A source emits a pulse every tick. Each pulse spreads as a sphere at the speed of sound a, centred on the point where it was emitted — while the source itself moves on at V = M a. All three regimes follow from that one race.
Each circle is one pulse, still expanding at a from where it was emitted — camera follows the source
One pulse of age t: it has grown to at while the source ran on Vt
How this is built. Pulse k of age tk has radius a tk and its centre sits V tk behind the source; the picture is that statement drawn for nine successive pulses. The fronts acquire a common tangent only when V > a, and the right triangle then gives sin μ = a t / V t = 1/M. Caveats. This is the linear acoustic construction — infinitesimally weak pulses, uniform air, no viscosity. A real shock has finite strength, so it stands detached ahead of a blunt nose and is stronger and steeper than the Mach wave drawn here; the regime bands (transonic ≈ 0.8–1.2) are conventional labels, not sharp physical boundaries.