Doppler effect

(PD) Image: John R. Brews
Boat traveling against waves experiences Doppler effect.

The Doppler effect (or Doppler shift, or Doppler's principle), named after Christian Doppler who proposed it in 1842, is the change in frequency of a wave for an observer moving relative to the source of the wave. If the observer moves oppositely to the direction of wave motion, the frequency appears higher, and if the observer moves in the same direction as the waves, the frequency appears lower.

The increase in frequency when moving against the waves can be explained using the figure. Waves blowing ashore with velocity c are spaced a distance λ apart. Therefore, the time between crests for an observer at a fixed location is λ / c, and the frequency with which a crest appears at a fixed location is f_w :

$f_{w}={\frac {1}{\lambda /c}}={\frac {c}{\lambda }}\ .$

The boat going to sea is running at velocity v in the opposite direction to the waves. Consequently the boat moves relative to the crests at a speed v+c. That means a crest is met at intervals of time τ :

$\tau ={\frac {\lambda }{(v+c)}}\ .$

In other words, the frequency with which the boat bumps over a crest f_b is:

$f_{b}={\frac {1}{\tau }}={\frac {1}{\lambda /(v+c)}}=f_{w}{\frac {v+c}{c}}=f_{w}\left(1+{\frac {v}{c}}\right)\ .$

so evidently f_b is a higher frequency than the frequency f_w of the waves themselves. This increase in frequency is the Doppler effect, and it is often expressed as the shift in frequency f_d, the Doppler shift, namely:

$f_{d}=f_{b}-f_{w}=f_{w}{\frac {v}{c}}\ .$

Of course, if the boat runs inshore with the wave, the opposite happens: it takes the boat longer between crests, and the frequency with which the boat bumps over a crest is lower than the actual frequency of the waves.

The same effect appears when the ear approaches an oncoming police siren: the sound of the siren to the ear is pitched higher than the actual pitch of the siren, and when the police pass, so the ear is then traveling away from the siren, and the waves from the siren are traveling in the same direction as the ear, the pitch to the ear drops to become lower than the pitch of the siren.