# Particle displacement

Sound measurements
Characteristic
Symbol
Sound pressure  p · SPL
Particle velocity  v · SVL
Particle displacement  ξ
Sound intensity  I · SIL
Sound power  P · SWL
Sound energy  W
Sound energy density  w
Sound exposure  E · SEL
Sound energy flux  Q
Acoustic impedance  Z
Speed of sound  c
Audio frequency  AF

Particle displacement or displacement amplitude is a measurement of distance of the movement of a particle from its equilibrium position in a medium as it transmits a sound wave.[1] 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.[2] A particle of the medium undergoes displacement according to the particle velocity of the sound wave traveling through the medium, while the sound wave itself moves at the speed of sound, equal to 343 m·s−1 in air at 20 °C.

## Mathematical definition

Particle displacement, denoted ξ and measured in m, is given by:[3]

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

where v is the particle velocity, measured in m·s−1.

## Equations in terms of other measurements

For sine waves with angular frequency ω, the amplitude of the particle displacement can be related to those of other sound measurements:

$\xi_\mathrm{m}(\mathbf r) = \frac{p_\mathrm{m}(\mathbf r)}{\omega \mathfrak{R}(z)} = \frac{1}{\omega} \sqrt{\frac{P_\mathrm{m}(\mathbf r)}{A \mathfrak{R}(z)}} = \frac{1}{\omega} \sqrt{\frac{I_\mathrm{m}(\mathbf r)}{\mathfrak{R}(z)}} = \frac{1}{\omega} \sqrt{\frac{c w_\mathrm{m}(\mathbf r)}{\mathfrak{R}(z)}}.$

It can also be related to the amplitude of the particle velocity and the particle acceleration:

$\xi_\mathrm{m}(\mathbf r) = \frac{v_\mathrm{m}(\mathbf r)}{\omega} = \frac{a_\mathrm{m}(\mathbf r)}{\omega^2}.$
Symbol Unit Meaning
c m·s−1 speed of sound
v m·s−1 particle velocity
z Pa·m−1·s specific acoustic impedance
A m2 area
p Pa sound pressure
P W sound power
I W·m−2 sound intensity
w J·m−3 sound energy density