# 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 $\alpha$ 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 angular 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's law for the friction force in fluid motion.

## Interpretation

Stokes's 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 is amended to include a contribution by the volume viscosity $\eta ^{\mathrm {v} }$ :

$\alpha ={\frac {(2\eta +3\eta ^{\mathrm {v} }/2)\omega ^{2}}{3\rho V^{3}}}$ The volume viscosity coefficient is relevant when the fluid's compressibility cannot be ignored, such as in the case of ultrasound in water. The volume viscosity of water at 15 C is 3.09 centipoise.

## Modification for very high frequencies Plot of reduced wave-vector, $kc\tau$ (blue), and attenuation coefficient, $\alpha c\tau$ (red), as functions of reduced frecuency $\omega \tau$ . Dotted lines are asymptotic regimes at low and high frequencies (Stoke's law is the dotted red line at the left.) In the labels, $\omega _{\mathrm {c} }=1/\tau$ 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 for water is about $2\times 10^{-12}\mathrm {s/rad}$ (one picosecond per radian), corresponding to a linear frequency of about 70 GHz. Thus Stokes's law is adequate for most practical situations.