# 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 .