Particle displacement

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

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

where v is the particle velocity.

The particle displacement of a progressive sine wave is given by

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

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

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

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

• Sound
• Sound particle
• Particle velocity
• Particle acceleration

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.

• 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