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.
Question
Question
Question
Show that the beam path is a parabola.
Think derive the previous differential equation over
.
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
).
A video giving the explanation of mirages :
A video about optical fiber :
Determine the expression of
using the fundamental relationship of fluid statics :
.
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 :
Gladstone's law is written :
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.
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.
