Stokes's law of sound attenuation

In acoustics, 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 Template:Mvar given by $$\alpha =\frac{2\eta {\omega }^2}{3\rho V^3}$$ where Template:Mvar is the dynamic viscosity coefficient of the fluid, Template:Mvar is the sound's angular frequency, Template:Mvar is the fluid density, and Template:Mvar is the speed of sound in the medium.

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 taking into account the effect of thermal conductivity was proposed by the German physicist Gustav Kirchhoff in 1868.

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

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 Template:Math at some point. After traveling a distance Template:Mvar from that point, its amplitude Template:Math will be $$A(d)=A_0{e}^{-\alpha d}$$

The parameter Template:Mvar is a kind of attenuation constant, dimensionally the reciprocal of length. In the International System of Units (SI), it is expressed in neper per meter or simply reciprocal of meter (mTemplate:Sup). That is, if Template:Mvar = 1 mTemplate:Sup, the wave's amplitude decreases by a factor of Template:Math for each meter traveled.

Importance of volume viscosity

The law is amended to include a contribution by the volume viscosity Template:Mvar: $$\alpha =\frac{(2\eta +\frac{3}{2}\zeta ){\omega }^2}{3\rho V^3}=\frac{(\frac{4}{3}\eta +\zeta ){\omega }^2}{2\rho V^3}$$ 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 and attenuation coefficient as functions of reduced frequency Template:Mvar. (In the labels, Template:Math) Template:Legend-line Template:Legend-line Template:Legend-line Template:Legend-line

Stokes's law is actually an asymptotic approximation for low frequencies of a more general formula involving relaxation time Template:Mvar: $$\begin{array}{cc}2{\left(\frac{\alpha V}{\omega }\right)}^2& =\frac{1}{\sqrt{1+{\left(\omega \tau \right)}^2}}-\frac{1}{1+{\left(\omega \tau \right)}^2}\\ \alpha & =\frac{\omega }{V}\sqrt{\frac{\sqrt{1+{\left(\omega \tau \right)}^2}-1}{2(1+{\left(\omega \tau \right)}^2)}}\\ \tau & =\frac{\frac{4\eta }{3}+\zeta }{\rho V^2}=\frac{4\eta +3\zeta }{3\rho V^2}\\ \alpha & =\omega \sqrt{\frac{3}{2}}{\left(\frac{\rho (\sqrt{{\left(\omega (4\eta +3\zeta )\right)}^2+{\left(3\rho V^2\right)}^2}-3\rho V^2)}{{\left(\omega (4\eta +3\zeta )\right)}^2+{\left(3\rho V^2\right)}^2}\right)}^{\frac{1}{2}}\\ \end{array}$$ The relaxation time for water is about Template:Convert per radianTemplate:Citation needed, corresponding to an angular frequency Template:Mvar of Template:Val radians (500 gigaradians) per second and therefore a frequency of about Template:Convert.

See also

• Acoustic attenuation