= Moving-point acoustic wavefronts
{title2=$(x-U\tau)^2+y^2=c^2\tau^2,\quad\tau<0$}
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.
\Image[/moving-source-acoustic-wavefronts.png]
{title=Subsonic and supersonic acoustic wavefronts from a moving point}
{description=Original axial cross-sections at $t=0$ for <Mach numbers> $0.5$ and $2$. With an arbitrary positive <time> scale $T$, the sampled release times are $\tau/T=-0.5,-1,\ldots,-3$. Each <circle> has center $(U\tau,0)$ and radius $-c\tau$; red lines give the <Mach cone> section. This weak <linear acoustics> model predicts geometry, not measured <shock wave> or <sonic boom> strength.}
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