Project PHYSNET Physics Bldg. Michigan State University East Lansing, MI MISN-0-204THE DOPPLER EFFECTEarEarEarSourceStati onaryMovi ngAwayMovi ngToward1THEDOPPLEREFFECTbyMary Lu LarsenTowson State University1. Introductiona. The Eect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1b. Questions to be Answered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. TheDopplerEectforSounda. Wave Source and Receiver Both Stationary. . . . . . . . . . . . . . 2b. Wave Source Approaching Stationary Receiver. . . . . . . . . . . 2c. Receiver Approaching Stationary Source. . . . . . . . . . . . . . . . . 4d. Source and Receiver Approaching Each Other . . . . . . . . . . . 5e. Relative Linear Motion: Three Cases . . . . . . . . . . . . . . . . . . . . 6f. Moving Source Not Equivalent to Moving Receiver . . . . . . . 6g. The Medium is the Preferred Reference Frame. . . . . . . . . . . 73. TheDopplerEectforLighta. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7b. Doppler Broadening of Spectral Lines . . . . . . . . . . . . . . . . . . . 7c. Receding Galaxies Emit Doppler Shifted Light . . . . . . . . . . . 84. LimitationsoftheResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92ID Sheet: MISN-0-204Title: TheDopplerEectAuthor: Mary Lu Larsen, Dept. of Physics, Towson State UniversityVersion: 4/17/2002 Evaluation: Stage 0Length: 1 hr; 24 pagesInputSkills:1. Vocabulary: wavefront (MISN-0-203) or (MISN-0-430).2. State the relationships between the wavelength, frequency, period,and speed of a wave (MISN-0-201) or (MISN-0-430).OutputSkills(Knowledge):K1. Vocabulary: relativespeed, Dopplerbroadening, Dopplereect,Doppler shift.K2. Describe how the Doppler eect is used by astronomers and cos-mologists to justify the expanding universe model.OutputSkills(ProblemSolving):S1. Solve any Doppler shift problem by deriving the shift for that par-ticular case alone(not by using the general Doppler shift formulaand not by deriving the general case and then using it).S2. Use the Doppler shift formula to determine the Doppler shift forgivenmotionsofasoundwavesourceandareceiverrelativetoeach other as well as to the acoustic medium.S3. GivenavaluefortheDopplershift, calculatetherelativespeedbetween receiver and source.3THIS IS A DEVELOPMENTAL-STAGE PUBLICATIONOF PROJECT PHYSNETThe goal of our project is to assist a network of educators and scientists intransferring physics from one person to another. We support manuscriptprocessinganddistribution,alongwithcommunicationandinformationsystems. Wealsoworkwithemployerstoidentifybasicscienticskillsaswell asphysicstopicsthatareneededinscienceandtechnology. Anumber of our publications are aimed at assisting users in acquiring suchskills.Our publicationsaredesigned: (i)tobe updatedquicklyin response toeld tests and new scientic developments; (ii) to be used in both class-roomandprofessionalsettings; (iii)toshowtheprerequisitedependen-ciesexistingamongthevariouschunksof physicsknowledgeandskill,as a guide both to mental organization and to use of the materials;and(iv) to be adapted quickly to specic user needs ranging from single-skillinstruction to complete custom textbooks.New authors, reviewers and eld testers are welcome.PROJECT STAFFAndrew Schnepp WebmasterEugene Kales GraphicsPeter Signell Project DirectorADVISORY COMMITTEED. Alan Bromley Yale UniversityE. Leonard Jossem The Ohio State UniversityA. A. Strassenburg S. U. N. Y., Stony BrookViews expressed in a module are those of the module author(s) and arenot necessarily those of other project participants.c 2002, Peter Signell for Project PHYSNET, Physics-Astronomy Bldg.,Mich. State Univ., E. Lansing, MI 48824; (517) 355-3784. For our liberaluse policies see:http://www.physnet.org/home/modules/license.html.4MISN-0-204 1THEDOPPLEREFFECTbyMaryLuLarsenTowsonStateUniversity1. Introduction1a. TheEect. You have probably had the experience of hearing anambulance approaching with its siren blaring, and you may have noticedthat there was a sharp drop in the pitch of the siren as it passed you. Thatwas because, as it passed you, the siren changed from coming toward youto going away from you. To a person in the ambulance the siren stayed ata frequency that was between the coming toward you (higher) one andthegoingawayfromyou(lower)one. Thesinglefrequencyheardbythe person in the ambulance was, in fact, the frequency you would haveheard if the ambulance had been stationary. This is true in general: anapproaching source of sound shows a higher frequency than a stationarysound, and a receding source shows a lower frequency. This change in theobservedfrequencyofthesound, duetothemotionofthesource, isaconsequence of the wave nature of sound and this phenomenon is calledthe Doppler eect.The change in the observed frequency is called theDoppler shift.The Doppler eect is useful for measuringthe velocities of wavesources. For example, it is used for measuring blood ow non-invasively,for measuring the speed of cars, and, since light is also a wave, for deter-mining the velocities of stars.1b. Questions to be Answered. In the search for an understandingof the Doppler eect, several interesting questions present themselves:1. Does the Doppler shift depend only on whether the source is moving,or on whether both the source and observer are moving, or does itperhaps depend only on their relative velocity?2. Does the Doppler shift depend on whether the air, the transporterof the sound, is moving?3. Canonederiveanexactformula, goodforall situations, whereinone can plug in all the relevant velocities, plus the frequency of thesource in its own rest frame, and get out the observed frequency?5MISN-0-204 2stationaryreceivervlstationarywave sourcepeakpeakFigure1. A stationary source producing sound waves thatreach a stationary receiver.2. TheDopplerEectforSound2a. Wave Source and Receiver Both Stationary. Here we reviewthe wavelength-frequency-speed equation for the non-Doppler case, wherethe sound wave source and receiver are both at rest with respect to theair(themediuminwhichthesoundpropagates). Thesourceproduceswaves of wavelength and frequencythat travel at wave speedv, thespeed of sound in air (see Fig. 1). The equation is: = v . (1)2b. WaveSourceApproachingStationaryReceiver. Nowwend the equation for the shift in frequency,the Doppler shift,when thewavesource ismovingatspeedvsrctowardastationaryreceiver.1InFig. 2 we show two frames of a movie of what is happening. At a cer-tain instant of time, wave peak #1 is emitted by the source (see Fig. 2a).2Exactly one wave periodTlater,peak #1 has moved a distance (onewavelength) and peak #2 is emitted (Fig. 2b). However, peak #2 is notemittedatthesamepositioninspaceaspeak#1wasemittedbecausein the timeTthe wave source has moved a distancevsrcT(assume thatvsrc