128.12.244.135: /* Castling */

2015-01-07T21:33:11Z

Castling

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	~~{{About\|sound attenuation in fluids\|\|Stokes (disambiguation){{!}}Stokes}}~~		I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Invoicing is what I do. For many years he's been living in Alaska and he doesn't strategy on changing it. What I love performing is soccer but I don't have the time recently.<br><br>Also visit my web blog; cheap psychic readings, [http://afeen.fbho.net/v2/index.php?do=/profile-210/info/ visit the following internet site],

	~~'''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~~
	~~:<math> \alpha = \frac{2 \eta\omega^2}{3\rho V^3}</math>~~
	~~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>~~

	~~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~~.

	~~==Interpretation==~~
	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~~
	:~~<math>A(d) = A_0e^{-\alpha d}<~~/~~math>~~

	~~The parameter <math>\alpha<~~/~~math> is [[dimensional analysis\|dimensionally]] the reciprocal of length~~.
	~~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~~.

	~~== Importance of volume viscosity ==~~
	~~The law has since been amended to include a contribution by the [[volume viscosity]] <math>\eta^\mathrm{v}</math>:~~
	~~:<math> \alpha~~ = ~~\frac{2 (\eta+3\eta^\mathrm{v}/4)\omega^2}{3\rho V^3}<~~/~~math>~~
	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>

	~~== Modification for very high frequencies ==~~
	~~Stokes's law is actually an [[asymptotic]] approximation for low frequencies of a more general formula:~~
	~~:<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>~~
	~~where~~ the ~~[[relaxation time~~]~~] <math>\tau</math> is given by:~~
	~~:<math> \tau = \frac{4 \eta/3 + \eta^\mathrm{v}}{\rho V^2}</math>~~
	~~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.~~

	~~==References==~~
	~~<references/>~~

	~~[[Category:Colloidal chemistry]]~~
	~~[[Category:Fluid dynamics]]~~
	~~[[Category:Acoustics]]~~

en>John: Reverted edits by Sol1 (talk) to last version by Rjwilmsi

2013-05-13T19:46:08Z

Reverted edits by Sol1 (talk) to last version by Rjwilmsi

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en>Rjwilmsi: /* References */Journal cites, added 2 DOIs, using AWB (8079)

2012-06-19T11:15:23Z

References: Journal cites, added 2 DOIs, using AWB (8079)

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128.12.244.135: /* Castling */

en>John: Reverted edits by Sol1 (talk) to last version by Rjwilmsi

en>Rjwilmsi: /* References */Journal cites, added 2 DOIs, using AWB (8079)