= Mach number
{c}
{title2=$\mathcal M$}
{wiki}
The <Mach number> is the ratio of <fluid flow> speed to local <sound speed>.
More precisely, the <Mach number> is the <dimensionless quantity>
$$
M=\frac{|\mathbf v_{\mathrm{body}}-\mathbf u_{\mathrm{gas}}|}{c}=\frac{U}{c},
$$
where $U$ is the magnitude of the body's <relative velocity> through the surrounding <gas>, and $c>0$ is the local <sound speed> measured in the <gas> rest <reference frame>. For a <fluid flow> past a stationary body, the same <Mach number> uses the local <fluid> speed in the body's rest <reference frame>. A <Mach number> below, equal to, or above one means <subsonic flow>, <sonic flow>, or <supersonic flow>, respectively. The denominator is a property of the local <gas>: its <sound speed> depends on <temperature> and composition.<Mach Number (NASA Glenn Research Center)>[Mach Number, paragraphs defining the speed ratio and dependence on gas type and temperature; flight-regime list]{ref}
The <relative velocity> matters even when the body is at rest in a <wind tunnel>. A uniform change of <reference frame> subtracts the same <velocity> from body and <gas>, so their <relative velocity> and the <Mach number> stay unchanged under a <Galilean transformation>. Ground speed alone cannot supply $U$ when there is wind. For an original collinear example, a body moving at $680\,\mathrm{m\,s^{-1}}$ over the ground through <gas> moving in the same direction at $170\,\mathrm{m\,s^{-1}}$ has $U=510\,\mathrm{m\,s^{-1}}$. With a stipulated <sound speed> $c=340\,\mathrm{m\,s^{-1}}$, its <Mach number> is $1.5$, rather than $2$. A moving observer changes apparent <acoustic wave> arrival times, as in the <radial Mach number>, without changing the body's <Mach number> relative to the <gas>.
Why is the threshold one so useful? <Acoustic waves> carry small <pressure> disturbances at the <sound speed>. In <subsonic flow>, such information can travel upstream relative to the body; in a uniform <supersonic flow>, it is swept downstream. The <Mach cone> turns this difference in communication into visible geometry. Matching <Mach number> between a <wind tunnel> and flight is therefore essential to matching <compressible flow> effects.<Mach Number (NASA Glenn Research Center)>[Mach Number, paragraphs on downstream propagation and matching wind-tunnel and flight Mach numbers]{ref}
An actual <supersonic wind tunnel> experiment makes the distinction between <wavefronts> and <shock waves> concrete. In NASA Glenn's 1-by-1-foot <wind tunnel>, interchangeable converging-diverging <nozzle> blocks set the <Mach number>. A strut-mounted exhaust-<nozzle> model had a nosecone, an air plenum and transparent sidewalls for <schlieren photography>. Before installing the model, the researchers photographed the empty test section: two <shock wave> patterns originated at a wall joint after the <nozzle> block. This control guided model placement to keep reflected <shock waves> away from the plume-study region. The rig included internal <total pressure> and <static pressure> sensing and a traversing external <static pressure> probe. A baseline <nozzle> and previous <pressure> data supplied a planned validation comparison; the report distinguishes this plan from the observed empty-tunnel control. This establishes an apparatus origin for those visible <shock waves>, not a measurement of the <Mach cone> from an infinitesimal source. The reported finite <shock wave> angle cannot simply be inserted into the <Mach angle> formula to calibrate $M$: <shock wave> strength and geometry also matter.<Wind Tunnel Model Design for the Study of Plume Effects on Sonic Boom for Isolated Exhaust Nozzles by Raymond S. Castner>[Wind Tunnel Model Design for the Study of Plume Effects on Sonic Boom for Isolated Exhaust Nozzles, NASA/TM-2010-216259, §II printed p. 2; §§IV–V printed pp. 8–9; Figs. 7–10 and concluding remarks printed p. 10]{ref}
In conventional <schlieren photography>, a bright slit source and two concave mirrors send parallel <light rays> through the <wind tunnel> and focus them at a knife edge before an <optical camera>. Spatial changes in <mass density> cause <refraction>; the knife edge converts the deflection into image contrast, making <shock waves> visible. This is the general optical principle, rather than a specification of every component in the preceding rig. A photograph projects a three-dimensional <fluid flow> into two dimensions; contrast alone does not provide a calibrated <pressure> jump or <acoustic wave> amplitude.<Schlieren Flow Visualization (NASA Glenn Research Center)>[Schlieren Flow Visualization, paragraphs describing the two-mirror slit-source and knife-edge system, density-gradient contrast, and two-dimensional recording of three-dimensional shocks]{ref}
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