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.

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

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

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

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:

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

where

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

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

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

${\displaystyle t}$ 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

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

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

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

${\displaystyle F=m\cdot a}$

Equations in terms of other measurements

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

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

Symbol	Units	Meaning
a	m/s^2	particle acceleration
v	m/s	particle velocity
δ	m, meters	particle displacement
${\displaystyle \omega }$ = 2 · ${\displaystyle \pi }$ · f	radians/s	angular frequency
f	Hz, hertz	frequency
p	Pa, pascals	sound pressure
Z	N·s/m^3	acoustic impedance
J	W/m^2	sound intensity
E	W·s/m^3	sound energy density
Pac	W, watts	sound power or acoustic power
A	m^2	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