C.5 Doppler effect

The Doppler effect describes how the observed frequency and wavelength of a wave depend on relative motion between the source and the observer.

Key ideas

Relative motion changes the spacing of wavefronts reaching an observer.
Approaching motion increases observed frequency; receding motion decreases it.
For light, frequency changes appear as blue shifts and red shifts.

Important formulas

v = f\lambda

Relationship between wave speed, frequency and wavelength.

f' = f\dfrac{v}{v \mp v_s}

Observed frequency for a moving source with a stationary observer (minus for approaching, plus for receding).

f' = f\dfrac{v \pm v_o}{v}

Observed frequency for a moving observer with a stationary source (plus when moving towards).

\dfrac{\Delta f}{f} \approx \dfrac{\Delta \lambda}{\lambda} \approx \dfrac{v_r}{c}

Approximate fractional Doppler shift for light when the relative speed is much less than the speed of light.

Practice problems

Nature of the Doppler effect

Explain qualitatively what is meant by the Doppler effect for sound and for electromagnetic waves. Describe what a stationary observer detects as a source approaches and then recedes.

Wavefront picture for a moving source

Describe the wavefront pattern produced by a stationary source and by a source moving at constant speed. Relate wavefront spacing to observed frequency.

Moving source (sound)

A siren emits sound at 600 Hz. The speed of sound is 340\ \mathrm{m\,s^{-1}}. Calculate the frequency heard by a stationary observer when the siren moves towards and then away from the observer at 20\ \mathrm{m\,s^{-1}}.

Moving observer (sound)

A stationary siren emits sound at 750 Hz. A cyclist rides towards the siren at 12\ \mathrm{m\,s^{-1}}. Calculate the frequency heard by the cyclist.

Small Doppler shift for light

A hydrogen spectral line is observed at 662 nm instead of 656 nm. Assuming the speed involved is much less than the speed of light, estimate the recession speed of the source.

Astronomical use of Doppler shifts

Explain how Doppler shifts in spectral lines are used to determine the motion of stars and galaxies.

No Doppler effect with no relative motion

A sound source and an observer move together at the same speed in the same direction through still air. Explain why no Doppler shift is observed.

Clarity tip: Always decide whether the source or observer is moving, choose the correct formula, and check that approaching motion leads to a higher observed frequency.