Moving-point acoustic wavefronts ($(x-U\tau {)}^2+y^2=c^2{\tau }^2,\tau <0$)

Take a steady moving point source in uniform resting gas, with constant sound speed $c$ and source velocity $U{\mathbf{e}}_x$. Place the source at the origin at the current time $t=0$. In the linear acoustics model, an emission at an earlier time $\tau <0$ has a spherical wavefront centered at $(U\tau ,0,0)$ with radius $-c\tau$. An axial cross-section is the circle in the displayed equation. It is centered at the old release position, not the source's present position. The circles describe the geometry of weak acoustic waves, not a measured shock wave or its pressure jump.

For $M=0.5$, every old circle contains the present source because its center is $U|\tau |$ from the origin while its radius is the larger distance $c|\tau |$. It reaches ahead of the source. For $M=2$, the rightmost point of every old circle is $(U-c)\tau <0$: the source outruns all of its old wavefronts. The continuously emitted wavefronts have a common geometric envelope, the Mach cone. The plotted discrete release times sample this continuous family; rotating the axial section about the motion axis recovers the three-dimensional construction.