Helmholtz resonance

A selection of Helmholtz resonators from 1870, at Hunterian Museum and Art Gallery in Glasgow.

Helmholtz described in his 1862 book On the Sensations of Tone an apparatus able to pick out specific frequencies from a complex sound. The Helmholtz resonator, as it is now called, consists of a rigid container of a known volume, nearly spherical in shape, with a small neck and hole in one end and a larger hole in the other end to emit the sound.

When the resonator's 'nipple' is placed inside one's ear, a specific frequency of the complex sound can be picked out and heard clearly. In his book Helmholtz explains: When we "apply a resonator to the ear, most of the tones produced in the surrounding air will be considerably damped; but if the proper tone of the resonator is sounded, it brays into the ear most powerfully…. The proper tone of the resonator may even be sometimes heard cropping up in the whistling of the wind, the rattling of carriage wheels, the splashing of water."

A set of varied size resonators was sold to be used as discrete acoustic filters for the spectral analysis of complex sounds. There is also an adjustable type, called a universal resonator, which consists of two cylinders, one inside the other, which can slide in or out to change the volume of the cavity over a continuous range. An array of 14 of this type of resonator has been employed in a mechanical Fourier sound analyzer. This resonator can also emit a variable-frequency tone when driven by a stream of air in the "tone variator" invented by William Stern, 1897.[2]

When air is forced into a cavity, the pressure inside increases. When the external force pushing the air into the cavity is removed, the higher-pressure air inside will flow out. Due to the inertia of the moving air 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 oscillations increasing and decreasing asymptotically after the sound starts and stops.

The port (the neck of the chamber) is placed in the ear, allowing the experimenter to hear the sound and to determine its loudness. The resonant mass of air in the chamber is set in motion through the second hole, which is larger and doesn't have a neck.

A gastropod seashell can form a Helmholtz resonator with low Q factor, amplifying many frequencies, resulting in the "sounds of the sea".

The term Helmholtz resonator is now more generally applied to include bottles from which sound is generated by blowing air across the mouth of the bottle. In this case the length and diameter of the bottle neck also contribute to the resonance frequency and its Q factor.

By one definition a Helmholtz resonator augments the amplitude of the vibratory motion of the enclosed air in a chamber by taking energy from sound waves passing in the surrounding air. In the other definition the sound waves are generated by a uniform stream of air flowing across the open top of an enclosed volume of air.

It can be shown[3] that the resonant angular frequency is given by:

${\displaystyle \omega _{H}={\sqrt {\gamma {\frac {A^{2}}{m}}{\frac {P_{0}}{V_{0}}}}}}$ (rad/s),

where:

• ${\displaystyle \gamma }$ (gamma) is the adiabatic index or ratio of specific heats. This value is usually 1.4 for air and diatomic gases.
• ${\displaystyle A}$ is the cross-sectional area of the neck;
• ${\displaystyle m}$ is the mass in the neck;
• ${\displaystyle P_{0}}$ is the static pressure in the cavity;
• ${\displaystyle V_{0}}$ is the static volume of the cavity.

For cylindrical or rectangular necks, we have

${\displaystyle A={\frac {V_{n}}{L_{eq}}}}$,

where:

• ${\displaystyle L_{eq}}$ is the equivalent length of the neck with end correction, which can be calculated as :${\displaystyle L_{eq}=L_{n}+0.3D}$, where ${\displaystyle L_{n}}$ is the actual length of the neck and ${\displaystyle D}$ is the hydraulic diameter of the neck;[4]
• ${\displaystyle V_{n}}$ is the volume of air in the neck,

thus:

${\displaystyle \omega _{H}={\sqrt {\gamma {\frac {A}{m}}{\frac {V_{n}}{L_{eq}}}{\frac {P_{0}}{V_{0}}}}}}$.

From the definition of mass density (${\displaystyle {\rho }}$): ${\displaystyle {\frac {V_{n}}{m}}={\frac {1}{\rho }}}$.

The speed of sound in a gas is given by:

${\displaystyle v={\sqrt {\gamma {\frac {P_{0}}{\rho }}}}}$ ,

thus, the resonance frequency is:

${\displaystyle f_{H}={\frac {v}{2\pi }}{\sqrt {\frac {A}{V_{0}L_{eq}}}}}$.

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.[5] 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.

Depending on the exact shape of the hole, the relative thickness of the sheet with respect to the size of the hole and the size of the cavity, this formula can have limitations. More sophisticated formulae can still be derived analytically, with similar physical explanations (although some differences matter). See for example the book by F. Mechel.[6] Furthermore, if the mean flow over the resonator is high (typically with a Mach number above 0.3), some corrections must be applied.

• Acoustic resonance#Resonance of a sphere of air (vented) for more detailed acoustics (physics perspective)
• Vessel flute for more detailed acoustics (musical perspective)
• Xun (instrument), an instrument that is a Helmholtz resonator with holes
• Resonance