C.5 Doppler effect

Core idea

The Doppler effect is the observed change in frequency, and therefore usually wavelength, caused by relative motion between a wave source and an observer. When the source and observer approach, wavefronts arrive more frequently and the observed frequency is higher. When they separate, the observed frequency is lower.

Apply it to sound

For sound, the wave travels through a medium. A moving source changes the spacing of emitted wavefronts in the medium; a moving observer changes how quickly the observer meets the wavefronts. In either case, motion toward one another gives a higher observed frequency and motion apart gives a lower one. The source frequency itself has not changed merely because the observer hears a different frequency.

Apply it to light

The Doppler effect also occurs for electromagnetic waves. A source moving toward an observer produces a shorter observed wavelength and a higher frequency (blueshift); moving away produces a longer wavelength and lower frequency (redshift). Unlike sound, the measured speed of light remains $c$; the observed change is in frequency and wavelength.

Use the shift as a measurement

Medical Doppler ultrasound uses a frequency shift in reflected sound to infer blood-flow speed. Radar uses a shift in reflected microwaves to infer the radial speed of a vehicle, aircraft or storm. In both cases the detected shift is tied to motion toward or away from the receiver, not to a change in the emitted frequency at the source.

Check the boundary

Do not explain light Doppler shift by adding the source speed to $c$. In this course the low-relative-speed approximation is used for the change in frequency or wavelength; the light speed remains $c$.

Start with the reference case

A stationary point source emits circular wavefronts whose centres remain at the source and whose spacing is the emitted wavelength. Use this as the comparison before adding motion. The wavefronts travel through the medium at the wave speed.

Move the source

If the source moves toward a stationary observer, successive wavefronts are emitted from progressively advanced positions, so the wavefronts are compressed in front of the source and spread behind it. The observer in front receives a shorter wavelength and higher frequency; behind it, the wavelength is longer and the frequency lower.

Move the observer

If the source is stationary and the observer moves toward it, the wavefront spacing in the medium is unchanged. The observer meets wavefronts more often, so the observed frequency increases. Moving away gives a lower observed frequency; it does not change the wavelength in the medium.

Common trap

For a moving source, shift the centres of successive wavefronts; for a moving observer, keep the wavefronts equally spaced and change only the rate at which the observer encounters them. Do not combine moving-source and moving-observer cases unless the question explicitly asks for both.

Use the low-speed model

This approximation applies when the relative source–observer speed $v$ is much smaller than the speed of light $c$. Here $f$ and $\lambda$ are emitted values, while $\Delta f$ and $\Delta\lambda$ are the magnitudes of the observed shifts.

Keep the direction

Motion away produces a redshift: $\Delta\lambda>0$ and the observed frequency decreases. Motion toward produces a blueshift: wavelength decreases and frequency increases. If a question asks for speed, use the absolute shift; if it asks for direction, use the sign of the wavelength or frequency change.

Worked example — spectral line

A line emitted at $4.86\times10^{-7}\,\text{m}$ is observed from a receding galaxy at $5.21\times10^{-7}\,\text{m}$. Then $\Delta\lambda=3.5\times10^{-8}\,\text{m}$. Substitution gives $v=(3.00\times10^8)(3.5\times10^{-8}/4.86\times10^{-7})=2.16\times10^7\,\text{m s}^{-1}$. The longer observed wavelength identifies recession.

Common trap

Do not put $\Delta\lambda$ in the denominator, and do not add $v$ to $c$. The approximation changes the observed frequency or wavelength, not the invariant speed of light.

Use the line pattern as a fingerprint

Elements produce characteristic sets of spectral lines. Compare the same line pattern measured in a laboratory with the pattern observed from a star or galaxy. If every characteristic line is displaced by the same fractional amount while the pattern remains recognisable, the displacement is evidence of relative motion along the line of sight.

Interpret redshift and blueshift

A shift toward longer wavelengths and lower frequencies is a redshift, indicating recession along the line of sight. A shift toward shorter wavelengths and higher frequencies is a blueshift, indicating approach. The line pattern itself identifies the element; the displacement carries the motion information.

State exactly what the shift reveals

A Doppler spectral shift gives the component of relative motion along the observer’s line of sight. Redshift indicates that separation is increasing; blueshift indicates that it is decreasing. The shift alone does not measure motion across the line of sight, and it does not imply that the emitting element or the speed of light has changed.

Common trap

Do not say that a redshift means the element has changed. The spectral identity remains in the line pattern; the wavelengths have shifted because of relative motion.

Set the scope

These equations apply to sound or other mechanical waves travelling at speed $v$ through a medium. Assume the source and observer move along the straight line joining them, and treat one moving object at a time as required by the syllabus. They do not apply to electromagnetic waves.

Choose the sign from the motion

For a moving source, subtract $u_s$ when it approaches and add it when it recedes. For a moving observer, add $u_o$ when it approaches and subtract it when it recedes. This makes approach give $f'>f$ and separation give $f'<f$.

Worked example — moving source

A $480\,\text{Hz}$ sound source approaches a stationary observer at $50.0\,\text{m s}^{-1}$ while sound travels at $350\,\text{m s}^{-1}$. Use the source-toward sign: $f'=480[350/(350-50.0)]=560\,\text{Hz}$. The result is above $480\,\text{Hz}$, matching the approach check.

Keep the syllabus boundary

Use one moving object at a time and motion along the line joining source and observer. Keep every speed in the same units. These mechanical-wave equations do not apply to electromagnetic waves; reflected medical or radar signals may involve a shift on both outward and return paths.