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.

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/s²)

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

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

$\mathbf {t}$ 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

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

Symbol	Units	Meaning
a	m/s²	particle acceleration
v	m/s	particle velocity
ξ	m, meters	particle displacement
$\omega$ = 2 · $\pi$ · 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
P_ac	W, watts	sound power or acoustic power
A	m²	area

Equations in terms of other measurements

See also