Acoustic Resonance

A resonant object will probably have more than one resonant 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 resonant 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 Resonator s, 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 Lute s, Harp s, Guitar s, Piano s, Violin 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 = {n\sqrt {T \over 
ho} \over 2 L} = {n\sqrt {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.

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 one that is open at both ends. Modern orchestral Flute s behave as open cylindrical pipes; Clarinet s behave as closed cylindrical pipes; and Saxophone s, Oboe s, and Bassoon s as closed conical pipes.

Vibrating air columns also have resonances at harmonics, like strings. Open cylindrical tubes resonate at the frequencies

:

f = {nv \over 2L}

where ''n'' = 1, 2, 3... This is similar to the string formula, except ''v'' now becomes the Speed Of Sound in air (which is approximately 340 meters per second at 20 °C and at sea level).

Note however that in practise the exact point at which a sound wave is reflecting at an open end is not perfectly at the end section of the tube. The wave in fact progresses over a small distance outside the tube and the reflection ratio is also not perfectly equal to one. This phenomenon is caused by the fact that the open end does not behave like an infinite acoustical Impedance . 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.

A stopped cylindrical tube will have resonances of

:

f = {nv \over 4L}

where in this case ''n'' = 1, 3, 5... This type of tube has its fundamental frequency an octave lower than that of an open cylinder (that is, half the frequency), and can produce only odd harmonics, ''f'', 3''f'', 5''f'', and so on.

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 = n\pi - 	an^{-1} kx

where the Wavenumber k is

:

k = 2\pi 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 n\pi

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.

RESONANCE IN MUSICAL COMPOSITION

Composers have begun to make resonance the subject of compositions. 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.

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Acoustic Resonance