Particle velocity

Sound measurements
Characteristic	Symbols
Sound pressure	p, SPL
Particle velocity	v, SVL
Particle displacement	δ
Sound intensity	I, SIL
Sound power	P, SWL
Sound energy	W
Sound energy density	w
Sound exposure	E, SEL
Acoustic impedance	Z
Speed of sound	c
Audio frequency	AF
Transmission loss	TL
	v; t; e;

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

Mathematical definition

Particle velocity, denoted v, is defined by

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

where δ is the particle displacement.

Progressive sine waves

The particle displacement of a progressive sine wave is given by

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

where

δ_m is the amplitude of the particle displacement;
$\varphi_{\delta, 0}$ is the phase shift of the particle displacement;
k is the angular wavevector;
ω 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

v(\mathbf{r},\, t) = \frac{\partial \delta}{\partial t} (\mathbf{r},\, t) = \omega \delta_\mathrm{m} \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}{\partial x} (\mathbf{r},\, t) = \rho c^2 k_x \delta_\mathrm{m} \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_m is the amplitude of the particle velocity;
$\varphi_{v, 0}$ is the phase shift of the particle velocity;
p_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)}.