The Doppler Effect and the Sound Barrier

Why the pitch changes. A sound source emits wave crests at equal time intervals T₀ = 1/f₀, and every crest spreads out at the speed of sound c = 340 m/s from the point where the source was at the moment of emission. The speed of sound is set by the medium, not by the source. When the source moves, the centres of the spheres shift: ahead of it the crests catch up with each other and the wavelength shortens, λ = (c − v)/f₀; behind it they stretch, λ = (c + v)/f₀. The crests reach an observer at rest with the same speed c, so the observer hears the frequency f = c/λ: f = f₀·c/(c − v) while the source approaches and f = f₀·c/(c + v) when it recedes. That is the Doppler effect — the familiar “eeee-oooo” of a passing ambulance.

What the simulator shows. The floor shows the air pressure in the plane of flight: orange is compression, blue is rarefaction, the light circles are the crests. Above the floor the same crests are shown in 3D as translucent domes. The green ball is the observer; click on the floor to move it. Every crest that reaches the observer makes it flash, so you can see that the flashes at the observer come more or less often than at the source. Below the scene there is a slice along the flight line: the pressure curve p(x) with the wavelengths ahead and behind, and rows of air particles that oscillate along the direction of the wave (sound is a longitudinal wave). Further down — an oscilloscope “what the source emits / what reaches the ear” and the graph of the observed frequency f(t) with dashed lines f₀·c/(c − v) and f₀·c/(c + v). Time is slowed down 100 times, but all numbers are real: the crest positions and the frequency come from exact formulas.

The observer can move too. If the source stands still and the observer runs towards it at v_o, the wavelength in the air does not change, but the crests are met more often: f = f₀·(c + v_o)/c. In general f = f₀·(c − v_o·cos θ)/(c − v·cos θ), where θ is the angle between the direction of flight and the line “point of emission → observer”. The Measurements tab shows the exact frequency and checks it against this formula. If the observer stands off the flight path, the frequency changes smoothly from high to low — exactly when the sound emitted during the pass reaches them.

The sound barrier and the Mach cone. The Mach number M = v/c tells how many times faster than sound the source is. At M = 1 the source keeps pace with its own crests: ahead of it they merge into a single wall of high pressure — a shock wave. At M > 1 the source outruns its sound and the fronts build up a cone with sin α = 1/M. Ahead of the cone lies a zone of silence: a supersonic plane flies past first, and only then does the cone reach the observer — a sharp bang, the sonic boom. After the boom the observer hears two sounds at once: waves emitted at different moments arrive together.

Try this. 1) Press “0.5c” and compare λ ahead and behind: 0.50 m and 1.50 m at f₀ = 340 Hz, with frequencies of 680 and 227 Hz. 2) Turn on “Listen as the observer” and hear the pitch drop during the pass; the “Fly-by in real time” button plays the same pass without slow motion. 3) Press “Speed up through the barrier” and watch a shock wave being born from the squeezed fronts. 4) At “1.5c” measure the cone angle: sin α = 1/1.5, α ≈ 41.8°. 5) Turn on “Test yourself” and predict the result before you see the experiment.