O θ O ' X,X ' Fig. 11. Propagation of a flat wave

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1 Y Y P Doppler s effe Professor Tonnela s mehod The Doppler s effe is manifesed in he hange in he veloi of he signals refleed (radiaed) b moving objes [5] [6] [8] [39] [43] [45]. I lies in he basis of he onsruion of he modern radioloaors whih are fouroordinaed i.e. he measure he hree spaial oordinaes and he speed of movemen of he objes. B objes from miliar viewpoin we shall undersand he arges (osmi earh waer) and he weapons poined a hem (rokes airrafs guided bombs). In order o deermine he influene of he new resuls on Doppler s effe we shall make use of he mehod proposed b M.A. Tonnela [9]. Le us suppose ha he soure of eleromagnei waves Radio loaor is an obje ha moves along he ОХ-ais and he radiaed signals are observed in he direion Оn (fig.). K K n H O O XX Fig.. Propagaion of a fla wave Le he observer P be relaed o an immobile oordinae ssem K wih oordinaes. In he ssem K relaed o he mobile soure he will have oordinaes. We assume ha he ssem К whose ais О Х moves along he ais OX wih a speed v oinides in he iniial momen wih he ssem K. The ime for whih he fron of he firs wave passes over from O o P is equal o: os os or if hen (.57)

2 48. (.58) The number of he waves reahing poin Р for he ime defined in he ssem K will be ( - ) where - frequen of he eleromagnei osillaions measured in he ssem K. Analogousl o (.58) we represen. (.59) where Sine for eah oordinae ssem he number of he waves reahing P is equal hen ( - ) ( ) (.6) - frequen in he relaed wih he soure ssem К. Therefore. (.6) B analog wih (.57) (.58) an be represened as a funion of he oordinaes of poin Р deermined in К : os (.6) where - an angle ha has o be deermined. From (.6) we obain os os (.63) Sine he values of and evidenl do no depend on he wa he posiion of poin P is hosen we subsiue in (.63) wih he needed (on X and Y) formulae from (.3) (.5):

3 49 v (.64) i.e. os v os os (.65) Formulae for ransformaion of he frequenies We obained an epression in whih he oeffiiens a and are equal o zero. Consequenl. os os os (.66) The firs wo equaions epress he relaionships beween and and in he lenghwise direion and he hird and he fourh in he widhwise direion of he movemen of he soure. From he seond and he hird equaions we have

4 5 and from he firs and he seond: os g (.67) os os From (.67) (.68) and (.69) is easil obained os (.68) os (.69) The epressions (.67) (.68) and (.69) oinide wih he well known equaions from Speial relaivi heor. From he firs equaion of (.66) aking ino aoun (.6) we wrie: 3 os (.7) This is a formula onneing and and haraerizing Doppler s effe aording o Vauum heor. The subsiuion of for orresponds o he aberraion phenomenon. If he observaion is arried along he direion of he movemen of he obje soure of eleromagnei waves (lenghwise Doppler s effe) hen and from (.66) i follows ha:. (.7) The phenomenon of aberraion is absen in his ase. If he observaion is arried ou in a direion perpendiular o he direion of movemen of he obje-soure (widh-wise Doppler s effe) hen π / os and. (.7) Thus a formula eplaining he aberraion (deviaion) of ligh oming from a sar around he zenih was obained. Wha is inereg is he presene of he fourh formula from (.66). If we ake ino aoun ha from he firs equaion is relaed wih he ОХ-ais and from he fourh wih he OYais hen aking ino aoun (.6) we have: os (.73) from where (.74)

5 5 (.75) os i.e. we obained he formula haraerizing Doppler s effe in SRT. And indeed if we reall Lorenz s formulae and he don hange in he widhwise ( z z ) direion of he movemen - onsequenl. Again (see p..) we obained formulae from Speial relaivi heor originaing from he more general epressions of he new Vauum heor. This gives us grounds o onsider ha he relaive moion is a pariular ase of he absolue one. GENERAL CONCLUSIONS TOWARDS CHAPTER TWO:. The speeds of movemen of he maerial bodies in spae should be regarded as speeds whih we are in a posiion o measure onduing opial eperimens. The new formulae for he ransformaion of he veloiies ake off he ban on speeds of movemen higher han he veloi of propagaion of ligh whih is he maimal measured hrough opial eperimens veloi Dopler s effe. Spae and ime onsidered as veors are absolue and separaed from one anoher. The relaionship beween hem is arried ou hrough heir omponens i.e. hrough he differen pae of ime in relaion o he respeive spaial sizes. There are direions oiniding wih he slan heighs of he ime ones along whih ime and he pae of ime preserve heir absolue haraer 3. The mass and he volume of he maerial bodies are onsan quaniies independen from heir speed of movemen i.e. he law of he onservaion of mass and he Seond Newon s law are preserved and he law for he preservaion of he volume is formulaed. 4. The appliaion of he new formulae haraerizing he speeds of movemen of he objes and Doppler s effe will lead o he inrease in he aura of he onemporar and he perspeive opial and radioloaion ssems.