= Mach cone
{c}
{title2=$\sin\theta=1/M$}
The <Mach cone> forms the <geometric envelope> of <acoustic wave> <wavefronts> from a source moving at a <supersonic> <Mach number> $M>1$. In the source <reference frame> the ray-<velocity> <sphere> is centered at $-Mc_0\mathbf e_x$ and has radius $c_0$. Its tangent rays make <angle> $\sin^{-1}(1/M)$ to the backward flight direction; all other emitted acoustic rays lie inside that <Mach cone>.
The <Mach cone> is the <geometric envelope> of the <spheres> in <moving-point acoustic wavefronts>. To obtain its axial boundary without a diagram, put $s=-\tau>0$ and impose the <geometric envelope> conditions $F=0$ and $\partial_sF=0$ on
$$
F(x,y,s)=(x+Us)^2+y^2-c^2s^2.
$$
The <partial derivative> condition gives $x=-(U^2-c^2)s/U$. Substituting into $F=0$ gives $y^2=c^2(U^2-c^2)s^2/U^2$, hence
$$
y=\pm\frac{-x}{\sqrt{M^2-1}},\quad x<0,
\qquad
\sqrt{y^2+z^2}=\frac{-x}{\sqrt{M^2-1}}quad\text{in three dimensions}.
$$
For $M=2$, the <Mach cone> section has slope magnitude $1/\sqrt3$, consistent with a <Mach angle> of $30^\circ$. If emission begins at a finite <time>, the reached region is limited by the oldest <wavefront>; the infinite cone describes the steady geometric boundary obtained with an indefinitely long source history. In a uniformly moving <gas>, <advection> shifts each emission's <sphere> by the gas <velocity> times its age; working with the source's <relative velocity> through the <gas> restores the same construction.
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