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 A₀ at some point. After traveling a distance d from that point, its amplitude A(d) will be

$${\displaystyle A(d)=A_{0}e^{-\alpha d}}$$

The parameter α 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^–1). That is, if α = 1 m^–1, the wave's amplitude decreases by a factor of 1/e for each meter traveled.

The law is amended to include a contribution by the volume viscosity ζ:

$${\displaystyle \alpha ={\frac {\left(2\eta +{\frac {3}{2}}\zeta \right)\omega ^{2}}{3\rho V^{3}}}={\frac {\left({\frac {4}{3}}\eta +\zeta \right)\omega ^{2}}{2\rho V^{3}}}}$$

The volume viscosity coefficient is relevant when the fluid's compressibility cannot be ignored, such as in the case of ultrasound in water.[4][5][6][7] The volume viscosity of water at 15 C is 3.09 centipoise.[8]

Plot of reduced wave-vector and attenuation coefficient as functions of reduced frequency ωτ. (In the labels, ω_c = 1/τ)

  Reduced wave-vector, kcτ.

  Asymptotic regime at low and high frequencies.

  Attenuation coefficient, αcτ.

  Asymptotic regime at low and high frequencies (Stokes' law is the leftmost line).

Stokes's law is actually an asymptotic approximation for low frequencies of a more general formula involving relaxation time τ:

$${\displaystyle {\begin{aligned}2\left({\frac {\alpha V}{\omega }}\right)^{2}&={\frac {1}{\sqrt {1+\left(\omega \tau \right)^{2}}}}-{\frac {1}{1+\left(\omega \tau \right)^{2}}}\\\alpha &={\frac {\omega }{V}}{\sqrt {\frac {{\sqrt {1+\left(\omega \tau \right)^{2}}}-1}{2\left(1+\left(\omega \tau \right)^{2}\right)}}}\\\tau &={\frac {{\frac {4\eta }{3}}+\zeta }{\rho V^{2}}}={\frac {4\eta +3\zeta }{3\rho V^{2}}}\\\alpha &=\omega {\sqrt {\frac {3}{2}}}\left({\frac {\rho \left({\sqrt {\left(\omega \left(4\eta +3\zeta \right)\right)^{2}+\left(3\rho V^{2}\right)^{2}}}-3\rho V^{2}\right)}{\left(\omega \left(4\eta +3\zeta \right)\right)^{2}+\left(3\rho V^{2}\right)^{2}}}\right)^{\frac {1}{2}}\\\end{aligned}}}$$

The relaxation time for water is about 2.0×10⁻¹² seconds (2 picoseconds) per radian, corresponding to an angular frequency ω of 5×10¹¹ radians (500 gigaradians) per second and therefore a frequency of about 3.14×10¹² hertz (3.14 terahertz).

• Acoustic attenuation