We will see the interest of using the phase wave in an important question in physics :
the formula which allows to know the kinetic energy of a particule in motion. It is amazing that such formulation of kinetic energy has been taken for granted
for over a century by the scientific community because, even though it was verified by the experiment,
it has never been demonstrated as such; even Einstein’s demonstration (1905) isn’t convincing (I will come back on it).

We set up our demonstration in two stages :

First stage : As per the hypothesis used to start with and the first elements of demonstration,
a comparatively rest particle is in the middle of a stationary, circular and concentric waves system due to the
interference of a dual wave sub-systems at a velocity c and a frequency ν_{0}, one being advanced and
the other retarded. The whole setting applies some inertia against any change in position or motion, due to
a repositioning at the frequency ν_{0}.
While in motion at a speed v, the two sub-systems are no longer in a precise superposition.
The sub-system which is retarded, moves with the particle towards the fore part of the motion, thus creating
a phase wave with the shift between the two waves. This phase wave travels at speed V
such as :

v.V = c^{2}

and which can be very high when v is low. It is not stationary but shifts along
with the particle.
A viewer travelling alongside the particle would notice that beats of this phase wave move past in front of him
at a frequency ν_{0}.

But the phase wave is in motion with the centre of the particle, in other words
the system in vibration shifts and an outside standstill viewer would notice a higher frequency ν_{v},
defined as follows:

such noticed inertia in the same direction as the motion by an outside viewer becomes :

I_{v} = hν_{0}/c^{2}(1 – v/V).

And knowing that, v/V = v^{2}/c^{2}, we obtain

I_{v}c^{2} = hν_{0}/(1 – v^{2}/c^{2}).

It is a mere Doppler-Fizeau effect, due to the phase wave existence and its shifting
with the particle.

Second stage : As we have just seen, this theory on repositioning gives to the special relativity a new interpretation,
but it keeps the changes, which allows to apply to frequency ν_{0} of a travelling particle the rule
of time slowing-down :

and this takes us back to the equation of energy applied to a particle in motion,
which is called kinetic energy :

This equation, by a difference with rest energy and by a sequential development
of powers to (v/c), allows us to easily meet again the classical expression for kinetic energy when the speed v
on the mobile is clearly lower than c :

˝mv^{2}.

From there, we can come back to classical Mechanic equations and fundamental
principles. The usual starting point for it consists, since Newton, to set a priori the relationship between strength,
mass and acceleration,

F = mγ,

as a fundamental principle. In the theory of repositioning, and after having
carried out this demonstration, such relationship will be duly deducted.

While more clear-sighted, Louis de Broglie had noticed the contradiction between:

the relativistic time slowing-down, due to velocity,

the frequency increase following the increase of kinetic mass, which would imply an acceleration of time.

He was preoccupied by this idea for a long time; it is the key element of his
thesis in 1925 onto which he will come back again in 1964, in The thermodynamic of the isolated particle.
It is where the idea of phase wave and the theory of the dual solution originated from. These last two paragraphs
cancel such contradiction: when the particle is in motion and when an outside viewer tries to stop it, the phase
wave increases, by Doppler effect, the inertia of the particle in the same direction as the motion, and well beyond
the kinetic inertia which is usually observed. But on the other hand the relativistic slowing-down of time, whose
equation we met above, restores the expression in terms we are familiar with. Kinetic inertia thus becomes clear
a phenomenon. When we stop a mass in motion,
the shock we feel and that doesn’t last, is due to repositioning last attempt alongside with the last passage
of the phase wave towards the front. The following instant, the particle returns to its comparative state of rest,
the same the viewer experiences when he no longer feels the Doppler effect.

This demonstration is interesting on several points :

it raises the veil on an important question in physics which seemed until now ambiguously and subject to contradictions,

it is exemplar what should be every demonstration in physics : associating the description of a phenomenon with its
explanation, in order to set up a formula with a clear physical signification for each of its terms, which will be checked by
experiment.

it shows the interest the phase wave carries by giving proof that it is such phase wave that dictates the position of the particle,
it is the position wave.