# Mechanical waves

# Reflection on a free mass

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.

At point of abscissa of a very long rope, linear density , a mass is attached on the rope.

An incident wave :

arrives from the side .

## Question

Determine the expression of the reflection coefficient (neglecting the weight of the mass).

What will happen if becomes very large ?

### Hint

What are the boundary conditions in ?

Apply Newton's second law to the mass

### Solution

The incident wave gives rise to a reflected wave :

And transmitted wave :

Where and are the complex coefficients of reflection and transmission.

The continuity of the position of the mass in gives the first equation :

The second law of Newton, applied to the mass (and neglecting the weight of the mass), is written (see figure) :

Where the speed of the mass can be expressed as :

And and denote the tension forces from the rope on both sides of the mass .

In projection along the (Ox) axis, we obtained (see figure to the definition of angles and ) :

The angles being small, the cosine is equal to . Thus :

The magnitude of tension is a constant .

In projection along the (Oy) axis, and assimilating the sines to the values of the angles expressed in radians :

Yet :

and

And :

and

That is to say :

Finally obtained the following system of two equations :

With :

The resolution gives :

For a very large mass ( ), the reflection coefficient approaches : there is total reflection in the opposite phase and without transmitted wave (transmission coefficient approaches ).

The inertia of the mass blocks the passage of the incident wave with creation of a standing wave for .