The Fizeau Experiment with Moving Water. Sokolov Gennadiy, Sokolov Vitali

Transcription

1 The Fizeau Experimet with Movig Water. Sokolov Geadiy, Sokolov itali I all papers o the Fizeau experimet with movig water, a aalysis cotais the statemet: "The beams travel relative to the iterferometer with differet speeds /+, /, cover the distace L for differet times t L/(/+ ), δ t L/( / ) (t t ) λ ad the estimated frige shift is." Already there is a pricipal error i calculatio because the estimated value δ is greater tha the experimetal value. I the ether hypothesis, this discrepacy is explaied by a partial etraimet of light by movig water ad the Fresel drag coefficiet ( ). The relativistic velocity addtio law ca oly approximately explai this discrepacy. The estimated value δ has ever bee questioed. As show below, the calculatio made by Fizeau i the iterferometer with movig water experimet usig oly the time differece t is wrog because the chage of the frequecies of the iterferig beams was ot take ito accout. The estimated frige shift calculated with the chage of the frequecies correspods exactly to the experimetal value.. The covetioal calculatio of the iterferometer. Istead of the real experimet, we cosider a simpler scheme i Fig. i which coheret beams eter two idetical pipes cotaiig water with couter motio. Beams travel i the air with frequecy ν ad wavelegth ν which decreases to i water. Fig.. Relative to the water beams travel with speed but because of the etraimet by movig with speed water their speeds relative to iterferometer are differet: L + δ

2 Beam travels with speed + ad covers distace L i the time t L +, Beam travels with speed ad covers distace L i the time t L. Because beams travel relative to the water with idetical speed, for differet time itervals t ad t they cover differet distaces relative to the water: Beam durig t covers the distace L t L ( Beam durig t covers the distace L t + ) <L, L ( ) >L. This allows us to cosider a simpler scheme i Fig. where coheret beams simultaeously eter two differet pipes with still water, coverig differet distaces L L δ L Fig.. ad L with a idetical speed ad exit from the water at idetical distaces from the scree. If we suppose, as Fizeau did, that beams travel with idetical frequecy ν, the frige shift i the iterferometer has to be equal to δ (t t ) L ( + )( ).. Ifluece of the frequecy of iterferig beams o the frige shift. We cosider a moochromatic beam as a sequece of wavefrots cosistig of photos of idetical frequecy ν ad movig i a vacuum with speed. The distaces betwee wavefrots cosistig of idetical-phase photos are equal to a wavelegth of ν T (Fig.3).

3 t ν π π.5π Fig.3. At the momet t let wavefrot of photos with zero phase eter two pipes with movig water. By eterig, the photos chage the frequecy ad travel i water with frequecies ν ν ( )<ν ad ν ν (+ )>ν. Relative to the iterferometer, photos move with differet speeds but relative to the water, their speeds are idetical ad equal. + ad Relative to water ad durig time T, photos of the frequecies ν ad ν cover the same distace T as photos ν ad at the momet tt whe wavefrot eters water, are i positio (Fig.4). t T A A ν A λ ν λ Fig.4 Durig time T photos ν arrive at positio ad their phase chages by π. Durig time T photos ν <ν cover the same distace but their phase chages

4 less tha π. Their phase chages by π oly up to momet T whe they cover the distace λ which is greater tha by λ λ (curve A). That is every oscillatio of the frequecy ν <ν is behid the oscillatio ν by the distace λ. The phase of photos ν >ν become equal to π at the momet T, whe they cover the distace λ, which is less tha by λ λ (curve A). That is, every oscillatio of the frequecy ν >ν is ahead the oscillatio ν by the distace λ. It should be oted that shifted by λ, the oscillatios ν i the pipe are sychroous with these shifted by λ oscillatios ν i pipe. Let us ext compare the situatios i the two pipes with still water. a) At the momet t the same wavefrot of frequecy ν simultaeously eters two idetical pipes of still water. That is, photos with idetical phases eter simultaeously i the pipes. It is obvious that photos cover the distace L i the idetical time t L L ad at the momet t L simultaeously exit from water (Fig.5). t ν L t L δ 3 4 Fig.5. Durig t L, N t L L T λ idetical umbers of photo oscillatios ad N of the wavelegths are cotaied i both pipes. That is, at momet t L sychroous photos exit from water at idetical distaces from the scree. The frige shift δ i the iterferometer is determied by these sychroous photos ad is equal to zero. The photos of the ext wavefrot cover the distace L i the same time t L. They remai sychroous whe they exit the water ad create friges i the same part of the scree. b) Now let us suppose that photos eterig the pipes with still water chage frequecies ad cover the distace L with differet frequecies ν <ν ad ν >ν. The photos move i water with the same speed ad cover the distace L i the same time t L L (here ad below we suppose that ν ad ν differ a isigificat amout N

5 ad therefore eglect the dispersio). Whe exitig from the water, the photos chage frequecies agai ad iterfere o the scree with a idetical frequecy ν (Fig.6). λ ' L ' 3' N λ N tl δ λ k 3 4 m λ " " 3" 4" N λ Fig.6. Simultaeously eterig the water, the photos of frequecies ν ad ν exit the water at the momet t L t. That is, at the same time as the photos of frequecy ν i Fig.5. However photos travel i water with differet frequecies ad their phases chage by π ot i the poits,, 3, 4 as i the beam of frequecy ν but i the poits ', ', 3' i beam ad i the poits ", ", 3", 4" i beam ad at the momet whe photos exit from water their phases are diferet ( N ). N L λ of wavelegths λ are cotaied i the distace L i water. Because every oscillatio of the frequecy ν <ν lags behid the oscillatio ν by λ, oscillatio N of frequecy ν is shifted relative to oscillatio N of frequecy ν by the λ distace N λ N λ N L N N N. L λ of wavelegths λ are cotaied i the distace L i water. Because every oscillatio of the frequecy ν >ν is ahead the oscillatio ν by λ, oscillatio of frequecy ν is shifted relative to oscillatio N of frequecy ν by distace λ λ L N. So, simultaeously eterig the water, photos of frequecy ν cover idetical distaces L i both pipes, ad at the momet t L simultaeously exit from the water with idetical phases ad at idetical distaces from the scree with, frige shift δ (curve A i Fig.7). Photos ν ad ν also eter water simultaeously, cover the same distace L i water ad at the same momet t L exit the water but they reach the exit with differet phases (curve B that passes through the poit "k" is shifted relative to the curve A by N λ ad curve B that passes through the poit "m" is shifted relative to the curve A by λ ). Whe exitig the water, the photos chage frequecies ad reach the

6 scree with idetical frequecy ν ad with differet phases. λ 3' A B N t L A k N λ N λ 3 A N λ m B 4" Fig.7. The iterferometer "sees" that at the momet t L istead of sychroous wavefrots N, wavefrot N which is shifted back relative to the iterferometer by N λ, exits from pipe (curve A) ad at the same momet t L the sychroous wave frot which is shifted forward relative to the iterferometer by λ exits from the pipe (curve A). The frige shift i the iterferometer turs out as i the case where sychroous wavefrots which are shifted relative to the pipes by N λ back ad λ forward (or relative to each other by λ N + λ ) simultaeously exit the water. So, if iterferig beams chage frequecies ad cover idetical distace L with differet frequecies ν <ν ad ν >ν, the frige shift δ λ N λ + λ λ arises as i the case where beams travel with differet speeds ad cover differet distaces L ' " L λ N ad L L + λn for a idetical time t L L. 3. The frige shift i the Fizeau iterferometr. I the Fizeau iterferometer, beams cover differet distaces L ad L i differet times t ad t i water (Fig.) ad, if we suppose that beams do ot chage

7 δ L ( + )( ). But the frige frequecies, the frige shift has to be equal to shift δ is less tha δ because beams travel with differet frequecies. I accordace with the Doppler effect, beams eterig movig water chage frequecy: i beam the frequecy of photos decreases to ν ν ( ), i beam the frequecy of photos icreases to ν ν (+ ), Photos cover the distaces L ad L i water with differet frequecies ν ad ν, Whe beams exit from movig water, frequecy of photos i beam chages by (+ ) ad becomes equal to νν ( ). Frequecy of photos i beam chages by ( ) ad becomes equal to νν ( frequecy νν ( ) too. Beams iterfere with idetical ) ν ad create a statioary iterferece patter. Whe water is at rest, photos of frequecy ν have wavelegth ν. Beam covers the distace L i water with frequecy ν ν ( ) ad wavelegth λ ν ν ( ) ( ) + ( ) greater tha L, there are λ N L ( + ). + λ. That is, the wavelegth is by λ ( ). Durig time t while beam covers the distace N L λ L( ) ( + ) oscillatios ad the wavefrot shifts by Beam covers the distace L i water with frequecy ν ν (+ ) ad wavelegth λ ν ν (+ ) (+ ) (+ ) λ less tha. That is the wavelegth is by λ (+ ). Durig the time t while beam cover the distace L, there are L λ L(+ ) ( ) oscillatios ad the wavefrot shifts by

8 λ L ( ). Because of chage of the frequecies, the total the shift of wave frots λ N + λ ( L ) + L + ( ) L decreases the frige shift by δ λ λ N + λ δδ δ ++ L ( )( + ) ( )( + ) ad the resultat frige shift i the L ( + )( ) L ( ( + )( ) δ ) iterferometer is which exactly coicides with the experimetal frige shift i Fizeau iterferometer. oclusio Fizeau made a mistake i the calculatio of his iterferometer experimet ad therefore the result was explaied improperly. By takig ito accout the chage of the frequecies of iterferig beams, the calculatio gives a value that exactly correspods to the experimetal value of the frige shift. Therefore Fizeau's experimet caot be cosidered as cofirmatio of special relativity or a ether hypothesis ad all attempts to explai this experimet with a partial etraimet of light by movig water or relativistic additio of velocities are wrog.