} = \frac{v}{2\pi}\sqrt{\frac{A}{V_0L}}$

The length of the neck appears in the denominator because the inertia of the air in the neck is proportional to the length. The volume of the cavity appears in the denominator because the spring constant of the air in the cavity is inversely proportional to its volume. The area of the neck matters for two reasons. Increasing the area of the neck increases the inertia of the air proportionately, but also decreases the velocity at which the air rushes in and out.

[edit] Applications

Helmholtz resonance finds application in internal combustion engines (see airbox), subwoofers and acoustics. In stringed instruments, such as the guitar and violin, the resonance curve of the instrument has the Helmholtz resonance as one of its peaks, along with other peaks coming from resonances of the vibration of the wood. An ocarina is essentially a Helmholtz resonator where the area of the neck can be easily varied to produce different tones. The West African djembe has a relatively small neck area, giving it a deep bass tone. The djembe may have been used in West African drumming as long as 3,000 years ago, making it much older than our knowledge of the physics involved.