3.1

Motion of a wave

Fermat's Principle

Consider the motion of a wave. It can be light, an electronic wave, a wave on water, a sound wave. In general, any wave, though classical, even quantum. So I drawn an arbitrary curved line from point A to point B:

1.2

Let this be the path of the wave. Perhaps you will not believe, but now, just looking at this picture, we will write the most fundamental equation of the wave motion. We do not need any knowledge of waves for this. Forget about Fermat’s principle and the principle of least action. We will not need them either.

Now draw a line from point A to point B, which is very close to our curve:

1.3

So that the distance between the curves was much smaller than the wavelength. Thus, the maximum on the additional curve will correspond to the maximum on the main curve, and the minimum on the additional curve will correspond to the minimum on the main curve. This means that along the two curves there is the same number of wave periods N. What is this number equal to? If the wavelength λ of our wave were constant, then this number would be equal to:

L

Here L is the traversed path.

The wavelength changes along the trajectory of motion. Therefore, it is necessary to take an infinitesimal change in the path dx and divide by the wavelength:

dN dx

Now we should take the integral of this expression along the trajectory from A to B:

dN = ∫dx

As we have already found out, the value of this integral (it is equal to the total number of wave oscillations that it made when moving from A to B) will be the same for all closely located trajectories. This means that the variation from this integral is zero:

3.1

We have obtained the basic equation of wave motion. According to quantum mechanics, all physical objects have a wave nature. Therefore, all physical objects obey this equation. This is the most fundamental equation in physics. We got it so easily.

What did we use?

We used only that in any wave the maximum passes into the minimum not immediately, but gradually. At very close distances (much less than the wavelength) there is also a maximum from the maximum. That’s all. That was enough.

We have obtained the basic equation of wave motion. According to quantum mechanics, all physical objects have a wave nature. Therefore, all physical objects obey this equation. This is the most fundamental equation in physics. We got it so easily.

What did we use?

We used only that in any wave the maximum transforms into the minimum not immediately, but gradually. At very close distances (much less than the wavelength) there is also a maximum from the maximum. That’s all. That was enough.

Vasily Yanchilin

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