# Mechanical waves

# A sound propagation model in the air

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.

An insulated pipe of cross section is divided into an infinite number of compartments ( ) by insulated pistons and of cross section and mass .

In each compartment is a mole of air, compared to a perfect gas evolving isentropically according to the law of Laplace, .

At equilibrium, the abscissa of the piston ( ) is :

and pressure has the same value in each compartment.

Besides the equilibrium position, the abscissa of the piston ( ) is :

With and the pressure in the compartment is .

## Question

Establish the expression of the pressure as the function of , , , and and the linearized using :

Deduce the linear differential equation determining the motion of the piston number ( ).

### Hint

Is Laplace checked ?

Do not forget that .

### Solution

The application of the Laplace law for the gas in the compartment leads to :

## Question

The approximation of a continuous medium is done by defining a function varying little in the scale of , as .

Establish the partial differential equation whose solution is .

Define a speed and comment expression.

### Hint

### Solution

The Newton's second law applied to the piston number gives:

The approximation of continuous medium resulting in :

Whence :

With :

increases if the environment is more rigid ( increases) and less inert ( decreases), which is natural for mechanical waves.

## Question

Evaluate the speed of sound in air assuming that the piston mass of the model are in fact constituted by the volume of air between two pistons in the model.

We give :

### Hint

### Solution

We find : (in good agreement with the expected value)