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:

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. The volume viscosity of water at 15 C is 3.09 centipoise.

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.