Particle displacement

From Wikipedia, the free encyclopedia

Sound measurements
Sound pressure p
Sound pressure level (SPL)
Particle velocity v
Particle velocity level (SVL)
(Sound velocity level)
Particle displacement ξ
Sound intensity I
Sound intensity level (SIL)
Sound power P_ac
Sound power level (SWL)
Sound energy density E
Sound energy flux q
Acoustic impedance Z
Speed of sound c

Particle displacement or particle amplitude (represented in mathematics by the lower-case Greek letter ξ) is a measurement of distance (in metres) of the movement of a particle in a medium as it transmits a wave. In most cases this is a longitudinal wave of pressure (such as sound), but it can also be a transverse wave, such as the vibration of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillations of air molecules with, and against, the direction in which the sound wave is travelling. A particle of the medium undergoes displacement according to the particle velocity of the wave traveling through the medium, while the sound wave itself moves at the speed of sound, equal to 343 m/s in air at 20 °C.

The instantaneous particle displacement ξ in m for a wave is:

$\xi = \int_{t} v\, \mathrm{d}t$

If the wave is a standing wave or a traveling wave containing a single frequency, the particle displacement is:

$\xi = \frac{1}{Z} \int_{t} p\, \mathrm{d}t$

This expression for $ξ$ undergoes simple harmonic oscillation, and as such is usually expressed as an rms time average.

Particle displacement for a traveling wave containing a single frequency can be represented in terms of other measurements:

$\xi = \frac{v}{\omega} = \frac{p}{Z_0 \cdot \omega} = \frac{a}{\omega^2} = \frac{1}{\omega}\sqrt{\frac{I}{Z_0}} = \frac{1}{\omega}\sqrt{\frac{E}{\rho}} = \frac{1}{\omega}\sqrt{\frac{P_{ac}}{Z_0 \cdot A}}$

where in the above equation, the quantities $ξ, v, a, I, E, P a c$ may be taken throughout as rms time-averages (or all as maximum values). The single frequency traveling wave has acoustic impedance equal to the characteristic impedance, $Z = Z 0$ . Further representations for $ξ$ can be found from the above equations using the replacement $ω = 2π f$ .

Symbol	Units	Meaning
ξ	m, meters	Particle displacement
v	m/s	particle velocity
ω = 2πf	radians/s	angular frequency
f	Hz, hertz	frequency
p	Pa, pascals	sound pressure
Z₀ = c · ρ	N·s/m³	characteristic impedance
Z = p / v	N·s/m³	acoustic impedance
c	m/s	Speed of sound
ρ	kg/m³	Density of air
I	W/m²	sound intensity
E	W·s/m³	sound energy density
P_ac	W, watts	sound power or acoustic power
A	m²	Area
a	m/s²	Particle acceleration