3) BASIC WAVE STRUCTURE OF THE ELECTRON IN UNIFORM MOTION ALONG A STRAIGHT LINE [ return ]

a) Introduction.

With clock1 (longitudinal), we have obtained the result that due to the classical Doppler effect, there is a time dilation of F=Fo.(1-bb) with b= v/c.

This law is changed into F=Fo.sqrt(1-bb) if the moving resonator is exactly contracted according to the classical Fitzgerald and Lorentz hypothesis L'1=L'.sqrt(1-bb) known to account very well of the result of the Michelson-Morley experiment.

And incredibly, due to a new change of frequency of the waves, there is a factor of the standing wave (in the rest frame) which is the de Broglie wave of the electron if the frequency at rest is chosen as to be mcc/h (where m is the rest mass of the electron and h the Planck's constant).

More, the frequency in the rest frame of the de Broglie wave increases according to the formula F=Fo.[1/sqrt(1-bb)] the same way as the mass of the electron in the Guye and Lavanchy experiment in 1915.

With clock 2, on the contrary, the correct time dilation has been obtained directly without the Fitzgerald and Lorentz' hypothesis, like also the de Broglie wave, the increase of the frequency like the mass of the electron. And more the Lorentz' hypothesis appears as a consequence of the theory, because the sinus-cosinus factors of the standing wave in the rest frame which is:

, has, as only fonction of x, the de Broglie's wave: with a wavelength: which contracts according to the Lorentz' contraction formula.

An important fact to understand already how an hypersonic electron works, is to know that this contraction of the de Broglie's wavelength is done with a true physical contraction of the classical hypersonic energy contained in the resonator of clock 2, simply because if a wave changes of direction, the place needed along the x-axis for the wave decreases according to L'=L.sqrt(1-bb).

This may be well understood in the figure below:

where we have seen that cos(A)= v/c and thus L'=L.cos(B)= L.sqrt(1-bb) with b=v/c.

We are also already able to understand partially here why and how the energy of the hypersonic electron increases with the velocity.

It is because, if S is the lateral area of the resonator (chosen rectangular), and because the classical sound-wave energy density of plane waves is proportionnal to the square of the frequency D=K.sq(f), the energy at rest is simply: E=K.sq(Fo).S.L, but the energy in motion is: E'=K.sq[Fo/sqrt(1-bb)].S.L.sqrt(1-bb) = K.sq[Fo].S.L. [1/sqrt(1-bb] , because it has been proved that the frequency of the waves increases according to [1/sqrt(1-bb)], and means finally that::

E'= E.[1/sqrt(1-bb)] in agreement with the Guye and Lavanchy experiment (increase of mass with velocity). More details about that is given here.

b) Hypothesis to continue.

We have thus already, a lot of good reasons to believe that the electron is made of sound-waves (hypersounds) in the gazeous ether, but this will be more convincing later when the mass-energy relation will be derived also from the classical energy density of scalar longitudinal sound-waves in a perfect gas (PV=nRT).

But even if there is a slight difference between the results from clock1 and clock2 where the longitudinal clock1 has needed the Lorentz' contraction hypothesis to obtain the correct time dilation we are going to consider that the electron is made of hypersound-waves which reflect on surfaces in motion at the velocity v, but with a frequency inside the electron which would be F'=Fo.sqrt(1-bb) in the whole wave structure seen from a Galileo's frame in motion with the electron, like this have been proved for the clocks. But when the electron wave structure will be established, it will be easy to understand why there is also a Lorentz' contraction for the longitudinal clock, and with the same consequence for the energy.

More, it is clear that we must certainly have a spherical-like wave which goes away from the center of the electron (divergent wave) and a wave which goes towards the center of it (convergent wave).

But obviously these waves must have a velocity c (hypersonic velocity) in the rest frame.

b) Analogy between the electron wave structure along the Y'-axis and the transversal clock.

To consider the electron made of a divergent wave and a convergent wave is certainly a good hypothesis, if we look at the figure below concerning the wave structure along the Y'-axis of the electron.

where we see that anywhere (nevertheless chosen) along the Y' axis, the divergent wave has been transmitted by the electron at the time t-delta(t) like the convergent wave will reach the electron after the same delta(t) time.

Thus, this leads to the fact that directions of the waves must have an angle a, with cos(a)=v/c, exactly the same as for the transversal clock2. And thus, it is clear that the de Broglie wave will have the same origin as in clock 2, and thus the energy of the hypersonic waves of the electron will certainly change the same way (as in fact we know it already).

C) Precisions concerning the frequency chosen.

The frequency chosen in the whole structure of the electron: F'=Fo.sqrt(1-bb), from a Galileo's frame in motion with it, is also a necessity in order to maintain the stucture of the electron. Because between two different points of the electron, if the frequency would not be the same, a continuous decreases of the number of equiphase surfaces would have as a consequence that the electron would be empty of energy after a short time between the points. The same, an increase of equiphase surfaces would increases the energy of the electron towards an infinite value.

Conclusion, if the electron is stable, the frequency must be the same everywhere in the structure (moving frame!).

d) Calculations of the anisotropic speeds of the hypersonic divergent and convergent waves of the electron in the moving frame in ether (also valid for the light speed obviously). Theses results for light will be used later to show that the round-trip speed of light is c, in the moving frame for any direction in space.

This is simply done geometrically, in considering that the speed of the waves must be c, in the rest frame (see figure below).

.

For the invariable direction determined by the angle " theta " in the moving frame, the velocity c' may be obtained by the obvious following formulas : and .

By elimination of cos(phi), we obtain the quadratic equation :, where the two c' solutions are the speeds of the divergent and convergent waves for the direction theta.

They are: for the divergent wave and for the convergent one.

E) Calculation of the wave equations of the two waves of the electron (an historical step!).

As and with :, , we obtain successively for the divergent wave: , and for the convergent wave: .

Or differently written: and:

Here, it is already possible to obtain the wave equations in the rest frame, if the Galileo's transformations are introduced into the equations above (x'=x-vt, t'=t, y'=y, z'=z).

F) Hypothesis relative to the amplitudes of the waves.

In fact, this equation is simply for the divergent wave, the equation of the wave produced by an oscillating source in motion in the propagating medium (perfect gas), like you may see it in a cup of tea with a moving oscilating spoon. But, it is better to ask me a computer animation program at roland.dewitte@pi.be (free!). But what is new, is that the convergent wave is exactly symetric relatively to the Z'Y' plane in the moving (electron) frame.

This strange fact, is due to a reflection (interaction) mechanism between the divergent wave and the convergent one which is produced by the waves themselves. With only one wave, such a mechanism would not exist, and the electron would be dispersed in space.

And the localization of energy towards the center of the electron is due to ellipsoidal partials reflectors produced by the waves in interaction with the ether very close to saturation. Recent calculations tend to lead to a [Volumic mass of frozen ether]/[Volumic mass of ether] ratio of only 1.64. A slight pressure tends to froze ether.

Thus, the hypersonic energy remains in the multy-cavity electron, like an electromagnetic wave may be trapped and accumulated in a resonant cavity.

But an important fact to tell here, is that for any position in space, we make the hypothesis, that the amplitude of the divergent wave is the same as the convergent wave.

It is a temporary hypothesis, wich will be derived later and even with the result that the amplitude is constant on ellipsoidal surfaces centered on the electron (symetry center).

G) De Broglie wave.

Due to the above temporary hypothesis, the signal (only accessible for a measurement) is the addition of the wave equations of the divergent and convergent waves with the following result in the moving frame: , which becomes in K with the Galileo's transformations introduced:

and with: which is also a wave (amplitude of the wave in motion at the velocity v in the rest frame).

But, we see immediately that the sinus factor is still the de Broglie wave, if the frequency at rest of the electron resonator is still chosen as to be: , like it was the case for clock1 and clock2.

The de Broglie wave is like an amplitude modulation of the structure and is produced by an interference between the two waves of the electron, which have different frequencies (variable also with the direction) due the classical Doppler effect after reflection on the partial mirrors where the interaction between the divergent and convergent wave takes place.

This thing will be precised later when the Maxwell's equations will be demonstrated .

But, what is incredible, is the fact that the de Broglie wave is fully plane, even if the wave equations are not simple (the de Broglie wave doesn't depend of z and y).

I am sure that Louis de Broglie would have found that very interesting, because in agreement with his intuitions.

The same way, the amplitude is variable, like he thought it.

An important fact, is also that the de Broglie wave exists in the moving frame of the electron and will be soon probably detected by experimentalist. But clearly in the wave equation above for the moving frame, the de Broglie wave has a frequency of , obviously in agreement with the time dilation chosen and is the only wave with a time dependant signal.

Finally, with the added wave fonctions in K, we see on the X axis (with y=z=o) that the other factor cosinus: is a wave of wavelength: which also contracts according to the Lorentz' contraction, like the de Broglie wave:

This is thus a double theoretical convincing result in favour of the existence of the Lorentz' contraction. Nevertheless an other explanation will be done below.

The new wave (cosinus factor) as a new wave discovered by me has been called obviously De Witte's wave and is certainly physically true, when we see that the ratio of the de Broglie 's wavelength by the De Witte's wavelength is 137.036 on the first orbit of the Bohr's atom for hydrogen.

Humour: if Louis de Broglie has obtained the Nobel Prize with his wave without to know the exact equation (temporary spinless electron) and with the erroneous theory of Einstein, it would be logic that I obtain the Nobel for the second wave. If not, the Nobel Foundation would not be equitable. Especially if I need money to live (unemployed) and for my researchs, on the contrary of Louis de Broglie.

But, the reader will see, that the discovery of my wave is a very little part of the incredible things that will be soon on my web-site.

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