C.5 Doppler effect

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.

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.

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.

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.

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.