Chapter 8

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.

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We want to prove that  longitudinal elastic wave of an angular frequency  and a wave vector can propagate along  an infinite chain of atoms, of mass , of a spring constant and rest positions :

In complex notation, the movement of the mass numbered is :

Dispersion on a chain of atoms

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.

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Sound propagation in solids
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Acoustic waves