# Particle velocity

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**Particle velocity** (denoted v or SVL) is the [velocity](/source/Velocity) of a [particle](/source/Particle) (real or imagined) in a [medium](/source/Transmission_medium) as it transmits a [wave](/source/Wave). The [SI unit](/source/International_System_of_Units) of particle velocity is the metre per second (m/s). In many cases this is a [longitudinal wave](/source/Longitudinal_wave) of [pressure](/source/Pressure) as with [sound](/source/Sound), but it can also be a [transverse wave](/source/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 [parcel of fluid](/source/Fluid_parcel) 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](/source/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](/source/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, [which depends mostly on the temperature and molecular mass](/source/Kinetic_theory_of_gases#Speed_of_molecules).

In applications involving sound, the particle velocity is usually measured using a logarithmic [decibel](/source/Decibel) scale called [particle velocity level](/source/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](/source/Green's_function).

## Mathematical definition

Particle velocity, denoted \mathbf v, is defined by

- \mathbf v = \frac{\partial \mathbf \delta}{\partial t}

where \delta is the [particle displacement](/source/Particle_displacement).

## Progressive sine waves

The particle displacement of a *progressive [sine wave](/source/Sine_wave)* is given by

- \delta(\mathbf{r},\, t) = \delta_\mathrm{m} \cos(\mathbf{k} \cdot \mathbf{r} - \omega t + \varphi_{\delta, 0}),

where

- \delta_\mathrm{m} is the [amplitude](/source/Amplitude) of the particle displacement;
- \varphi_{\delta, 0} is the [phase shift](/source/Phase_shift) of the particle displacement;
- \mathbf{k} is the [angular wavevector](/source/Angular_wavevector);
- \omega is the [angular frequency](/source/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

- 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}),
- 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}),

where

- v_\mathrm{m} is the amplitude of the particle velocity;
- \varphi_{v, 0} is the phase shift of the particle velocity;
- p_\mathrm{m} is the amplitude of the acoustic pressure;
- \varphi_{p, 0} is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields

- \hat{v}(\mathbf{r},\, s) = v_\mathrm{m} \frac{s \cos \varphi_{v,0} - \omega \sin \varphi_{v,0}}{s^2 + \omega^2},
- \hat{p}(\mathbf{r},\, s) = p_\mathrm{m} \frac{s \cos \varphi_{p,0} - \omega \sin \varphi_{p,0}}{s^2 + \omega^2}.

Since \varphi_{v,0} = \varphi_{p,0}, the amplitude of the specific acoustic impedance is given by

- 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}.

Consequently, the amplitude of the particle velocity is related to those of the particle displacement and the sound pressure by

- v_\mathrm{m} = \omega \delta_\mathrm{m},
- v_\mathrm{m} = \frac{p_\mathrm{m}}{z_\mathrm{m}(\mathbf{r},\, s)}.

## Particle velocity level

For other uses, see [Sound level (disambiguation)](/source/Sound_level_(disambiguation)).

**Sound velocity level** (SVL) or **acoustic velocity level** or **particle velocity level** is a [logarithmic measure](/source/Level_(logarithmic_quantity)) of the effective particle velocity of a sound relative to a reference value. Sound velocity level, denoted *L**v* and measured in [dB](/source/Decibel), is defined by[1]

- 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},

where

- *v* is the [root mean square](/source/Root_mean_square) particle velocity;
- *v*0 is the *reference particle velocity*;
- is the [neper](/source/Neper);
- is the [bel](/source/Decibel);
- is the [decibel](/source/Decibel).

The commonly used reference particle velocity in air is[2]

- v_0 = 5 \times 10^{-8}~\mathrm{m/s}.

The proper notations for sound velocity level using this reference are *L**v*/(5 × 10−8 m/s) or *L**v* (re 5 × 10−8 m/s), but the notations dB SVL, dB(SVL), dBSVL, or dBSVL are very common, even though they are not accepted by the SI.[3]

## See also

- [Sound](/source/Sound)
- [Sound particle](/source/Sound_particle)
- [Particle displacement](/source/Particle_displacement)
- [Particle acceleration](/source/Particle_acceleration)

## References

1. ["Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units"](http://webstore.iec.ch/webstore/webstore.nsf/artnum/028981), *IEC 60027-3 Ed. 3.0*, International Electrotechnical Commission, 19 July 2002.

1. Ross Roeser, Michael Valente, *Audiology: Diagnosis* (Thieme 2007), p. 240.

1. 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 [PDF](http://physics.nist.gov/cuu/pdf/sp811.pdf)

## External links

- [Ohm's Law as Acoustic Equivalent. Calculations](http://www.sengpielaudio.com/calculator-ak-ohm.htm)
- [Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave](http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf)
- [The particle Velocity Can Be Directly Measured with a Microflown](https://www.microflown.com)
- [Particle velocity measured with Weles Acoustics sensor - working principle](http://www.weles-acoustics.com/en/technologies/particle-velocity-sensor/)
- [Acoustic Particle-Image Velocimetry. Development and Applications](https://oro.open.ac.uk/44496/1/ali_tonddast_navaei_thesis.pdf)

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Adapted from the Wikipedia article [Particle velocity](https://en.wikipedia.org/wiki/Particle_velocity) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Particle_velocity?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.