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/second&sup2;. 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
 * $$\mathbf{\bar{a}}$$ is the average acceleration (m/s&sup2;)

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

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

$$t$$ 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
 * $$ \mathbf{a} = - \frac{v^2}{r} \frac{\mathbf{r}}{r} = - \omega^2 \mathbf{r}$$

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

In classical mechanics, acceleration $$ a \ $$ is related to force $$F \ $$ and mass $$m \ $$ (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} $$