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. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation

a = d v d t {\displaystyle \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:

a ¯ = v − u Δ t {\displaystyle \mathbf {\bar {a}} ={\frac {\mathbf {v} -\mathbf {u} }{\Delta t}}}
where

  • a ¯ {\displaystyle \mathbf {\bar {a}} } is the average acceleration (m/s2)
  • u {\displaystyle \mathbf {u} } is the initial velocity (m/s)
  • v {\displaystyle \mathbf {v} } is the final velocity (m/s)
  • Δ t {\displaystyle \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

a = − v 2 r r r = − ω 2 r {\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 a {\displaystyle a} is related to force F {\displaystyle F} and mass m {\displaystyle m} (assumed to be constant) by way of Newton's second law:

F = m a {\displaystyle 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 = δ ⋅ ω 2 = v ⋅ ω = p ⋅ ω Z = ω J Z = ω E ρ = ω P ac Z ⋅ A {\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_{\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

References

External links

  • Relationships of acoustic quantities associated with a plane progressive acoustic sound wave - pdf