II. Properties of the sound-wave energy contained in a simple resonator in motion (simplified electron).__[ Return home ]

We consider a perfect sound-wave (without loss) produced by an ether loudspeaker and trapped between two mirrors (see figure below).

I am going to show that the wave trapped has all the properties of the electron and this fact is a good way to understand easily what is the intimate fonctionning of the elementary particules.

The electron is also composed of self-produced cavities with sounds of very high frequency in the order of 1.23E(+20) Hz. If you like to see how are the cavities and the waves, just send me an E-mail at roland.dewitte@ping.be to ask it and you will receive a free animation program.

But to understand the simplified electron considered here, we have to ignore here the energy needed to accelerate the mirrors, but not obviously the part needed to accelerate the wave contained in the resonator.

I) Cavity at rest.

First, we consider the wave produced in the resonator at rest, made of two counter-travelling waves: y1(t)=Ao.sin[2.pi.f(t-y/c)] and y2(t)=Ao.sin[2.pi.f(t+y/c)].

The frequency f is chosen in order to have an integer number of wavelength along the round-trip in the resonator. That is to say with n.Lambda= n.[c/f]=2.H.

The whole sound-wave energy contained in the resonator is thus:

If we define the volumic mass of this simplified electron as to be:

we obtain the well-known Einstein's equation: .

The same way, we can define something which plays the same role as the Plank's constant:

with .

But here, for this simplified electron, the constant may be variable because the frequency or the amplitude of the waves contained in the resonator may be mofified when the wave is built.

But when the wave is built, I am going to show, that even if the wave is accelerated or in gravity, the Plank's constant defined is invariable.

By a mistery of the nature that I have not yet understood, the frequency and the amplitude allowed for the elementary particules have only a limited possible values. It is a goal of my life to find a logical explanation for that which is certainly related to the spin.

But the important fact here, is to see that the defined Planck's constant contains the frequency of the electron and that means that the Planck's constant is related to the electron or to other electron-like particules .

The consequence of such a result is that when an electron interacts with light (frequency f) and extracts the energy E=hf, the energy extracted may not be due to a photon of energy hf, because h is an electron property, but due to the fact that the electron extracts the energy hf from the non-photonic electromagnetic field.

In clear, that means that I don't believe in the existence of the photon, nor aniway in the Einstein's interpretation of the phoelectric effect for which he has obtained the Nobel Prize in 1921. The same way, I firmly believe that the Compton effect is not a collision between a photon and an electron, but a simple classical Doppler phenomenon where a part of non-photonic light energy is transferred to the electron.

II) Cavity in motion.

Obviously if the mirrors are simply accelerated , the sound-waves escape from the resonator. Thus, to maintain the soud-wave energy between the mirrors in motion, it is very important to tilt slightly a mirror during the acceleration (see figure above).

But such an action is not without influence on the frequency of the wave, because the classical Doppler is going to increase the frequency of the wave which will be again invariable when the tilted mirror will be again in the same direction as at rest (parallel to the X axis).

The way the frequency and direction change is explained in the description of clock2.

The result of the acceleration is represented above and will be enough to calculate the new frequencies of the travelling waves.

For that, we have to note that to produce a sound-wave, we need an oscillation of the mirrors.

But as the wave has been accelerated without periodic oscillation of the tilted mirror, the number of wavelengths between the mirrors may not have changed.

For the secondary school students I have to show how to write the wave equation which doesn't propagate along the Y or X axis. To write a wave equation of the form: x(t)=Ao.sin[2.pi.f(t-x/c)] for such a wave, we need the length L (to replace x) along the direction of propagation (see figure below).

And this length L is the scalar product of the vector R=(x,y) by the unit vector E=[cos(a), sin(a)];, that is to say: L=[x.cos(a)+y.sin(a)].

The wave equations of the waves in the cavity in motion are thus (the angle has changed of letter):

with the same amplitude not represented.

As the velocity [c.cos(a)] of the wave along the X axis is the velocity V of the resonator, we have cos(a)=v/c.

Thus, for any of these waves the wavelength along Y (x fixed) is: .

Now the hypothesis above is used. If the number of wavelengths along the Y axis remains unchanged after the acceleration, the wavelength obtained here is the same as at rest:

, that is to say the new frequency is: .

Now we can derive easily the energy inside the simplified electron-resonator in motion, if we note that the length were the energy is along the x axis has decreased according to the formula:

.

This simple classical phenomenon produced by the change of direction of the waves (see figure above) with the angle given by has not been understood by Einstein, nor by Lorentz even if this phenomenon is called the Lorentz-Fitzgerald contration.

With this result the energy in the resonator in motion is:

.

And we have obtained for this simplified electron the Einstein's formula where the mass m (defined for the wave at rest) and the energy in motion increases according to the factor: .

We have also, with the Planck's constant defined (invariable when the wave is built), the formula: E=h.f1

These results shows already to the reader that the high level university course on relativity doesn't need a high level at all to understand very simple classical phenomenons.

III.) Classical de Broglie's wave and time dilation.

As the up and down wave equations (without the amplitude factor) in the moving resonator are:

and . It may be interesting to calculate the sum, that is to say the interference:.

Incredibly, the first factor is a wave which has a wavelength of

and is the de Broglie wavelength of the simplified electron which uses a particular Planck's constant. For a similar, but false calculation based on the logically impossible Einstein's theory of relativity, de Broglie has obtained the Nobel Prize in 1929. A fact which proves that it is very easy on the Earth to obtain it.

An other important result to obtain with the simplified electron is the time dilation phenomenon.

To build a clock, you need an oscillator and a counter. If you use the resonator at rest, the signal which exist in the cavity is the sum (the interference) of the up and down waves, that is to say: where the frequency is f outside the nodes.

For the cavity in motion, the signal in the rest frame is: .

But we need to know the signal in the frame in motion with the resonator which is different of f1 due to the classical Doppler effect.

To calculate it, it is very easy. You know that when a detector is in motion relatively to a source, the frequency detected decreases according to: f'=Fo.[(c-v)/c].

But this result is obtained without any complicated calculation in introducing the Galileo's transformation

(x'=x-vt) into the wave equation x(t)=A0.sin[2.pi.f.(t-x/c)] which yields:

x(t)=Ao.sin[2.pi.f.((c-v)/c)[t-x'/(c-v)] with the frequency f.[(c-v)/c] and the velocity [c-v].

The same way,the Galileo's transformation introduced in the equation above, yields the frequency:

This equation means that the frequency of the oscillator decreases while measured in the cavity in motion. Thus, if the frequency at rest is 1000 Hz, you need an electronic devider of 1000 to have one pulse by second.

Thus, when the cavity is in motion, if you use the same divider, as the frequency of the oscillator decreases, the period increases from 0.001 to [0.001]/sqrt(1-bb) second and thus, the moving second is a bit longer than at rest (depending of the velocity of the resonator).

This classical phenomenon is known in high level relativity courses as to be the time dilation phenomenon.

[ Return home ]