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Orange source emits blue wavefront circles while moving right. Below M=1 the fronts bunch ahead; at M=1 they form a wall; above M=1 the yellow tangent lines form the Mach cone. Green: fixed observer; red cross: cone-ground crossing.
The curve is mu=arcsin(1/M), for M from1 to5. The marker shows the input only for M>=1; below1 the angle is undefined and no marker is drawn. Play advances wavefront time without changing M.
Mach angle (cone half-angle):
$$\sin\mu = \frac{1}{M}, \qquad \mu = \arcsin\!\left(\frac{1}{M}\right)$$Flight speed and ground reach distance of the cone:
$$V = M\,c, \qquad \ell = \frac{h}{\tan\mu}$$Time delay of the boom (between the aircraft passing overhead and the cone reaching the observer below):
$$t_{\text{delay}} = \frac{\ell}{V} = \frac{h}{V\tan\mu} = \frac{h}{M\,c\,\tan\mu}$$For $M\lt 1$ disturbances propagate ahead of the body and no cone forms; $M=1$ is a plane wave; only for $M\gt 1$ does the Mach cone appear. With the defaults $M=2,\ c=343\ \text{m/s},\ h=1000\ \text{m}$ this gives $\mu=30^\circ$, $V=686\ \text{m/s}$, $\ell=1732\ \text{m}$ and $t_{\text{delay}}=2.52\ \text{s}$.
