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| {{About|sound attenuation in fluids||Stokes (disambiguation){{!}}Stokes}}
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| '''Stokes law of sound attenuation''' is a formula for the [[attenuation]] of [[sound]] in a [[Newtonian fluid]], such as water or air, due to the fluid's [[viscosity]]. It states that the [[amplitude]] of a [[plane wave]] decreases [[exponential decay|exponentially]] with distance traveled, at a rate <math>\alpha</math> given by
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| :<math> \alpha = \frac{2 \eta\omega^2}{3\rho V^3}</math>
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| where <math>\eta</math> is the [[dynamic viscosity|dynamic viscosity coefficient]] of the fluid, <math>\omega</math> is the sound's [[frequency]], <math>\rho</math> is the fluid [[density]], and <math>V</math> is the [[sound speed|speed of sound]] in the medium:<ref>Stokes, G.G. "On the theories of the internal friction in fluids in motion, and of the equilibrium and motion of elastic solids", ''Transaction of the Cambridge Philosophical Society'', vol.8, 22, pp. 287-342 (1845</ref>
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| The law and its derivation were published in 1845 by physicist [[George Gabriel Stokes|G. G. Stokes]], who also developed the well-known [[Stokes' law]] for the [[friction]] force in fluid motion.
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| ==Interpretation==
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| Stokes' law applies to sound propagation in an [[isotropic]] and [[homogeneous]] Newtonian medium. Consider a plane [[sinusoidal]] [[pressure wave]] that has amplitude <math>A_0</math> at some point. After traveling a distance <math>d</math> from that point, its amplitude <math>A(d)</math> will be
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| :<math>A(d) = A_0e^{-\alpha d}</math> | |
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| The parameter <math>\alpha</math> is [[dimensional analysis|dimensionally]] the reciprocal of length.
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| In the [[International System of Units]] (SI), it is expressed in [[neper]] per [[metre|meter]] or simply [[reciprocal]] of meter (<math>\mathrm{m}^{-1}</math>). That is, if <math>\alpha = 1 \mathrm{m}^{-1}</math>, the wave's amplitude decreases by a factor of <math>1/e</math> for each meter traveled.
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| == Importance of volume viscosity ==
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| The law has since been amended to include a contribution by the [[volume viscosity]] <math>\eta^\mathrm{v}</math>:
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| :<math> \alpha = \frac{2 (\eta+3\eta^\mathrm{v}/4)\omega^2}{3\rho V^3}</math>
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| The volume viscosity coefficient is relevant when the fluid's [[compressibility]] cannot be ignored, such as in the case of ultrasound in water.<ref>Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", ''Prentice-Hall'', (1965)</ref><ref>Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", ''Pergamon Press'',(1959)</ref><ref>Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", ''Princeton University Press''(1986)</ref><ref>Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", ''Elsevier'', (2002)</ref> The volume viscosity of water at 15 [[celsius|C]] is 3.09 [[centipoise]].<ref>Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, ''Academic Press'', NY, (1964)</ref>
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| == Modification for very high frequencies ==
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| Stokes's law is actually an [[asymptotic]] approximation for low frequencies of a more general formula:
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| :<math> 2\left(\frac{\alpha V}{\omega}\right)^2 = \frac{1}{\sqrt{1+\omega^2 \tau^2 }}-\frac{1}{1+\omega^2 \tau^2}</math>
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| where the [[relaxation time]] <math>\tau</math> is given by:
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| :<math> \tau = \frac{4 \eta/3 + \eta^\mathrm{v}}{\rho V^2}</math>
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| The relaxation time<!--FOR WHAT FLUID??--> is about <math>10^{-12} \mathrm{s}</math> (one [[picosecond]]), corresponding to a frequency of about 1000 [[GHz]]. Thus Stokes' law is adequate for most practical situations.
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| ==References==
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| <references/>
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| [[Category:Colloidal chemistry]]
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| [[Category:Fluid dynamics]]
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| [[Category:Acoustics]]
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