How the Doppler Effect Works: The Formula, the Sign Convention, and Why Moving the Source Is Not the Same as Moving the Listener

One formula, one sign convention, stated before any numbers

Write the sound Doppler effect once, as f' = f(v + v_o)/(v − v_s). Here f is the frequency the source emits, f' the frequency the observer hears, and v the speed of sound in the medium — 343 m/s in air at 20 °C. The two remaining symbols are signed speeds measured relative to the medium, not relative to each other: v_o is the observer's speed component toward the source, positive when approaching and negative when moving away; v_s is the source's speed component toward the observer, again positive when approaching. Substituting a positive v_s makes the denominator smaller and the pitch higher, which is the physical check you should run before trusting any answer.

Run the arithmetic on a concert-pitch A, f = 440 Hz. A source approaching at 30 m/s gives f' = 440 × (343 + 0)/(343 − 30) = 440 × 343/313 = 482.17 Hz. The same source receding is v_s = −30, so f' = 440 × 343/373 = 404.61 Hz. Across the pass-by the pitch falls by 77.56 Hz, a frequency ratio of 1.1917, which on a musical scale is 304 cents — almost exactly three semitones. That is the interval you actually hear when a car passes at 108 km/h, and it is worth memorising as a plausibility check: if a calculator hands you a shift of half a semitone or of an octave for a road vehicle, something has gone wrong with the units.

Moving the source and moving the listener give different numbers

The single combined formula hides a real asymmetry. When the source moves, it chases its own waves and genuinely shortens the wavelength in the air: a 440 Hz source has λ = 343/440 = 0.7795 m at rest, and approaching at 30 m/s it emits λ' = (343 − 30)/440 = 0.7114 m. The listener stands still, meets those shorter waves at the unchanged 343 m/s and hears 343/0.7114 = 482.17 Hz. When the listener moves instead, the wavelength in the air is untouched at 0.7795 m; the listener simply runs into the crests at a closing speed of 343 + 30 = 373 m/s and hears 373/0.7795 = 478.48 Hz. Two different mechanisms, two different numbers, from the same 30 m/s.

The gap grows with speed. At 10 m/s it is 0.39 Hz, hardly audible; at 30 m/s it is 3.69 Hz; at 50 m/s it is 10.95 Hz, 515.09 against 504.14, which is 37 cents — around a third of a semitone and clearly audible to a trained ear. One subtlety worth stating carefully, because it is easy to overclaim: the ratio between the approaching and receding pitches is the same in both cases, (v + speed)/(v − speed), so a pass-by spans the same musical interval whichever party is moving. What differs is where that interval sits relative to the emitted note. Say which of the two is in motion, or the answer is only approximately right.

Where the sound formula stops: Mach 1, the weather, and light

Push v_s toward v and the denominator collapses. A 440 Hz source approaching at 300 m/s is heard at 3 510 Hz; at 340 m/s, 50 307 Hz; at exactly 343 m/s the expression divides by zero. Physically the source is now keeping pace with its own wavefronts, they pile up into a single shock, and the smooth Doppler picture no longer describes anything. Beyond that speed the formula returns negative frequencies, which is the algebra's way of telling you it has left its domain. A second, quieter dependence is the medium itself: v = 331.3 × √(1 + T/273.15) for dry air, so the speed of sound is 325.18 m/s at −10 °C and 351.89 m/s at 35 °C. The same 30 m/s approach then yields 484.72 Hz on a cold day and 481.01 Hz on a hot one. A steady wind matters too, since v_o and v_s are defined relative to the air, not to the ground.

Light is a different problem, not the same formula with a bigger v. Sound needs a medium, which is what lets the source case and the observer case be told apart; light has none, so only the relative velocity between source and observer exists and the two cases must give identical answers. The relativistic longitudinal Doppler formula does exactly that: f' = f√((1 + β)/(1 − β)) for approach, with β = v/c, and the reciprocal for recession. At β = 0.1 the exact factor is 1.105542, whereas the sound-shaped shortcuts give 1/(1 − β) = 1.111111 for a moving source and 1 + β = 1.100000 for a moving observer. Neither is right, and they disagree with each other — which is the clearest possible sign that the acoustic equation does not transfer. At β = 0.5 an approaching source is blue-shifted by a factor of 1.732 and a receding one red-shifted to 0.577.

Frequently asked questions

Why does a siren drop in pitch as the ambulance passes?

Because the sign of v_s flips at the moment of passing. Approaching, the ambulance is chasing its own sound and the wavelength reaching you is compressed, so the pitch is raised; once it is past, it is running away from the sound it emits, the wavelength is stretched, and the pitch drops. A 700 Hz siren on a vehicle at 25 m/s is heard at 755.0 Hz on approach and 652.4 Hz afterwards, a fall of 102.6 Hz. The siren itself never changes; nothing about the sound leaving the vehicle is different before and after. The drop is not gradual either — it happens fastest as the vehicle passes closest, because that is when the component of its velocity along the line to your ear swings from positive to negative.

Which speed of sound should I put into the formula?

The speed in the medium the wave is actually travelling through, at its actual temperature. For dry air, v = 331.3 × √(1 + T/273.15) m/s, which gives 331.3 m/s at 0 °C, 343.2 m/s at 20 °C and 351.9 m/s at 35 °C — the 343 m/s that textbooks use is a room-temperature value, not a constant. The effect on the answer is modest but real: a 440 Hz source approaching at 30 m/s is heard at 484.72 Hz at −10 °C and 481.01 Hz at 35 °C. Pressure and altitude barely matter for air, but the medium does: sound travels near 1 480 m/s in water, so an underwater Doppler shift at the same source speed is about 4.6 times smaller.

Can I use the same Doppler formula for light?

No. The acoustic formula distinguishes a moving source from a moving observer because sound propagates through a medium that defines a rest frame; light has no medium, so only the relative velocity is physically meaningful and the two cases must agree. The correct longitudinal expression is f' = f√((1 + β)/(1 − β)) for approach and f√((1 − β)/(1 + β)) for recession, with β = v/c. At β = 0.1 that gives a factor of 1.105542, while the acoustic shortcuts give 1.111111 and 1.100000 depending on which body you imagine moving — different from each other and both wrong. At everyday speeds the three agree to a few parts in a million, which is why radar guns can use the simple form; at any appreciable fraction of c they do not.