# Particle velocity

Sound measurements
Sound pressure p, SPL
Particle velocity v, SVL
Particle displacement ξ
Sound intensity I, SIL
Sound power Pac
Sound power level SWL
Sound energy
Sound exposure E
Sound exposure level SEL
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound
Audio frequency AF

Particle velocity is the velocity of a particle (real or imagined) in a medium as it transmits a wave. 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 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.

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. Only since recent years it is possible to directly measure particle velocity with a Microflown sensor.Template:Fact

## Mathematical definition

Particle velocity, denoted v and measured in m·s−1, is given by:

${\displaystyle \mathbf {v} ={\frac {\partial \mathbf {\xi } }{\partial t}}}$

where ξ is the particle displacement, measured in m.

## Equations in terms of other measurements

Particle velocity can be related to other sound measurements:

${\displaystyle v={\frac {p}{{\mathfrak {R}}(z)}}={\sqrt {\frac {P}{A{\mathfrak {R}}(z)}}}={\sqrt {\frac {I}{{\mathfrak {R}}(z)}}}={\sqrt {\frac {cw}{{\mathfrak {R}}(z)}}},}$

For sine waves with angular frequency ω, the amplitude of the particle velocity can be related to those of the particle displacement and the particle acceleration:

${\displaystyle v_{\mathrm {m} }(\mathbf {r} )=\omega \xi _{\mathrm {m} }(\mathbf {r} )={\frac {a_{\mathrm {m} }(\mathbf {r} )}{\omega }}.}$
Symbol Unit Meaning
c m·s−1 speed of sound
v m·s−1 particle velocity
z Pa·m−1·s specific acoustic impedance
A m2 area
p Pa sound pressure
P W sound power
I W·m−2 sound intensity
w J·m−3 sound energy density
ξ m particle displacement
a m·s−2 particle acceleration

## Particle velocity level

Sound velocity level (SVL) or acoustic velocity level is a logarithmic measure of particle velocity in comparison to a reference level.
Particle velocity level, denoted Lv and measured in dB, is given by:

${\displaystyle L_{v}=20\,\log _{10}\left({\frac {v}{v_{0}}}\right)\mathrm {dB} }$

where:

• v is the particle velocity, measured in m·s−1;
• v0 is the reference particle velocity, measured in m·s−1.

If v0 is the standard reference particle velocity:{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}

${\displaystyle v_{0}=5.0\times 10^{-8}~\mathrm {m} \cdot \mathrm {s} ^{-1},}$

then dB SVL (sound velocity level) are used.