Doppler Shift in Sound
When an ambulance rushes past you with its siren wailing, you notice something curious: the pitch of the siren drops sharply as it passes. The siren itself never changes its frequency — yet your ears tell a different story. This phenomenon is the Doppler effect, named after Austrian physicist Christian Doppler. It describes how the observed frequency of a wave changes whenever there is relative motion between the source of the wave and the observer. It applies to all waves, but is most vividly experienced with sound in everyday life. The key insight is that motion compresses or stretches the wave pattern in space. When a source moves towards you, it is effectively chasing its own wavefronts, squashing them closer together. A shorter wavelength means more wave crests arrive per second — so you hear a higher pitch. When the source moves away, wavefronts are stretched apart, fewer crests arrive per second, and the pitch drops. Crucially, it is the relative motion that matters: the same shift occurs if the observer moves and the source stays still. The Doppler shift in sound is not merely a curiosity — it underpins technologies such as Doppler radar for weather forecasting, speed cameras used by police, and medical ultrasound imaging of blood flow. Understanding the mathematics behind it allows physicists and engineers to extract precise velocity information from frequency measurements alone.
- Wavefronts and wavelength compression
- Moving source towards a stationary observer
- Moving source away from a stationary observer
- Moving observer
- General Doppler formula
Wavefronts and wavelength compression
Sound travels as a pressure wave through a medium at speed v (approximately 343 m/s in air at 20 °C). A stationary source emitting frequency f₀ produces spherical wavefronts equally spaced by λ = v/f₀ in all directions. When the source moves, it partially 'catches up' with the wavefronts it has already emitted in the forward direction, bunching them closer together and stretching them out behind it.
Moving source towards a stationary observer
If the source approaches at speed vₛ, the effective wavelength in front of the source becomes λ' = (v − vₛ)/f₀. Since the observer receives crests at speed v but with compressed wavelength, the observed frequency is f' = v/λ' = f₀ × v/(v − vₛ). The denominator is smaller than v, so f' > f₀ — you hear a higher pitch. Think of a speedboat churning up a bow wave: the waves in front are packed tightly together.
Moving source away from a stationary observer
When the source recedes at speed vₛ, the wavefronts in the backward direction are stretched: λ' = (v + vₛ)/f₀, giving f' = f₀ × v/(v + vₛ). Now f' < f₀ — a lower pitch. This is the characteristic drop you hear after the ambulance passes you. The faster the source, the more dramatic the drop.
Moving observer
If instead the observer moves at speed v₀ towards a stationary source, they intercept wavefronts more frequently: f' = f₀ × (v + v₀)/v. Moving away gives f' = f₀ × (v − v₀)/v. Notice that numerator changes (not denominator) — a subtle but important distinction from the moving-source case, though the qualitative effect is the same.
General Doppler formula
Both effects are combined in the general formula. The sign convention is: use '+' in the numerator when the observer moves towards the source, '−' when moving away; use '−' in the denominator when the source moves towards the observer, '+' when moving away. A helpful mnemonic: motions that bring source and observer together raise the frequency, motions that separate them lower it.
The Key Equation
- f' — Observed frequency (Hz): The frequency detected by the observer, which differs from the source frequency due to relative motion.
- f_0 — Source frequency (Hz): The actual frequency emitted by the source, which remains constant regardless of motion.
- v — Speed of sound in medium (m/s): The speed at which sound propagates through the medium (≈ 343 m/s in air at 20 °C).
- v_o — Speed of observer (m/s): Speed of the observer relative to the medium. Use '+' when moving towards the source, '−' when moving away.
- v_s — Speed of source (m/s): Speed of the source relative to the medium. Use '−' in denominator when source moves towards observer, '+' when moving away.
Worked Example
Problem. A police car travels at 30 m/s towards a stationary pedestrian, with its siren emitting sound at 800 Hz. The speed of sound in air is 340 m/s. What frequency does the pedestrian hear? After the car passes and is moving away at the same speed, what frequency does the pedestrian hear?
Answer. The pedestrian hears approximately 877 Hz as the car approaches and approximately 735 Hz as it recedes — a drop of about 142 Hz as the car passes.
Origins · 1 minChristian Doppler and how this was discovered
Christian Doppler first described the effect bearing his name in his 1842 treatise 'Über das farbige Licht der Doppelsterne' ('On the coloured light of binary stars'). He proposed that the observed frequency of waves depends on the relative motion of source and observer — originally applying the idea to explain the colour differences of binary stars. Although his stellar application was later shown to be incorrect (stars are too slow for visible colour shifts), the underlying principle was correct and was soon confirmed for sound experimentally. Doppler's effect was first tested for sound in 1845 by the Dutch scientist Buys Ballot, who hired a group of trumpet players on a moving train to play a sustained note while trained musicians on the platform compared the pitch — a gloriously analogue experiment!
Related Concepts
- Wave Speed, Frequency and Wavelength — The Doppler effect is fundamentally about changes in observed wavelength and frequency; understanding the relationship v = fλ is essential groundwork.
- Properties of Waves — Grasping how wavefronts propagate and what wavelength represents physically is necessary to visualise why motion compresses or stretches wave patterns.
- Sound Waves and the Speed of Sound — The Doppler equation for sound depends directly on the speed of sound in the medium; knowing how this speed varies with temperature and medium is important for applying the formula correctly.
- Resonance and Standing Waves — Both topics involve frequency analysis of sound; understanding resonance deepens appreciation of how sources produce and sustain particular frequencies.