Particle acceleration

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second2. In acoustics or physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time2. In SI units, this is m/s2.

To accelerate an object (air particle) is to change its velocity over a period. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation [math]\displaystyle{ \mathbf{a} = \frac{d\mathbf{v}}{dt} }[/math] where

* a is the acceleration vector
* v is the velocity vector expressed in m/s
* t is time expressed in seconds.

This equation gives a the units of m/(s·s), or m/s2 (read as "metres per second per second", or "metres per second squared").

An alternative equation is: [math]\displaystyle{ \mathbf{\bar{a}} = \frac{\mathbf{v} - \mathbf{u}}{\Delta t} }[/math] where

* [math]\displaystyle{ \mathbf{\bar{a}} }[/math] is the average acceleration (m/s2)
* [math]\displaystyle{ \mathbf{u} }[/math] is the initial velocity (m/s)
* [math]\displaystyle{ \mathbf{v} }[/math] is the final velocity (m/s)
* [math]\displaystyle{ \Delta t }[/math] is the time interval (s)

Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have [math]\displaystyle{ \mathbf{a} = - \frac{v^2}{r} \frac{\mathbf{r}}{r} = - \omega^2 \mathbf{r} }[/math]

One common unit of acceleration is g-force, one g being the acceleration caused by the gravity of Earth.

In classical mechanics, acceleration [math]\displaystyle{ a }[/math] is related to force [math]\displaystyle{ F }[/math] and mass [math]\displaystyle{ m }[/math] (assumed to be constant) by way of Newton's second law: [math]\displaystyle{ F = m a }[/math]

## Equations in terms of other measurements

The Particle acceleration of the air particles a in m/s2 of a plain sound wave is: [math]\displaystyle{ a = \delta \cdot \omega^2 = v \cdot \omega = \frac{p \cdot \omega}{Z} = \omega \sqrt \frac{J}{Z} = \omega \sqrt \frac{E}{\rho} = \omega \sqrt \frac{P_\text{ac}}{Z \cdot A} }[/math]

Symbol | Units | Meaning

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a | m/s2 | particle acceleration
v | m/s | particle velocity
δ | m, meters | particle displacement
ω = 2πf | radians/s | angular frequency
f | Hz, hertz | frequency
p | Pa, pascals | sound pressure
Z | N·s/m3 | acoustic impedance
J | W/m2 | sound intensity
E | W·s/m3 | sound energy density
Pac | W, watts | sound power or acoustic power
A | m2 | area

## See also

* Sound
* Sound particle
* Particle displacement
* Particle velocity

## External links

* Relationships of acoustic quantities associated with a plane progressive acoustic sound wave - pdf

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