1The idea in one paragraph

When a source of waves and an observer move towards each other, the observer receives a higher frequency than the source emits; when they move apart, a lower one. That change is the Doppler effect. For sound, it matters who is moving: a moving source squeezes its wavefronts together in front of it, shortening the wavelength, while a moving observer simply runs into the wavefronts more often. For light there is no medium, so only the relative speed matters, and when that speed is much less than c the fractional shift is Δf/f = Δλ/λ ≈ v/c. Astronomers read the shift from spectral lines: a redshift means the body is moving away, a blueshift means it is approaching. Radar speed guns and medical ultrasound measure speeds the same way, from waves reflected off a moving target. At HL, you calculate the sound shifts exactly.

2What changes, and what does not

Two statements settle almost every Doppler question.

1. The source sets how many waves are emitted per second, its frequency f.
2. The medium sets how fast the waves travel, the wave speed v. For sound in air this is about 340 m s⁻¹ whatever the source is doing.

So when something moves, v stays the same. What changes is either the wavelength in the medium (if the source moves) or the rate at which the observer meets the waves (if the observer moves). Either way, the observed frequency f′ differs from f.

This is why sound and light differ. Sound moves through air, so a source moving through the air and an observer moving through it are physically different situations, with slightly different formulas at HL. Light needs no medium and its speed is c for every observer (A.5), so only the relative speed matters, and one formula covers both.

3A moving source: the wavefronts bunch up

Figure 1 compares a source at rest with a source moving to the right at half the speed of the waves. Each circle is a wavefront, drawn one period apart.

In panel (a) each wavefront is a circle centred on the source, and the spacing, the wavelength, is the same in every direction.

In panel (b) the source has moved on between emitting one wavefront and the next. Each circle is centred on the point where it was emitted and grows at speed v. So:

• Ahead of the source, towards observer A, the wavefronts are squeezed together. The wavelength is shorter. The waves still travel at v, so more of them reach A each second: A hears a higher frequency.
• Behind the source, at B, the wavefronts are spread out. The wavelength is longer, fewer arrive each second, and B hears a lower frequency.

That is the pitch drop you hear as an ambulance passes. The siren itself vibrates at its usual frequency, and the driver hears it unchanged; what changes is the wavelength in the air, and so the frequency received.

4A moving observer: meeting the waves more often

Now keep the source still and move the observer. Figure 2 shows the observer O moving towards the source through the wavefronts.

The wavefronts are spaced exactly as they would be anyway: the wavelength is unchanged. But the observer is running into them, so relative to the observer the waves go past at v + u_o, where u_o is the observer's speed. More wavefronts pass each second, and the frequency received is higher. Move away from the source and the waves pass at v − u_o, and the frequency is lower.

Moving source: the wavelength in the medium changes. Moving observer: the wavelength is unchanged, but the rate of meeting wavefronts changes. Either way, approaching raises the frequency and receding lowers it.

5The Doppler effect for light

For light and all electromagnetic waves, the guide gives the approximate equation, valid when the relative speed v is much less than the speed of light c:

Here λ (or f) is the value emitted, measured in a laboratory at rest; Δλ (or Δf) is the size of the shift; and v is the relative speed of source and observer along the line joining them.

• If source and observer are moving apart, the received wavelength is longer and the frequency lower. For visible light this moves every line towards the red end of the spectrum: a redshift.
• If they are moving together, the wavelength is shorter: a blueshift.

Figure 3 shows the same set of absorption lines in a laboratory, from a receding body and from an approaching one. The shifts are exaggerated so you can see them.

Each line moves by Δλ = λ × v/c, a fixed fraction of its own wavelength, so the line at 656 nm moves further than the one at 410 nm, and the pattern stays recognisable.

Worked example: a receding galaxy. A hydrogen line with laboratory wavelength 656.3 nm is observed in the light of a galaxy at 662.9 nm. Determine the galaxy's speed and direction relative to Earth.

Divide by the laboratory wavelength (nanometres over nanometres is fine), and state the direction. If v were not much less than c, the approximation would fail and a relativistic formula would be needed; that is beyond this course, but checking v/c is not.

6What spectral lines tell us about stars and galaxies

A star's light carries absorption lines from the elements in its outer layers, at wavelengths fixed by atomic physics and measurable on Earth. When those lines are found shifted, the shift gives the star's speed along the line of sight, towards or away from us. Motion across the sky produces no first-order shift at all.

Three uses follow.

Stars and galaxies moving through space. A single shift gives a single speed along the line of sight. The light from almost all distant galaxies is redshifted, and more distant galaxies show larger redshifts: the evidence that the universe is expanding.

Rotation of an extended body. A star, a planet or a galaxy is not a point. If it rotates, one edge moves towards us and the opposite edge moves away. Figure 4 shows what that does to a single spectral line.

The approaching edge is blueshifted, the receding edge redshifted. If the body is also moving as a whole, both edges share that shift too; so take half the difference between the two edge shifts to get the rotation speed, and the average of the two to get the speed of the whole body.

Binary stars. Two stars orbiting each other alternately approach and recede, so their lines shift back and forth with the orbital period. Even when the two stars cannot be seen separately, the periodic shift reveals the pair and its orbital speed.

7Radar and ultrasound: speed from a reflected wave

The guide asks for the Doppler effect in radar and in medical physics as examples. Both send a wave of known frequency at a moving target and measure the frequency of the reflection.

Figure 5 shows why the shift from a reflection is twice the shift for one-way travel.

1. The target moves towards the emitter. It receives the wave as a moving observer, at a raised frequency.
2. It reflects that raised frequency, and in doing so it acts as a moving source, so the frequency is raised again on the way back.

For target speeds u much less than the wave speed, each step contributes about u/(wave speed), so the total fractional shift is about twice that. For radar, which uses microwaves travelling at c:

Worked example: a speed gun. A police radar gun emits microwaves at 24.0 GHz. The reflection from an approaching car is 4.80 kHz higher. Find the car's speed.

Doppler ultrasound. A probe on the skin sends ultrasound, a few megahertz, into the body. Red blood cells moving in an artery reflect it with a small shift, and the machine turns the shift into a blood speed. A narrowed artery shows up as a region of unusually fast flow, and the same method monitors a baby's heartbeat before birth. The wave speed here is the speed of sound in soft tissue, about 1540 m s⁻¹, and blood moves at well under 1 m s⁻¹, so the small-speed rule still applies with the speed of sound in place of c:

A 2 kHz shift is audible, so many machines let the operator listen to the flow.