How to use it
The dot labelled “source” is a sound source moving from left to right. Each pressure crest it sends out spreads as a circle at the speed of sound, centred on the spot where the source was when the crest left. Press Sound to hear what the observer, the green ear marker, receives at this instant. Try a preset, then drag the ear somewhere else.
Controls
- Scene: press or drag to place the ear; with the scene focused, arrow keys move it by 5 m.
- Buttons: Play/Pause (Space), Step (S), Reset (R) and Sound on/off.
- Time scale: from real time down to 1/100. The pitch stays true in slow motion; only the timing is stretched.
- Sliders: source speed (also shown as a Mach number), observer speed (positive is to the right), emitted frequency and the ear’s distance from the path.
- Medium and sound: air, carbon dioxide, helium or water; pure tone, two-tone siren or buzzy engine.
How it works
The Doppler effect is the change in received frequency when a wave source, an observer, or both move through the medium that carries the wave. A moving source partly catches up with the crests it sends forward, so they are bunched together ahead of it and spread out behind. Every crest still travels at the speed of sound c, so more of them reach a listener in front each second and the pitch is higher. Ahead of the source the wavelength is (c − vs)/f, behind it (c + vs)/f.
Here vs,r and vo,r are the parts of the source and observer velocities along the line from the emission point to the observer, counted positive in the direction the sound travels. A source heading straight at a resting listener gives f′ = fc/(c − vs), straight away fc/(c + vs). A 700 Hz siren at 30 m/s in air (c = 343 m/s) is heard at 700 × 343/313 ≈ 767 Hz approaching and 700 × 343/373 ≈ 644 Hz receding, a drop of about three semitones.
The simulator finds the emission time te from |robs(t) − rsrc(te)| = c(t − te), a quadratic equation for straight-line motion, then applies the formula.
Unless the source runs right over you, the pitch does not jump. Only the velocity component pointing at you counts, and it shrinks gradually to zero at closest approach. The sound emitted at closest approach arrives unshifted, a moment after the source has gone by. The nearer the path, the faster the drop.
Sound needs a medium, so it matters who moves. A listener driving at 30 m/s towards a resting 700 Hz source hears 700 × 373/343 ≈ 761 Hz, not 767 Hz.
At Mach 1 the source keeps pace with its own crests, which pile up at its nose. Faster still, it outruns them and the circles pile up along a shock front, the Mach cone, whose half-angle θ obeys sin θ = 1/M. Outside the cone there is silence; when the cone sweeps over a listener the pressure jumps: the sonic boom. The supersonic audio is simplified. After the boom a real listener receives sound from two earlier source positions at once, one of them reversed in time; this page plays only the later-emitted one, replaces the boom by a short thump of filtered noise and caps the pitch at 15 kHz. The picture shows only one crest in every N, and the model ignores wind, echoes and absorption.
Light shows the same effect: light from a star moving away from us is shifted towards red, from an approaching star towards blue. But light needs no medium, so only the relative velocity matters and the relativistic formula must be used. Radar and medical ultrasound measure speeds from Doppler shifts too. Christian Doppler proposed the effect in 1842; in 1845 Buys Ballot confirmed it for sound with musicians playing on a moving train.
Things to try
- Run “Passing siren”, then move the ear closer to the path: the S-shaped curve in the graph gets steeper.
- Set the distance to 0 m: the pitch now falls from the upper limit to the lower one in a single step.
- Compare a 30 m/s source and a resting ear with a resting source and an ear moving at 30 m/s.
- Keep the speed and switch the medium to helium or water: the Mach number shrinks, and so does the shift.
- In “Supersonic”, step until the cone line crosses the ear, then check the angle against sin θ = 1/M.
Frequently asked questions
Does the pitch rise while the vehicle approaches?
No. At constant speed on a straight path the observed pitch never rises: it falls, slowly at first and fastest around closest approach. Only the loudness rises, which is easily mistaken for rising pitch.
Does the shift depend on distance?
Not directly. It depends on the speeds along the line of sight compared with the speed of sound. Distance sets the loudness and how quickly the pitch changes.
What is the small ring on the path?
It marks where the source was when it emitted the sound arriving now. Sound takes about 0.3 s per 100 m in air, so you always hear the past.
Why is it silent in front of a supersonic source?
The source moves faster than its own sound, so no wave can get ahead of it. A listener first receives the shock front.