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{{Sound measurements}}
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'''Particle displacement''' or '''displacement amplitude''' (represented in [[mathematics]] by the [[Lower case|lower-case]] [[Greek alphabet#Main table|Greek letter]] [[Xi (letter)|&xi;]]) is a [[measurement]] of [[distance]] of the movement of a particle from its equilibrium position in a medium as it transmits a [[wave]].<ref>
{{cite book
| title = Microsensors, MEMS, and Smart Devices
| author = Julian W. Gardner, V. K. Varadan, Osama O. Awadelkarim
| publisher = John Wiley and Sons
| year = 2001
| isbn = 978-0-471-86109-6
| pages = 321–322
| url = http://books.google.com/books?id=ht69jE9ypSwC&pg=PA321
}}</ref> 
In most cases this is a [[longitudinal wave]] of pressure (such as [[sound]]), but it can also be a [[transverse wave]], such as the [[oscillation|vibration]] of a taut string.  In the case of a [[sound wave]] travelling through [[air]], the '''particle displacement''' is evident in the [[oscillation]]s of air [[molecule]]s with, and against, the direction in which the sound wave is travelling.<ref>{{cite book
| author= Arthur Schuster
| title=An Introduction to the Theory of Optics
| publisher=London: Edward Arnold
| year=1904
| url = http://books.google.com/books?id=Zb4KAAAAIAAJ&printsec=frontcover&dq=An+Introduction+to+the+Theory+of+Optics+By+Arthur+Schuster
}} </ref>
A particle of the medium undergoes displacement according to the [[particle velocity]] of the wave traveling through the medium, while the sound wave itself moves at the [[speed of sound]], equal to 343&nbsp;m/s in air at 20 °C.


The instantaneous particle displacement ''ξ'' for a wave is:<ref>
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{{cite book
  | author = John Eargle
  | authorlink = John M. Eargle
  | title = The Microphone Book: From mono to stereo to surround – a guide to microphone design and application  | publisher = Focal Press  | date = January 2005
  | location = Burlington, Ma  | pages = 27
  | url =http://books.google.com/books?id=w8kXMVKOsY0C&pg=PA27&dq=instantaneous+particle+displacement
  | isbn =978-0-240-51961-6
}}</ref>
 
:<math>
\xi = \int_{t} v\, \mathrm{d}t
</math>
 
If the wave is a standing wave or a traveling wave containing a single frequency, the particle displacement is:
:<math>
\xi = \frac{1}{Z} \int_{t} p\, \mathrm{d}t
</math>
This expression for <math>\xi</math> undergoes simple harmonic oscillation, and as such is usually expressed as an [[root mean square|RMS]] time average.
 
'''Particle displacement''' for a traveling wave containing a single frequency can be represented in terms of other measurements:
:<math>
\xi = \frac{v}{\omega}
    = \frac{p}{Z_0 \cdot \omega}
    = \frac{a}{\omega^2}
    = \frac{1}{\omega}\sqrt{\frac{I}{Z_0}}
    = \frac{1}{\omega}\sqrt{\frac{E}{\rho}}
    = \frac{1}{\omega}\sqrt{\frac{P_{ac}}{Z_0 \cdot A}}
</math>
where in the above equation, the quantities <math>\xi, v, a, I, E, P_{ac}</math> may be taken throughout as rms time-averages (or all as maximum values).  The single frequency traveling wave has acoustic impedance equal to the characteristic impedance, <math>Z=Z_0</math>. Further representations for <math>\xi</math> can be found from the above equations using the replacement <math>\omega=2\pi{f}</math>.
 
{| class="wikitable"
! Symbol !! Units !! Meaning
|-
! ''ξ''
| m, meters || Particle displacement
|-
! ''v''
| [[Meters per second|m/s]] || [[particle velocity]]
|-
! ω = 2π''f''
| [[radian]]s/[[second|s]] || [[angular frequency]]
|-
! ''f''
| Hz, [[hertz]] || [[frequency]]
|-
! ''p''
| Pa, [[pascal (unit)|pascal]]s || [[sound pressure]]
|-
! ''Z<sub>0</sub>'' = ''c'' · ''ρ''
| [[Newton (unit)|N]]·[[second|s]]/m<sup>3</sup> || [[characteristic impedance]]
|-
! ''Z'' = ''p'' / ''v''
| [[Newton (unit)|N]]·[[second|s]]/m<sup>3</sup> || [[acoustic impedance]]
|-
! ''c''
| m/[[second|s]] || [[Speed of sound]]
|-
! ''ρ''
| [[Kilogram|kg]]/m<sup>3</sup> || [[Density of air]]
|-
! ''I''
| [[Watt|W]]/m<sup>2</sup> || [[sound intensity]]
|-
! ''E''
| [[Watt|W]]·[[second|s]]/m<sup>3</sup> || [[sound energy density]]
|-
! ''P''<sub>ac</sub>
| W, [[watt]]s || [[sound power]] or [[acoustic power]]
|-
! ''A''
| m<sup>2</sup> || [[Area]]
|-
! ''a''
| m/[[second|s]]<sup>2</sup> || [[Particle acceleration]]
|}
 
== See also ==
 
* [[Wave]]
* [[Sound particle]]
 
==References and notes==
{{reflist}}
 
'''Related Reading:'''
*{{cite book | author=Wood, Robert Williams
| title=Physical optics  | publisher=New York: The Macmillan Company | year=1914}}
*{{cite book  | author=Strong, John Donovan; and  Hayward, Roger
| title=Concepts of Classical Optics  | publisher=Dover Publications | year=January 2004
| isbn= 978-0-486-43262-5 | page= }}
*{{cite book  | last = Barron  | first = Randall F.
  | authorlink =  | coauthors =    | title = Industrial noise control and acoustics
  | publisher = CRC Press  | date = January 2003
  | location = NYC, New York  | pages = 79, 82, 83, 87
  | url =http://books.google.com/books?id=k1tXPl2hC-cC&pg=PA82&dq=instantaneous+particle+displacement
  | doi =  | id =
  | isbn =978-0-8247-0701-9}}
 
== External links ==
*[http://acoustics.open.ac.uk/802574C70048F266/%28httpAssets%29/EC0466002A53AFDD802574E200381B0C/$file/ali_tonddast_navaei_thesis.pdf Acoustic particle-image velocimetry - Development and applications]
*[http://www.sengpielaudio.com/calculator-ak-ohm.htm Ohm's law as acoustic equivalent  - calculations]
*[http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf Relationships of acoustic quantities associated with a plane progressive acoustic sound wave - pdf]
 
[[Category:Sound measurements]]

Revision as of 03:43, 5 February 2014

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