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Particle displacement
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<h2 id="mathematical-definition">Mathematical definition</h2> <p>Particle displacement, denoted <a href="/facts/Delta_(letter)/otPsSYj6">δ</a>, is given by<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> δ = ∫ t v d t {\displaystyle \mathbf {\delta } =\int _{t}\mathbf {v} \,\mathrm {d} t} <p>where v is the <a href="/facts/Particle_velocity/V2SSGkXU">particle velocity</a>. </p> <h2 id="progressive-sine-waves">Progressive sine waves</h2> <p>The particle displacement of a <i>progressive <a href="/facts/Sine_wave/mD1CB09M">sine wave</a></i> is given by </p> 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}),} <p>where </p> <ul><li> δ {\displaystyle \delta } is the <a href="/facts/Amplitude/aEckuXJM">amplitude</a> of the particle displacement;</li> <li> φ δ , 0 {\displaystyle \varphi _{\delta ,0}} is the <a href="/facts/Phase_shift/cpGtuVll">phase shift</a> of the particle displacement;</li> <li> k {\displaystyle \mathbf {k} } is the <a href="/facts/Angular_wavevector/Jk8wUEft">angular wavevector</a>;</li> <li> ω {\displaystyle \omega } is the <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a>.</li></ul> <p>It follows that the particle velocity and the sound pressure along the direction of propagation of the sound wave <i>x</i> are given by </p> 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}),} 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}),} <p>where </p> <ul><li> v {\displaystyle v} is the amplitude of the particle velocity;</li> <li> φ v , 0 {\displaystyle \varphi _{v,0}} is the phase shift of the particle velocity;</li> <li> p {\displaystyle p} is the amplitude of the acoustic pressure;</li> <li> φ p , 0 {\displaystyle \varphi _{p,0}} is the phase shift of the acoustic pressure.</li></ul> <p>Taking the Laplace transforms of <i>v</i> and <i>p</i> with respect to time yields </p> 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}}},} 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}}}.} <p>Since φ v , 0 = φ p , 0 {\displaystyle \varphi _{v,0}=\varphi _{p,0}} , the amplitude of the specific acoustic impedance is given by </p> 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 }}.} <p>Consequently, the amplitude of the particle displacement is related to those of the particle velocity and the sound pressure by </p> δ = v ω , {\displaystyle \delta ={\frac {v}{\omega }},} δ = p ω z ( r , s ) . {\displaystyle \delta ={\frac {p}{\omega z(\mathbf {r} ,\,s)}}.} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Sound/DlKXlPuF">Sound</a></li> <li><a href="/facts/Sound_particle/eCaV3ldQ">Sound particle</a></li> <li><a href="/facts/Particle_velocity/V2SSGkXU">Particle velocity</a></li> <li><a href="/facts/Particle_acceleration/gRHpkT4v">Particle acceleration</a></li></ul> <h2 id="references-and-notes">References and notes</h2> <p>Related Reading: </p> <ul><li>Wood, Robert Williams (1914). <i>Physical optics</i>. New York: The Macmillan Company.</li> <li>Strong, John Donovan & Hayward, Roger (January 2004). <i>Concepts of Classical Optics</i>. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-43262-5.</li> <li>Barron, Randall F. (January 2003). <a href="https://books.google.com/books?id=k1tXPl2hC-cC&q=instantaneous+particle+displacement&pg=PA82"><i>Industrial noise control and acoustics</i></a>. NYC, New York: CRC Press. pp. 79, 82, 83, 87. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8247-0701-9.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="http://acoustics.open.ac.uk/802574C70048F266/%28httpAssets%29/EC0466002A53AFDD802574E200381B0C/$file/ali_tonddast_navaei_thesis.pdf">Acoustic Particle-Image Velocimetry. Development and Applications</a></li> <li><a href="http://www.sengpielaudio.com/calculator-ak-ohm.htm">Ohm's Law as Acoustic Equivalent. Calculations</a></li> <li><a href="http://www.sengpielaudio.com/RelationshipsOfAcousticQuantities.pdf">Relationships of Acoustic Quantities Associated with a Plane Progressive Acoustic Sound Wave</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Gardner, Julian W.; Varadan, Vijay K.; Awadelkarim, Osama O. (2001). Microsensors, MEMS, and Smart Devices John 2. Wiley. pp. 23–322. ISBN 978-0-471-86109-6. <a href="978-0-471-86109-6" target="_blank">978-0-471-86109-6</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Arthur Schuster (1904). An Introduction to the Theory of Optics. London: Edward Arnold. An Introduction to the Theory of Optics By Arthur Schuster. <a href="https://archive.org/details/bub_gb_Zb4KAAAAIAAJ" target="_blank">https://archive.org/details/bub_gb_Zb4KAAAAIAAJ</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="978-0-240-51961-6" target="_blank">978-0-240-51961-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>