The Doppler Effect: When the Source Moves
Imagine you're standing by a road. A car with its horn blaring races toward you. The pitch sounds high and urgent. The instant it passes you and speeds away, the pitch drops to a lower, flatter note. The horn itself hasn't changed — the car is playing the same note the whole time. What changed? The motion of the source.
That shift in pitch is the Doppler effect. It happens whenever a wave source moves relative to an observer. Sound waves, light waves, even water ripples — all of them get compressed or stretched by motion.
The Intuition: Why Does the Frequency Change?
Think of a stationary source — say, a speaker — sending out sound waves. Each wave crest leaves the source and travels outward at the speed of sound v. The distance between successive crests is the wavelength λ, and the number of crests leaving per second is the frequency f. For a stationary source, the waves are evenly spaced in all directions.
Now imagine the source moves toward you. Between one wave crest and the next, the source itself has moved a little closer to you. So the second crest doesn't have to travel as far to reach you as the first one did. The crests bunch up — the wavelength shortens. Since the wave speed v hasn't changed (sound speed depends on the medium, not the source), a shorter wavelength means more crests hit your ear per second. You hear a higher frequency.
When the source moves away, the opposite happens. Each successive crest is launched from a point farther from you. The crests stretch out — wavelength increases — and fewer crests reach you per second. The frequency drops.
The wave speed v is fixed by the medium (air, water, etc.). The source's motion does not change how fast the wave travels. It only changes the spacing of the wave crests — the wavelength — and therefore the frequency you detect.
The Precise Statement
Let:
- f = frequency emitted by the source (the "true" frequency)
- f′ = frequency heard by the observer
- v = speed of sound in the medium
- vs = speed of the source (relative to the medium)
Case 1: Source moving toward a stationary observer
The source chases its own waves. In one second, the source emits f waves. But because it moves toward you, those f waves are packed into a distance v−vs instead of v. The observed wavelength is λ′=fv−vs. Since v=f′λ′, we get:
f′=λ′v=(v−vs)/fv=f⋅v−vsv
Because v−vs<v, the denominator is smaller than the numerator, so f′>f. The pitch rises.
Case 2: Source moving away from a stationary observer
Now the source recedes. The waves stretch out. In one second, the f emitted waves are spread over a distance v+vs. The observed wavelength is λ′=fv+vs, so:
f′=λ′v=(v+vs)/fv=f⋅v+vsv
Here v+vs>v, so f′<f. The pitch falls.
f′=f⋅v∓vsv
Upper sign (minus): source toward observer → frequency rises
Lower sign (plus): source away from observer → frequency falls
What the Formula Tells You
The ratio v∓vsv is always greater than 1 when the source approaches, and less than 1 when it recedes. The faster the source moves, the more extreme the shift. If vs gets close to v (the source approaches the speed of sound), the denominator v−vs becomes tiny, and f′ becomes huge — the sound gets extremely high-pitched. That's the buildup just before a sonic boom. …