Optical Fizeau Experiment with Moving Water is Explained without Fresnel ' s Hypothesis and Contradicts Special Relativity

In 1886 Michelson and Morley repeated Fizeau experiment and with higher accuracy confirmed the decrease of the fringe shift in moving medium. Taking into account the dispersion of the medium, Lorentz derived a formula for the drag coefficient. To confirm this formula, Zeeman in experiments with moving water and Harres in the experiment with the linearly moving quartz cylinder determined drag coefficients for the red and green light. However, as we know, in the calculation of the interferometer with moving water, nobody investigated and nobody considered the change of the the frequencies and the phase deviations, arising in interfering beams when they enter into moving water [3].


Introduction
200 years ago, Fresnel, trying to explain results of the optical Arago's experiments by the aether wave hypothesis, suggested that a moving at speed V medium drags the light only partially and the speed of the light changes by drag coefficient To test this hypothesis, in 1851 Fizeau carried out interference experiment with moving water.
At the speed of the water V=7,059 m/s, the length of pipe L=2.974 m, The length of light wave λ o =526·10 -9 m and refraction index n=1.33, He expected to receive the fringe shift 0.47099 in the case if the light was completely dragging by moving water. However, the shift in the experiment was less and equal to 0.23, that is it differed almost by  [1,2]. Although that time Doppler has already showed that the light changes its frequency when enters moving medium, Fizeau did not tried to explainn that result somehow differently and decided that he confirmed Fresnel's aether drag hypothesis about partial dragging of the light by moving medium.
In 1886 Michelson and Morley repeated Fizeau experiment and with higher accuracy confirmed the decrease of the fringe shift in moving medium. Taking into account the dispersion of the medium, Lorentz derived a formula for the drag coefficient. To confirm this formula, Zeeman in experiments with moving water and Harres in the experiment with the linearly moving quartz cylinder determined drag coefficients for the red and green light. However, as we know, in the calculation of the interferometer with moving water, nobody investigated and nobody considered the change of the the frequencies and the phase deviations, arising in interfering beams when they enter into moving water [3].
The erroneous explanation of Fizeau's experiment, a significant role of which in the creation of the special relativity was repeatedly emphasized by Einstein, is still considered as one of the most important confirmation of the special relativity [1].
As shown below, the beams in Fizeau interferometer travel at speeds that is complete but not partial dragging takes place. The fringe shift is less than 0.47099 not because of Fresnel's hypothesis about "partial dragging" but because of phase deviations arising in interfering beams in moving water and therefore Fizeau experiment does not confirm but, on the contrary, disproves special relativity.

The Conventional Calculation of the Fizeau Interferometer
In Fizeau interferometer, the beam 1 travels in direction of moving water and the beam 2 travels toward moving water. Instead real scheme of Fizeau interferometer in which the beams travel in the same pipe and pass the same distance L in opposite directions, we consider more simple equivalent scheme explained in Figure 1, where the beams 1 and 2 pass identical distances in two pipes in which water moves at speed V in opposite directions. Just as in the experiment Fizeau, photons exit from moving water at the same distance from the screen, as in immovable water. At the moment t=0, the photons of the frequency ν 0 with identical initial phase equal to zero simultaneously enter in both pipes. If water is at rest, photons travel with identical frequency ν 0 and speed  When water moves at speed V, the speeds of the photons and their frequencies change.
In the beam 1, photons move at speed C n relative to water and at For the time t 1 , every photon passes relative to water the distance In the beam 2, photons move at speed C n relative to water and relative to pipe. They pass the distance L for the time For the time t 2 every photon passes relative to water the distance Photons of the beam 1 come to the screen earlier than photons of the beam 2 and the fringes shift in interferometer.
In usual two-beam interferometers, the interfering beams pass the distances with identical frequency ν 0 and the fringe shift is determined simply by the difference of the times : ∆t=t 2 -t 1 ( ) If L=2.974 m, C=299 792 458 m/s, V=7,059 m/s, λ 0 =526·10 -9 m and n=1.33, the expression (1) gives a value of the fringe shift δ V =0.471 ∆t=0.0000000826372807322078835068034 Such fringe shift, according to the Fizeau, should be in his interferometer. But in experiment the fringe shift was less than 0.471 and equal to 0.23.

The Change of the Frequencies in Moving Water
In Fizeau interferometer, the light is completely dragged by moving water. But because the beams enter the moving water from immovable light source, in accordance with Doppler effect, their frequencies change and an observer moving together with water will see different frequency. Additional phase deviations arise in interfering beams. Because of these deviations, the resulting fringe shift in the interferometer with moving water cannot be determined by the expression (1) and is less than δ V =0.471.Photons of the beam 1 entering moving water with a speed V change the frequency from ν 0 to and with speed C n and frequency ν 1 <ν 0 pass relative to water the distance L 1 .
At the moment t 1 they exit from moving water changing the frequency from ν 1 to in air to the screen with the speed C and frequency and interfere with photons of the beam 2.
Photons of the beam 2 entering moving water change the frequency from ν 0 to and with speed C n relative to water and frequency ν 2 >ν 0 pass the distance L 2 . At the moment t 2 they exit from moving water changing their frequency from ν 2 to In the air photons travel to the screen with the speed C and frequency and interfere with photons of the beam 1.

The Change of the Distances between the Wave Fronts
Besides the fact that photons change their frequency in moving water, the distances between the wave fronts change too, which leads to an additional change of resultant fringe shift.
When photons enter the pipe with immovable water Figure 2a Photons also do not change frequency if to suppose that the source S moves in interferometer together with water ( Figure 2b). Moving with water, the observer will see the same frequency ν 0 and wavelength In the case when the source S is at rest relative to pipe Figure 2c frequencies of the photons change but the same time the distances between the wave fronts change too. Moving with water, the observer will see that the beam changes in water not only its frequency but, depending on the direction of water movement, it "stretches" or "contracts".
In all situations, photons travel relative water at speed both pipes with moving water as shown in Figure 2b and 2c, photons are dragged completely and travel at speed C V n + . For each period T 0 , they are ahead at identical distances VT 0 relative to photons in pipe with immovable water.
In the pipe Figure 2b where the source moves together with water, photons travel with frequency ν 0 and the distances between the wave fronts are equal to wavelength 0 In the pipe Figure 2c photons change frequency and travel in water with frequency ν 1 less than ν 0 . For the time T 0 , each wavefront passes relative to water the distance 0 C T n . Because water moves at speed V, at the moment T 0 , when next wave front enters water, previous wavefront is at the same distance as the wave front in Figure 2b.
Though photons travel with frequency ν 1 , the distance between the wave fronts is The distances are the same as in Figure 2b where photons travel with frequency ν 0 . As shown below, because of the "stretching" or "contraction" additional phase shift arises and resulting fringe shift in interferometer changes.

Additional Fringe Shifts
Imagine that in the Fizeau interferometer, as well as in Figure  2, there is an additional pipe with immovable water, and consider propagation of photons in the pipes 1 and 2. The introduction of additional pipe simplifies analysis of the interferometer, as it allows to consider the motion of photons in each pipe separately, comparing the positions of the photons in the pipe with moving water with positions of the photons in the pipe with immovable water.
In usual two-beam interferometer, synchronous photons pass different distances with the same speed or the same distance with different speeds and therefore the fringe shift arises. Since synchronous photons travel with the same frequency and come to the screen with the same phase, the fringe shift can be determined by the difference In interferometer with moving water, because wave lengths and distances between wavefronts change, the fringe shift change and is less than δ V .

The decrease of the fringe shift because of the wavelengths change
In Fizeau interferometer, the beam 1 and 2 travel with frequencies and with different wavelengths.
During the time while the beams travel in water, a phase deviation arises and because of it the fringe shift decreases.
In Figure 3, for example beam 1, it is shown the phase shift in the case when photons travel with different frequencies ν 0 and ν 1 and different wavelengths is the number of oscillations in photons ν 1 for the time t 1 . That is, at the moment t 1 when synchronous photons exit from water, photon ν 0 equivalent to photon ν 1 is behind in water at distance ∆λ 1 N 1 . In Fizeau interferometer, photons ν 0 also enter in imaginary pipe with immovable water. When water in the pipe 1 is at rest, photons exit from both pipes with identical frequency ν 0 and create interference fringes. When the water moves at speed V and photons travel with frequency ν 1 in pipe 1, the fringe shift is measured relative to these fringes. Interferometer works with frequency ν 0 . It "does not know" that frequency changes in moving water and reacts with equivalent photon ν 0 which is behind in water at ∆λ 1 N 1 . During the time The fringes shift in interferometer as if photons pass the distance L 1 at a speed less than C n .

The increase of the fringe shift because of the periods oscillations change
In Figure 4, also for example beam 1, a movement of the photons ν 1 in pipe 1 is compared with the movement of the photons ν 0 in pipe with immovable water. At the same time as the fringe shift decreases To the moment T 0 when first wavefront has not yet passed relative to water the distance λ 1 and is at distance from the entrance in the pipe, next wavefront already enters the water.
The frequency with which wave fronts, with phases 2π,4π and so on, travel in water, is determined by the "clock" frequency ν 0 and therefore the distances between wave fronts are less than their

C V T T n
During the time t 1 , N 1 oscillations occur in the photons of the beam 1. Each wave front passes relative to water the distance which is by less than the distance L 1 which is passed by photons in additional pipe with immovable water shown in Figure 5.
Thus, for the time while photons in additional pipe pass the distance ∆ T1, photons exiting from pipe pass in the air the distance n∆ T1 and are ahead at(n-1) ∆ T1 . That is, because of decrease of the distance passed in water by wave fronts of the beam 1, interference fringes additionally shift ahead by ( ) ( ) ( )

Resulting fringe shift in Fizeau interferometer
The fringe shift in Fizeau interferometer with moving water is determined by three components: ( ) earlier than in additional pipe.
Because of time difference ∆t 1 the fringe shift δ V1 has to arise:  arises. Because of δ λ1 , resulting fringe

Conclusion
The fringe shift in the interferometer with moving water is less than fringe shift in usual two-beam interferometers because of the change of the frequencies and additional phase deviations arising in interfering beams. Fresnel's explanation of the fringe shift decrease by hypothesis that the light is dragged partially by nonexistent aether is wrong and cannot be considered as confirmation of Einstein's special relativity.