Sketch · 2026-09-14
Doppler, and the Mach cone.
A point source moves across the canvas and emits a circular wavefront at a steady tempo. Each ring expands at the wave speed c, independent of the source. Slide the source speed past c and the ordinary Doppler pattern collapses into the conical shock that sonic booms and bow waves both wear.
The source loops left to right. Every emitted ring is fixed in the medium the moment it leaves the source; it does not care where the source goes next.
Note
Below Mach 1 the rings still overtake the source, so an observer ahead of it meets crests more often than the source emits them; an observer behind meets them less often. That is the whole Doppler effect in one picture: same wave speed, same emission rate, but the spacing of the crests depends on where you are relative to the motion.
At Mach 1 the source keeps pace with its own wavefront. Every ring it has ever emitted is tangent to the same point ahead of it; the pressure piles up on a wall you can hear.
Above Mach 1 the source outruns its rings. Each ring's centre sits behind the source now; the common tangent to the whole family is a cone with half-angle sin⁻¹(1 / M), and every wavefront the source has ever emitted lies inside that cone. Ernst Mach worked this out in 1887 from photographs of a bullet crossing a candle-lit gap.
Source
Mach, E. and Salcher, P. (1887). Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge. Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften 95: 764–780.