Prehomogeneous vector space

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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:

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 has since been 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.