Special Relativity in Acoustic and Electromagnetic Waves Without Phase Invariance and Lorentz Transformations 1. Introduction n k.
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1 Speial Relativit in Aousti and Eletomagneti Waves Without Phase Invaiane and Loentz Tansfomations Benhad Rothenstein Abstat. Tansfomation equations fo the phsial quantities intodued in ode to haateize the popagation of waves (aousti and eletomagneti) ae deived without invoking invaiane of the phase o using tansfomation equations fo the spaetime oodinates. 1. Intodution The tuism that leanes benefit fom seeing a subjet developed in moe than one wa has patiula elevane fo speial elativit. [1] If the ase when the new development is moe tanspaent and time saving, the benefit ineases. Molle [] deives tansfomation equations fo the phsial quantities intodued in ode to haateize the popagation of a plane aousti wave. The phsial quantities ae: unit veto n that defines the dietion in whih the wave popagates with phase veloit w, fequen ν, wave length λ and wave veto k. The phsial quantities ae measued b obseves fom the inetial efeene fame I. The same phsial quantities measued b obseves fom a efeene fame I ae n, w, ν, λ and k. The two fames ae in the standad onfiguation with I moving with veloit with espet to I. Let O and O be the spatial oigins of the fames. O and O oinide at the oigin of time in the two fames t=t =0. Deiving tansfomation equations fo the paametes mentioned above, Molle [1] invokes the invaiane of the phase of the wave and pefoms the Loentz-Einstein tansfomations of the spae-time oodinates whih intevene thee. The appoah leads to tansfomation equations fo all the phsial quantities mentioned above. The pupose of ou pape is to show that the same esults ould be eoveed without invoking the invaiane of the phase and without using the Loentz-Einstein tansfomations fo the spae-time oodinates intodued in ode to haateize it.. Dopple Effet, abeation and wave veto in the aousti wave Conside a plane aousti wave that popagates with wave veloit w along a dietion that makes an angle θ with the positive dietion of the OX ais as deteted fom I. The loks of that fame ae snhonized following a poedue poposed b Einstein. [3] We ae patiulal inteested in the eading of a lok K 0 loated at the oigin O. When lok K 0 eads t e a wave est intesets the oigin. The wave est aives at a point M haateized b the pola oodinates (,θ) when the lok K loated thee eads t. As a onsequene of the fat that the loks ae Einstein snhonized, the two lok eadings ae elated b 1
2 t = te +. (1) w Diffeentiating both sides of (1) the esult is d dt = dte + () dw o dte 1 d =. (3) dt w dt Let K be a lok of I momentail loated in font of lok K eoding a pope time inteval dt elated to dt b the fomula whih aounts fo the time dilation effet dt dt = (4) with whih (3) beomes osθ dte = u (5) dt whee we have taken into aount that b definition d = osθ (6) dt epesents the instantaneous adial omponent of the veloit of lok K mentioned above. Equation (5) elates the pope emission peiod measued as a diffeene between the eadings of lok K 0 and the pope eeption peiod dt measued as a diffeene between the eadings of the moving lok K. Intoduing the onept of emission fequen ν=1/dt e and of eeption fequen ν =1/ dt (5) elates them as f = f osθ u (7) Equation (7) aounts fo the elativisti Dopple Effet. Consideing the same epeiment fom fame I, we obtain that ν and ν ae elated b
3 osθ ν = ν w. (8) Multipling (7) and (8) side b side the esult is 1 osθ 1 osθ = + (9) u u that leads to w osθ osθ =. (10) u u osθ u Multipling both sides of (10) with ν it leads to w osθ ν osθ ν =. (11) w u Intoduing the onept of wave veto its magnitude in I is 1 ν k = = (1) λ w and 1 w k = = (13) λ u being pesented as vetos k=(1/λ)n=(ν/w)n (14) and as k =(1/λ )n =(ν /w )n. (15) With the new phsial onept (11) leads to w ν osθ k k osθ = k = (16) and equation that pefoms the tansfomation of the, omponents of the wave veto. Equation (7) an be pesented as ν k ν =. (17) Inspeting equations (16) and (17) we see that the ould hold onl if the, omponents of the wave veto have the same magnitude in all inetial efeene fames i.e. 3
4 k = k (18) i.e. (ν/,k) is a fou veto [4] The esult is that the magnitude of the wave veto tansfoms as k = k + k = k w w u osθ + sin θ. (19) The involved angles tansfom as tgθ k = = k sinθ w osθ (0) equation (0) aounting fo the abeation in the aousti wave. The wave lengths tansfom as λ = λ w w + sin θ. (1) The esults pesented above wee obtained b Molle [] and b Huang [4] stating with the invaiane of the phase and pefoming the Loentz-Einstein tansfomations fo the spae-time oodinates used in ode to define the phase. We have obtained the same esults without invoking the invaiane of the phase and without involving the Loentz- Einstein tansfomations, the single elativisti ingedient used being the fomula that aounts fo the time dilation effet. 3. Tansfomation equations fo the paametes haateizing a plane eletomagneti wave An eletomagneti wave popagates in vauum with the invaiant phase veloit. We obtain the oesponding tansfomation equations fom those deived above b simpl making w=w =, i.e. tgθ = sinθ osθ () aounting fo the abeation in the eletomagneti wave and 4
5 osθ ν = ν. (3) Equation (3) suggests defining a wave veto ν k = n (4) the omponents of whih tansfom as k ν k = (5) k = k. (6) Epessed as a funtion of the pojetions of k (3) beomes ν k ν =. (7) We have eoveed that wa the esults obtained following the phase invaiane stateg [5],[6],[7]. Conlusions We have shown that the tansfomation equations fo the paametes whih haateize the popagation of a plane aousti and eletomagneti wave ould be deived without invoking the invaiane of the phase and the Loentz-Einstein tansfomations fo the spae-time oodinates intodued in ode to haateize the phase. In the ase of the eletomagneti wave the tansfomation equations ae deived as a onsequene of the invaiane of the speed at whih the popagate in empt spae. Refeenes [1] L. Kannenbeg, Altenative appoah to the onepts of speial elativit, Am.J.Phs. 51, (1983) [] C. Molle, The Theo of Relativit, (Ofod at the Claendon Pess, 195) pp [3] Thomas A. Mooe, A Tavele s Guide to Spaetime, An Intodution to the Theo of Relativit, (MGaw-Hill, In. 1995) pp.9-31 [4] Young-Sea Huang, The invaiane of the phase of waves among inetial fames is questionable, EPL, 79 (007, [5] Robet Resnik, Intodution to Speial Relativit, (John Wile and Sons, In. New Yok, London, Sdne, 1968) pp [6] W.G.. Rosse, An Intodution to the theo of Relativit, (Buttewoth, London, 1964) pp [7] Hans C. Ohanian, Speial Relativit: A moden Intodution, (Phsis Cuiulum and Instution, In. Lakeville USA, 001) pp
6 Refeenes [1] L. Kannenbeg, Altenative appoah to the onepts of speial elativit, Am.J.Phs. 51, (1983) [] C. Molle, The Theo of Relativit, (Ofod at the Claendon Pess, 195) pp [3] Thomas A. Mooe, A Tavele s Guide to Spaetime, An Intodution to the Theo of Relativit, (MGaw-Hill, In. 1995) pp.9-31 [4] Young-Sea Huang, The invaiane of the phase of waves among inetial fames is questionable, EPL, 79 (007,
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