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 α given by
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.[2][3]
Sound attenuation in fluids is also accompanied by acoustic dispersion, meaning that the different frequencies are propagating at different sound speeds.[1]
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
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:
Stokes's law is actually an asymptotic approximation for low frequencies of a more general formula involving relaxation time τ: