View source for Particle velocity

{{short description|Velocity of a particle in a medium as it transmits a wave}}
{{Sound measurements}}

'''Particle velocity''' (denoted {{mvar|v}} or {{math|SVL}}) is the [[velocity]] of a [[particle]] (real or imagined) in a [[Transmission medium|medium]] as it transmits a [[wave]]. The [[International System of Units|SI unit]] of particle velocity is the metre per second (m/s). In many cases this is a [[longitudinal wave]] of [[pressure]] as with [[sound]], but it can also be a [[transverse wave]] as with the vibration of a taut string.

When applied to a sound wave through a medium of a fluid like air, particle velocity would be the physical speed of a [[Fluid parcel|parcel of fluid]] as it moves back and forth in the direction the sound wave is travelling as it passes.

Particle velocity should not be confused with the speed of the [[wave]] as it passes through the medium, i.e. in the case of a sound wave, particle velocity is not the same as the [[speed of sound]]. The wave moves relatively fast, while the particles oscillate around their original position with a relatively small particle velocity. Particle velocity should also not be confused with the velocity of individual molecules, [[Kinetic theory of gases#Speed of molecules|which depends mostly on the temperature and molecular mass]].

In applications involving sound, the particle velocity is usually measured using a logarithmic [[decibel]] scale called [[particle velocity level]]. Mostly pressure sensors (microphones) are used to measure sound pressure which is then propagated to the velocity field using [[Green's function]].

==Mathematical definition==
Particle velocity, denoted <math>\mathbf v</math>, is defined by
:<math>\mathbf v = \frac{\partial \mathbf \delta}{\partial t}</math>
where <math>\delta</math> is the [[particle displacement]].

==Progressive sine waves==
The particle displacement of a ''progressive [[sine wave]]'' is given by
:<math>\delta(\mathbf{r},\, t) = \delta_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}),</math>
where
*<math>\delta_\mathrm{m}</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_\mathrm{m} \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_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{p, 0}),</math>
where
*<math>v_\mathrm{m}</math> is the amplitude of the particle velocity;
*<math>\varphi_{v, 0}</math> is the phase shift of the particle velocity;
*<math>p_\mathrm{m}</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 <math>v</math> and <math>p</math> with respect to time yields
:<math>\hat{v}(\mathbf{r},\, s) = v_\mathrm{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},</math>
:<math>\hat{p}(\mathbf{r},\, s) = p_\mathrm{m} \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_\mathrm{m}(\mathbf{r},\, s) = |z(\mathbf{r},\, s)| = \left|\frac{\hat{p}(\mathbf{r},\, s)}{\hat{v}(\mathbf{r},\, s)}\right| = \frac{p_\mathrm{m}}{v_\mathrm{m}} = \frac{\rho c^2 k_x}{\omega}.</math>

Consequently, the amplitude of the particle velocity is related to those of the particle displacement and the sound pressure by
:<math>v_\mathrm{m} = \omega \delta_\mathrm{m},</math>
:<math>v_\mathrm{m} = \frac{p_\mathrm{m}}{z_\mathrm{m}(\mathbf{r},\, s)}.</math>

==Particle velocity level==
{{Other uses|Sound level (disambiguation){{!}}Sound level}}

'''Sound velocity level''' (SVL) or '''acoustic velocity level''' or '''particle velocity level''' is a [[Level (logarithmic quantity)|logarithmic measure]] of the effective particle velocity of a sound relative to a reference value.<br>
Sound velocity level, denoted ''L''<sub>''v''</sub> and measured in [[Decibel|dB]], is defined by<ref name=IEC60027-3>[http://webstore.iec.ch/webstore/webstore.nsf/artnum/028981 "Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units"], ''IEC 60027-3 Ed. 3.0'', International Electrotechnical Commission, 19 July 2002.</ref>
:<math>L_v = \ln\!\left(\frac{v}{v_0}\right)\!~\mathrm{Np} = 2 \log_{10}\!\left(\frac{v}{v_0}\right)\!~\mathrm{B} = 20 \log_{10}\!\left(\frac{v}{v_0}\right)\!~\mathrm{dB},</math>
where
*''v'' is the [[root mean square]] particle velocity;
*''v''<sub>0</sub> is the ''reference particle velocity'';
*{{no break|1=1 Np = 1}} is the [[neper]];
*{{no break|1=1 B = {{sfrac|1|2}} ln 10}} is the [[Decibel|bel]];
*{{no break|1=1 dB = {{sfrac|1|20}} ln 10}} is the [[decibel]].

The commonly used reference particle velocity in air is<ref>Ross Roeser, Michael Valente, ''Audiology: Diagnosis'' (Thieme 2007), p. 240.</ref>
:<math>v_0 = 5 \times 10^{-8}~\mathrm{m/s}.</math>
The proper notations for sound velocity level using this reference are {{nobreak|''L''<sub>''v''/(5 × 10<sup>−8</sup> m/s)</sub>}} or {{nobreak|''L''<sub>''v''</sub> (re 5 × 10<sup>−8</sup> m/s)}}, but the notations {{nobreak|dB SVL}}, {{nobreak|dB(SVL)}}, dBSVL, or dB<sub>SVL</sub> are very common, even though they are not accepted by the SI.<ref name=NIST2008>Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel", ''Guide for the Use of the International System of Units (SI) 2008 Edition'', NIST Special Publication 811, 2nd printing (November 2008), SP811 [http://physics.nist.gov/cuu/pdf/sp811.pdf PDF]</ref>

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

==References==
{{Reflist}}

==External links==
*[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]
*[https://www.microflown.com The particle Velocity Can Be Directly Measured with a Microflown]
*[http://www.weles-acoustics.com/en/technologies/particle-velocity-sensor/ Particle velocity measured with Weles Acoustics sensor - working principle]
*[https://oro.open.ac.uk/44496/1/ali_tonddast_navaei_thesis.pdf Acoustic Particle-Image Velocimetry. Development and Applications]

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