The 1851 Fizeau Water Experiment
"The most incomprehensible thing about the world is that it is comprehensible"  Albert Einstein ( 1879  1955 )
Some Common Refractive Indices Before Fresnel's Convection Equation is Applied u = v_{p} = c/n, phase velocity^{∗}, in a stationary medium where v = 0 


Intervening Medium  Physical Material  n  u 
vacuum space  pure rarified  1 ( exactly )  1c 
Gases 0 °C & 1 atm 
hydrogen  1.000132  0.999868c 
helium  1.000036  0.999964c  
atmosphere at STP ( Standard Temperature and Pressure ) 
1.0002926  0.9997c  
water vapor  1.000256  0.99974c  
oxygen  1.000271  0.99973c  
Other Optical Materials room temp 
diamond  2.4235  0.41263c 
amber  1.55  0.645c  
crown glass ( pure )  1.50  1.54  0.66667c  0.64935c  
crown glass ( impure )  1.485  1.755  0.6734c  0.5698c  
silicon  3.4170  0.29265c  
Liquids 20°C 
fresh water  1.333  0.75018c 
benzene  1.501  0.66622c  
glycerol  1.47  0.68027c  
liquid paraffin  1.48  0.67568c  

During most of pre  Galileo and Newton and for subsequent eras as well, it was supposed that in the interstitial spaces between objects of matter that there existed a "carrying medium" or aether for the transmission of light from source to reflecting object and thence to the human eye for perception. Two French physicists, Jean Bernard Léon Foucault ( 1819  1868 ) and Armand Hippolyte Louis Fizeau ( 1819  1896 ), attempted the determination for the finite speed of light; Fizeau did so singly in 1849 and again in 1850 together with Foucault but thereafter independently sought the speed of light in his famous 1851 Fizeau Water Experiment whenever light was transmitted thru a high velocity flowing medium such as water. In essence, therefore, Fizeau attempted to confirm Augustin  Jean Fresnel ( 1788  1827 )'s "velocity drag coefficient"
for light transmitted thru high  velocity ( at least approx. 30 m/sec ) flowing water. It should be thus noted that Augustin  Jean Fresnel, French mathematical theorist and experimenter in optical wave physics, is the original mathemtical discoverer in 1818 of the velocity drag coefficient.
In such a situation, Fizeau discovered that each different flowing liquid acting as a carrying medium or aether for light, in both positive and negative directions of velocity flow, exhibited different refraction indices in the speed of light as demonstrated by measuring the positions of interference fringes for light.
In the 1851 water experiment, Fizeau also discovered that simple Galilean addition of light velocity plus the velocity of the transmitting carrying medium did not fully suffice to explain the resulting interference light fringes. For any explanation of this phenomenon it would take the equations of special relativity ( Albert Einstein, 1905 ) for the addition of relativistic velocities in order to make apparent deeper theoretical understanding.
Fizeau did however successfully confirm that the speed of light, , is attenuated by a refractive index, , unique to each different carrying medium as well as effected by the velocity, either positive or negative, of the local transmitting medium. Fresnel's convection equation
which Fizeau's 1851 water experiment confirmed, indicates that for increasingly rarified medium such as a vacuum space, approaches 1 and the observed speed of light, , becomes since velocity, , simply disappears as the dependence of on becomes increasingly neglible.
Interestingly Fizeau rather accurately determined the speed of light, , as 3.13 x 10^{8} m/sec and became the first to do so using an earth  based apparatus as opposed to Danish Olaf Roëmer ( 1644  1710 )'s astronomical determination in 1676.
Although no other deeper theoretical explanation existed for the results of the 1851 Fizeau experiment other than the Fresnel factor
representing an "aether drag coefficient" for transmitted light going thru a high velocity moving fluid, it still nevertheless remained a puzzle, especially after the null result of the famed Michelson  Morley experiment ( 1887 ) completely failed to discover an equivalent relationship to the Fizeau  Fresnel equation for light transmitted thru vacuum space.
On the other hand, the FitzGerald  Lorentz transformation equations theoretically resurrected the "existence" of the aether transmitting medium by postulating an "aether drag" in the very instrumentation of the Michelson  Morley experiment resulting in a longitudinal contraction of their equipment apparatus and thereby salvaging the concept of the "aether". In other words, the idea of the "aether" as a transmitting medium for vacuum light similar to the transmitting fluid ( water ) in the Fizeau experiment is thereby sustained, even though undetected, since there became a physical contraction in the length of the Michelson  Morley apparatus in the longitudinal direction of earth's motion in its orbit about the sun due to an "aether drag", thus mathematically shown by the FitzGerald  Lorentz transformation equations.
Lorentz Transformation Solution to the Fizeau Experiment
By associating ( relatively stationary ) system with earth and connecting the high velocity water with ( relatively moving ) inertial frame of reference system , the equation for the light will be
and by applying the FitzGerald  Lorentz transformations
we get
But when we let time , starting time for fluid motion, this then gives
This then is the Fizeau  Fresnel's convection equation by means of FitzGerald  Lorentz transformation equations. Q.E.D.
Relativistic Addition of Velocities as a Solution to the Fizeau Experiment
In several of Einstein's writings this is the proffered solution while still using system for earth and system for the flowing liquid where the relativistic addition of velocities is given by
Let , phase velocity of light in a stationary transmitting medium ( fluid liquid ), therefore
This again is the Fizeau  Fresnel's convection equation by means of the Einstein relativistic addition of velocities. Q.E.D.!!
Why the historic 1851 Fizeau Experiment is important
The historic 1851 Fizeau Experiment is important because it demonstrated experimentally the validity of both the FitzGerald  Lorentz transformation equations ( developed 1889  1892 ) and Einstein's relativistic addition of velocities whereby Einstein's special relativity mathematics could explain prior conundrums in 19th century physics as well as bringing illumination into 20th century astronomical and atomic physics.
§ References:
"On the Effect of the Motion of a Body upon the Velocity with which it is traversed by Light", by M.H. Fizeau, published Philosophical Magazine and Journal of Science [ Fourth Series ], London, Edinburgh, and Dublin, April 1860 for the English translation from Annales de Chimie et de Physique, December 1859, where the original paper was presented to the Parisian Academy of Sciences on September 29, 1851.
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