# Stokes's law of sound attenuation

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Stokes's 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 ${\displaystyle \alpha }$ given by

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

where ${\displaystyle \eta }$ is the dynamic viscosity coefficient of the fluid, ${\displaystyle \omega }$ is the sound's angular frequency, ${\displaystyle \rho }$ is the fluid density, and ${\displaystyle V}$ is the speed of sound in the medium.[1]

The law and its derivation were published in 1845 by the Anglo-Irish physicist G. G. Stokes, who also developed Stokes's law for the friction force in fluid motion. A generalisation of Stokes attenuation (and dispersion), taking into account the effect of thermal conductivity was proposed by the German physicist Gustav Kirchhoff in 1868.[2][3]

Sound attenuation in fluids is also accompanied by acoustic dispersion, meaning that the different frequencies are propagating at different sound speeds.[1]

## Interpretation

Stokes's law of sound attenuation applies to sound propagation in an isotropic and homogeneous Newtonian medium. Consider a plane sinusoidal pressure wave that has amplitude ${\displaystyle A_{0}}$ at some point. After traveling a distance ${\displaystyle d}$ from that point, its amplitude ${\displaystyle A(d)}$ will be

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

The parameter ${\displaystyle \alpha }$ 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 (${\displaystyle \mathrm {m} ^{-1}}$). That is, if ${\displaystyle \alpha =1\mathrm {m} ^{-1}}$, the wave's amplitude decreases by a factor of ${\displaystyle 1/e}$ for each meter traveled.

## Importance of volume viscosity

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

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

The volume viscosity coefficient is relevant when the fluid's compressibility cannot be ignored, such as in the case of ultrasound in water.[4][5][6][7] The volume viscosity of water at 15 C is 3.09 centipoise.[8]

## Modification for very high frequencies

Plot of reduced wave-vector, ${\displaystyle kc\tau }$ (blue), and attenuation coefficient, ${\displaystyle \alpha c\tau }$ (red), as functions of reduced frequency ${\displaystyle \omega \tau }$. Dotted lines are asymptotic regimes at low and high frequencies (Stoke's law is the dotted red line at the left.) In the labels, ${\displaystyle \omega _{\mathrm {c} }=1/\tau }$

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

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

where the relaxation time ${\displaystyle \tau }$ is given by:

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

The relaxation time for water is about ${\displaystyle 2\times 10^{-12}\mathrm {s/rad} }$ (one picosecond per radian), corresponding to a linear frequency of about 70 GHz. Thus Stokes's law is adequate for most practical situations.

## References

1. ^ a b Stokes, G.G. "On the theories of the internal friction in fluids in motion, and of the equilibrium and motion of elastic solids", Transactions of the Cambridge Philosophical Society, vol.8, 22, pp. 287-342 (1845)
2. ^ G. Kirchhoff, "Ueber den Einfluss der Wärmeleitung in einem Gase auf die Schallbewegung", Ann. Phys. , 210: 177-193 (1868). Link to paper
3. ^ S. Benjelloun and J. M. Ghidaglia, "On the dispersion relation for compressible Navier-Stokes Equations," Link to Archiv e-print Link to Hal e-print
4. ^ Happel, J. and Brenner , H. "Low Reynolds number hydrodynamics", Prentice-Hall, (1965)
5. ^ Landau, L.D. and Lifshitz, E.M. "Fluid mechanics", Pergamon Press,(1959)
6. ^ Morse, P.M. and Ingard, K.U. "Theoretical Acoustics", Princeton University Press(1986)
7. ^ Dukhin, A.S. and Goetz, P.J. "Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound", Edition 3, Elsevier, (2017)
8. ^ Litovitz, T.A. and Davis, C.M. In "Physical Acoustics", Ed. W.P.Mason, vol. 2, chapter 5, Academic Press, NY, (1964)