Physics › 🌊 Waves & Sound

The Doppler Effect

Why a siren sounds high on the way in and low on the way out — even though nothing about the sound itself has changed.

High school
💡
The big idea: When a sound source and a listener are moving relative to each other, the listener hears a different frequency than the source is actually emitting. This isn't a trick of the ear — it's a direct geometric consequence of wavefronts being squeezed together on the approaching side and stretched apart on the receding side.
🎯 By the end, you'll be able to
  • Explain why a moving source or moving observer changes the frequency you hear, in terms of wavefronts bunching up or spreading out.
  • Calculate the observed frequency for a moving source and for a moving observer using the Doppler equations.
  • Identify which quantity in the Doppler equation represents the source's motion and which represents the observer's motion, and get the sign right.
  • Recognize the same principle at work in police radar guns and in the redshift astronomers use to measure how galaxies move.
📎 You should already know
  • Wave basics: frequency, wavelength, and wave speed (v = fλ)
  • Speed of sound in air (~340 m/s)

The Sound That Changes as It Passes You

Stand by the side of a road as an ambulance goes past. As it approaches, the siren sounds noticeably higher-pitched. The instant it passes and starts moving away, the pitch drops — like someone turned a dial. The siren itself never changed. The ambulance's electronics are producing the exact same frequency the whole time. What changed is your position relative to a moving source of sound waves, and that's enough to shift the frequency your ear receives.

This is the Doppler effect, and once you see the geometry behind it, the pitch-drop of a passing siren, the beep of a radar gun, and the redshifted light from a distant galaxy all turn out to be the same idea.

🔑 It's Not the Sound Speeding Up — It's the Wavefronts Bunching Together

Picture the ambulance emitting sound wavefronts like ripples from a pebble dropped in a pond, once every period \(T\). But the ambulance is moving, so each new ripple starts from a slightly different spot — a bit closer to you than the last one. Ahead of the ambulance, the wavefronts get crowded closer together: a shorter wavelength reaches you, which means a higher frequency. Behind the ambulance, each wavefront starts a bit farther back, so the wavefronts stretch apart: a longer wavelength, a lower frequency. The speed of sound through the air itself never changes — only the spacing between wavefronts does.

\[ v = f\lambda \]
The basic wave relationship: speed equals frequency times wavelength. This holds for the sound waves in the air the whole time — it's the wavelength that gets squeezed or stretched, not the speed.
\[ f_{o} = f_{s}\dfrac{v}{v \mp v_{s}} \]
Moving source, stationary observer. Use the minus sign while the source approaches (frequency goes up) and the plus sign while it recedes (frequency goes down). \(v\) is the speed of sound, \(v_s\) is the source's speed.
\[ f_{o} = f_{s}\dfrac{v \pm v_{o}}{v} \]
Stationary source, moving observer. Use the plus sign when the observer moves toward the source, minus when moving away. \(v_o\) is the observer's speed.
🎮 Interactive: The Doppler Effect LIVE
Drag the source's speed and direction and watch the wavefronts bunch up ahead of it and spread out behind it — then compare the pitch an observer hears on each side.

Why the Direction of Motion Is Everything

Only the part of the motion that points directly along the line between source and observer matters. A source flying straight at you produces the maximum shift; a source moving past you but not toward or away from you — like a plane crossing directly overhead at the instant it's closest — produces almost no shift at that instant, even though it's moving fast. This is why the pitch of a passing siren doesn't jump instantly from high to low at one point; it slides smoothly through the moment of closest approach, when the along-the-line component of velocity passes through zero.

Notice also that the source and observer versions of the equation aren't quite symmetric — dividing by \((v-v_s)\) versus multiplying by \((v+v_o)\). At everyday speeds the difference is tiny, but it's a real feature of sound: it needs a physical medium to travel through, and the source and observer play different roles relative to that medium.

📝 Worked example: An ambulance siren emits a steady 600 Hz tone. It approaches you at 20 m/s, then passes and recedes at the same speed. Take the speed of sound as 340 m/s. What frequency do you hear on each side?
  1. While approaching, the source is moving toward you, so use the minus sign: \(f_o = f_s \dfrac{v}{v - v_s}\).
  2. Substitute: \(f_o = 600 \times \dfrac{340}{340 - 20} = 600 \times \dfrac{340}{320}\).
  3. Compute: \(600 \times 340 = 204{,}000\), and \(204{,}000 / 320 = 637.5\) Hz.
  4. While receding, switch to the plus sign: \(f_o = 600 \times \dfrac{340}{340 + 20} = 600 \times \dfrac{340}{360} = 204{,}000/360 \approx 566.7\) Hz.
✓ About 637.5 Hz on approach and about 566.7 Hz while receding — a jump of roughly 71 Hz across the moment it passes you, even though the siren itself never changed.
📝 Worked example: A stationary loudspeaker emits a steady 500 Hz tone. A cyclist rides toward it at 10 m/s. Speed of sound is 340 m/s. What frequency does the cyclist hear?
  1. The source is stationary and the observer is moving toward it, so use the plus sign: \(f_o = f_s \dfrac{v + v_o}{v}\).
  2. Substitute: \(f_o = 500 \times \dfrac{340 + 10}{340} = 500 \times \dfrac{350}{340}\).
  3. Compute: \(500 \times 350 = 175{,}000\), and \(175{,}000 / 340 \approx 514.7\) Hz.
✓ About 514.7 Hz — noticeably higher than the 500 Hz emitted, because the cyclist is riding into the oncoming wavefronts and meeting them more often per second.
⚠️ Common Trap: Don't Confuse Loudness With Pitch

A passing siren also gets louder as it approaches and quieter as it recedes — but that's a separate effect (distance changing intensity), not the Doppler shift. The Doppler shift is specifically about frequency and pitch, and it happens even at constant distance if the motion has a component along the line of sight. Also remember these equations describe steady, constant-velocity motion in one medium; they break down as a source's speed approaches the speed of sound itself (that's where sonic booms come from), and light's Doppler shift — used for redshift in astronomy — needs a different, relativistic version of the formula rather than this sound-wave one.

Same Idea, Different Waves: Radar and Redshift

Police radar guns send out a radio wave, let it bounce off your moving car, and measure how much the reflected wave's frequency has shifted — that shift converts directly into your speed. Astronomers do something similar with light from distant galaxies: light from galaxies moving away from us arrives at a longer wavelength than emitted, shifted toward the red end of the spectrum, while light from things moving toward us shifts toward blue. It's the identical wavefront-spacing idea you just worked through with sound, just applied to a different kind of wave.

Check your understanding

1. Why does a car horn sound higher-pitched as the car approaches you?
The speed of sound through the air doesn't change. What changes is the spacing between successive wavefronts: a moving source emits each new wavefront a little closer to you than the last, compressing the wavelength and raising the frequency you hear.
2. A train horn emits 500 Hz. The train approaches a stationary observer at 34 m/s, and the speed of sound is 340 m/s. What frequency does the observer hear?
Use \(f_o = f_s \dfrac{v}{v - v_s} = 500 \times \dfrac{340}{340-34} = 500 \times \dfrac{340}{306} \approx 555.6\) Hz, which rounds to about 556 Hz.
3. Which scenario produces the largest Doppler shift?
The shift depends only on the component of velocity directed along the line between source and observer. Moving straight toward the observer maximizes that component; at the moment of closest approach on a straight flyby, that along-the-line component is momentarily zero, so the shift briefly vanishes.
4. Which real-world technology relies directly on the Doppler effect?
Radar guns bounce a radio wave off a moving vehicle and measure the frequency shift of the reflection to compute its speed — the same wavefront-spacing principle used for sound, applied to radio waves.
✅ Key takeaways
  • Pitch shifts because relative motion between source and observer compresses or stretches the spacing of wavefronts — the speed of sound in the medium itself never changes.
  • An approaching source (or an observer moving toward a source) raises the frequency you hear; a receding one lowers it, and the shift is greatest when the motion is directly along the line between them.
  • Two related equations cover the everyday cases: \(f_o = f_s \frac{v}{v \mp v_s}\) for a moving source, and \(f_o = f_s \frac{v \pm v_o}{v}\) for a moving observer.
  • The same wavefront-spacing principle drives police radar speed guns and the redshift/blueshift astronomers use to measure how fast distant galaxies are moving.