Acoustic resonance

Acoustic resonance is the tendency of an acoustic system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration (its "resonance frequency") than it does at other frequencies.

A resonant object will probably have more than one resonance frequency, especially at harmonics of the strongest resonance. It will easily vibrate at those frequencies, and vibrate less strongly at other frequencies. It will "pick out" its resonance frequency from a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its resonance.

Acoustic resonance is an important consideration for instrument builders, as most acoustic instruments use resonators, such as the strings and body of a violin, the length of tube in a flute, and the shape of a drum membrane.

Resonance of a string

Strings under tension, as in instruments such as lutes, harps, guitars, pianos, violins and so forth, have resonant frequencies directly related to the mass, length, and tension of the string. The wavelength that will create the first resonance on the string is equal to twice the length of the string. Higher resonances correspond to wavelengths that are integer divisions of the fundamental wavelength. The corresponding frequencies are related to the speed "v" of a wave traveling down the string by the equation

:$f\; =\; \{nv\; over\; 2L\}$

where "L" is the length of the string (for a string fixed at both ends) and "n" = 1, 2, 3... The speed of a wave through a string or wire is related to its tension "T" and the mass per unit length ρ:

:$v\; =\; sqrt\; \{T\; over\; ho\}$

So the frequency is related to the properties of the string by the equation

:$f\; =\; \{nsqrt\; \{T\; over\; ho\}\; over\; 2\; L\}\; =\; \{nsqrt\; \{T\; over\; m\; /\; L\}\; over\; 2\; L\}$

where "T" is the tension, ρ is the mass per unit length, and "m" is the total mass.

Higher tension and shorter lengths increase the resonant frequencies. When the string is excited with an impulsive function (a finger pluck or a strike by a hammer), the string vibrates at all the frequencies present in the impulse (an impulsive function theoretically contains 'all' frequencies). Those frequencies that are not one of the resonances are quickly filtered out—they are attenuated—and all that is left is the harmonic vibrations that we hear as a musical note.

tring resonance in music instruments

String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental or overtone frequencies when other strings are sounded. For example, an A string at 440 Hz will cause an E string at 330 Hz to resonate, because they share an overtone of 1320 Hz (3rd overtone of A and 4th overtone of E).

Resonance of a tube of air

The resonance of a tube of air is related to the length of the tube, its shape, and whether it has closed or open ends. Musically useful tube shapes are "conical" and "cylindrical" (see bore). A pipe that is closed at one end is said to be "stopped" while an "open" pipe is open at both ends. Modern orchestral flutes behave as open cylindrical pipes; clarinets and lip-reed instruments (brass instruments) behave as closed cylindrical pipes; and saxophones, oboes, and bassoons as closed conical pipes. Vibrating air columns also have resonances at harmonics, like strings.

Cylinders

By convention a rigid cylinder that is open at both ends is referred to as an "open" cylinder; whereas, a rigid cylinder that is open at one end and has a rigid surface at the other end is referred to as a "closed" cylinder.

Open

Open cylindrical tubes resonate at the approximate frequencies

:$f\; =\; \{nv\; over\; 2L\}$

where "n" is a positive integer (1, 2, 3...) representing the resonance node, "L" is th the length of the tube and "v" is the speed of sound in air (which is approximately 344 meters per second at 20 °C and at sea level).

A more accurate equation considering an end correction is given below:

:$f\; =\; \{nv\; over\; 2(L+0.8d)\}$

where d is the diameter of the resonance tube. This equation compensates for the fact that the exact point at which a sound wave is reflecting at an open end is not perfectly at the end section of the tube, but a small distance outside the tube.

The reflection ratio is slightly less than 1; the open end does not behave like an infinite acoustic impedance; rather, it has a finite value, called radiation impedance, which is dependent on the diameter of the tube, the wavelength, and the type of reflection board possibly present around the opening of the tube.

Closed

A closed cylinder will have approximate resonances of

:$f\; =\; \{nv\; over\; 4L\}$

where "n" here is an odd number (1, 3, 5...). This type of tube produces only odd harmonics and has its fundamental frequency an octave lower than that of an open cylinder (that is, half the frequency).

A more accurate equation is given below:

$f\; =\; \{nv\; over\; 4(L+0.4d)\}$.

Cones

An open conical tube, that is, one in the shape of a frustum of a cone with both ends open, will have resonant frequencies approximately equal to those of an open cylindrical pipe of the same length.

The resonant frequencies of a stopped conical tube — a complete cone or frustum with one end closed — satisfy a more complicated condition:

:$kL\; =\; npi\; -\; an^\{-1\}\; kx$

where the wavenumber k is

:$k\; =\; 2pi\; f/v$

and "x" is the distance from the small end of the frustum to the vertex. When "x" is small, that is, when the cone is nearly complete, this becomes

:$k(L+x)\; approx\; npi$

leading to resonant frequencies approximately equal to those of an open cylinder whose length equals "L" + "x". In words, a complete conical pipe behaves approximately like an open cylindrical pipe of the same length, and to first order the behavior does not change if the complete cone is replaced by a closed frustum of that cone.

Rectangular box

For a rectangular box, the resonant frequencies are given by

:$f\; =\; \{v\; over\; 2\}\; sqrt\{left(\{ell\; over\; L\_x\}\; ight)^2\; +\; left(\{m\; over\; L\_y\}\; ight)^2\; +\; left(\{n\; over\; L\_z\}\; ight)^2\}$

where "v" is the speed of sound, "L_x" and "L_y" and "L_z" are the dimensions of the box, and $scriptstyleell,$ "n", and "m" are the nonnegative integers. However, $scriptstyleell$, "n", and "m" cannot all be zero.

Resonance in musical composition

Composers have begun to make resonance the subject of compositions. Alvin Lucier has used acoustic instruments and sine wave generators to explore the resonance of objects large and small in many of his compositions. The complex inharmonic partials of a swell shaped crescendo and decrescendo on a tam tam or other percussion instrument interact with room resonances in James Tenney's "Koan: Having Never Written A Note For Percussion". Pauline Oliveros and Stuart Dempster regularly perform in large reverberant spaces such as the two million gallon cistern at Fort Warden, WA, which has a reverb with a 45-second decay.

References

* Nederveen, Cornelis Johannes, "Acoustical aspects of woodwind instruments". Amsterdam, Frits Knuf, 1969.
* Rossing, Thomas D., and Fletcher, Neville H., "Principles of Vibration and Sound". New York, Springer-Verlag, 1995.

See also

* Harmony
* Music theory
* Resonance
* Sympathetic string