Stokes' law of sound attenuation

This article is about sound attenuation in fluids. For other uses, see Stokes.

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 exponentially with distance traveled, at a rate $\alpha$ given by

\alpha ={\frac {2\eta \omega ^{2}}{3\rho V^{3}}}

where $\eta$ is the dynamic viscosity coefficient of the fluid, $\omega$ is the sound's frequency, $\rho$ is the fluid density, and $V$ is the speed of sound in the medium:^[1]

The law and its derivation were published in 1845 by physicist 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 $A_{0}$ at some point. After traveling a distance $d$ from that point, its amplitude $A(d)$ will be

A(d)=A_{0}e^{{-\alpha d}}

The parameter $\alpha$ is dimensionally the reciprocal of length. In the International System of Units (SI), it is expressed in neper per meter or simply reciprocal of meter ( ${\mathrm {m}}^{{-1}}$ ). That is, if $\alpha =1{\mathrm {m}}^{{-1}}$ , the wave's amplitude decreases by a factor of $1/e$ for each meter traveled.

Importance of volume viscosity

The law is amended to include a contribution by the volume viscosity $\eta ^{{\mathrm {v}}}$ :

\alpha ={\frac {2(\eta +3\eta ^{\mathrm {v} }/2)\omega ^{2}}{3\rho V^{3}}}

The volume viscosity coefficient is relevant when the fluid's compressibility cannot be ignored, such as in the case of ultrasound in water.^[2]^[3]^[4]^[5] The volume viscosity of water at 15 C is 3.09 centipoise.^[6]

Modification for very high frequencies

Stokes's law is actually an asymptotic approximation for low frequencies of a more general formula:

2\left({\frac {\alpha V}{\omega }}\right)^{2}={\frac {1}{{\sqrt {1+\omega ^{2}\tau ^{2}}}}}-{\frac {1}{1+\omega ^{2}\tau ^{2}}}

where the relaxation time $\tau$ is given by:

\tau ={\frac {4\eta /3+\eta ^{{\mathrm {v}}}}{\rho V^{2}}}

The relaxation time is about $10^{{-12}}{\mathrm {s}}$ (one picosecond), corresponding to a frequency of about 1000 GHz. Thus Stokes' law is adequate for most practical situations.

References

↑ 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
↑ Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965)
↑ Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press,(1959)
↑ Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", Princeton University Press(1986)
↑ Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)
↑ Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, Academic Press, NY, (1964)

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