2) Mechanism for the localisation of energy.-----[ Return home ]

__A. Existence of a retrodiffusion mechanism.__

If we consider the simple structure of the electron at rest with the wave equations:

and , we may do a new hypothesis on it.

First, it is cristal clear that the two waves may not have an amplitude of the form:

Ao/r, which would be the case with especially a sustained divergent spherical wave.

It is because, the energy density of hypersonic longitudinal travelling waves is like:

and leads to a total integral of energy in the whole space which is infinite (even outside the nucleus where in fact there is no energy).

Such a fact is not logically possible, because the electron appears always with a finite energy.

Thus, we need absolutely an amplitude decrease of the waves more rapid than in 1/r.

Unfortunaly, we have the immediate problem that a wave like A=A(r).sin[2.pi(t+r/c)] is not solution of the wave equation: Laplacian A-[1/cc]ddA/dtdt=0, if the amplitude A(r) smoothly decreases more rapidly than 1/r.

But we are going to see later that there is an other problem.

But now, we comme back first with the whole energy.

For example for an amplitude decrease in , the whole energy: (with Ro, the radius of the nucleus) is finite if a>1.5.

Nevertheless, nothing permits to say that the amplitude has really that simple fonction, but what is true is that the whole energy must be finite (for both waves).

But also, by the fact that the energy of the convergent wave must have its obvious origin in the loss of the divergent wave whose amplitude has to verify the whole finite energy criterion, the principle of preservation of the energy leads us naturally to affirm that it is the divergent wave which is the origin of the convergent one, by a **mechanism of retrodiffusion** (or reflection).

Yes, without loss of energy, if we consider the energy contained in a spherical layer of thickness dr of the divergent wave which is: , where D(R1) is the energy density for this radius, the principle of preservation of the energy leads to find again (after propagation), the same energy in any other layer of same thickness and of radius R2, which implies that the energy density would be in , or more precisely that: , and thus that the whole energy would be infinite.

Mechanisms for retrodiffusion exist already in the nature, as it is the case notably for the molecular diffusion of light in an optically pure medium.

Here, such an effect must be particularly active for the special conditions of mass, concentration, speed and energy of the ether particules.

This mechanism for retrodiffusion must also be subjected to a new fully plausible hypothesis that the divergent wave must be the only one which retrodiffuses.

In fact, if the convergent wave would be retrodiffusing also, the same way as the divergent one, it would lose all its energy before to reach the geometric center of the electron, in crossing with the same energy density, the layers where the divergent wave loses its energy while scattering in space. The result would be that the energy density would be very weak in the center (where there are still waves) like at long distances. Thus not really plausible.

Finally the existense of the nucleus of frozen ether would be difficult to justify, nor the numerous interesting consequences that this nucleus implies, notably about the interpretation of the Compton effect as a Doppler effect.

__B. Rejection of the existence of a continuous retrodiffusion mechanism.__

It has been supposed that the divergent wave may lose its energy in favour of the convergent one in a smoothly maneer. Unfortunately this hypothesis may not lead us to a decreasing fonction identical for the two waves, which is fully a necessity according to the principle of preservation of the energy.

This is because on any spherical surface, the incoming energy must balance the outgoing one. Without that, the electron decomposes.

The following calculation which doesn't permit to reach an acceptable result is a necessity for the reader to convince himself that the consideration of a coherent and discontinuous decrease of the amplitudes of the composite waves of the electron is the most plausible (see farther).

Here we still consider the electron at rest.

We have seen that the principle of preservation of the energy applied to a divergent wave which scatters in space without attenuation, leads to a decrease of the energy in .

Thus the amplitude decreases in 1/r and we may represent it by , where A(Ro) is the amplitude in Ro. We have also:.

Now, we consider an unknown fonction with the decrease which represents the real amplitude of the composite waves of the electron.

If from a radius r the energy of the divergent wave would be scattered in space without attenuation by retrodiffusion, the rate of decrease would be for r : .

Thus, the rate of decrease which is the origin of the retrodiffusion is obviously: , or , a term which is always negative, because the decrease of the real amplitude must be more rapid that the simple decrease by scattering of the energy, in order to produce a retrodiffusion.

As the direct (divergent) wave is by hypothesis of the form: , the elementary convergent wave in r and retrodiffused between Xo and Xo + dXo with Xo > r is:

where the factor Xo/r comes from the fact that the amplitude of the convergent wave increases in 1/r from Xo towards r, because there is not a retrodiffusion for it.

La phase constant is determined by the fact that we must have in Xo an identical phase between the infinitesimal retrodiffused wave and the divergent wave, that is to say: , or better: .

Finally, the whole retrodiffused wave (convergent) would be of the form:

.

Unfortunately, even if the present calculation is very rigorous, all the computer trials

of integration of this wave from appropriate fonctions F(Xo) and especially with sq(Xo) (which would be in agreement with the intensity of the electric field) lead to the fact that the amplitude is too weak for a great r and too strong for a weak r, in comparison of the correct amplitude must be , in order to verify the principle of preservation of the energy and to allow the electron to continue to exist.

The same way, I had a few hopes to determine with this fonction, where the amplitude must be the one of the divergent wave, the frequency Fo of the electron.

Unfortunately, even if a particular frequency is needed, it is ridiculously weak in comparison of the internal frequency of the electron: .

All these problem disappear when we consider that the amplitude of the waves of the electron decrease in 1/r, except for certain values of r where an abrupt decrease occurs and where there is always synchronism (simple phase relationship) between the divergent wave and the retrodiffused one (reflected).

__C. Possibility for the existence of a discontinuous and coherent retrodiffusion.__

If the divergent wave may not decrease smoothly (by retrodiffusion), we need then to consider the case where an abrupt decrease occurs for particular values of r outside of which the decrease would be in 1/r. Obviously, in this case, the term of speed decrease for retrodiffusion: is nil, because F(r)=r.

The problem has been to know where this abrupt retrodiffusion occurs. The choice has been relatively easy when I have seen that the interference of the composite waves yields the standing wave: where certain discontinuous values determine a nul amplitude (nodes).

If we consider that on the nucleus surface, the amplitude of this wave is nil, the discontinuous values of r are obviously: where n is an integer number, positive or nil.

If the fonction decreases abruptly after all the positions determined by an integer value of n, then there is a fonction: of the discontinuous values r(n) which represents the decrease of amplitude of the divergent wave.

Obviously, all the informations for the decrease and for the non-discontinuous values of r are also determined by this fonction, because it is in 1/r outside the zones where the abrupt retrodiffusion occurs. In fact in the semi-open interval: , the fonction of the amplitude of the divergent wave is: and then, the whole fonction may be represented by: where the fonction if and zero else.

The wave fonction (without any similarity with QM) of the divergent wave is thus:

.

Now we derive the convergent wave.

As directly after the point r(n), the abrupt decrease of the amplitude of the divergent wave is: , its opposite represents the amplitude of one of the multiple composite waves of the convergent wave which in the interval: is: and where the phase constant may be simply derived relatively to the divergent wave in considering that in , that is to say on the surface of the nucleus, the divergent wave after reflection is in phase opposition with the convergent wave.

Ans this means that: , or .

A remark here is that we obtain the same result if we consider that the retrodiffusion in r(n) with the hypothesis that the phase of the retrodiffused wave experiences a phase rotation of pi like it is the case in general when a sound-wave reflects on a rigid surface.

In fact implies that , that is to say: if we take into account that: .

Then the equation of the whole retrodiffused wave is:

This time, the amplitude of the retrodiffused wave may be rigorously the same as for the divergent wave, under a particular condition that the amplitude of the divergent wave must meet: it must decrease more rapidly than in 1/r and in such a manner that: .

For that, we have to note first that the convergent wave is always nil if: (decrease in 1/r), because the generic factor of the amplitude of the composite waves is nil: .

It remains thus to verify that the above hypothesis for convergence leads to have the same amplitude for the waves:

where the common factor 1/r in both members is not written.

Now we consider the right sum for any value of . Thus we have: which is the same as:, or also:, where the delta fonction is equal to one when the inequality is not verified. Thus for K great enough the expression becomes:

, that is to say: , because when the limit is performed, the last term is nil (by hypothesis).

If we consider now the left member we obtain immediately: when .

Even if no powerful, the hypothesis: is a first criterion of validity of the amplitude of the waves of the electron, and is notably verified for a decrease in 1/rr like the intensity of the electric field in electrostatic. A field which is certainly a proportional expression to the amplitude of the waves.

An other fortunate remark in favour of the whole which has been said, is that a pure decrease of the amplitude in 1/rr is not solution of the sound-wave equation, but that the parts of the discontinuous waves considered here in 1/r are good solutions.

An other thing to consider finally is that the phase constant of the convergent wave

changes a bit the positions (at a given time) in space of the different waves, like the positions relatively to the nucleus, without modifying their propagation. This difference will be considered elsewhere.

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