Q.A police car travels towards a stationary observer at a speed of 15 m/s. The siren on the car emits a sound of frequency 250 Hz. Calculate the recorded frequency. The speed of sound is 340 m/s.

The Doppler effect (Christian Doppler, 1842) is the apparent change in a wave's frequency, perceived by a listener, whenever there is relative motion between the wave's source and the listener -- it applies to both sound and light, though this chapter develops it for sound. Three cases are distinguished, all built from tracking how the spacing between successively-emitted wave crests changes due to relative motion.

Source moving, listener stationary: $n=n_0\dfrac{v}{v\mp v_s}$ (minus for approach, plus for recede) -- an approaching source compresses the wavelength ahead of it, raising the apparent frequency. Listener moving, source stationary: $n=n_0\dfrac{v\pm v_L}{v}$ (plus for approach, minus for recede) -- here the wavelength in the medium is unchanged, but the listener's own changing rate of encountering crests shifts the apparent frequency. Both moving: the general formula $n=n_0\dfrac{v\pm v_L}{v\mp v_s}$ applies, with each sign decided independently (upper for that party's own relative approach, lower for recede, always judging the other party as if momentarily at rest …