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

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

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

${\displaystyle \mathbf {\delta } =\int _{t}\mathbf {v} \,\mathrm {d} t}$

where v is the particle velocity.

Progressive sine waves

The particle displacement of a progressive sine wave is given by

${\displaystyle \delta (\mathbf {r} ,\,t)=\delta \sin(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}),}$

where

• ${\displaystyle \delta }$ is the amplitude of the particle displacement;
• ${\displaystyle \varphi _{\delta ,0}}$ is the
  phase shift
  of the particle displacement;
• ${\displaystyle \mathbf {k} }$ is the
  angular wavevector
  ;
• ${\displaystyle \omega }$ 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

${\displaystyle v(\mathbf {r} ,\,t)={\frac {\partial \delta (\mathbf {r} ,\,t)}{\partial t}}=\omega \delta \cos \!\left(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}+{\frac {\pi }{2}}\right)=v\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{v,0}),}$

${\displaystyle p(\mathbf {r} ,\,t)=-\rho c^{2}{\frac {\partial \delta (\mathbf {r} ,\,t)}{\partial x}}=\rho c^{2}k_{x}\delta \cos \!\left(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}+{\frac {\pi }{2}}\right)=p\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{p,0}),}$

where

• ${\displaystyle v}$ is the amplitude of the particle velocity;
• ${\displaystyle \varphi _{v,0}}$ is the phase shift of the particle velocity;
• ${\displaystyle p}$ is the amplitude of the acoustic pressure;
• ${\displaystyle \varphi _{p,0}}$ is the phase shift of the acoustic pressure.

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

${\displaystyle {\hat {v}}(\mathbf {r} ,\,s)=v{\frac {s\cos \varphi _{v,0}-\omega \sin \varphi _{v,0}}{s^{2}+\omega ^{2}}},}$

${\displaystyle {\hat {p}}(\mathbf {r} ,\,s)=p{\frac {s\cos \varphi _{p,0}-\omega \sin \varphi _{p,0}}{s^{2}+\omega ^{2}}}.}$

Since ${\displaystyle \varphi _{v,0}=\varphi _{p,0}}$, the amplitude of the specific acoustic impedance is given by

${\displaystyle z(\mathbf {r} ,\,s)=|z(\mathbf {r} ,\,s)|=\left|{\frac {{\hat {p}}(\mathbf {r} ,\,s)}{{\hat {v}}(\mathbf {r} ,\,s)}}\right|={\frac {p}{v}}={\frac {\rho c^{2}k_{x}}{\omega }}.}$

Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by

${\displaystyle \delta ={\frac {v}{\omega }},}$

${\displaystyle \delta ={\frac {p}{\omega z(\mathbf {r} ,\,s)}}.}$

See also

• Sound
• Sound particle
• Particle velocity
• Particle acceleration

References and notes

• 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
  .

• 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