Stokes's law of sound attenuation

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 at some point. After traveling a distance from that point, its amplitude will be A(d) = A_0e^{-\alpha d} The parameter 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 (m). That is, if  = 1 m, the wave's amplitude decreases by a factor of for each meter traveled. ==Importance of volume viscosity==

The law is amended to include a contribution by the volume viscosity : \alpha = \frac{\left( 2\eta + \frac{3}{2}\zeta \right)\omega^2}{3\rho V^3} = \frac{\left( \frac{4}{3}\eta + \zeta \right)\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.{{cite book | last1=Dukhin | first1=A.S. | last2=Goetz | first2=P.J. |title=Characterization of liquids, nano- and micro- particulates and porous bodies using Ultrasound | publisher= Elsevier | year=2017 ==Modification for very high frequencies==

Stokes's law is actually an asymptotic approximation for low frequencies of a more general formula involving relaxation time : \begin{align} 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\left(1+\left(\omega\tau\right)^2\right)}}\\ \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\left(\sqrt{\left(\omega\left(4\eta+3\zeta\right)\right)^2+\left(3\rho V^2\right)^2}-3\rho V^2\right)}{ \left(\omega\left(4\eta+3\zeta\right)\right)^2+\left(3\rho V^2\right)^2} \right)^\frac{1}{2}\\ \end{align} The relaxation time for water is about per radian, corresponding to an angular frequency of radians (500 gigaradians) per second and therefore a frequency of about . ==See also==