Helmholtz resonance

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

Helmholtz resonance 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, the author of the classic study of acoustic science, used to identify the various frequencies or musical pitches present in music and other complex sounds.[1]

Qualitative explanation[edit]

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. 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.

The air in the port (the neck of the chamber) has 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.

Quantitative explanation[edit]

It can be shown[2] that the resonant angular frequency is given by:

\omega_{H} = \sqrt{\gamma\frac{A^2}{m} \frac{P_0}{V_0}} (rad/s),

where:

  • \gamma (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 neck;
  • P_0 is the static pressure in the cavity;
  • V_0 is the static volume of the cavity.

For cylindrical or rectangular necks, we have

 A = \frac{V_n}{L_{eq}} ,

where:

  • L_{eq} is the equivalent length of the neck with end correction, which can be calculated as : L_{eq} = L_n + 0.6 D , where L_n is the actual length of the neck and D is the hydraulic diameter of the neck;[3]
  • V_n is the volume of air in the neck,

thus:

\omega_{H} = \sqrt{\gamma\frac{A}{m} \frac{V_n}{L_{eq}} \frac{P_0}{V_0}}.

From the definition of mass density ({\rho}):  \frac{V_n}{m} = \frac{1}{\rho} , thus:

\omega_{H} = \sqrt{\gamma\frac{P_0}{\rho} \frac{A}{V_0 L_{eq}}} ,

and

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

where:

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_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. 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.[4] Furthermore, if the mean flow over the resonator is high (typically with a Mach number above 0.3), some corrections must be applied.

Applications[edit]

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, 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 combined area of the opened finger holes determines the note played by the instrument.[5] The West African djembe is a Helmholtz resonator with a small neck area, giving it a deep bass 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.)

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 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:

  • a simple sheet of metal (or other material) perforated with little holes spaced out in a regular or irregular pattern; this is called a resistive sheet;
  • a series of so-called honeycomb cavities (holes with a honeycomb shape, but in fact only their volume matters).

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%.[6]

Helmholtz resonance sometimes occurs when a slightly open single car window makes a very loud sound, also called "side window buffeting".[7]

Notes[edit]

Sources[edit]