Sympathetic resonance
Encyclopedia
Sympathetic resonance is a 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...

 phenomenon wherein a formerly passive string or vibratory body responds to external vibrations to which it has a harmonic likeness. The classic example is demonstrated with two similar tuning-forks of which one is mounted on a wooden box. If the other one is struck and then placed on the box, then muted, the un-struck mounted fork will be heard. In similar fashion, strings will respond to the external vibrations of a tuning-fork when sufficient harmonic relations exist between the respective vibratory modes. A unison or octave will provoke the largest response as there is maximum likeness in vibratory motion. Other links through shared resonances occur at the fifth and third though with less effect. The principle of sympathetic resonance has been applied in musical instruments from many cultures and times. Apart from the basic principle at work on instruments with many undamped strings, such as harps, flutes, guitars and pianos with the dampers raised, other instruments are fitted with extra choirs of 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...

s, which respond with a silvery halo to the tones played on the main strings.

Lewcock et al.(2006) states that:

The property of sympathetic vibration is encountered in its direct form in room acoustics in the rattling of window panes, light shades and movable panels in the presence of very loud sounds, such as may occasionally be produced by a full organ. As these things rattle (or even if they do not audibly rattle) sound energy is being converted into mechanical energy, and so the sound is absorbed. Wood paneling and anything else that is lightweight and relatively unrestrained have the same effect. Absorptivity is at its highest at the resonance frequency, usually near or below 100 Hz.


Arden Wilken on his website provides a significant example of the power of resonance:

An example of proper sympathetic resonance is a windowpane rattling steadily at the very low powerful sound of a bus or truck engine going stationary. The rattling will usually occur at a higher harmonic of the sound made by the engine. As soon as the driver changes into gear the rattling will stop, often changing its rhythm before it stops altogether. Powerful sopranos bursting wineglasses fits in to the same category - sympathetic resonance at a distance.

Failure of the original Tacoma Narrows Bridge

The dramatically visible, rhythmic twisting that resulted in the 1940 collapse of the original Tacoma Narrows Bridge
Tacoma Narrows Bridge (1940)
The 1940 Tacoma Narrows Bridge was the first incarnation of the Tacoma Narrows Bridge, a suspension bridge in the U.S. state of Washington that spanned the Tacoma Narrows strait of Puget Sound between Tacoma and the Kitsap Peninsula. It opened to traffic on July 1, 1940, and dramatically collapsed...

, has sometimes been characterized in physics textbooks as a classical example of resonance; however, this description is insufficient. The catastrophic vibrations that destroyed the bridge were not due only to mechanical resonance, but to a more complicated oscillation between the bridge and the winds passing through it — a phenomenon known as aeroelastic flutter. Robert H. Scanlan
Robert H. Scanlan
Robert H. Scanlan was a civil and aeronautical engineer who came to be widely recognized as a leader in the analysis of wind effects on large structures. Scanlan created the concept of flutter derivatives to aid in the representation of self-excited forces in theoretical models...

, father of the field of bridge aerodynamics, wrote an article about this misunderstanding.

String resonance in music instruments

String resonance occurs on string instruments. Strings or parts of strings may resonate at their fundamental
Fundamental
Fundamental may refer to:* Foundation of reality* Fundamental frequency, as in music or phonetics, often referred to as simply a "fundamental"...

 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).

According to Grove Music Online (2007) article on Duplex Scaling, Steinway
Steinway & Sons
Steinway & Sons, also known as Steinway , is an American and German manufacturer of handmade pianos, founded 1853 in Manhattan in New York City by German immigrant Heinrich Engelhard Steinweg...

 progressed a system of aliquot scaling
Aliquot stringing
Aliquot stringing is the use of extra, unstruck strings in the piano for the purpose of enriching the tone.-History:Julius Blüthner invented the aliquot stringing system in 1873....

 to provide sympathetic resonance with the intention of enriching the treble register of the pianoforte. In the 'octave duplex' piano by Hoerr of Toronto
Toronto
Toronto is the provincial capital of Ontario and the largest city in Canada. It is located in Southern Ontario on the northwestern shore of Lake Ontario. A relatively modern city, Toronto's history dates back to the late-18th century, when its land was first purchased by the British monarchy from...

, each note had four strings, of which two, three or four could potentially be struck by the hammer depending on the depression of one of four pedals. Steinway
Steinway & Sons
Steinway & Sons, also known as Steinway , is an American and German manufacturer of handmade pianos, founded 1853 in Manhattan in New York City by German immigrant Heinrich Engelhard Steinweg...

’s duplex scale was precipitated a half century earlier by an experiment conducted by the German piano maker Wilhelm Leberecht Petzoldt, in which a small bridge was placed behind the standard larger one with the intention of maximizing the potential additional 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...

of a sympathetically vibrating additional length of string.
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