MOVING SIGNALS AND THEIR MEASURED FREQUENCIES

Transcription

1 MOVING SIGNALS AND THEIR MEASURED FREQUENCIES Auors: Candru Iyer and * G. M. Prabu Teink Indusries, C-4, pase-ii, Noida, 0305, India, andru.iyer@luxoroie.om Deparmen o Compuer Siene, Iowa Sae Uniersiy, Ames, IA 500, USA prabu@iasae.edu Publised: In. J. Engg. Res. & Si & Te. ISSN: Vol (Isssue 3) (03) pp 4-36 Link o paper: p:// ABSTRACT In deermining e lassial Doppler Ee, wo assumpions are used or ompuing e dierene in disane raelled by onseuie signals: (a) e reepor is saionary, and (b) e emier is saionary. Te alulaed Doppler Ee under e wo assumpions are idenial, proided e eloiy o propagaion wi respe o soure and e eloiy o propagaion wi respe o e reepor dier exaly by e eloiy o relaie moion. We sow a, in e ase o lig, e raio o e wo alulaed lassial Doppler Ees, wi propagaion speed in e soure and reepor inerial rames respeiely, remains onsan in all geomeries and orienaions. Furermore, e obsered Doppler Ee, as predied by speial relaiiy, is e geomeri mean o e wo expeed lassial Doppler Ees in all geomeries and orienaions. Tis leads o wo simulaneous onlusions: () by e reepor a e lok assoiaed wi e emier runs slow, and () by e emier a e lok assoiaed wi e reepor runs slow. Tese dierenes an be resoled i we eorize a lig raels a speed wi respe o e emier as i leaes e emier and raels a speed wi respe o e reepor as i approaes e reepor. KEY WORDS: Doppler Ee, lassial pysis, relaiisi pysis INTRODUCTION Te Doppler Ee, originally proposed in 84 by e Ausrian pysiis Crisian Doppler, arises ou o e dierene in disane raelled by onseuie ress o a wae, wile raelling rom a soure o a reepor, wen e reepor and soure are in relaie moion []. I was urer inesigaed by oer sieniss [] in e 9 enury and is known by e name o Doppler as e was e irs o propose e exisene o e penomenon in a eoreial ramework. Te Doppler Ee is widely used in medial sanning appliaions [3]. I is

2 pysially perepible o e normal uman aulies in e ase o an approaing or leaing soure o sound. In lassial pysis e Doppler Ee an be alulaed by knowing: e orienaion o e line joining e soure and reepor wi respe o e line o relaie moion o e soure and reepor, e emiing requeny, and e speed o propagaion wi respe o e soure. In e reepor s assessmen, e moemen o e emier during e ime o propagaion o e signal rom emier o reepor does no igure in e relean alulaions. In e emier s assessmen, e moemen o e reepor during e ime o propagaion o e signal rom emier o reepor is relean and signiian. Tis perepion gies rise o e obsered disanes raelled by a signal beween emier and reepor o be dieren as obsered by e emier and reepor respeiely. Te dierene in disanes is ompensaed, in lassial pysis, by e dierene in speed o propagaion o e signal as obsered by e wo inerial rames assoiaed wi e soure and reepor, and one alulaes idenial resuls rom eier inerial rame. Te expeed Doppler Ee under lassial pysis alulaed by e reepor and emier are dieren, wen one assumes a e speed o lig is e same wi respe o bo inerial rames. Any se o obseraions wi assoiaed spae-ime oordinaes is essenially lassial as ar as obserers in one inerial rame are onerned. Tus ese obserers will inerpre e obseraions under lassial pysis. I is in is onex a su obserers obsere a moing loks run slow, assuming a e propagaion o lig is a onsan speed weer e soure o lig is saionary or moing. Howeer, e orienaion is dynami and as e relaie moion progresses, e orienaion oninually anges. Hene e Doppler Ee also anges as e relaie moion progresses. Tis is e reason a e wisle o an approaing rain is sriller o a person on e ground. Wen e rain is ar away, e orienaion angle beween e line o moion and e line joining e reepor and e wisle is almos zero. I is neer exaly zero unless e reepor is a poin on e line o moion, wi is no e ase as e person on e ground will no usually posiion imsel along e line o moion. As e rain approaes loser e orienaion sis rom zero o 80 degrees in a oninuous asion (see Fig. ). Te Doppler Ee depends on e orienaion [4]. Tis is wy an approaing siren is sriller and slides o aual requeny as e siren passes you (orienaion 90 0 ) and beomes low pied as e siren leaes e obserer. In priniple, beween wo subsequen (emissions o) ress e orienaion anges sligly. Bu is ange an be negleed wen e requeny is ery ig ompared o e reiproal o e ime aken by e wae o rael beween soure and reepor. Wen e orienaion is 90 0 and d in Fig. is ery small, is ange in orienaion, a is, e ange in θ beween wo onseuie ress, anno be negleed.

3 Line o Relaie Moion Train d Signal Propagaion Obserer s as as as alm os Zero, Wen s beom es Zero, s beom es e s d and s e s d Figure. Doppler Ee and dynami orienaion In lassial pysis e orienaion θ, a any gien insan, is e same as obsered by e reepor and soure. Tis is beause e deiniion o an insan o ime is same in bo rames and bo agree on a snapso o eens a a gien insan as e same. In relaiisi pysis, e wo inerial rames do no agree on an insan and us ey disagree on e orienaion a any insan o emission o a res. Te disagreemen in e deiniion o e insan arises rom e long debaed onenion in deining simulaneiy a is elaboraed in is isorial, sienii and pilosopial aspes by Janis [5]. In relaiisi pysis e loaion o e reepor wi respe o e soure, wen a res is emied by e soure, is no e same as obsered by e inerial rame o-moing wi e reepor and as obsered by e inerial rame o-moing wi e soure. Sine e Doppler Ee depends on orienaion, is perepion o dierene in orienaion leads o dieren resuls as alulaed by obserers in bo e inerial rames. Moreoer bo ses o obserers assume a lig raels a a speed wi respe o e inerial rame a ey are a res. We sow a e obsered Doppler Ee is e geomeri mean o e wo alulaed (lassial) ones by e wo ses o obserers. Table enumeraes e pysial aspes a onribue o e Doppler Ee in lassial and relaiisi pysis.

4 Relaiisi Pysis Classial Pysis. Tere is a onribuion o e perepion o dierene in disane raelled by a signal (res) as obsered by Reepor and Soure on aoun o e perepions a Reepor moes (aording o soure) and e Reepor is saionary (aording o Reepor) during e propagaion o e wae/ signal In agreemen a e Reepor moes (aording o Soure) and e Reepor is saionary (aording o Reepor) during e propagaion o e wae/ signal. In agreemen a e disane raelled by e res is obsered o be dieren by e Reepor and Soure beause o ese dieren perepions.. Speed o propagaion (o lig) is idenial as obsered by Soure and Reepor. 3. Disanes raelled by e signal are dieren as obsered by e wo inerial rames beause o e dierenes in perepion o e insan isel. 4. Insan o Reeip is dieren in bo e rames as e soure and reepor are spaially separaed and ey anno agree on e insan snapso o e pysial world. (Remarks: (a) Insan o emission is synronized beween e wo rames as = 0; = 0. (b) I e insan o reeip is synronized beween e wo rames, en e insan o emission is pereied o be dieren!) 5. Loaion o reeip is pereied / obsered o be dieren (i e loaion o emission is aken as x = 0; x = 0). Loaion o Emission is pereied / obsered o be dieren (i e loaion o reeip is aken as x = 0; x = 0). Speed o propagaion as obsered by e Soure and Reepor are dieren and dier exaly by e relaie eloiy beween e Soure and Reepor. Tis ompensaes or e ee in number aboe, resuling in idenial resuls (Doppler Ee) as alulaed by e wo inerial rames assoiaed wi e Soure and Reepor. Tere is agreemen in wa is pereied as an insan and ereore no dierene in rael disane is aribuable on is aoun. Te Soure and Reepor agree on e insan o ime and no dierenes arise on is aoun ( = ). Loaion o Reeip is obsered o be dieren only o e exen o e relaie moion (x = x-). Tis is aouned or in Number as aboe and ompensaed by Number as aboe. Tus bo e inerial rames agree on e alulaed Doppler Ee. Howeer, i bo e inerial rames assume e speed o propagaion as same and idenial (as ), en e dierene reaed by Number is no ompensaed and bo e inerial rames alulae e Doppler Ee o be dieren.

5 6. In relaiisi pysis, e disane beween e soure and e reepor a e insan o emission o e signal is obsered o be dieren by e wo inerial rames, arising ou o e disagreemen on e onep o insan isel due o asynronizaion o spaially separaed objes/loks (soure and reepor). So all alulaions based on onsruie meods su as ime a insan o signal emission + ime duraion o rael o signal rom soure o reepor = ime a insan o reeip, anno be used in a simple manner espeially wen wo onseuie signals, separaed by a ime ineral (reiproal o requeny), are originaing rom e soure and reaing e reepor. So e relaiisi Doppler Ee is alulaed by using e Lorenz ransormaions and no by e equaion Time a insan o signal emission + Time duraion o rael o signal rom soure o reepor = Time a insan o reeip. Tis is beause ere is disagreemen on e insan o emission and insan o reeip beween e wo inerial rames. In lassial pysis, e disane beween e soure and e reepor a e insan o emission o e signal is obsered o be e same by e wo inerial rames. Te insan o reeip is also obsered o be e same. Only e disane raelled and e propagaion speed are obsered o be dieren; bu ey ompensae ea oer resuling in e ime aken o rael rom e soure o reepor o be idenial. One we assume e propagaion speed o be same, e alulaed Doppler Ees by bo e rames beome dieren. Te relaiisi Doppler Ee is e geomeri mean o ese wo lassial Doppler Ees in all geomeries and orienaions. In all ases, e lassial Doppler Ee an be alulaed rom irs priniples, a is, Time a insan o signal emission + Time duraion o rael o signal rom soure o reepor = Time a insan o reeip. Table. Faors a ae Doppler Ee in lassial and relaiisi pysis Wi e aboe onsideraions in mind, i is possible o onsruiely alulae e Doppler Ee only in e ase o lassial pysis. We may use eier inerial rame o alulae e Doppler Ee, bu we need o know e speed o propagaion wi respe o a inerial rame. Wen we alulae e lassial Doppler Ee by wo oies, namely, (a) based on obseraions by obserers o-moing wi e soure, and (b) based on obseraions by obserers o-moing wi e reepor, e wo resuls are idenial only wen e speed o propagaion assumed in (a) and e speed o propagaion assumed in (b) dier exaly by e relaie eloiy beween e inerial rames. Tereore, in one o e inerial rames, e speed o propagaion is direion dependen (anisoropi) wereas in e oer rame e speed o propagaion is isoropi. As a onsequene wen we assume bo e rames o be isoropi, e alulaed Doppler Ees in ases (a) and (b) are no idenial. Tereore, one may sae a e lassial Doppler Ee gies wo separae resuls: one by assuming a e inerial rame assoiaed wi e soure is isoropi, and e oer by assuming a e inerial rame assoiaed wi e reepor is isoropi. In is onex wen we say an inerial rame is isoropi, we mean a e propagaion o lig rom one loaion o anoer loaion wiin a inerial rame is

6 a speed in all direions irrespeie o weer e lig soure is saionary or moing wi respe o a inerial rame. Te relaiisi Doppler Ee is alulaed by using e Lorenz Transormaions in sandard exs [6, 7, 8]; a onsruie deriaion is preluded by e relaiisi onep o relaiiy o simulaneiy. Sine ea inerial rame adops is own onenion o simulaneiy [5], e insan a wi reeip o a signal akes plae is obsered o be dieren and e ime ineral beween emission and reeip is obsered o be o dieren duraions by e inerial rames assoiaed wi e res rames o e soure and e reepor. In is paper we ormulae e wo lassial Doppler ees () by assuming a e inerial rame assoiaed wi e soure is isoropi, and () by assuming a e inerial rame assoiaed wi e reepor is isoropi. We deelop ormulaions or () and () in e orienaions longiudinal, ranserse, and arbirary orienaion and sow a e raio o e obsered (by e reepor) requenies alulaed in () and () remain onsan in all e orienaions. Furer, we sow a e relaiisi Doppler Ee is e geomeri mean o e wo lassial Doppler Ees in () and () in all orienaions. We also presen some onlusions rom e a a e obsered Doppler Ee is e mean o e wo lassial Doppler Ees. Classial Longiudinal Doppler Ee: Lig speed is wi respe o Soure Consider (as illusraed in Fig. ) a soure o lig moing a speed wi respe o e laboraory rame o reerene. Assume also a lig is emanaing rom e soure raelling in all direions a speed wi respe o e soure. Te speed o e lig rays wi respe o e Laboraory reerene rame is o be ealuaed by e Galilean / Newonian ormulas or addiie eloiies, as done in lassial pysis. (Soure o Lig) Reepor Laboraory reerene rame Figure. Soure moing wi respe o laboraory reerene rame We use primed noaion or e inerial rame o-moing wi e soure (K ) and unprimed noaion or e saionary laboraory reerene rame (K) in wi e reepor is siuaed.

7 Tis is onsisen wi e way e subje maer is deal wi in sandard exs [6] werein i is oneied a a moing lig soure is obsered by a saionary reerene rame in wi a laboraory wi neessary insrumens is loaed. Te requeny o e soure o lig is e number o peaks (o wae rons) emanaing per seond. Te wo onseuie peaks may be onsidered as wo signals separaed by a ime ineral (/ ). For a lig wae, is requeny is o e order o 0 5 and us i is a ery large number and in our ealuaions we will be jusiiably negleing quaniies o e order o magniude /( ). We use as e requeny o e soure o lig onsisen wi e assumpion a a moing soure (primed rame) is being obsered by a saionary laboraory. For e longiudinal Doppler Ee, we onsider only eens on e x- axis (Fig. 3). In is ase, e line joining e soure and reepor is ollinear wi e line o relaie moion. R Reepor S Soure Figure 3. Soure moing away rom reepor on x-axis Assume a e soure o lig is moing wi respe o e insrumen reeiing e signals a a speed and away rom e insrumen. Le us also say a a a gien insan = 0, e disane beween e soure and e insrumen is s. A is insan e irs signal emanaes rom e soure raelling a speed wi respe o e soure and a speed ( ) wi respe o e insrumen, in aordane wi lassial pysis. s Te irs signal reaes e insrumen a. A e insan, e seond signal is emanaed rom e soure and a is insan e separaion beween e soure and insrumen is s and e signal sars a and reaes e s s insrumen aer a ime ineral o and e insan i reaes e insrumen is. Tereore e requeny measured by e Laboraory reerene rame is deried as ollows: s s () () (3)

8 Equaion (3) gies e lassial longiudinal Doppler Ee wen e speed o lig is wi respe o e soure. Classial Longiudinal Doppler Ee: Lig speed is wi respe o Reepor For is ase we reer o Fig. 3, bu we will ake e speed o e lig ray o be wi respe o e reepor. Two signals or wo onseuie peaks are separaed by a ime ineral (/ ) in e reerene rame o-moing wi e soure. Assume a e soure o lig is moing wi respe o e insrumen reeiing e signals a a speed o and away rom e insrumen. Le us also say a a gien insan = 0, e disane beween e soure and e insrumen is s. A is insan e irs signal emanaes rom e soure raelling a speed wi respe o e laboraory reerene rame. Te insan a wi is signal reaes e insrumen is s. A e insan e seond signal is emanaed rom e soure and a is insan e separaion beween e soure and e insrumen is s. Te insan a wi is signal s reaes e insrumen is. Tereore, e requeny measured by e laboraory reerene rame is deried as ollows: s s (4) (5) (6) Equaion (6) gies e lassial longiudinal Doppler Ee wen e speed o lig is wi respe o e reepor. In bo equaions (3) and (6) e soure and e reepor are moing away rom ea oer. Te relaiisi Doppler Ee is: (7)

9 (see also Equaion 9, page 90 o [6]). We obsere a e relaiisi Doppler Ee is e geomeri mean o e wo longiudinal Doppler Ees gien by equaions (3) and (6) respeiely. We now proeed o sow a e relaiisi ranserse Doppler Ee is also e geomeri mean o ) e lassial ranserse Doppler Ee wen e speed o lig is wi respe o e reepor and ) e lassial ranserse Doppler Ee wen e speed o lig is wi respe o e soure. Classial Transerse Doppler Ee: Lig speed is wi respe o Reepor Te lassial ranserse Doppler Ee is wen e Laboraory insrumen reeies e signal a 90 0 wi respe o e line o moion o e soure. For e ase wen e speed o lig is wi respe o e reepor we may reer o Fig. 4. A B Soure Reepor Figure 4. Signal reeied perpendiular o line o moion Poin A signiies e emanaion o e irs signal a = 0. Poin B signiies e emanaion o e seond signal a. Te disane AB =.

10 Te irs signal reaes e reepor a ; e seond signal reaes e reepor a and ereore e requeny obsered by e reepor is gien by (8) Considering a and are large, e aboe equaion redues o (9) Or (0) Tis is reerred o as e inabiliy o lassial pysis o predi a ranserse Doppler ee [6]. Howeer, we will sow in e nex seion a lassial pysis does predi a ranserse Doppler ee wen e speed o lig is wi respe o e soure. Classial Transerse Doppler Ee: Lig speed is wi respe o Soure In e ase wen e speed o lig is wi respe o e soure, e ranserse Doppler Ee an be isualized as sown in Fig. 5. O A Soure B O Reepor Figure 5. Signal reeied perpendiular o line o moion A = 0, a peak o e wae ron o lig (e irs signal) emanaes rom e soure and reaes e reepor B a =. A is insan =, e soure as moed and is now (a = ) loaed a poin

11 A, and e disane raelled by e lig ray is AB rom e soure. Tereore (AB/) =, wen e lig is raelling a wi respe o e soure o lig. (Please noe a all obserers saionary wi respe o e inerial rame o-moing wi e Reepor obsere a e lig raels along e line OB, wereas all obserers saionary wi respe o e inerial rame o-moing wi e soure o lig obsere a e lig ray raels along e line AB). In order o deermine e lassial ranserse Doppler Ee wen e speed o lig is wi respe o soure, i is reommended o ae e reerene rame o-moing wi e soure as e preerred reerene rame. Tis ailiaes e easy deerminaion o disanes raelled by e lig signals rom e soure as obsered by e rame o-moing wi e soure. Tis is depied in Fig. 6. O Soure Signal II Signal I C B Reepor A Figure 6. Speed o lig is wi respe o e soure To be onsisen wi our ormulaions in preious seions were we ad e reepor saionary and e soure moing a along posiie x axis, we now ae e soure saionary and e reepor moing a. Poin O is e soure. A ime = 0 signal I (a peak in e wae ron) emanaes rom e soure. A is insan e reepor is loaed a A. As e lig signal moes rom O o B, e reepor moes rom A o B. Te irs signal is reeied by e reepor a loaion B. Eidenly, i is e ime a wi signal I is reeied by e reepor, en AB = () OB = () BAO = (3) Te doed lines indiae a, as obsered by e reepor, e lig signal OB raels perpendiular o e line o relaie moion.

12 Te seond lig signal emanaes rom O a = ; being e requeny o e soure lig beam. Tis signal reaes poin C a ime. Eidenly AC = and e requeny obsered by e reepor is gien by. OB (4) OC (5) ; ime beinge sar beingeimeelapsed OC ) ( ) ( (6) By applying Pyagoras eorem or riangle OAB, we ge (7) By applying Pyagoras eorem or riangle OAC, and noing rom equaion (5) a OC = [ (/ )] we ge (8) Tereore, rom equaions (7) and (8) we obain (9) (0) () Te requeny o lig being normally a ery large number, is negligible, being a square o e ime dierene beween wo signals (peaks o e wae ron). Tereore,

13 () (3) (4) ] [ (5) By ompleing e square on e LHS by adding and subraing e erm /[ (- / ) ], we obain (6) Te seond erm on e LHS an be ignored as is a seond order erm o small quaniy (sine is large) (7) (8) or ) / ( (9) were we ollow e usual noaion = / [-( / )] (30) Tereore, (3)

14 Equaion (3) gies e lassial ranserse Doppler Ee wen e speed o lig is wi respe o e soure. Equaion (0) gies e lassial ranserse Doppler Ee wen e speed o lig is wi respe o e reepor. Te relaiisi ranserse Doppler Ee is (3) (see also Eqn -30, page 90 o.[6]). Tus e relaiisi ranserse Doppler Ee as gien by equaion (3) is e geomeri mean o e wo lassial Doppler ees gien by equaions (0) and (3). Doppler Ees a arbirary angle o emission and reeip From e preeding our seions, we an see a or bo e longiudinal and ranserse ases e raio o e ree quaniies Doppler ee predied by lassial eory wi speed o lig as wi respe o soure : Doppler ee predied by relaiiy eory : Doppler ee predied by lassial eory wi speed o lig as wi respe o reepor remains (/ γ) : : γ. In is seion we sow a e raio o ese ree quaniies remains (/ γ): : γ or all angles o emission and reeip. Speed o lig is wi respe o soure R R Reepor moing a - Lig ray II Lig ray I S Line o Moion Figure 7. Obseraions by inerial rame o-moing wi soure (speed o lig wi respe o soure) S is e soure, R is e posiion o reepor a reeip o irs signal, and R is e posiion o reepor a reeip o seond signal. is e angle a wi lig ray is emied rom soure wi reerene o e line o moion (as obsered by inerial rame o-moing wi soure).

15 Te primed noaion is used onsisen wi ormulaion o e problem a e soure is moing. Furer we ae designed e diagram (Fig. 7) in su a way a is aue and posiie so a sign onenions o rigonomeri unions are auomaially aken are o. By e ormulaions in geomery we ae S R S R R R S R R R Cos Wi requeny o lig being ery large, R R is ery small. Te square o R R may be negleed and we ae S R S R SR R R Cos S R R R Cos R R Cos Te las erm an be negleed as R R is ery small. Tereore we ae S R SR R R Cos Tereore e seond signal akes a lile less ime in reaing e reepor gien by R R Cos. Furer e seond signal leaes a lile laer gien by e quaniy. Tereore e requeny o reeip is gien by e equaion R R Cos Now RR ; e disane raelled beween e wo reepions. Tereore Cos or os (33) Equaion (33) gies e lassial Doppler Ee a arbirary angle o emission θ wen e speed o lig is wi respe o e soure. is e expeed reepor requeny and is e soure requeny. Comparing wi e relaiisi prediion gien in [6] equaion -5a pp89, Cos Reerene [6] equaion -5a. Comparing e aboe wi equaion (33) we ae e raio o e lassial prediion wi lig raelling a wi respe o e soure o e relaiisi prediion o be (/ γ ) :.

16 Speed o lig is wi respe o reepor R Reepor Lig ray I Lig ray II S S Soure moing a Figure 8. Obseraions by inerial rame o-moing wi reepor (speed o lig wi respe o reepor) R is e reepor. S is e posiion o e soure a e insan o emission o irs signal. S is e posiion o e soure a e insan o emission o seond signal. Is e angle a wi lig ray is emied rom soure wi reerene o e line o moion (as obsered by e inerial rame o-moing wi reepor). Te noaion (no primed), is used or e angle obsered by e inerial rame o-moing wi e reepor, onsisen wi e ormulaion o e problem a e reepor is a res and soure is moing. Moreoer we ae designed e diagram (Fig. 8) in su a way a is aue and posiie so a sign onenions o rigonomeri unions are auomaially aken are o. ( S R) ( S R) S S S R S S By e ormulaions in geomery we ae Negleing (S S ) as S S is ery small we ae S R S S Cos ( S R) ( S R) Cos S S Cos or S R ) S R SS ( Cos S R S R S S Cos Negleing e las erm as S S is ery small we ae. Tereore signal II reaes reepor aser bu i is emanaed aer a ime lag o reiproal o e soure requeny. e Tereore, e obsered requeny is gien by = S S Cos

17 S S Cos Te erm is e ime dierene due o e sorening o e rael pa. Now S S a is e disane raelled beween wo onseuie signals emanaing rom e soure. Tereore Cos or Cos (34) Equaion (34) gies e lassial Doppler Ee a arbirary angle o reeip θ wen e speed o lig is wi respe o e reepor. is e obsered reepor requeny and is e (alulaed) soure requeny. Comparing wi e relaiisi prediion gien in Equaion - 5b pp 89 o reerene [6] Cos ( / ) Reerene [6] equaion -5b wi an be re-wrien as (rearranged) Cos Reerene [6] equaion -5b (rearranged) Reerene [6] equaion -5b Cos Comparing e relaiisi prediion wi e one predied by lassial pysis (equaion 34), e raio o e relaiisi prediion o e requeny o reeip o e lassial prediion wi lig raelling a wi respe o e reepor is : Tus we ae or all angles o emission and reeip, e raio o e ree quaniies, namely, requeny o reeip predied by lassial pysis wi speed o lig wi respe o soure : relaiisi prediion o requeny o reeip : requeny o reeip predied by lassial pysis wi speed o lig wi respe o reepor = :: (35)

18 CONCLUSION Tere are wo possible inerpreaions or e raio desribed by Equaion (35) in e preious seion. Relaiisi Inerpreaion: Te relaiisi inerpreaion o e raio is as ollows. Obserers o-moing wi e soure expe e reepor o obsere a pariular requeny, wereas e reepor obseres a iger requeny inreased by e aor γ. Tis is inerpreed by e obserers o-moing wi e soure due o e slow running o e loks assoiaed wi e moing reepor. Similarly, assuming a e emission requeny as obsered by obserers o-moing wi e soure o be orre, obserers o-moing wi e reepor expe o obsere a pariular requeny, wereas ey obsere a redued requeny by e aor γ. Te obserers o-moing wi e reepor inerpre is resul due o e slow running o loks assoiaed wi e inerial rame o-moing wi e soure. Tus ey rekon a e aual requeny o emission is lower by a aor γ an a obsered by e obserers o-moing wi e soure. Tese resuls are onsisen wi e speial relaiiy eory a wo loks in relaie moion appear o run slow o ea oer. We ae sown a is raio remains inarian or all angles o emission and reeip o e lig ray meaning a e muual slow running o e loks in relaie moion depends only on e magniude o e relaie eloiy, onsisen wi e eory o speial relaiiy. Alernaie Inerpreaion: As e relaiisi (obsered) Doppler Ee is e geomeri mean o e wo lassial Doppler Ees, one assuming a e speed o lig is wi respe o e soure and e oer assuming a e speed o lig is wi respe o e reepor, a ommon sense inerpreaion will be a neier o ese assumpions are rue bu lig raelled a an aerage speed a is in beween e aboe wo speeds. Mainaining e equialene and isoropy o e inerial rames assoiaed wi e soure and reepor and indeed all inerial rames, one may sugges a lig raels a speed wi respe o maerial objes in e iiniy o ose maerial objes (like e soure o lig and reepor). Tis resoluion will mainain e equialene o all inerial rames a are in relaie moion wi respe o ea oer. Te oer possible resoluion an be a preerred inerial rame as enisaged originally by Lorenz werein e moing objes aually onra along e direion o moion by a aor (- / ) / and moing loks aually run slow by a aor (- / ) /. Tis resoluion will no old all inerial rames o be equialen.

19 Reerenes. Doppler, C. J. (84). Über das arbige Li der Doppelserne und einiger anderer Gesirne des Himmels (Abou e oloured lig o e binary sars and some oer sars o e eaens). Publiser: Abandlungen der Königl. Böm. Gesellsa der Wissensaen.. So Russell, Jon (848). "On erain ees produed on sound by e rapid moion o e obserer". Repor o e Eigeen Meeing o e Briis Assoiaion or e Adanemen o Siene (Jon Murray, London in 849) 8 (7): Rerieed Eans, D. H. and MDiken, W. N. (000). Doppler Ulrasound (Seond ed.). New York: Jon Wiley and Sons. ISBN Benson T (008) Beginners Guide Home Page, p:// /airplane/doppler.ml 5. Janis, A (00). Conenionaliy o Simulaneiy in e Enylopedia o Pilosopy edied by Edward N Zala. p://plao.sanord.edu/enries/spaeime-onensimul/ 6. Resnik, R. (968). Inroduion o Speial Relaiiy. Jon Wiley and Sons. 7. Bom, D. (965). Te Speial Teory o Relaiiy. W.A. Benjamin, New York. 8. Møller, C. (95). Te Teory o Relaiiy. Oxord Uniersiy Press.