Particle acceleration
From Wikipedia, the free encyclopedia
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
- <math>
\mathbf{a} = {d\mathbf{v}\over 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>
\mathbf{\bar{a}} = {\mathbf{v} - \mathbf{u} \over t} </math>
where
- <math>\mathbf{\bar{a}}</math> is the average acceleration (m/s²)
<math>\mathbf{u}</math> is the initial velocity (m/s)
<math>\mathbf{v}</math> is the final velocity (m/s)
<math>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> \mathbf{a} = - \frac{v^2}{r} \frac{\mathbf{r}}{r} = - \omega^2 \mathbf{r}</math>
One common unit of acceleration is g, one g being the acceleration caused by the gravity of Earth.
In classical mechanics, acceleration <math> a \ </math> is related to force <math>F \ </math> and mass <math>m \ </math> (assumed to be constant) by way of Newton's second law:
- <math>
F = m \cdot a </math>
[edit] Equations in terms of other measurements
The Particle acceleration of the air particles a in m/s² of a plain sound wave is:
- <math>
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} </math>
| Symbol | Units | Meaning |
|---|---|---|
| a | m/s² | particle acceleration |
| v | m/s | particle velocity |
| ξ | m, meters | particle displacement |
| <math>\omega</math> = 2 · <math>\pi</math> · 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 |
