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 of time. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation

$$

\mathbf{a} = {d\mathbf{v}\over dt}

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:

$$

\mathbf{\bar{a}} = {\mathbf{v} - \mathbf{u} \over t}

where

$\backslash mathbf\{\backslash bar\{a\}\}$ is the average acceleration (m/s²)

$\backslash mathbf\{u\}$ is the initial velocity (m/s)

$\backslash mathbf\{v\}$ is the final velocity (m/s)

$\backslash mathbf\{t\}$ 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

$\backslash mathbf\{a\}\; =\; -\; \backslash frac\{v^2\}\{r\}\; \backslash frac\{\backslash mathbf\{r\}\}\{r\}\; =\; -\; \backslash omega^2\; \backslash mathbf\{r\}$

One common unit of acceleration is g, one g being the acceleration caused by the gravity of Earth at sea level at 45° latitude (Paris), or about 9.81 m/s².

In classical mechanics, acceleration $a\; \backslash$ is related to force $F\; \backslash$ and mass $m\; \backslash$ (assumed to be constant) by way of Newton's second law:

$$

F = m \cdot a

Equations in terms of other measurements

The Particle acceleration of the air particles a in m/s² of a plain sound wave is:

$$

a = \xi \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_{ak}}{Z \cdot A}

Symbol	Units	Meaning
a	m/s²	particle acceleration
v	m/s	particle velocity
ξ	m, meters	particle displacement
$\backslash omega$ = 2 · $\backslash pi$ · f	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 pressure
• particle displacement
• particle velocity

Acoustics

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