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Particle velocity
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<h2 id="mathematical-definition">Mathematical definition</h2> <p>Particle velocity, denoted v {\displaystyle \mathbf {v} } , is defined by </p> v = ∂ δ ∂ t {\displaystyle \mathbf {v} ={\frac {\partial \mathbf {\delta } }{\partial t}}} <p>where δ {\displaystyle \delta } is the <a href="/facts/Particle_displacement/FjGlAwP1">particle displacement</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 m cos ( k ⋅ r − ω t + φ δ , 0 ) , {\displaystyle \delta (\mathbf {r} ,\,t)=\delta _{\mathrm {m} }\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{\delta ,0}),} <p>where </p> <ul><li> δ m {\displaystyle \delta _{\mathrm {m} }} 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 m 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_{\mathrm {m} }\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 m 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_{\mathrm {m} }\cos(\mathbf {k} \cdot \mathbf {r} -\omega t+\varphi _{p,0}),} <p>where </p> <ul><li> v m {\displaystyle v_{\mathrm {m} }} 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 m {\displaystyle p_{\mathrm {m} }} 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 v {\displaystyle v} and p {\displaystyle p} with respect to time yields </p> v ^ ( r , s ) = v m s cos φ v , 0 − ω sin φ v , 0 s 2 + ω 2 , {\displaystyle {\hat {v}}(\mathbf {r} ,\,s)=v_{\mathrm {m} }{\frac {s\cos \varphi _{v,0}-\omega \sin \varphi _{v,0}}{s^{2}+\omega ^{2}}},} p ^ ( r , s ) = p m s cos φ p , 0 − ω sin φ p , 0 s 2 + ω 2 . {\displaystyle {\hat {p}}(\mathbf {r} ,\,s)=p_{\mathrm {m} }{\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 m ( r , s ) = | z ( r , s ) | = | p ^ ( r , s ) v ^ ( r , s ) | = p m v m = ρ c 2 k x ω . {\displaystyle z_{\mathrm {m} }(\mathbf {r} ,\,s)=|z(\mathbf {r} ,\,s)|=\left|{\frac {{\hat {p}}(\mathbf {r} ,\,s)}{{\hat {v}}(\mathbf {r} ,\,s)}}\right|={\frac {p_{\mathrm {m} }}{v_{\mathrm {m} }}}={\frac {\rho c^{2}k_{x}}{\omega }}.} <p>Consequently, the amplitude of the particle velocity is related to those of the particle displacement and the sound pressure by </p> v m = ω δ m , {\displaystyle v_{\mathrm {m} }=\omega \delta _{\mathrm {m} },} v m = p m z m ( r , s ) . {\displaystyle v_{\mathrm {m} }={\frac {p_{\mathrm {m} }}{z_{\mathrm {m} }(\mathbf {r} ,\,s)}}.} <h2 id="particle-velocity-level">Particle velocity level</h2> <p class="note">For other uses, see <a href="/facts/Sound_level_(disambiguation)/Pyof5AyU">Sound level</a>.</p> <p>Sound velocity level (SVL) or acoustic velocity level or particle velocity level is a <a href="/facts/Level_(logarithmic_quantity)/0Yt12Pfr">logarithmic measure</a> of the effective particle velocity of a sound relative to a reference value. Sound velocity level, denoted <i>L</i><i>v</i> and measured in <a href="/facts/Decibel/0LyxAPXh">dB</a>, is defined by<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> L v = ln ( v v 0 ) N p = 2 log 10 ( v v 0 ) B = 20 log 10 ( v v 0 ) d B , {\displaystyle L_{v}=\ln \!\left({\frac {v}{v_{0}}}\right)\!~\mathrm {Np} =2\log _{10}\!\left({\frac {v}{v_{0}}}\right)\!~\mathrm {B} =20\log _{10}\!\left({\frac {v}{v_{0}}}\right)\!~\mathrm {dB} ,} <p>where </p> <ul><li><i>v</i> is the <a href="/facts/Root_mean_square/Cfmg7dws">root mean square</a> particle velocity;</li> <li><i>v</i>0 is the <i>reference particle velocity</i>;</li> <li>1 Np = 1 is the <a href="/facts/Neper/nzHnA6ev">neper</a>;</li> <li>1 B = 1/2 ln 10 is the <a href="/facts/Decibel/0LyxAPXh">bel</a>;</li> <li>1 dB = 1/20 ln 10 is the <a href="/facts/Decibel/0LyxAPXh">decibel</a>.</li></ul> <p>The commonly used reference particle velocity in air is<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> v 0 = 5 × 10 − 8 m / s . {\displaystyle v_{0}=5\times 10^{-8}~\mathrm {m/s} .} <p>The proper notations for sound velocity level using this reference are <i>L</i><i>v</i>/(5 × 10−8 m/s) or <i>L</i><i>v</i> (re 5 × 10−8 m/s), but the notations dB SVL, dB(SVL), dBSVL, or dBSVL are very common, even though they are not accepted by the SI.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <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_displacement/FjGlAwP1">Particle displacement</a></li> <li><a href="/facts/Particle_acceleration/gRHpkT4v">Particle acceleration</a></li></ul> <h2 id="external-links">External links</h2> <ul><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> <li><a href="https://www.microflown.com">The particle Velocity Can Be Directly Measured with a Microflown</a></li> <li><a href="http://www.weles-acoustics.com/en/technologies/particle-velocity-sensor/">Particle velocity measured with Weles Acoustics sensor - working principle</a></li> <li><a href="https://oro.open.ac.uk/44496/1/ali_tonddast_navaei_thesis.pdf">Acoustic Particle-Image Velocimetry. Development and Applications</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Letter symbols to be used in electrical technology – Part 3: Logarithmic and related quantities, and their units", IEC 60027-3 Ed. 3.0, International Electrotechnical Commission, 19 July 2002. <a href="http://webstore.iec.ch/webstore/webstore.nsf/artnum/028981" target="_blank">http://webstore.iec.ch/webstore/webstore.nsf/artnum/028981</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Ross Roeser, Michael Valente, Audiology: Diagnosis (Thieme 2007), p. 240. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Thompson, A. and Taylor, B. N. sec 8.7, "Logarithmic quantities and units: level, neper, bel", Guide for the Use of the International System of Units (SI) 2008 Edition, NIST Special Publication 811, 2nd printing (November 2008), SP811 PDF <a href="http://physics.nist.gov/cuu/pdf/sp811.pdf" target="_blank">http://physics.nist.gov/cuu/pdf/sp811.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>