Matthew Schwartz Lecture 21: The Doppler effect

1 Matthew SchwartzLecture 21:The Doppler effect1 Moving sourcesWe d like to understand what happens when waves are producedfrom a moving source. Let ssay we have a source emitting sound with the frequency . In this case, the maxima of theamplitude of the wave produced occur at intervals of the periodT=1 . If the source is at rest,an observer would receive these maxima spaced byT. If we draw the waves, the maxima areseparated by a wavelength =Tcs, withcsthe speed of , say the source is moving at velocityvs. After the source emits one maximum, it movesa distancevsTtowards the observer before it emits the next maximum. Thus the two successivemaxima will be closer than apart. In fact, they will be ahead= (cs vs)Tapart. The secondmaximum will arrive in less thanTfrom the first blip.

2 It will arrive with periodTahead= aheadcs=(cs vscs)T(1)The frequency of the blips/maxima directly ahead of the siren is thus ahead=1 Tahead=(cscs vs)1T=(cscs vs) .(2)In other words, if the source is traveling directly towards us, the frequency we hear is shiftedupwards by a factor ofcscs can do a similar calculation for the case in which the source is traveling directly awayfrom us with velocityv. In this case, in between pulses, the source travels a distanceTand theold pulse travels outwards by a distancecsT. The physical spacing between maxima is therefore behind= (cs+v)T. The frequency as perceived by an observer behind the siren is thus behind=(cscs+vs) (3) lower than for a stationary other words, the frequency goes up when the source is approaching us, and goes downwhen it is traveling away from sound waves produced by the moving source are depicted waves emitted by a source moving to the left.

3 The peaks in pressure ahead of thesource are more closely spaced than the peaks behind the can summarize these two results by saying that for a stationary observer directly aheador behind the moving source, =(cscs+vs) (4)wherevsis positive if the source is moving away from the observer, and negative if the source ismoving towards the is not hard to include also the case when the observer is in motion . Say the source ismoving away from the observer. Then the spacing between adjacent peaks are spread out asbefore behind= (cs+vs)T=cs+vs . If the observer is moving towards the source with velocityvr, then the she passes these peaks faster, at the rate =cs+vr ahead=(cs+vrcs+vs) (5)In the cases where the direction of either the source of observer is flipped, the sign ofvsorvrflips, but this equation still holds.

4 Note that if the observer and source are moving at the samespeed in the same direction, no frequency change is type of change in frequency due to motion is called theDoppler motion at an angleWhat happens if the source is not moving directly towards or away from the receiver? Say thesource is moving at an angle with respect to a stationary receiver, as shown in moving at angle relative to the axis connecting the source and the thispicture, the observer is along the angled blue line, say on the top of the s easiest to see what the observed frequency is in this case by looking at the picture. Nowthe maxima are spaced = (cs vscos ). Check that for = 0, this reduces to the ahead case,for = it reduces to the behind case and for = 2, where the observer is orthogonal to thedirection, there is no change.

5 Following the same logic as before, the frequency of the sound thereceiver hears is given by =(cscs vscos ) (6)This angular dependence explains why the siren of a police car or ambulance sounds the wayit does when it passes you. While approaching at a distance, the car is basically going towardsyou and the frequency is increased. When it s going away, thefrequency is lowered. As the carpasses us, the angle transitions pretty quickly, and the sound transitions from high to low as thecar goes through the intermediate that the angle denotes the angle to the velocity vector of the police car when thesource was emitted, when the sound is received. To see this, imagine that we are very far awayfrom the source, and it took a couple of days for the sound to get to us.

6 Clearly what the sourcedid during those days, such as the position it ended up it, is irrelevant to the frequency we 2 Thus the only possibly relevant piece of information is where the source was and how it wasmoving when it emitted the sound. So the fastest frequency change is not when the car ismoving perpendicular to your line of sight, but slightly before the car gets to that Source moving at or faster than soundWhat happens if the source approaches the speed of sound, or surpasses it? Again, the result iseasiest to understand with pictures. Here are some wavefronts as the speed of sound isapproached and given off by a source moving to the right at various speeds. The older pulses are depicted asdimmer. All the circles are growing with first two panels we ve already discussed: stationary source and a source movingsubsoni-cally(less than the speed of sound).

7 As the source approaches the speed of sound, you can see from the picture that the wave-fronts in the forward direction start bunching up. When the speed of sound is hit, they all cometogether. Remember for sound these are wavefronts of pressure. The large increase in pressurefrom the accumulation of maxima is followed by a large decrease in pressure from the rapid change in pressure is very loud. It is called asonic boom. You heard it as thecracking of a the source goessupersonically(faster than the speed of sound), the sonic boom goesfrom a straight wavefront perpendicular to the motion of thesource to being bent backwards ina cone. The sonic boom is now at an angle to the motion of the source, but it is still there. So,to be clear, the sonic boom is a sign that the speed of sound hasbeen surpassed; it is not some-thing that happens only exactly whenv= RedshiftThe picture with the wavefronts works just as well for light as for sound, at least for the casevs c.

8 Of course, the source can never go faster than the speed of light, due to special rela-tivity, so these equations need some nice thing about relativity is that you can pick whateverinertial reference frame youwant. So let s work in the rest frame of the source, and callvthe velocity of the observer. Sincethe source is at rest, the wave crests are spaced =c apart. If the moving observer is headingaway from the source, she passes the crests at a rate move=c v =c vc , as in Eq. (5). How-ever, since the observer is moving very fast there is also a time dilation effect. Time is slower forher, so she really sees successive crests at the lower frequency = move1 v2c2 =1 vc1 +vc (7)Redshift3 More generally,vis the relative velocity of the source and the observer:v=vs vrin the nota-tion from before.

9 Ifv >0the two are moving away from each other andv <0if they aremoving towards each that for smallv c, we can Taylor expand Eq. (7) giving = (1 +vr vsc+ ).Taylor expanding Eq. (5) gives = (1 +vr vscs+ ). So in the small velocity limit, this rela-tivistic analysis reduces to our previous astronomy, it is useful to define theredshiftof a signal asz = = = 1 .(8)Negative redshift is referred to asblueshift. These names are used because a signal which isredshifted is shifted to longer wavelengths, and red is longest-wavelength light visible to signals are shifted to shorter wavelengths, and blue is the shortest-wavelength lightvisible to humans. Astronomers commonly use red as a synonym for long-wavelength,and blue as a synonym for short-wavelength.

10 For objects atlow velocity compared to thespeed of light, plugging Eq. (8) into the above formula for redshift yieldsz vc(9)wherevis positive if the source is moving away from the receiver. Sources receding from theobserver thus appear redshifted, while sources moving towards the observer appear is an incredibly useful fact in astrophysics, as it allows us to measure the velocity withwhich distant sources of light are receding from or approaching what do we find? Everywhere we look, the objects are redshifted. We can sometimesmeasure the distance to an object by how bright it is, or usingparallax. When we do this wefind an essentially linear relationship between distance and (10)whereH0is a constant , calledHubble s constant , named after Edwin Hubble who firstobserved this relation.