Particle Displacement

Particle displacement or displacement amplitude (represented in mathematics by the lower-case Greek letter ξ) is a measurement of distance of the movement of a particle from its equilibrium position 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 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 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 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, . Further representations for can be found from the above equations using the replacement .

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
Pac W, watts sound power or acoustic power
A m2 Area
a m/s2 Particle acceleration

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