Doppler Effect — Apparent Frequency Formula

The Doppler effect formula for a moving sound source — why sirens change pitch — with a worked ambulance example.

f' = f\frac{v \pm v_o}{v \mp v_s}

Observed frequency equals emitted frequency times the ratio of wave speed adjusted for observer and source motion.

What it means

Motion squeezes or stretches waves: approaching motion bunches wavefronts together (higher frequency), receding motion spreads them (lower frequency). The top sign is for motion toward; the bottom for motion away. This shift reveals speeding cars, blood flow in ultrasound, and receding galaxies.

Variables

Symbol	Name	Unit	Meaning
$f'$	Observed frequency	Hz	Frequency the observer hears
$f$	Source frequency	Hz	True emitted frequency
$v$	Wave speed	m/s	Wave speed in the medium
$v_o$	Observer speed	m/s	Observer speed (positive toward source)
$v_s$	Source speed	m/s	Source speed (positive toward observer)

Assumptions

Source and observer move along the line joining them
Speeds are far below the speed of sound
Medium (air) is still

Worked example

Problem

An ambulance siren emits 700 Hz while driving toward you at 30 m/s. What frequency do you hear? (Sound speed 343 m/s.)

Given:

f = 700 Hz (siren)
v = 343 m/s (sound in air)
v_s = 30 m/s toward observer
observer stationary (v_o = 0)

Write the formula (source approaching): $f' = f\frac{v}{v - v_s}$
Substitute: $f' = 700 \times \frac{343}{343 - 30}$
Calculate: $f' = 700 \times \frac{343}{313} = 700 \times 1.0958 \approx 767\ \mathrm{Hz}$

≈ 767 Hz

The pitch rises from 700 Hz to 767 Hz as the ambulance approaches — waves bunch up ahead of the source. After passing, the same formula with +v_s gives a lower pitch.