Acoustic resonance

# Acoustic resonance

Overview
Acoustic resonance is the tendency of an acoustic system
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

to absorb more energy when it is forced or driven at a frequency that matches one of its own natural frequencies of vibration (its resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

frequency
) than it does at other frequencies.

The term acoustic resonance is sometimes used to narrow mechanical resonance
Mechanical resonance
Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies...

to the frequency range of human hearing, but since acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

is defined in general terms concerning vibrational waves in matter acoustic resonance can occur at frequencies outside the range of human hearing.

An acoustically resonant object usually has more than one resonance frequency, especially at harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

s of the strongest resonance.
Discussion

Recent Discussions
Encyclopedia
Acoustic resonance is the tendency of an acoustic system
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

to absorb more energy when it is forced or driven at a frequency that matches one of its own natural frequencies of vibration (its resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

frequency
) than it does at other frequencies.

The term acoustic resonance is sometimes used to narrow mechanical resonance
Mechanical resonance
Mechanical resonance is the tendency of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies...

to the frequency range of human hearing, but since acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

is defined in general terms concerning vibrational waves in matter acoustic resonance can occur at frequencies outside the range of human hearing.

An acoustically resonant object usually has more than one resonance frequency, especially at harmonic
Harmonic
A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental...

s 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
Musical instrument
A musical instrument is a device created or adapted for the purpose of making musical sounds. In principle, any object that produces sound can serve as a musical instrument—it is through purpose that the object becomes a musical instrument. The history of musical instruments dates back to the...

use resonator
Resonator
A resonator is a device or system that exhibits resonance or resonant behavior, that is, it naturally oscillates at some frequencies, called its resonant frequencies, with greater amplitude than at others. The oscillations in a resonator can be either electromagnetic or mechanical...

s, such as the strings and body of a violin
Violin
The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....

, the length of tube in a flute
Flute
The flute is a musical instrument of the woodwind family. Unlike woodwind instruments with reeds, a flute is an aerophone or reedless wind instrument that produces its sound from the flow of air across an opening...

, and the shape of a drum membrane. Acoustic resonance is also important for hearing. For example, resonance of a stiff structural element, called the basilar membrane
Basilar membrane
The basilar membrane within the cochlea of the inner ear is a stiff structural element that separates two liquid-filled tubes that run along the coil of the cochlea, the scala media and the scala tympani .-Function:...

within the cochlea
Cochlea
The cochlea is the auditory portion of the inner ear. It is a spiral-shaped cavity in the bony labyrinth, making 2.5 turns around its axis, the modiolus....

of the inner ear
Inner ear
The inner ear is the innermost part of the vertebrate ear. In mammals, it consists of the bony labyrinth, a hollow cavity in the temporal bone of the skull with a system of passages comprising two main functional parts:...

allows hair cells on the membrane to detect sound. (For mammals the membrane by having different resonance on either end so that high frequencies are concentrated on one end and low frequencies on the other.)

Like mechanical resonance, acoustic resonance can result in catastrophic failure of the vibrator. The classic example of this is breaking a wine glass with sound at the precise resonant frequency of the glass; although this is difficult in practice.

## Resonance of a string

Strings under tension, as in instruments such as lute
Lute
Lute can refer generally to any plucked string instrument with a neck and a deep round back, or more specifically to an instrument from the family of European lutes....

s, harp
Harp
The harp is a multi-stringed instrument which has the plane of its strings positioned perpendicularly to the soundboard. Organologically, it is in the general category of chordophones and has its own sub category . All harps have a neck, resonator and strings...

s, guitar
Guitar
The guitar is a plucked string instrument, usually played with fingers or a pick. The guitar consists of a body with a rigid neck to which the strings, generally six in number, are attached. Guitars are traditionally constructed of various woods and strung with animal gut or, more recently, with...

s, piano
Piano
The piano is a musical instrument played by means of a keyboard. It is one of the most popular instruments in the world. Widely used in classical and jazz music for solo performances, ensemble use, chamber music and accompaniment, the piano is also very popular as an aid to composing and rehearsal...

s, violin
Violin
The violin is a string instrument, usually with four strings tuned in perfect fifths. It is the smallest, highest-pitched member of the violin family of string instruments, which includes the viola and cello....

s and so forth, have resonant frequencies
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's 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
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...

wavelength. The corresponding frequencies are related to the speed v of a wave traveling down the string by the equation

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:

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

where T is the tension
Tension (mechanics)
In physics, tension is the magnitude of the pulling force exerted by a string, cable, chain, or similar object on another object. It is the opposite of compression. As tension is the magnitude of a force, it is measured in newtons and is always measured parallel to the string on which it applies...

, ρ is the mass per unit length, and m is the total mass
Mass
Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

.

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.

### String resonance in music instruments

String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...

or overtone
Overtone
An overtone is any frequency higher than the fundamental frequency of a sound. The fundamental and the overtones together are called partials. Harmonics are partials whose frequencies are whole number multiples of the fundamental These overlapping terms are variously used when discussing the...

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
Bore (wind instruments)
The bore of a wind instrument is its interior chamber that defines a flow path through which air travels and is set into vibration to produce sounds. The shape of the bore has a strong influence on the instruments' timbre.-Bore shapes:...

). A pipe that is closed at one end is said to be stopped while an open pipe is open at both ends. Modern orchestral flute
Flute
The flute is a musical instrument of the woodwind family. Unlike woodwind instruments with reeds, a flute is an aerophone or reedless wind instrument that produces its sound from the flow of air across an opening...

s behave as open cylindrical pipes; clarinet
Clarinet
The clarinet is a musical instrument of woodwind type. The name derives from adding the suffix -et to the Italian word clarino , as the first clarinets had a strident tone similar to that of a trumpet. The instrument has an approximately cylindrical bore, and uses a single reed...

s and lip-reed instruments (brass instrument
Brass instrument
A brass instrument is a musical instrument whose sound is produced by sympathetic vibration of air in a tubular resonator in sympathy with the vibration of the player's lips...

s) behave as closed cylindrical pipes; and saxophone
Saxophone
The saxophone is a conical-bore transposing musical instrument that is a member of the woodwind family. Saxophones are usually made of brass and played with a single-reed mouthpiece similar to that of the clarinet. The saxophone was invented by the Belgian instrument maker Adolphe Sax in 1846...

s, oboe
Oboe
The oboe is a double reed musical instrument of the woodwind family. In English, prior to 1770, the instrument was called "hautbois" , "hoboy", or "French hoboy". The spelling "oboe" was adopted into English ca...

s, and bassoon
Bassoon
The bassoon is a woodwind instrument in the double reed family that typically plays music written in the bass and tenor registers, and occasionally higher. Appearing in its modern form in the 19th century, the bassoon figures prominently in orchestral, concert band and chamber music literature...

s 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

where n is a positive integer (1, 2, 3...) representing the resonance node, L is the length of the tube and v is the speed of sound
Speed of sound
The speed of sound is the distance travelled during a unit of time by a sound wave propagating through an elastic medium. In dry air at , the speed of sound is . This is , or about one kilometer in three seconds or approximately one mile in five seconds....

in air (which is approximately 343 meters per second at 20 °C and at sea level).

A more accurate equation considering an end correction
End correction
In physics, end correction is the anomal difference between the frequency of a tuning fork and the corresponding sound waves inside of a tube. It is caused because generally there is space between the fork and the pipe end, causing the air column to vibrate a short distance beyond the edge of the...

is given below:

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 infinitesimal acoustic impedance
Acoustic impedance
The acoustic impedance at a particular frequency indicates how much sound pressure is generated by a given air vibration at that frequency. The acoustic impedance Z is frequency dependent and is very useful, for example, for describing the behaviour of musical wind instruments...

; 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

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:

.

### Cones

An open conical tube, that is, one in the shape of a frustum
Frustum
In geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....

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:

where the wavenumber
Wavenumber
In the physical sciences, the wavenumber is a property of a wave, its spatial frequency, that is proportional to the reciprocal of the wavelength. It is also the magnitude of the wave vector...

k is

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

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.

### Closed Rectangular box

Sound waves in a rectangular box include such examples as loudspeaker enclosure
Loudspeaker enclosure
A loudspeaker enclosure is a purpose-engineered cabinet in which speaker drivers and associated electronic hardware, such as crossover circuits and amplifiers, are mounted...

s and buildings. Rectangular building have resonances described as room modes. For a rectangular box, the resonant frequencies are given by

where v is the speed of sound, Lx and Ly and Lz are the dimensions of the box, and n, and m are the nonnegative integers. However, , n, and m cannot all be zero.

## Resonance of a sphere of air (Vented)

The diameter of a sphere with a sound hole is given by

where: (in meters)
D = diameter of sphere
d = diameter of sound hole
f = frequency

The diameter of a sphere with a necked sound hole is given by

where: (in meters)
D = diameter of sphere
d = diameter of sound hole
C = speed of sound
L = length of neck
f = frequency

## False tones

Some large conical instruments like tubas have a strong and useful resonance that is not in the well-known harmonic series. For example, most large B-flat tubas have a strong resonance at low E-flat (E-flat1, 39 Hz), which is between the fundamental and the second harmonic (an octave higher than the fundamental). These alternative resonances are often known as false tones or privileged tones.

The most convincing explanation for false-tones is that the horn is acting as a 'third of a pipe' rather than as a half-pipe. The bell remains an anti-node, but there would then be a node 1/3 of the way back to the mouthpiece. If so, it seems that the fundamental would be missing entirely, and would only be inferred from the overtones. However, the node and the anti-node collide in the same spot and cancel out the fundamental.

## Resonance in musical composition

Several composers have begun to make resonance the subject of compositions. Alvin Lucier
Alvin Lucier
Alvin Lucier is an American composer of experimental music and sound installations that explore acoustic phenomena and auditory perception. A long-time music professor at Wesleyan University, Lucier was a member of the influential Sonic Arts Union, which included Robert Ashley, David Behrman, and...

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
Crescendo
-In music:*Crescendo, a passage of music during which the volume gradually increases, see Dynamics * Crescendo , a Liverpool-based electronic pop band* "Diminuendo and Crescendo in Blue", one of Duke Ellington's longer-form compositions...

and decrescendo on a tamtam
TamTam
The tamtam is a percussion instrument that is similar to a gong. It is sometimes spelled tam-tam.TamTam, Tam-Tam, tamtam, or tam-tam may also refer to:* Tam-Tams, a weekly drum circle held Sundays in the summer in Montreal...

or other percussion instrument interact with room resonances in James Tenney
James Tenney
James Tenney was an American composer and influential music theorist.-Biography:Tenney was born in Silver City, New Mexico, and grew up in Arizona and Colorado. He attended the University of Denver, the Juilliard School of Music, Bennington College and the University of Illinois...

's Koan: Having Never Written A Note For Percussion. Pauline Oliveros
Pauline Oliveros
Pauline Oliveros is an American accordionist and composer who is a central figure in the development of post-war electronic art music....

and Stuart Dempster
Stuart Dempster
Stuart Dempster is a trombonist, didjeridu player, improvisor, and composer.-Biography:After Dempster completed his studies at San Francisco State College, he was appointed assistant professor at the California State College at Hayward, and instructor at the San Francisco Conservatory...

regularly perform in large reverberant
Reverberation
Reverberation is the persistence of sound in a particular space after the original sound is removed. A reverberation, or reverb, is created when a sound is produced in an enclosed space causing a large number of echoes to build up and then slowly decay as the sound is absorbed by the walls and air...

spaces such as the 2 million USgals (7,570.8 m³) cistern at Fort Worden, WA, which has a reverb
Reverberation
Reverberation is the persistence of sound in a particular space after the original sound is removed. A reverberation, or reverb, is created when a sound is produced in an enclosed space causing a large number of echoes to build up and then slowly decay as the sound is absorbed by the walls and air...

with a 45-second decay.

• Ear Damage due to Acoustic Resonance
• Harmony
Harmony
In music, harmony is the use of simultaneous pitches , or chords. The study of harmony involves chords and their construction and chord progressions and the principles of connection that govern them. Harmony is often said to refer to the "vertical" aspect of music, as distinguished from melodic...

• Music theory
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...

• Resonance
Resonance
In physics, resonance is the tendency of a system to oscillate at a greater amplitude at some frequencies than at others. These are known as the system's resonant frequencies...

• Reverberation
Reverberation
Reverberation is the persistence of sound in a particular space after the original sound is removed. A reverberation, or reverb, is created when a sound is produced in an enclosed space causing a large number of echoes to build up and then slowly decay as the sound is absorbed by the walls and air...

• Sympathetic string
Sympathetic string
Sympathetic strings or resonance strings are auxiliary strings found on many Indian musical instruments, as well as some Western Baroque instruments and a variety of folk instruments...