Boundary-value analysis

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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. 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:

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

where

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

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

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

$t$ 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:

$F=m\cdot a$

Equations in terms of other measurements

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

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

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
Pac	W, watts	sound power or acoustic power
A	m²	area

See also

• Sound pressure
• Particle displacement
• Particle velocity

External links

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