Doppler effect
A Socratic walk-through of the Doppler effect — reasoned out one step at a time, not lectured.
The question we started with
THE QUESTION #Why does a siren change pitch as it passes you, when the siren itself never changes?
The driver of the ambulance hears one unchanging note for the whole journey. You, on the pavement, hear the note fall as the vehicle goes by. Nothing inside the siren has altered. So where exactly does the change live — in the air, in your ear, or in something about the geometry between the two of you?
Worth asking a sharper version first, because the sharper version is where most people's picture is quietly wrong: when does the pitch fall? Most of us would say it slides down steadily as the vehicle approaches and keeps sliding as it leaves. Hold that thought.
Reasoning it through
REASONING #Take the siren apart into what it actually does: it pushes out a pressure crest at some fixed rate, say a thousand times a second. Each crest, once released, is no longer the siren's business. It travels through the air at the speed of sound — about 343 metres per second in room-temperature air — and that speed is a property of the air, not of whatever made the crest. A crest from a parked ambulance and a crest from a speeding one travel at exactly the same speed.
Now let the source move toward you while it emits. The first crest sets off. A thousandth of a second later the siren, having moved a little closer to you, releases the second crest — from a nearer starting line. So the gap between the two crests, measured along the road toward you, is smaller than it would have been. The crests bunch up ahead of the vehicle and stretch out behind it. Your ear does not measure spacing; it measures how often a crest arrives. Bunched crests arrive more often, and "more often" is what pitch means.
So no one changed the note. The emission rate is constant; the arrival rate is not.
How large is the effect? For an approaching source the arriving frequency is multiplied by c divided by (c minus the source's speed), and for a receding one by c divided by (c plus it), where c is the speed of sound. A vehicle at 20 metres per second — 72 km/h — gives about 1.062 on approach and about 0.945 going away. The step between them is a factor of roughly 1.12, which on a piano is close to two semitones. Small, and yet unmistakable, because the ear is far better at hearing a change in pitch than at naming an absolute one.
Now return to the sharper question. What actually enters the formula is not the vehicle's speed but the part of it directed along the line between you and the vehicle — the radial component. While the ambulance is a hundred metres up the road, almost all of its motion is straight at you, so that component is nearly the full 20 metres per second. It stays nearly the full amount right up until the vehicle is close. Then, in the second or two it takes to pass, the component swings through zero and out to nearly the full amount the other way.
So the pitch is not a slide. It is a plateau, a rapid swoop, and another plateau — and the swoop is located at the pass, not spread over the approach. That mismatch between what people report and what happens is worth sitting with, because it says the sensation belongs to a geometry, not to a machine winding down.
One more thing falls out of taking the air seriously. A moving source and a moving listener do not give the same answer even at the same relative speed, because the air sits still in the middle. For light there is no such medium, only relative motion — and relativity adds a shift with no classical analogue: a transverse one, present even at closest approach when nothing is moving toward or away from you, arising purely from time dilation.
The analogy
THE ANALOGY #Imagine posting one letter a day to a friend while walking steadily toward their house. You never change your habit — one letter, every day. But each letter starts its journey from a little closer than the last, so each spends a little less time in transit. Your friend receives letters slightly faster than one a day. Turn around and walk away, and they arrive slightly slower.
letters are separate objects that each travel independently, whereas crests are features of one continuous medium; and the analogy has nothing whatever to say about the transverse case for light, where a shift appears with no change in distance at all.
Clarifying the model
THE MODEL #Three refinements, each aimed at a place the picture usually goes soft.
The shift is in arrival rate, not in the sound being "squeezed" in some material way. Nothing is compressed except the spacing between crests already in flight, and that spacing was set at the moment of emission.
Loudness is a separate effect that rides along and gets confused with pitch. The siren does get louder as it nears, for the ordinary reason that it is closer. Louder-then-quieter and higher-then-lower arrive together, which is why people describe the whole event as one sensation of winding down.
And the plateau-swoop-plateau shape depends on the geometry. Pass very close and the swoop is nearly instantaneous; pass far off to one side and the two plateaus sit closer together, because the radial component never reaches the full speed.
A picture of it
THE PICTURE #How to readTime runs left to right, with zero at the moment the vehicle draws level with you. The curve is the note you actually hear: a flat high shelf for the whole approach, a drop concentrated in about two seconds around the pass, then a flat low shelf. The siren's own frequency would be a horizontal line at 1000 the whole way. Notice how little happens while the ambulance is coming toward you — precisely the part most listeners remember as a gradual slide.
What became clearer
WHAT CLEARED #The pitch change is not something the siren does, and not really something your ear does either. It is a fact about geometry: crests emitted at a steady rate arrive at a rate set by how fast the gap between source and listener is closing. Because that closing rate stays nearly constant during a long approach and reverses within seconds at the pass, the note is steady-high, then steady-low, with the whole drama compressed into the moment of passing.
Where to go next
ONWARD #- What happens when the source reaches and exceeds the speed of sound, and the bunched crests pile into a shock front.
- How the same arrival-rate reasoning, applied to light from distant galaxies, became the evidence for an expanding universe.
Key terms
TERMS #| Term | What it means |
|---|---|
| Wavefront | a surface of constant phase, such as one pressure crest, travelling outward at the medium's own speed once emitted. |
| Radial velocity | the component of a source's motion directly along the line of sight, and the only component the classical shift depends on. |
| Transverse Doppler shift | a relativistic redshift observed even when the source is moving purely sideways, caused by time dilation and absent from the classical account. |
Every term the collection defines is gathered in the glossary.