Samiksha Jaiswal (Editor)

Stokes' law of sound attenuation

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

Stokes 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

Contents

α = 2 η ω 2 3 ρ V 3

where η is the dynamic viscosity coefficient of the fluid, ω is the sound's frequency, ρ 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' law for the friction force in fluid motion.

Interpretation

Stokes' 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 − α d

The parameter α 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 ( m − 1 ). That is, if α = 1 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 η v :

α = 2 ( η + 3 η v / 2 ) ω 2 3 ρ 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

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

2 ( α V ω ) 2 = 1 1 + ω 2 τ 2 − 1 1 + ω 2 τ 2

where the relaxation time τ is given by:

τ = 4 η / 3 + η v ρ V 2

The relaxation time is about 10 − 12 s (one picosecond), corresponding to a frequency of about 1000 GHz. Thus Stokes' law is adequate for most practical situations.

References

Stokes' law of sound attenuation Wikipedia
(Text) CC BY-SA


Similar Topics