# Stokes' law of sound attenuation

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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 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

$A(d) = A_0e^{-\alpha d}$

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 $\eta^\mathrm{v}$:

$\alpha = \frac{2 (\eta+3\eta^\mathrm{v}/4)\omega^2}{3\rho V^3}$

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

1. ^ 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
2. ^ Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965)
3. ^ Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press,(1959)
4. ^ Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", Princeton University Press(1986)
5. ^ Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, (2002)
6. ^ Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, Academic Press, NY, (1964)