You already know this one. An ambulance screams toward you and the siren sounds high and urgent; the instant it passes, the pitch sags into a lower wail. The driver hears one steady note the whole time — only you, standing on the curb, hear it change. Nothing is wrong with the siren. What changed is how the sound waves arrive at your ear.
Picture the source flinging out a fresh ripple — a wavefront — on a steady beat. Sitting still, it makes evenly spaced rings, like dropping pebbles in a pond. But if it drives toward you, it sneaks a little closer to its own last ripple before launching the next one, so the rings in front get squashed together and the ones trailing behind get stretched apart. Squashed waves mean a shorter wavelength and a higher pitch ahead of the source; stretched waves mean a lower pitch behind it.
Putting numbers on it
Name the players: the emitted frequency $f_0$ is how many wavefronts leave the source each second, and the sound speed $v$ is how fast they race through the air. When the source rushes toward a still listener at speed $v_s$, the observed frequency climbs to
Source approaching a still listener
$$f' = f_0\,\frac{v}{v - v_s}$$
Try a real horn: $f_0 = 440$ Hz on a car doing $v_s = 30$ m/s in $v = 343$ m/s air gives $f' = 440\times 343/(343-30) \approx 482$ Hz — about a semitone sharp. Flip the minus to a plus and the receding car drops to roughly $405$ Hz.
The full picture
Let either party move and the law becomes $f' = f_0\,(v + v_o)/(v - v_s)$, where $v_o$ is the observer's speed and the signs are arranged so that "approaching" always raises $f$. Push the source all the way up to the sound speed itself ($v_s = v$) and the denominator collapses to zero — the wavefronts pile into one violent shock front, the sonic boom. Light has no air to travel through, so for galaxies only relative speed matters and the relativistic form $f' = f_0\sqrt{(1-\beta)/(1+\beta)}$ takes over; the stretch toward red is the "redshift" astronomers use to clock the expanding universe. (The formal derivation below labels the sound speed $v_s$ and the source speed $v_{src}$ — identical physics, just different letters; the sliders use $v$ and $v_s$.)
Try this in the sim above
Set Observer speed to 0 and slowly raise Source speed: watch $f'$ climb while the front wavefronts crowd together. Then push Source speed up to 343 — equal to Sound speed — and see the rings collapse onto a single front; that pile-up is the boom. Finally switch to the Redshift mode and drag $\beta$ toward 1: the emitted colors slide bodily toward red as the galaxy appears to flee.
Physical mechanism. When a source moves toward an observer, each wavefront is emitted slightly closer — compressing the wavelength. Observer hears higher frequency: $f'>f_0$. When source recedes: wavelength stretches, $f'
2
Sign convention. Numerator: $+v_o$ when observer moves toward source. Denominator: $-v_{src}$ when source moves toward observer. Memorise by: approach = higher $f$ (numerator up or denominator down).
3
Sonic boom. When $v_{src}=v_s$: wavelengths pile up — divergence. When $v_{src}>v_s$ (supersonic): Mach cone forms with half-angle $\theta=\arcsin(v_s/v_{src})$. The shock wave is heard as a sonic boom as the cone sweeps past.
4
Relativistic Doppler. For light: $f'=f_0\sqrt{(1-\beta)/(1+\beta)}$. Redshift $z=(\lambda'-\lambda)/\lambda$ measures recessional velocity of galaxies. Hubble's Law: $v=H_0 d$ — the universe is expanding.
Ref: Halliday, Resnick & Walker 10th Ed., §17-5; Serway & Jewett 8th Ed., §17.4; French — Vibrations and Waves.
Section 04
Frequently Asked Questions
Both play a role. The source emits wavefronts at its own frequency $f_0$. When moving, each successive wavefront is emitted from a different position, altering the spacing (wavelength) of wavefronts arriving at the observer. The observer counts more (or fewer) wavefronts per second than the source emits per second.
Key takeaway: Doppler effect: motion of source compresses/stretches wavelength; observer counts different f.
Police radar guns (Doppler shift of reflected microwaves gives speed), weather radar (velocity of raindrops), medical ultrasound (blood flow velocity), sonar (submarine detection), bat echolocation, GPS satellite corrections, redshift measurement in astronomy, and exoplanet detection (stellar radial velocity).
Key takeaway: Doppler is used for speed measurement, medical imaging, astronomy, and navigation.
The classical Doppler effect (sound) is asymmetric — moving source and moving observer with the same relative velocity give different results. The relativistic Doppler effect (light) is symmetric — only relative motion matters. This asymmetry for sound reveals that motion relative to the medium (air) is what counts for classical waves.
Redshift $z=(\lambda'-\lambda)/\lambda$ is measured from spectral lines (e.g., hydrogen 656 nm). For low $v$: $z\approx v/c$. Hubble's Law: $v=H_0 d$ where $H_0\approx70$ km/s/Mpc. So $d=cz/H_0$. A galaxy with $z=0.1$ is receding at $3\times10^4$ km/s and is about $430$ Mpc away.
Key takeaway: Hubble's Law + Doppler redshift: galaxy distance $d=cz/H_0$.
Resources: Khan Academy; HyperPhysics; MIT OCW.
Section 05
Common Misconceptions
❌ The Doppler effect only occurs when the source is moving.
✅ The Doppler effect occurs whenever there is relative motion between source and observer — either or both can be moving. Moving observer with stationary source: $f'=f_0(v_s+v_o)/v_s$. However, the results for "source moving" vs "observer moving" are different for sound, because the medium (air) defines an absolute rest frame.
📖 HRW 10th Ed., §17-5.
❌ The Doppler effect shifts all frequencies by the same amount in Hz.
✅ Doppler shifts multiply frequency by a factor, not add a constant. A 440 Hz tone from a car at 30 m/s gives $f'=440\times343/(343-30)\approx482$ Hz (Δ≈42 Hz). A 880 Hz tone from the same car gives Δ≈84 Hz. The ratio is constant, but the Hz shift depends on the original frequency.
📖 HRW 10th Ed., §17-5.
❌ A sonic boom is heard only when the aircraft passes through the sound barrier.
✅ A sonic boom is heard every time the Mach cone sweeps past the observer — continuously as the aircraft flies overhead. It's not a one-time event. The "breaking the sound barrier" phrase is misleading: the boom is heard at ground level as the cone moves, not at the moment the plane first exceeds Mach 1.
📖 HRW 10th Ed., §17-5.
Misconception research: Arons — A Guide to Introductory Physics Teaching.