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.[1] 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.[2]

Sound measurements
 Sound pressure p, SPL, LPA
 Particle velocity v, SVL
 Particle displacement δ
 Sound intensity I, SIL
 Sound power P, SWL, LWA
 Sound energy W
 Sound energy density w
 Sound exposure E, SEL
 Acoustic impedance Z
 Audio frequency AF
 Transmission loss TL

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 in air at 20 °C.

Mathematical definition edit

Particle displacement, denoted δ, is given by[3]


where v is the particle velocity.

Progressive sine waves edit

The particle displacement of a progressive sine wave is given by



  •   is the amplitude of the particle displacement;
  •   is the phase shift of the particle displacement;
  •   is the angular wavevector;
  •   is the angular frequency.

It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave x are given by



  •   is the amplitude of the particle velocity;
  •   is the phase shift of the particle velocity;
  •   is the amplitude of the acoustic pressure;
  •   is the phase shift of the acoustic pressure.

Taking the Laplace transforms of v and p with respect to time yields


Since  , 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


See also edit

References and notes edit

  1. ^ Gardner, Julian W.; Varadan, Vijay K.; Awadelkarim, Osama O. (2001). Microsensors, MEMS, and Smart Devices John 2. pp. 23–322. ISBN 978-0-471-86109-6.
  2. ^ Arthur Schuster (1904). An Introduction to the Theory of Optics. London: Edward Arnold. An Introduction to the Theory of Optics By Arthur Schuster.
  3. ^ John Eargle (January 2005). The Microphone Book: From mono to stereo to surround – a guide to microphone design and application. Burlington, Ma: Focal Press. p. 27. ISBN 978-0-240-51961-6.

Related Reading:

  • Wood, Robert Williams (1914). Physical optics. New York: The Macmillan Company.
  • Strong, John Donovan & Hayward, Roger (January 2004). Concepts of Classical Optics. Dover Publications. ISBN 978-0-486-43262-5.
  • Barron, Randall F. (January 2003). Industrial noise control and acoustics. NYC, New York: CRC Press. pp. 79, 82, 83, 87. ISBN 978-0-8247-0701-9.

External links edit

  • Acoustic Particle-Image Velocimetry. Development and Applications
  • Ohm's Law as Acoustic Equivalent. Calculations
  • Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave