2) CLOCK2 (Transversal wave clock) .

More complex case with the mirrors along the Y' axis of the moving frame, but more interesting for the results.

A) INTRODUCTION.

We consider first the clock at rest in the rest frame fully the same as the clock1, in chosing a frequency of resonance fo in order to have obviously 2.L'=n[c/fo] where L' is the distance between the mirrors and n an integer number of wavelengths along the round-trip. It is also in the purpose to have always a well-defined frequency, with no discontinuity, which would not exist if the number of wavelengths would not be an integer.

Now, if the resonator is put in motion at the velocity v along the X-axis and because I have a logical mind, it is clear for me that the hypersound-wave energy will go out of the resonator, because I have no means to carry along the waves with no walls (mirrors) along the X' axis.

But, nevertheless, in order to have the energy of the hypersound-wave in motion, there is the possibility to tilt slightly and precisely during the acceleration, one the two a mirrors, which will change the orientation in space of the equiphase surfaces of the waves.

And the result will be the figure below, with a direction orthogonal to the equiphase surfaces making an angle with the X'-axis given by the formula:

or the same: .

In this case, the velocity of the waves along the X-axis will be v, and the hypersound-energy remains between the mirrors.

FIG 1

As, I am going to show it, to have tilt the mirror (1 or 2) during the acceleration is not without influence on the waves, because due to the classical Doppler effect, the frequencies have been also changed.

b) Increase (decrease) of frequency by the classical Doppler effect when a wave reflects on a tilted moving mirror .

As the figure 2 shows it below, the equation of the downward wave (negative Y) in the rest frame is: (a is the incident angle) and the one for the upward wave is: (with the reflected angle).

FIG.2

It is also clear that the mirror equation is: with the tilted miror angle of the figure.

But now, if we introduce this equation into the wave equations above by elimination of x, the two remaining parameters: y and t determine the phases of the waves on the mirror.

But as they must be the same, even at a about a constant number, we obtain:

This equation may be also rewritten:

And thus ,because it must be valid for all the values of t and y and in particular for t=y=0, the constant Cte must be nil, like also subsequently for the coefficients of t and y, as we may convince ourself of it in giving a nil value to one parameter (ex: t) and not to the other (y)!.

These two nil coefficients are very important and will be used rigorously several times later.

They are in fact a set of two equations in two unknows which permits to calculate the frequency and the reflected angle of the reflected wave when these data are known for the incident wave.

In particular, when the incident angle is 90°, the nil coefficients of t and Y becomes the equations: and .

The first equation shows, as announced, that the output frequency F2 is greater than the incident F1, because cos (phi)=cos(angle of reflection) must be positive when the tilted mirror angle is positive. When the same inclination as in the figure above is used during the acceleration of the resonator.

c) Important result.

The solution of this new classical Doppler effect relations, shows immediately that the incident wave doesn't reflect according to the geometrical optics law.

This thing may be seen in the particular case where the incident angle is 90° and the mirror tilted angle 45°.

By a member-member division of the two fondamental equations, we have:

, and finally:

which is not nil, if the velocity v is not nil, and is thus a result different of the one of classical and geometrical optics.

Such a law may explain the result of the Esclangon experiment in 1927 where a reflection on a mirror is performed. A sidereal period has been detected for variations of angles.

d) Calculation of the frequency inside the clock 2 of fig. 2.

Now according to the fact that the frequencies of the waves in the resonator have been increased during the acceleration due to the classical new Doppler effect, thus in the rest frame, the equiphase planes are at a smaller distance than when the resonator is at rest.

But, it is also clear that the number of equiphase planes (upward and downward included) has not changed during the acceleration like in the clock1, because we condider hypersound-waves which are like sound-waves and which may only be produced by an oscillation like a loudspeaker works. With a non-oscillating acceleration, no equiphase plane may has been added.

Thus, as the cyclic distance 2.L= n.[c/fo] (resonator at rest above) has not changed, the wavelength along the axis Y' in the K' moving frame has not changed.

Thus, as the velocity of the intersection of the equiphase planes of the waves with the Y' axis is: (don't forget that the Y' axis is in motion), we obtain:

and finally: where F' is the clock frequency

inside the moving resonator.

A result for time dilation which is in agreement with the one obtained for the longitudinal clock along the X'-axis (clock 1), but this time without using the Fitzgerald and Lorentz classical hypothesis. It is an important result.

Now, we are going to obtain an other important result, if we consider the waves from the rest frame where the frequency has increased due to the Doppler effect during the acceleration.

As it is clear that the waves (upward and downward) have the same increased F1 frequency,

the equations are (figure 1 above):

and .

The addition yields: .

Now, if we introduce the Galileo's transformations (x'=x-vt, t'=t, y'=y) into this expression to see what is the wave in the moving frame K', with the additionnal relation , we are led to the wave equation: where we see that the frequency F' already calculated above is: . And finally with the value above of F', we obtain: .

It is an increase of frequency like the energy of the electron in relativity , but which is valid here, only in the rest frame like all the present electron theory.

But, an important result is that when we introduce this frequency in the sinus factor of the wave above (calculated for the rest frame), and if the frequency at rest of the resonator is chosen as to be: , like for the longitudinal wave clock1, the sinus factor is also the de Broglie wave: which is fully plane, with a wavelength: which contract according to the Fitzgerald and Lorentz hypothesis, and with a velocity sq[c]/v.

Now, we have all we need to establish the wave structure of the electron.

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