C.5 Doppler effectIB Physics HL: Revision notes

The Doppler effect is the change in observed frequency and wavelength caused by relative motion between source and observer. Approaching: higher frequency; receding: lower frequency. For sound and other mechanical waves, the wave speed is fixed by the medium, so a moving source and a moving observer are different situations. Light needs no medium and travels at c for every observer, so only the relative velocity matters.

Moving source: wavefronts are circles emitted from successive positions, so they are crowded in front (shorter λ) and spread out behind (longer λ). Wave speed in the medium is unchanged.

Moving observer: wavefronts stay concentric and evenly spaced, so λ is unchanged; the observer meets them at a different relative speed, so the number arriving per second changes.

For v ≪ c: $\frac{\Delta f}{f} = \frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}$. Redshift (longer λ) means receding; blueshift (shorter λ) means approaching. Every spectral line of a star or galaxy shifts by the same fraction, revealing the line-of-sight velocity. Distant galaxies are redshifted, so they are receding; binary stars show periodic shifts; rotating bodies are blueshifted on one side and redshifted on the other.

For a source moving at speed us relative to the medium:

$f' = f\frac{v}{v \pm u_s}$

Use v − us (smaller denominator, higher frequency) when the source approaches and v + us when it recedes. The wavelength in front is λ′ = (v − us)/f.

For an observer moving at speed uo relative to the medium:

$f' = f\frac{v \pm u_o}{v}$

Use v + uo when the observer moves towards the source and v − uo when moving away. The wavelength is unchanged; the relative speed of the waves changes.

When a wave reflects from a moving object, apply the Doppler effect twice: first the reflector is an observer receiving f′, then it is a source re-emitting f′. For a bat or car moving at u towards a stationary wall, the echo frequency is $f'' = f\frac{v + u}{v - u}$. When u ≪ v, both sound equations reduce to Δf/f ≈ u/v, the same form as for light.