Chapter 8

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.

The transmission line is considered below :

LC network

Question

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

What is the cutoff frequency ?

Hint

Solution

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

Transmission along an LC network

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.

Previous
Soundproofing
Next
Multiple choice quiz