So, from the most general considerations, we strictly deduced the equation of wave motion. In my opinion, this equation is very beautiful, because it is simple and fundamental.
The magnitude of the integral is simply the number of wave periods. In other words, it is the phase difference of the wave at points B and A:
The wavelength λ is in the integrand. Instead of it you can use the period of the wave T. If the period were constant, then the total number N of oscillations of the wave would be equal to:
N = t/T
Here t is the time of motion of the wave from point A to point B. Since T changes, it is necessary to rewrite this equation as follows
dN = dt/T(t)
Then we will integrate it:
∫dN = ∫dt/T(t)
As a result, we will obtain the wave motion equation written in another form:
If the variation of the integral is zero, then the integral takes an extreme value: either the minimum or maximum. In our case, obviously, this is the minimum. Therefore, wave equations can be represented in the following form:
Any wave moves to spend a minimum of its own oscillations on the traversed path. If we assume that the natural oscillations of the wave are its own clock, then we can say that the wave always moves to spend a minimum of own time on the passed way.
In this formulation, Fermat’s principle is applicable to the motion of light and to the movement of a stone.
Consider a stone that moves in a gravitational field along a parabola:
The stone moves along the trajectory ACB. At point C, At point C, the velocity of the stone decreases, its kinetic energy also decreases. Consequently, the quantum frequency of its wave decreases, and its period increases. Therefore, moving from point A to point B through point C, the stone spends a minimum of its proper time (the number of oscillations of the quantum wave associated with it) on the traversed path.
We have “rehabilitated” Fermat’s principle and now it is just as fundamental as the principle of least action. Only Fermat’s principle is simpler, clearer and more understandable.
Let us formulate it again.
Any object (light, stone, electron…) has wave properties, so it always moves to spend a minimum of its own time on the path traveled. Why does it need this? It does not need anything. Just on any other trajectory neighboring waves will not be in phase and will mutually cancel each other. Neighboring waves do not extinguish each other only if they move with the same phase. It is this condition that is expressed in the wave equations.