Particle displacement
From Free net encyclopedia
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 of air in and against the direction in which the sound wave is travelling with the speed of sound with 343 m/s at 20 °C.
Particle displacement ξ in m for a plane wave is:
- <math>
\xi = \int_{t} v\, \mathrm{d}t = \int_{t} \frac{p}{Z}\, \mathrm{d}t </math>
Particle displacement can be represented in terms of other measurements:
- <math>
\xi = \frac{v}{\omega} = \frac{v}{2 \cdot \pi \cdot f} = \frac{p}{Z \cdot \omega} = \frac{p}{Z \cdot 2 \cdot \pi \cdot f} = \frac{a}{\omega^2} = \frac{1}{\omega}\sqrt{\frac{I}{Z}} = \frac{1}{\omega}\sqrt{\frac{E}{\rho}} = \frac{1}{\omega}\sqrt{\frac{P_{ac}}{Z \cdot A}} </math>
We get for the sound pressure p:
- <math>
p = {\xi \cdot Z \cdot \omega} = {\xi \cdot Z \cdot 2 \cdot \pi \cdot f} </math>
- <math>
p = \frac{a \cdot Z}{\omega} = c \cdot \sqrt{\rho \cdot E} </math>
where:
| 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 |
| Z = c · ρ | 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 |
| Pac | W, watts | sound power or acoustic power |
| A | m2 | Area |
| a | m/s2 | Particle acceleration |