What the Doppler effect is

The Doppler effect is a wave phenomenon where an observer detects a frequency different from the frequency emitted by a source because the source and observer move relative to each other along the line of wave travel.

That last part matters. If a source moves sideways across your view, the Doppler shift depends on the component of its velocity towards or away from you, not on the full velocity. In simple IB diagrams, we usually take the motion to be directly along the line joining source and observer.

For sound, think of a siren: it sounds higher-pitched as it approaches and lower-pitched as it moves away. The siren hasn’t changed its emitted frequency; the observer is receiving wavefronts at a different rate.

Sound compared with light

A mechanical wave is a wave that transfers energy through a material medium by oscillations of the particles of that medium. Sound is mechanical, so it needs air, water, metal, or some other material. For sound, you can sensibly ask whether the source or the observer is moving relative to the medium.

An electromagnetic wave is a transverse wave made of oscillating electric and magnetic fields that can transfer energy through a vacuum. Light, microwaves, radio waves and X-rays are electromagnetic waves. They do not need a medium, and in a vacuum they travel at the same speed for all inertial observers.

This is why Doppler calculations for sound and light don’t quite match. For sound, wavefronts are laid down in a medium, so a moving source changes the spacing of wavefronts in that medium. For light, there is no medium to be “stationary relative to”, so the shift is described using the relative motion between source and observer. At low relative speeds the light result becomes pleasantly simple, but at speeds that are not small compared with the speed of light a relativistic treatment is needed.

Similarities and differences between light and sound waves

Light and sound both show the familiar wave behaviours from Theme C: reflection, diffraction, interference and Doppler shifts. An echo is reflection of sound; optical reflection is reflection of light. Both can be focused by suitable materials, and both can carry information from a source to an observer.

The physical mechanisms are different. Sound reflection involves changes in pressure and particle motion at a boundary. Light reflection is an electromagnetic interaction with charges in the material. The main Doppler difference is this: sound has a wave speed measured relative to a medium, while light in a vacuum has speed $c$, where $c$ is the speed of electromagnetic radiation in a vacuum ($m\,s^{-1}$), for all inertial observers. That invariant speed is why light Doppler shifts fit naturally with special relativity.

Reading a wavefront diagram

A wavefront is a surface joining points on a wave that are at the same phase of oscillation. In two-dimensional textbook diagrams, spherical wavefronts appear as circles. The gap between neighbouring wavefronts represents the wavelength: closer wavefronts mean a shorter wavelength and, if the wave speed is fixed, a higher frequency.

If the source and observer are both stationary relative to the medium, a point source sends out evenly spaced circular wavefronts centred on the source. The observer receives wavefronts at the same rate as the source emits them, so the observed frequency equals the emitted frequency.

Moving source, stationary observer

When the source moves towards a stationary observer, each new wavefront is emitted from a position slightly closer to the observer than the one before it. In front of the source, the wavefronts get compressed. More wavefronts cross the observer each second, so the observed frequency is higher and the observed wavelength is shorter.

Behind the moving source, the reverse happens. Successive wavefronts are emitted from positions farther away, spreading the wavefronts out. An observer behind the source detects a lower frequency and a longer wavelength.

For the stationary observer, the wave speed in the medium has not changed. The spacing of the wavefronts in the medium has changed.

Stationary source, moving observer

When the source is stationary, the wavefronts remain symmetrical circles centred on the source. If the observer moves towards the source, they meet wavefronts more often than they would while standing still, so the observed frequency increases.

If the observer moves away, the wavefronts take longer to catch up, so the observed frequency decreases. Here, the wavelength in the medium is unchanged; what changes is the rate at which the moving observer crosses wavefronts.

The qualitative distinction is simple: a moving source changes the wavelength pattern in the medium; a moving observer changes the encounter rate with an already-existing pattern.

The low-speed Doppler approximation for light

For light, the syllabus uses this low relative speed approximation:

Use this approximation only when $v$ is much smaller than $c$. In school physics terms, “much smaller” means that the ratio $v/c$ is small enough for higher-order relativistic terms to make no difference to the accuracy needed.

The equation gives the size of the fractional shift. The direction of the shift comes from the physical situation:

• approaching source and observer: observed frequency increases, observed wavelength decreases;
• receding source and observer: observed frequency decreases, observed wavelength increases.

A redshift is a spectral shift to longer wavelength, usually shown towards the red end of the visible spectrum. A blueshift is a spectral shift to shorter wavelength, shown towards the blue end. These names still work when the radiation is not visible light.

What if the relative speed is not small compared with the speed of light?

When the relative speed is not much smaller than $c$, the simple approximation stops being reliable. A full treatment has to include special relativity, since time intervals and measured wavelengths do not transform according to ordinary Newtonian ideas at these speeds.

For IB, the expectation is qualitative: recognise that the simple fractional relation is a low-speed approximation. If speeds become a significant fraction of $c$, a relativistic Doppler expression is needed instead of $\Delta f/f \approx v/c$ or $\Delta\lambda/\lambda \approx v/c$.

Calculating speed from light Doppler shifts

The Doppler effect for light is a powerful measuring tool because spectroscopy can measure very small wavelength changes accurately. If a laboratory line has a known wavelength, and the same line is observed from a moving astronomical object, the relative speed along the line of sight can be estimated using

$$v\approx \frac{\Delta \lambda}{\lambda}c$$

with the same symbols as defined above.

This shows how physics often uses indirect measurement. We don't need to chase a star or galaxy; we compare the observed spectrum with a laboratory reference and infer its line-of-sight speed.

Why spectral lines are useful

A spectral line is a narrow range of wavelength in an emission or absorption spectrum, produced when atoms, ions or molecules move between quantised energy levels. Different elements have different spacings between their energy levels, so their line patterns work like fingerprints.

An emission spectrum is a set of bright spectral lines produced when excited atoms, ions or molecules emit photons at specific wavelengths. An absorption spectrum is a set of dark spectral lines produced when cooler gas absorbs selected wavelengths from a continuous spectrum.

In astronomy, the pattern of spectral lines from a star or galaxy is compared with the same pattern measured in a laboratory. If the whole pattern has shifted, the source is moving relative to us along the line of sight. A shift to longer wavelengths shows recession; a shift to shorter wavelengths shows approach.

Stars, galaxies and astronomical distances

For nearby stars and rotating bodies, Doppler shifts can show speeds of approach, recession, or rotation. For distant galaxies, redshift is one of the key observations behind the large-scale expansion of the universe. In the low-speed approximation, a measured fractional wavelength shift gives a recessional speed using $v\approx (\Delta\lambda/\lambda)c$.

If a relation between recessional speed and distance has been established, such as Hubble’s law for sufficiently distant galaxies, redshift can also be used to estimate distance. The chain of reasoning is: measure spectral shift $\to$ infer recessional speed $\to$ use the speed–distance relation to infer distance. The observation is indirect, but it’s powerful.

Not every redshift is just a Doppler shift caused by motion through space. Gravitational fields can redshift light, and the expansion of spacetime produces cosmological redshift. For this topic, keep the distinction clear: Doppler redshift is about relative motion; other redshifts have different physical origins.

Rotational speed of extended bodies

An extended body is an object whose different parts can have different velocities relative to an observer. A rotating star is a good example. One limb of the star may move towards Earth while the opposite limb moves away.

Light from the approaching limb is blueshifted; light from the receding limb is redshifted. Radiation from a point whose surface velocity is perpendicular to the line of sight has no line-of-sight Doppler shift. Across the disk, this produces broadening, or splitting in careful measurements, of spectral lines.

Once the tangential speed at the edge is known, the rotational period can be estimated from

Moving source, stationary observer

For sound and other mechanical waves, the medium matters.

The emitted frequency $f$ has already been defined.

For a moving source and stationary observer,

Use the minus sign when the source moves towards the observer. The wavefronts ahead of the source are closer together, so the observed frequency is higher. Use the plus sign when the source moves away; the wavefronts are farther apart, so the observed frequency is lower.

A useful way to remember the sign is to avoid memorising “plus” or “minus” first. Ask yourself: should $f'$ be larger or smaller than $f$? Then choose the denominator that gives that result.

The observed wavelength for the stationary observer can then be found using

The wave speed in the medium has not changed here. The motion of the source has changed the spacing between wavefronts.

Moving observer, stationary source

For a stationary source and moving observer,

Use the plus sign when the observer moves towards the source, because the observer meets wavefronts more often. Use the minus sign when the observer moves away, because the wavefronts catch up less often.

In this case, the observer’s motion does not change the wavelength in the medium. The observed frequency changes because the observer crosses the wavefront pattern at a different rate.

IB problems for this statement will not combine a moving source and a moving observer in the same calculation. They can, however, ask you to find the speed of the source or the observer by rearranging these equations.

Comparison of the two mechanical Doppler cases in this section.

Medical ultrasound

Ultrasound is sound with frequency above the upper limit of normal human hearing, about 20 kHz. In medical Doppler ultrasound, a transducer sends ultrasound into the body and receives echoes from moving blood cells.

There is a double Doppler shift. First, the moving blood cell acts as a moving observer of the transmitted ultrasound. After reflection, the same moving blood cell acts as a moving source, sending sound back to the transducer. For blood speeds much smaller than the speed of sound in tissue, the fractional shift is approximately

The cosine appears because the Doppler shift measures only the component of velocity along the beam.

The medical value is that blood-flow speed can be estimated without inserting a probe into the vessel. This makes the method non-invasive and reduces risks such as infection or damage to the vessel.

Radar and reflected waves

Radar is a detection method that uses reflected radio or microwave electromagnetic waves to locate objects or measure their motion. Doppler radar sends a wave to a moving object and compares the reflected frequency with the transmitted frequency.

For a reflected signal from a moving object at speeds much smaller than $c$, the shift is doubled in the same spirit as ultrasound: the object first receives a shifted wave and then behaves like a moving source of the reflected wave. For motion directly along the radar beam,

$$\Delta f\approx \frac{2v}{c}f$$

using the previously defined $\Delta f$, $v$, $c$ and $f$. This is the basis of vehicle speed measurements, weather radar measurements of rain motion, and flow measurements in rivers, oceans and industrial systems.

Do not mix the mechanical-wave equations with light or radar without thinking. Sound has a speed relative to a medium, while radar uses electromagnetic waves and is usually treated with the low-speed light approximation in this course.