Chapter 9

# Interpretation of mirages

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.

We study the propagation of a light beam in a vertical plane (Oxz) where the index of the medium (air) varies according to the law :

Where and are constant.

Interpretation of mirages
Comment courber un rayon lumineux ?

## Question

Show that :

### Hint

Apply Descartes relationship for refraction, valid here for a stratified continuous medium.

### Solution

The index equal lines are given by (these are horizontal planes).

The law of refraction of Descartes writes :

So :

## Question

Determine the differential equation of the trajectory of the light beam :

### Hint

Use the fact that

### Solution

We can write the relationship between the angle and the elementary variations and of geometric coordinates of the light beam on its path :

Therefore :

## Question

Show that the beam path is a parabola.

### Hint

Think derive the previous differential equation over .

### Solution

Using the expression of , the above differential equation becomes :

This equation is derived in relation to  :

Therefore :

This equation is solved by :

Where we used the boundary conditions :

and

The trajectory of the light beam in the in-homogeneous medium is a parabola.

## Question

Show that this theory applies to mirages.

We will justify the change in the index of a gas with altitude from the relationship of Gladstone :

Where is the density of the gas. We can assume that the temperature varies with altitude as follows :

Where is a constant (and we can assume that ).

### Hint

Determine the expression of using the fundamental relationship of fluid statics : .

### Solution

The gas is air, likened to an ideal gas. The equation of state can be written as :

Where is the molar mass of air.

Moreover, the basic relationship of the hydrostatic gives :

Then obtained a differential equation satisfied by the pressure :

So :

We deduce by integration :

As , one can perform the limited development :

So :

By using the obtained relationship between pressure and temperature, we get :

Development in the first order :

Hence the expression :

Can thus be identified :

And :

Finally :

• Inferior Mirage (or hot mirage) :

A ground plane, arid, overheated by the sun creates temperature conditions given above.

Bitumen of a road creates the same phenomenon.

Light rays become parabolic and give the impression to be reflected, analogous to the reflection on a mirror.

Hot mirage
• Superior Mirage (or cold mirage) :

A reverse phenomenon can be created when atmospheric layers are warmer and the temperature gradient decreases with altitude drop : it is a phenomenon typically observed at sea.

Cold mirage
Previous
Fermat's principle
Next
Scalar theory of light, light intensity (illumination)