Particle displacement

Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. 
The SI unit of particle displacement is the metre (m). 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 sound wave traveling through the medium, while the sound wave itself moves at the speed of sound, equal to in air at .
Mathematical definition.
Particle displacement, denoted δ, is given by
where v is the particle velocity.
Progressive sine waves.
The particle displacement of a "progressive sine wave" is given by
where
It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave "x" are given by
where
Taking the Laplace transforms of "v" and "p" with respect to time yields
Since formula_10, the amplitude of the specific acoustic impedance is given by
Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by
References and notes.
Related Reading: