Chapter 8

Mechanical waves

Soundproofing

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

To study the sound attenuation introduced by a wall, we adopt the following summary model :

In a pipe of cross section , a harmonic incident plane progressive sound wave of angular frequency arrives on a piston of surface , of thickness and of density , free to move around .

Note the speed of sound in air and density of air at rest.

It seeks the form of velocity field :

And :

Question

Justify these expressions and write the corresponding overpressures .

Hint

Solution

The piston created a reflected wave and, using acoustic impedances :

Question

Write the boundary conditions on the non-deformable piston and deduce that :

Hint

Solution

The continuity of speeds gives :

The Newton's second law applied to the piston gives (the piston speed is for example given by ) :

Is :

We deduce then :

Question

Deduce the factor of transmittance in power of the wall.

We give :

What should be the minimum thickness of the wall if you want an attenuation of at least decibels for then for  ?

Hint

Solution

The factor of transmittance in power of the wall here :

We wish attenuation (at least). Therefore, .

We obtain at and at .

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Reflection and transmission of sound waves
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Transmission along an LC network