The Doppler Effect
When a sound source moves toward you, it keeps catching up a little to its own sound waves, squeezing them closer together in front — so you hear a higher pitch. When it moves away, the waves stretch out behind it, so the pitch drops. This is why a siren sounds higher as it approaches and lower as it passes and speeds away.
The formula
f_obs = f · c / (c − v_s)
- f_obs — observed frequency (Hz): the pitch you actually hear
- f — source frequency (Hz): the pitch the source emits when standing still
- c — speed of sound (m/s): how fast sound waves travel through the air
- v_s — source speed (m/s): how fast the source moves toward you
Worked example
An ambulance siren emits a steady 320 hertz tone while driving toward you at 20 meters per second. Sound travels at 340 meters per second. What frequency do you hear?
- f = 320 Hz
- c = 340 m/s
- v_s = 20 m/s
- f_obs = f · c / (c − v_s)
- f_obs = 320 Hz · 340 / (340 − 20) = 320 Hz · 340 / 320
f_obs = 340 Hz
Test yourself
A quieter siren emits 160 hertz while approaching you at the same 20 meters per second, with sound still traveling at 340 meters per second. What frequency do you hear?
- Correct answer: 170 Hz
- 150 Hz
- 180 Hz
Right! f_obs = 160 Hz · 340 / (340 − 20) = 160 Hz · 340 / 320 = 170 Hz.
If that same ambulance instead drove away from you at 20 meters per second, would the pitch you hear be higher, lower, or the same as the siren's true 320 hertz?
- Higher than 320 Hz
- Correct answer: Lower than 320 Hz
- The same, 320 Hz
Correct — a receding source stretches the sound waves out, so you hear a pitch below 320 hertz.
Where you see this
The passing siren is the textbook case: pitch high while the ambulance approaches, dropping abruptly as it passes and recedes. Astronomers read the same effect in light — a distant galaxy's spectrum shifted toward the red means its wavelengths are stretched out behind it as it recedes, the same stretching that lowers a siren's pitch.
Common mistakes
The deepest mistake is thinking the siren itself changes pitch — it keeps emitting a steady 320 hertz; only the frequency reaching your ear changes, because the moving source catches up to its own wavefronts and squeezes them. Watch the formula's signs too: f_obs = f · c / (c − v_s) has the source speed in the denominator, so an approaching source (v_s = 20 m/s toward you, sound at 340 m/s) raises 320 Hz to 340 Hz — a receding source swaps the sign and drops it.
How it connects
This builds directly on the wave equation: the wave still moves at the speed the medium allows (340 m/s in air, set by the air and not the siren), and all the source's motion does is crowd or stretch the wavelengths it emits. Beats, next, are the other classic way a heard frequency differs from the emitted ones — two steady sources interfering at your ear.