Particle acceleration

This article is about small-scale particle acceleration in acoustics. For acceleration of charged particles to very high energies, see particle accelerator.

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

$$\mathbf {a} ={\frac {d\mathbf {v} }{dt}}$$

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:

$$\mathbf {\bar {a}} ={\frac {\mathbf {v} -\mathbf {u} }{\Delta t}}$$

where

• $\mathbf {\bar {a}}$ is the average acceleration (m/s²)
• $\mathbf {u}$ is the initial velocity (m/s)
• $\mathbf {v}$ is the final velocity (m/s)
• $\Delta t$ 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:

$$F=ma$$

Equations in terms of other measurements

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

$$a=\delta \cdot \omega ^{2}=v\cdot \omega ={\frac {p\cdot \omega }{Z}}=\omega {\sqrt {\frac {J}{Z}}}=\omega {\sqrt {\frac {E}{\rho }}}=\omega {\sqrt {\frac {P_{\text{ac}}}{Z\cdot A}}}$$

Symbol	Units	Meaning
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