# Mechanical waves

# Transversal oscillations of a leaded rope

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 elastic cord of negligible mass is at equilibrium, tense with a force between two fixed points and of a distance from each other.

The rope supports, regularly spaced, three weights , and of the same mass .

Neglecting the weight of the leads, each string section of length is in the initial equilibrium state is characterized by the stiffness of the string and natural length .

Let .

Small transverse movements of the leads are studied ; ordinate of lead is at time ( ).

It is assumed that its abscissa is constantly equal to .

## Question

Establish a differential system of the second order relative to the studied movement.

### Hint

Apply the Newton's second law to each lead.

Remember, the amplitudes are low.

### Solution

## Question

We seek the solution of type (all weights vibrating in phase with the same frequency).

Determine the possible values of for such movements (proper modes of the system).

### Hint

### Solution

The linear system is obtained :

This system has nontrivial solutions if the determinant is zero :

We then obtain three angular frequencies of the oscillations :

#### Simulation : Amazing pendulum wave effect

Do you know how to build this wave pendulum ?