# Mechanical waves

# Transmission along an LC network

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

Determine the dispersion relation for a periodic wave crossing the line.

What is the cutoff frequency ?

### Hint

Write Kirchhoff's laws.

One can also use the Millman's theorem.

### Solution

Is the voltage across capacitor of the cell number and the current in the coil follows (see figure), we can write :

The last equation is derived with respect to time :

And used two loop laws :

Whence :

We are interested in the propagation of plane waves that we write in the form :

where is the "width" of a cell.

The previous differential equation then becomes :

Or, finally :

It is requested dispersion relation.

The cut-off frequency is obtained when is equal to :

## Question

Determine the propagation equation satisfied by the voltage across the capacitors within the approximation of continuous medium.

### Hint

### Solution

It is assumed that the distance between the cells is small compared to the wavelength ( .

We can then define a function such as (Taylor-Young Development of the second order) :

And the propagation equation becomes :

Is :

We find a d'Alembert equation, characteristic of a propagation without dissipation at the speed .

#### Complément :

We take the dispersion relation obtained in the first question :

And assume that (assumption of continuous medium).

So :

Consequently :

The phase velocity (equal here also to the group velocity) is then :

We find the speed of the wave obtained in the second question.