Helmholtz resonance

Helmholtz resonance or wind throb is the phenomenon of air resonance in a cavity, such as when one blows across the top of an empty bottle. The name comes from a device created in the 1850s by Hermann von Helmholtz, the "Helmholtz resonator", which he used to identify the various frequencies or musical pitches present in music and other complex sounds.
History.
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.
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 external meatus of 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.
Quantitative explanation.
It can be shown that the resonant angular frequency is given by:
where:
For cylindrical or rectangular necks, we have
where:
thus:
From the definition of mass density (formula_14): formula_15.
The speed of sound in a gas is given by:
thus, the resonance frequency is:
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.
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. Mechels. Furthermore, if the mean flow over the resonator is high (typically with a Mach number above 0.3), some corrections must be applied.
Applications.
Helmholtz resonance finds application in internal combustion engines (see airbox), subwoofers and acoustics. Intake systems described as 'Helmholtz Systems' have been used in the Chrysler V10 engine built for both the Dodge Viper and the Ram pickup truck, and several of the Buell tube-frame series of motorcycles. In stringed instruments as old as the veena or sitar, or as recent 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 combined area of the opened finger holes determines the note played by the instrument. The West African djembe is the original Helmholtz resonator with a small neck area, giving it a deep bass tone. It has been in use for thousands of years.
Conversely, the human mouth is effectively a Helmholtz resonator when it is used in conjunction with a jaw harp, shepherd's whistle, nose whistle, nose flute. The nose blows air through an open nosepiece, into an air duct, and across an edge adjacent to the open mouth, creating the resonator. The volume and shape of the mouth cavity augments the pitch of the tone.
The theory of Helmholtz resonators is used in motorcycle and car exhausts to alter the sound of the exhaust note and for differences in power delivery by adding chambers to the exhaust. Exhaust resonators are also used to reduce potentially loud and obnoxious engine noise where the dimensions are calculated so that the waves reflected by the resonator help cancel out certain frequencies of sound in the exhaust.
In some two-stroke engines, a Helmholtz resonator is used to remove the need for a reed valve. A similar effect is also used in the exhaust system of most two-stroke engines, using a reflected pressure pulse to supercharge the cylinder (see Kadenacy effect.)
Vitruvius, a 1st-century B.C. Roman architect, described the use of bronze or pottery resonators in classical theater design.
Helmholtz resonators are used in architectural acoustics to reduce undesirable low frequency sounds (standing waves, etc.) by building a resonator tuned to the problem frequency, thereby eliminating it.
Helmholtz resonance is also used in Bass-reflex speaker enclosures, with the compliance of the air mass inside the enclosure and the mass of air in the port forming a helmholtz resonator. By tuning the resonant frequency of the helmholtz resonator to the lower end of the loudspeaker's usable frequency range, the speaker's low-frequency performance is improved.
Helmholtz resonators are also used to build acoustic liners for reducing the noise of aircraft engines, for example. These acoustic liners are made of two components: 
Such acoustic liners are used in most of today's aircraft engines. The perforated sheet is usually visible from inside or outside the airplane; the honeycomb is just under it. The thickness of the perforated sheet is of importance, as shown above. Sometimes there are two layers of liners; they are then called "2-DOF liners" (DOF meaning Degrees Of Freedom), as opposed to "single DOF liners".
This effect might also be used to reduce skin friction drag on aircraft wings by 40%.
Helmholtz resonance sometimes occurs when a slightly open single car window makes a very loud sound, also called side window buffeting or wind throb.
Helmholtz resonance is one of the principles behind the way piezoelectric buzzers work: a piezoelectric disc acts as the excitation source, but it relies on the acoustic cavity resonance to produce an audible sound.