3) Hypersonic calculation. [ Return home ]

1) Introduction.

The calculation of the energy will be done with the energy density of the sound-waves derived in an other chapter: .

As we need the amplitudes A and the frequencies F of the waves everywhere in the structure, we are going to calculate them first.

2) Doppler effect.

In the direction (cosa, sina), the electron tranmits an hypersonic signal whose equation is: . The frequency f depends of the direction, that is to say: f=f(cos(a),sin(a))

In the moving frame, according to the Galileo's transformations X'=x-vt, t'=t, the equation is:

with a frequency:

But as, according to the time dilation,: in the whole structure in the moving Galileo frame, we have in the rest frame:.

3) Retarded frequency.

According to the figure below, it is easy to understand that when the frequency of a part of the hypersonic structure of the electron is needed, this frequency calculation with the formula above need to know cos(a) at the retarded time when the wave has been emitted in that direction.

4) Calculation of cos(a).

First wa have immediately: , t being the retarded time.

But as it has been established that the velocity for the divergent wave, we have:

But as and , we obtain:

5) Frequency calculation.

With cosa in the fomula , we obtain:

6) Amplitude calculation.

In the derivation of the Maxwell equations and at several other places, it has been established that the averaged amplitudes of the two hypersonic components of the electron obey the following equation: where Ao is the maximum amplitude on the nucleus surface and Ro the radius of the nucleus at rest, the center of the electron is considered at the origin of the coordinate system with t=0.

7) Energy calculation.

Thus, the integration of the energy density outside the ellipsoidal nucleus yields:

with and

A factor 2 appears because we have two waves of identical energy and which are fully symetrical in amplitude and frequency.

By the same Jacobi transformation as for the electromagnetic calculation: with , the integral become more simple:

This time the mathematical nucleus is spherical due to the change of variable.

By multiplication of the numerator and the denominator by: this integral simplifies into:

or more:

The second integral is nil and because with we have finally:

fully identical as in the electromagnetic calculation where the result was: .

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