Editing Particle displacement

{{Sound measurements}}
'''Particle displacement''' or '''displacement amplitude''' is a [[measurement]] of [[distance]] of the movement of a [[sound particle]] from its [[mechanical equilibrium|equilibrium]] position in a medium as it transmits a sound wave.<ref>
{{cite book
 | title = Microsensors, MEMS, and Smart Devices John 2 | year = 2001
 | isbn = 978-0-471-86109-6
 | pages = 23–322
 | url = https://books.google.com/books?id=ht69jE9ypSwC&pg=PA321
 | last1 = Gardner
 | first1 = Julian W.
 | last2 = Varadan
 | first2 = Vijay K.
 | last3 = Awadelkarim
 | first3 = Osama O.
 | publisher = Wiley
 }}</ref>  
The [[International System of Units|SI unit]] of particle displacement is the [[metre]] (m). 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 = https://archive.org/details/bub_gb_Zb4KAAAAIAAJ
 | quote= 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 sound wave traveling through the medium, while the sound wave itself moves at the [[speed of sound]], equal to {{nobreak|343 m/s}} in air at {{nobreak|20 °C}}.

==Mathematical definition ==
Particle displacement, denoted '''[[Delta (letter)|δ]]''', is given by<ref>
{{cite book
  | author = John Eargle
  | author-link = 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 =https://books.google.com/books?id=w8kXMVKOsY0C&q=instantaneous+particle+displacement&pg=PA27 
  | isbn =978-0-240-51961-6
 }}</ref>
:<math>\mathbf \delta = \int_{t} \mathbf v\, \mathrm{d}t</math>
where '''v''' is the [[particle velocity]].

==Progressive sine waves==
The particle displacement of a ''progressive [[sine wave]]'' is given by
:<math>\delta(\mathbf{r},\, t) = \delta \sin(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}),</math>
where
*<math>\delta</math> is the [[amplitude]] of the particle displacement;
*<math>\varphi_{\delta, 0}</math> is the [[phase shift]] of the particle displacement;
*<math>\mathbf{k}</math> is the [[angular wavevector]];
*<math>\omega</math> is the [[angular frequency]].

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave ''x'' are given by
:<math>v(\mathbf{r},\, t) = \frac{\partial \delta(\mathbf{r},\, t)}{\partial t} = \omega \delta \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = v \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{v, 0}),</math>
:<math>p(\mathbf{r},\, t) = -\rho c^2 \frac{\partial \delta(\mathbf{r},\, t)}{\partial x} = \rho c^2 k_x \delta \cos\!\left(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0} + \frac{\pi}{2}\right) = p \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),</math>
where
*<math>v</math> is the amplitude of the particle velocity;
*<math>\varphi_{v, 0}</math> is the phase shift of the particle velocity;
*<math>p</math> is the amplitude of the acoustic pressure;
*<math>\varphi_{p, 0}</math> is the phase shift of the acoustic pressure.

Taking the Laplace transforms of ''v'' and ''p'' with respect to time yields
:<math>\hat{v}(\mathbf{r},\, s) = v \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},</math>
:<math>\hat{p}(\mathbf{r},\, s) = p \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}.</math>

Since <math>\varphi_{v,0} = \varphi_{p,0}</math>, the amplitude of the specific acoustic impedance is given by
:<math>z(\mathbf{r},\, s) = |z(\mathbf{r},\, s)| = \left|\frac{\hat{p}(\mathbf{r},\, s)}{\hat{v}(\mathbf{r},\, s)}\right| = \frac{p}{v} = \frac{\rho c^2 k_x}{\omega}.</math>

Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by
:<math>\delta = \frac{v}{\omega},</math>
:<math>\delta = \frac{p}{\omega z(\mathbf{r},\, s)}.</math>

==See also==
*[[Sound]]
*[[Sound particle]]
*[[Particle velocity]]
*[[Particle acceleration]]

==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  |author1=Strong, John Donovan |author2=Hayward, Roger |name-list-style=amp | title=Concepts of Classical Optics  | publisher=Dover Publications | date=January 2004
 | isbn= 978-0-486-43262-5 }}
*{{cite book   | last = Barron  | first = Randall F.
  | title = Industrial noise control and acoustics
  | publisher = CRC Press   | date = January 2003
  | location = NYC, New York   | pages = 79, 82, 83, 87
  | url =https://books.google.com/books?id=k1tXPl2hC-cC&q=instantaneous+particle+displacement&pg=PA82 
  | 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]

[[Category:Acoustics]]
[[Category:Sound]]
[[Category:Sound measurements]]
[[Category:Physical quantities]]