whatOne rule: a moving source rings a circular wavefront every tick, and every front expands at the speed of sound, ignorant of who made it. The camera rides with the source; the medium -- and a listener -- slide past.

whyThe claim: everything Doppler is already in the picture. Ahead, the fronts bunch (wavelength squeezed by 1-M); behind, they stretch (1+M) -- the HUD measures both spacings off the drawn circles and checks the textbook ratio. The listener counts the fronts that actually cross it: pitch up approaching, down after. Drag the speed through Mach 1 and the fronts ahead pile onto their maker, then fold into the cone whose half-angle is asin(1/M) -- measured live off the same circles. When the cone sweeps the listener, every held-back front arrives at once: the boom.

howPure kinematics -- no integrator to trust, just geometry: front positions are exact circles, and every number in the HUD (wavelength ratio, cone angle, what the listener hears) is measured from them, not from a formula. Left alone the piece ramps itself through the sound barrier; the slider hands you the throttle. One integer seeds the ticks.