= Mach angle
{c}
{title2=$\mu=\arcsin(M^{-1})$}
{wiki}
The <Mach angle> of a <supersonic flow> with <Mach number> $M>1$ is $\mu=\arcsin(M^{-1})$. Infinitesimal <pressure> disturbances lie on the corresponding <Mach cone>, and a weak <oblique shock> approaches this angle as its strength tends to zero.
The <Mach angle> is the half-angle between a <Mach cone> generator and its backward motion axis for a steady point source in a uniform three-dimensional medium with <isotropy> and constant <sound speed>. It obeys
$$
\sin\mu=\frac{c}{U}=\frac{1}{M},\qquad 0<\mu<\frac{\pi}{2},\quad M>1.
$$
The relation concerns weak <acoustic waves> in <linear acoustics>. It is not the angle of a finite <oblique shock> produced by finite turning of the <fluid flow>; that angle also depends on the deflection and the <gas> properties.<Mach Angle (NASA Glenn Research Center)>[Mach Angle, geometric construction and displayed sin(mu)=1/M and mu=asin(1/M) equations]{ref}
For a short proof, choose an old emission in the <moving-point acoustic wavefronts> construction and set $s=-\tau>0$. Its <circle> has center $(-Us,0)$ and radius $cs$. A <tangent line> from the present source at the origin meets the <circle> at a <right angle> to its radius. The resulting <right triangle> has hypotenuse $Us$ and opposite side $cs$, giving the displayed <sine> ratio. The emission age cancels: every <wavefront> has the same tangent directions, so a continuum of expanding <spheres> produces one simple <Mach cone>.
In an original example, stipulate uniform <gas> with <sound speed> $c=340\,\mathrm{m\,s^{-1}}$ and a source moving through it at $U=680\,\mathrm{m\,s^{-1}}$. Then $M=2$ and $\mu=\arcsin(1/2)=30^\circ$. An emission one second earlier is now a <sphere> centered $680\,\mathrm m$ behind the source with radius $340\,\mathrm m$. The selected <sound speed> defines this example; it is not a constant for air at every <temperature>. The <Mach angle> decreases as $M$ increases: its <derivative> is $-1/(M\sqrt{M^2-1})$ in <radians>, and its <limits> are $90^\circ$ as $M\downarrow1$ and $0$ as $M\to\infty$.
\Image[/mach-angle-versus-speed-ratio.png]
{title=Mach half-angle decreases as the gas-relative speed ratio increases}
{description=Original plot of the <Mach angle> $\mu=\arcsin(1/M)$ for $M>1$, with the constructed $M=2$, $\mu=30^\circ$ example. The open point at $M=1$ marks a <limit>, not a finite <supersonic> cone. This weak-wave prediction gives neither a finite-deflection <oblique shock> angle nor measured <sonic boom> strength.}
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