The source keeps its beat. Why does the pitch change?
Why does an ambulance sound different as it approaches and moves away? The source keeps the same vibration rhythm. Count how many wave crests reach the listener each second: their spacing and the listener’s motion change the arriving rhythm.
Try two small comparisons
- Move the source toward the receiver. Are the wavefronts closer together?
- Move it away and compare the arrival rhythm.
Solve the journey, then draw the rhythm
Look closer: observations and controls
An approaching siren can sound higher without changing how quickly it vibrates. Follow one crest: it starts at the source’s position when it was emitted and spreads through the air at the speed of sound. By the next emission, the source has moved.
Ahead of the source, successive crests are closer together. Behind it, they are farther apart. A receiver in front waits less time for the next crest; a receiver behind waits longer. Frequency counts arriving cycles. It does not measure loudness.
The upper ruler, S, marks equally spaced emissions. The lower ruler, O, marks calculated arrival times. The vertical line is now; the pale region to its right is the future. The two emphasized circles share their identities with the arrivals immediately before and after now.
Each chapter starts an independent constant-velocity comparison. The source has been emitting since before the displayed time origin; earlier crests follow the same propagation geometry.
Watch the film, then enter exploration. Arrow keys adjust each slider; Home and End select its endpoints. Scrubbing time reconstructs emission positions, expanding fronts and arrival records together.
- Walk into the crests: keep the source stationary and move the receiver left. Wavelength stays fixed, but arrivals become more frequent.
- Move together: give both bodies the same rightward velocity. The forward wavelength is shorter, yet the receiver gets exactly the original frequency.
- Lateral distance: change the distance from the source’s path. Watch the gradual shift during a pass. The gold line points back to emission point P; the dashed line points to the source’s current position.
- Source speed: move closer to the speed of sound and watch the fronts crowd ahead. Exploration stops at 90% of the sound speed. This is not a shock-wave demonstration.
Calculations, sources and scope
Use the rest frame of the air. Sound speed is c, the emission period is T₀, and f₀ = 1/T₀. Crest k leaves at tₑ = kT₀. Its center is S(tₑ); its radius at time t is c(t−tₑ). Later motion of the source does not move that center.
For a receiver following O(t), an arrival must satisfy:
|O(t) − S(tₑ)| = c(t − tₑ), with t ≥ tₑ.
Straight, constant-velocity trajectories turn this into a quadratic equation. The displayed frequency ratio is T₀/Δt, where Δt is the interval between the two actual arrivals bracketing now. It is an average over that interval.
For a source moving directly toward a stationary receiver, the forward wavelength is (c−vₛ)T₀, giving a received frequency of f₀c/(c−vₛ). Recession replaces the minus sign with a plus. Receiver motion works differently: it changes how fast the receiver crosses existing crests, without changing their wavelength in the air.
Away from a straight line, use the direction the sound actually traveled. If n points from emission to reception, the instantaneous frequency ratio is (c−n·vₒ)/(c−n·vₛ). This local derivative differs from the finite-interval average on the display. At the moment the source reaches its nearest position, the receiver is still getting sound emitted earlier. A source that looks sideways now does not automatically give zero frequency shift now.
The model assumes uniform, still air and constant velocities. Circles are planar sections of spherical phase fronts, slowed down and measured in source periods; they do not depict air parcels, pressure amplitude or loudness. Reflection, absorption, wind and dispersion are omitted. Exactly sonic and supersonic motion are excluded. The relativistic Doppler effect for light requires a different model.
Technical basis: OpenStax’s acoustic Doppler derivation, UNSW Physclips experiments with moving sources and receivers, and OpenStax’s discussion of shock waves.