Stokes' law of sound attenuation

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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[edit]

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[edit]

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[edit]

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[edit]

  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)