Chapter 8

Mechanical waves

Influence of viscosity on the propagation of sound

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In the case of a viscous fluid, the equation satisfied by the velocity field is the Navier-Stokes :

Where is the dynamic viscosity of the fluid.

We assume that the density and pressure fluctuations are small and that evolution is isentropic.

In the following, we place in one dimension along the (Ox) axis.

Question

Establish the propagation equation :

Hint

  • Linearize the Euler equation and mass conservation equation.

  • Using the coefficient of isentropic compressibility.

Solution

The acoustic overpressure is noted and the change in density with respect to the equilibrium value, denoted .

The mass conservation equation linearized gives in one dimension :

Similarly, in connection with the acoustic hypothesis, the Navier-Stokes becomes :

Furthermore, the isentropic compressibility coefficient is written as :

The mass conservation equation can be written as :

Is :

Referring to the Navier-Stokes :

The above equation is derived with respect to space and, using the equality between the two derivatives :

Finally leads to the equation :

Question

We seek a solution in the form of a monochromatic progressive plane wave of the type :

  • To determine the relationship between and .

  • We set . For a low viscosity fluid, .

    Give the expression of in the first order of .

    What is its physical meaning ?

Hint

In complex notation, a derivative with respect to time becomes multiplying by and derivate with respect to becomes multiplying by .

Solution

  • The dispersion relation is obtained from the propagation equation, noting that a derivative with respect to time is equivalent to multiplying by  and derivate with respect to is equivalent to multiplying by .

    So :

    Whence :

    With :

  • To a low viscosity fluid, . Therefore :

    For identification :

    and

    In real notation, the pressure is expressed in the form :

    A characteristic distance can be defined :

    that represents the attenuation characteristic length of the pressure wave in the viscous medium.

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Acoustic waves
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Acoustic impedances