Particle displacement (original) (raw)

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

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]

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

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

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

where v is the particle velocity.

Progressive sine waves

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The particle displacement of a progressive sine wave is given by

δ ( r , t ) = δ sin ⁡ ( k ⋅ r − ω t + φ δ , 0 ) , {\displaystyle \delta (\mathbf {r} ,\,t)=\delta \sin(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}),} {\displaystyle \delta (\mathbf {r} ,\,t)=\delta \sin(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}),}

where

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

v ( r , t ) = ∂ δ ( r , t ) ∂ t = ω δ cos ( k ⋅ r − ω t + φ δ , 0 + π 2 ) = v cos ⁡ ( k ⋅ r − ω t + φ v , 0 ) , {\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 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}),}

p ( r , t ) = − ρ c 2 ∂ δ ( r , t ) ∂ x = ρ c 2 k x δ cos ( k ⋅ r − ω t + φ δ , 0 + π 2 ) = p cos ⁡ ( k ⋅ r − ω t + φ p , 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}),} {\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

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

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

p ^ ( r , s ) = p s cos ⁡ φ p , 0 − ω sin ⁡ φ p , 0 s 2 + ω 2 . {\displaystyle {\hat {p}}(\mathbf {r} ,\,s)=p{\frac {s\cos \varphi _{p,0}-\omega \sin \varphi _{p,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 φ v , 0 = φ p , 0 {\displaystyle \varphi _{v,0}=\varphi _{p,0}} {\displaystyle \varphi _{v,0}=\varphi _{p,0}}, the amplitude of the specific acoustic impedance is given by

z ( r , s ) = | z ( r , s ) | = | p ^ ( r , s ) v ^ ( r , s ) | = p v = ρ c 2 k x ω . {\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 }}.} {\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

δ = v ω , {\displaystyle \delta ={\frac {v}{\omega }},} {\displaystyle \delta ={\frac {v}{\omega }},}

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

References and notes

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  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.

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