Doppler effect

You have heard it every time an ambulance passes: the pitch of the siren is higher as it comes toward you and lower once it has gone past. That change is the Doppler effect — named after Christian Doppler, who described it in 1842.

The one-sentence definition: the Doppler effect is the change in the observed frequency (and wavelength) of a wave whenever there is relative motion between the source of the wave and the observer.

Two crucial ideas to hold onto from the start:

1. It is the observed frequency that changes, not the emitted one. The siren's horn keeps vibrating at exactly the same frequency the whole time. What changes is how many wave crests reach your ear each second. A microphone travelling with the ambulance would measure no change at all.
2. Only relative motion matters. It makes no difference whether the source moves toward a stationary observer, or the observer moves toward a stationary source — either way the crests arrive more often, so the observed frequency is higher. (For sound the two cases give slightly different formulae because sound travels through a medium; for light only the relative velocity matters.)

The effect works for any wave: sound, water ripples, and electromagnetic waves such as light, microwaves and radio. The size of the shift depends on how fast the source and observer move relative to the wave speed: fast relative motion gives a big shift, slow relative motion gives a tiny one.

A quick feel for the sign of the shift:

Relative motion	Wavefronts	Observed wavelength	Observed frequency	Pitch / colour
Approaching	Bunched up	Shorter	Higher	Higher pitch / bluer
Receding	Stretched out	Longer	Lower	Lower pitch / redder

Everything else in this topic is just making that table quantitative — first for sound, then for light.

To understand the Doppler effect — rather than just memorise a formula — picture the wavefronts (the crests) spreading out from a source.

A stationary source. A source at rest emits a crest every period $T$. Each crest spreads outward as a circle centred on the source. Because the source stays put, the circles are concentric and evenly spaced: the wavelength is the same in every direction, so every observer measures the same frequency $f$.

A moving source. Now let the source move to the right at speed $u_s$ (less than the wave speed $v$). It still emits a crest every period $T$ — but between emitting one crest and the next, the source has moved forward. So each new crest is emitted from a point slightly ahead of the last.

• Ahead of the source, the source is chasing its own waves. Each crest is released closer to the one in front, so the crests bunch up → the wavelength is shortened. An observer ahead therefore receives crests more often → higher observed frequency.
• Behind the source, the source moves away from the waves it has already emitted, so successive crests are further apart → the wavelength is stretched. An observer behind receives crests less often → lower observed frequency.

Turning the picture into the sound equation. Consider an observer directly ahead of a source approaching at $u_s$. In one period $T = 1/f$ the wave would normally advance one wavelength $\lambda = vT$. But in that same time the source moves forward $u_s T$. So the effective wavelength ahead is the normal wavelength minus the distance the source has gained: $\lambda' = vT - u_s T = (v - u_s)T$ The observed frequency is the wave speed divided by this shortened wavelength: $f' = \frac{v}{\lambda'} = \frac{v}{(v - u_s)T} = f\,\frac{v}{v - u_s}$ Because $v - u_s < v$, the fraction is greater than 1, so $f' > f$ — approach raises the frequency, exactly as the picture promised. If the source recedes, replace $-u_s$ with $+u_s$ and the observed frequency drops. This is the origin of the moving-source formula in the next section.

The same reasoning for light (Δf/f ≈ v/c). For light there is no medium, but the geometry still tells us the fractional change in wavelength equals the fractional change caused by the relative motion. When the relative speed $v$ is much smaller than the speed of light $c$, this reduces to the simple magnitude relation $\frac{\Delta \lambda}{\lambda} = \frac{\Delta f}{f} \approx \frac{v}{c}$ which is all you need at SL.

For sound, the wave travels through a medium (usually air) at a fixed speed $v$ — around $340\,\text{m s}^{-1}$ in air. The IB data booklet provides both equations; your job is to pick the right one and, above all, the right sign.

Case 1 — source moving, observer stationary $f' = f\,\frac{v}{v \pm u_s}$

• Source moving toward the observer → use − in the denominator → smaller denominator → higher $f'$.
• Source moving away → use + → larger denominator → lower $f'$.

Here $u_s$ is the speed of the source and $v$ is the speed of sound.

Case 2 — observer moving, source stationary $f' = f\,\frac{v \pm u_o}{v}$

• Observer moving toward the source → use + in the numerator → larger numerator → higher $f'$.
• Observer moving away → use − → lower $f'$.

Here $u_o$ is the speed of the observer.

The sign trap — and the foolproof check. The two cases put the ± in different places (denominator for a source, numerator for an observer) and use opposite signs for "toward". Do not try to memorise four sign rules. Instead:

Whatever the situation, if the source and observer are getting closer, the observed frequency must be HIGHER than $f$. If they are getting further apart, it must be LOWER. After every calculation, check your answer against this — if approach gave you a lower frequency, you used the wrong sign.

Worked mini-example (moving source). A train sounds a $400\,\text{Hz}$ horn and approaches a stationary observer at $u_s = 30\,\text{m s}^{-1}$; speed of sound $v = 340\,\text{m s}^{-1}$. $f' = 400 \times \frac{340}{340 - 30} = 400 \times \frac{340}{310} = 439\,\text{Hz}$ As it recedes: $f' = 400 \times \frac{340}{340 + 30} = 400 \times \frac{340}{370} = 368\,\text{Hz}$ The pitch heard therefore drops from $439\,\text{Hz}$ to $368\,\text{Hz}$ as the train passes — a jump of about $71\,\text{Hz}$, which is exactly the audible "neeee-aaww" of a passing vehicle.

Consistency of speeds. Always keep $v$ (the speed of sound in the medium) and the source/observer speed in the same units (m s⁻¹) and never confuse the two — $v$ is a property of the medium, $u_s$ or $u_o$ is how fast the object moves.

Light is an electromagnetic wave, so it shows the Doppler effect too — but with an important difference. Light needs no medium, and it always travels at $c = 3.00 \times 10^{8}\,\text{m s}^{-1}$. For everyday and most astronomical speeds the relative speed $v$ is far smaller than $c$, so at SL we use the simple approximation: $\frac{\Delta f}{f} = \frac{\Delta \lambda}{\lambda} \approx \frac{v}{c}$ where $\Delta \lambda$ is the change in wavelength, $\lambda$ the emitted wavelength, and $v$ the relative speed of source and observer along the line of sight. (The full relativistic formula is HL only — at SL this magnitude relation is all that is required.)

Do NOT use the sound equations for light. The sound formulae $f' = f v/(v \pm u_s)$ contain the speed of sound; they are meaningless for light, which has no medium. For light, always reach for $\Delta f/f \approx v/c$.

Red-shift and blue-shift. Because visible light runs from violet/blue (short wavelength) to red (long wavelength):

• Source receding → wavelength increases (stretched) → light shifts toward the red end → red-shift.
• Source approaching → wavelength decreases (squashed) → light shifts toward the blue end → blue-shift.

Astronomers spot the shift by looking at the dark spectral (absorption) lines in a star's spectrum. These lines occur at wavelengths fixed by the atoms (e.g. hydrogen), so we know their true, lab-measured wavelengths precisely. If the whole pattern of lines is shifted toward longer wavelengths, the star is receding; toward shorter wavelengths, it is approaching.

Worked mini-example. A hydrogen line normally at $\lambda = 656\,\text{nm}$ is observed from a distant galaxy at $671\,\text{nm}$, so $\Delta \lambda = +15\,\text{nm}$ (a red-shift). The recession speed is $v = c\,\frac{\Delta \lambda}{\lambda} = (3.00 \times 10^{8}) \times \frac{15}{656} = 6.9 \times 10^{6}\,\text{m s}^{-1}$ i.e. the galaxy is moving away from us at about $6.9 \times 10^{6}\,\text{m s}^{-1}$ (roughly $2\%$ of the speed of light, comfortably in the range where the approximation holds).

The Doppler effect is not just a curiosity — it is one of the most useful measuring tools in physics, because a frequency (or wavelength) shift is easy to measure precisely and turns directly into a speed.

1. Speed of stars and galaxies — evidence for an expanding universe. By measuring the red-shift or blue-shift of a star's spectral lines and using $v = c\,\Delta\lambda/\lambda$, astronomers find how fast it is approaching or receding. The landmark discovery (Edwin Hubble, 1920s) was that almost all distant galaxies are red-shifted, and the further away a galaxy is, the greater its red-shift — so the more distant galaxies are receding faster. The natural explanation is that space itself is expanding, carrying the galaxies apart. Galactic red-shift is therefore a key piece of evidence for the expanding universe and, ultimately, the Big Bang.

2. Radar speed guns. A speed gun (used by police, and in sport) emits microwaves of known frequency toward a moving car. The car reflects them, and the reflected waves come back Doppler-shifted. Because the car acts first as a moving observer (receiving shifted waves) and then as a moving source (re-emitting them), the shift is applied twice, giving $\Delta f = \frac{2 f v}{c}$ The gun measures $\Delta f$ (an approaching car gives a positive, blue-type shift) and computes the car's speed $v$. A microwave photograph or radar of an approaching object shows a blue-shift; a receding one a red-shift.

3. Medical ultrasound (Doppler blood-flow). A probe sends high-frequency ultrasound (typically a few MHz) into the body. Sound reflected from moving red blood cells returns Doppler-shifted; the shift is proportional to the blood's flow speed (again a double shift, $\Delta f = 2 f v \cos\theta / v_{\text{sound}}$, where $\theta$ is the angle between the beam and the flow). Measuring $\Delta f$ lets doctors map how fast — and in which direction — blood is flowing, revealing blockages, narrowed arteries or heart-valve problems, entirely non-invasively.

The common thread: a measured Doppler shift is converted straight into a speed, whether that speed belongs to a galaxy, a car or a red blood cell.

The classic exam scenario is a source passing a stationary observer at constant speed — an ambulance, a racing car, a train. Understanding the graph of observed frequency against time ties every idea in this topic together.

Approaching (before it reaches you). While the source moves toward you along the line of sight, you hear a constant frequency higher than the true frequency: $f' = f\,v/(v - u_s)$. It is roughly constant because, as long as the source is still some way off and heading toward you, almost all of its velocity is directed at you.

The instant it passes (closest approach). For the brief moment the source is level with you, it is moving neither toward nor away along your line of sight — its velocity is at right angles to the line joining you. At that instant the Doppler shift is zero and you hear the true frequency $f$.

Receding (after it has passed). Now the source moves away from you, so you hear a constant lower frequency: $f' = f\,v/(v + u_s)$.

Key features examiners look for:

• The heard frequency is above $f$ throughout the approach and below $f$ throughout the recession — it is never simply "$f$ getting steadily lower".
• The change is sudden, happening as the source passes, not a slow slide from start to finish.
• The true frequency $f$ lies between the two heard values, but not at their simple average — because $v/(v-u_s)$ and $v/(v+u_s)$ are not symmetric about 1.

Numerical illustration. A car sounds a $500\,\text{Hz}$ horn and passes at $u_s = 25\,\text{m s}^{-1}$ ($v = 340\,\text{m s}^{-1}$). Approaching: $f' = 500 \times 340/315 = 540\,\text{Hz}$. Receding: $f' = 500 \times 340/365 = 466\,\text{Hz}$. So the pitch drops from $540\,\text{Hz}$ to $466\,\text{Hz}$ — a fall of $74\,\text{Hz}$ — as it goes by, while the horn itself never changed.