# Mechanical waves

# Dispersion on a chain of atoms

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.

## Question

Why doesn't depend on ?

### Hint

### Solution

The wave is neither stationary nor damped : all atoms are therefore the same amplitude for a progressive wave.

The system is generally translational invariant.

Thus, is independent of the considered atom.

## Question

Determining, as a function of , the possible values of the wave frequency that can propagate on the chain.

What does we deduce from the nonlinearity of this relationship ?

### Hint

### Solution

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

The solution proposed in this equation is injected and we obtain :

Is :

This dispersion relation is non-linear : there is dispersion.

## Question

Calculate the phase and group velocities .

Give their limits for and and comment.

### Hint

### Solution

The phase velocity is : (one located in the following in the case )

And the group velocity :

For (thus big wavelengths) :

We find the velocity obtained in the approximation of continuous medium.

And for :

And :

The wave doesn't pass anymore.