Particle acceleration

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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 = \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} = - \mathbf \over t}]

where

[\mathbf}] is the average acceleration (m/s²)

[\mathbf] is the initial velocity (m/s)

[\mathbf] is the final velocity (m/s)

[\mathbf] 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 = - \frac \frac} = - \omega^2 \mathbf]

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 = \omega \sqrt \frac = \omega \sqrt \frac = \omega \sqrt \frac}]

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

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

Particle acceleration

Equations in terms of other measurements

See also