Doppler Effect for Light
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Direct answer
Sound needs a medium, so its Doppler effect is asymmetric between source and observer; light needs none, so only relative motion counts and the formula is symmetric. For speeds much smaller than c, the fractional shift is Δλ/λ = Δν/ν = v/c: a source receding at speed v stretches every wavelength by the fraction v/c (redshift), an approaching one compresses it (blueshift). The exact relativistic expression for recession is ν' = ν√((1 − β)/(1 + β)) with β = v/c, which reduces to the linear law for β << 1. Astronomers define z = Δλ/λ₀ and use it as a speedometer — Hubble's law turns it into a distance ladder — and a radar speed gun exploits the double shift of a wave that travels out and back: Δν = 2v/λ.
What you must remember
- Non-relativistic law: Δλ/λ = Δν/ν = v/c; a galaxy receding at 0.1 per cent of c shifts every spectral line redward by 0.1 per cent.
- Sign convention: receding → wavelength increases, frequency decreases (redshift); approaching → the opposite (blueshift); the colour words encode the whole logic.
- Relativistic exact form: ν' = ν√((1 − β)/(1 + β)) for recession; use it whenever v is a sizable fraction of c, and check that β → 1 drives ν' → 0.
- Symmetry: swapping "source moves" and "observer moves" changes nothing — no medium, no asymmetry; this is the sharpest contrast with sound.
- Astronomical redshift: z = (λ_observed − λ₀)/λ₀ ≈ v/c; combined with Hubble's law (recession speed proportional to distance), z becomes a cosmic distance indicator.
- Radar double shift: a wave reflected from a moving target is shifted twice, so the beat frequency against the transmitter is Δν = 2v/λ — the working principle of speed guns.
- Spectral-line fingerprint: the shift is always measured against known laboratory lines (Hα at 656.3 nm, say), which is why the spectrometer is the Doppler instrument of astronomy.
Redshift arithmetic
A star's Hα line, laboratory value 656.3 nm, arrives at 656.7 nm. The shift is Δλ = 0.4 nm, so v = c(Δλ/λ) = 3 × 10⁸ × (0.4/656.3) ≈ 1.83 × 10⁵ m/s — about 183 km/s of recession, a typical value for a Milky Way star drifting relative to the Sun. Now the galaxy version: the same line appears at 721.9 nm, a 10 per cent redshift; at z = 0.1 the linear rule gives 0.1c as a first estimate, and the exact formula refines it to about 0.095c — the boundary where JEE stops and cosmology begins.
The radar case closes the practical loop. A speed gun transmits at wavelength 3 cm (10 GHz); a car approaching at 30 m/s (108 km/h) returns a signal shifted twice: Δν = 2v/λ = 2 × 30/0.03 = 2000 Hz. The receiver beats echo against transmitter, and a countable 2 kHz audio tone stands in for highway speed. The factor of two is the trap and the teaching point: the car first meets more wavefronts as a moving observer, then re-emits the reflection as a moving source, and both shifts have the same sign. Weather radar and medical Doppler ultrasound run the identical arithmetic — one formula spanning traffic policing to blood-flow measurement.
Light Doppler in exams
The first separator from sound is symmetry: a question that distinguishes source-motion from observer-motion for light is testing whether you know no medium exists — for sound the two cases give different formulas, for light they cannot. Second, sign errors: receding means longer wavelength and lower frequency simultaneously, and options pair them correctly only in one row; reddening the frequency instead of the wavelength picks the mirrored wrong answer. Third, the radar factor of two is omitted in a classic distractor — the single-shift value sits among the options. Fourth, validity: the linear law needs v << c; a 0.8c problem requires the relativistic square-root form, and plugging 0.8 into v/c overestimates grossly. Main tests the fractional-shift plug-in and radar; Advanced has used Hubble-law combinations (distance from z via H₀ ≈ 70 km/s per megaparsec) and the distinction between Doppler, gravitational and expansion redshift in statement questions.
Frequently asked questions
How does the Doppler effect for light differ from that for sound?
Light needs no medium, so only relative motion matters and source-versus-observer motion makes no difference; sound's two cases give different formulas because air anchors the asymmetry.
What is the approximate Doppler shift formula for light?
Δλ/λ = Δν/ν = v/c for speeds much smaller than c, with recession producing a longer wavelength (redshift) and approach a shorter one (blueshift).
Why does a radar speed gun see a double Doppler shift?
The moving vehicle acts first as a moving observer of the incoming wave and then as a moving source of the reflection, so the echo's frequency shifts by Δν = 2v/λ.
How do astronomers measure a galaxy's recession speed?
By comparing its spectral lines (like Hα at 656.3 nm) with laboratory wavelengths, converting the redshift z = Δλ/λ into speed through v ≈ zc, or the relativistic formula for large z.
When must the relativistic Doppler formula be used?
Whenever v is a significant fraction of c: ν' = ν√((1 − β)/(1 + β)) then replaces the linear approximation, which increasingly overestimates the shift as β grows.