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The Mach number is the ratio of fluid flow speed to local sound speed.

More precisely, the Mach number is the dimensionless quantitywhere $U$ 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.^[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 $680\mathrm{m}{\mathrm{s}}^{-1}$ over the ground through gas moving in the same direction at $170\mathrm{m}{\mathrm{s}}^{-1}$ has $U=510\mathrm{m}{\mathrm{s}}^{-1}$. With a stipulated sound speed $c=340\mathrm{m}{\mathrm{s}}^{-1}$, 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.^[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.^[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.^[ref]

Put ${\rho }^{\prime }=\rho -{\rho }_0$, and take the retarded acoustic Green function for ${\partial }_t^2-c_0^2{\nabla }^2$:This selects the causal, outgoing acoustic density perturbation, with no additional incoming homogeneous wave equation solution. Convolving the acoustic dipole forcing with this Green function and moving its spatial derivative outside the integral givesThe surface delta distribution converts the spatial integral to the moving surface. At fixed surface labels $(p,q)$, setThe radial Mach number enters the moving-surface retarded Jacobian, because $dr/d\tau =-\hat{r}\cdot v$ and henceLet ${\tau }^{\ast }$ be a root of the retarded time equationThe delta change-of-variable rule now givesHere $h_p{h}_q$ is the surface-area factor from the orthogonal surface coordinates. For a subsonic surface, $|v|<c_0$, the retarded equation is monotone in $\tau$ and has one root when the motion is defined for the required past times. For more general motion, sum the displayed contribution over all simple retarded roots. A root with $1-M_r=0$ requires a separate limiting treatment; the simple-root formula does not apply there.

Write $\alpha =\Omega {\tau }_0^{\ast }$ and $s=\sin \theta$, where ${\tau }_0^{\ast }=t-R/c_0$. Label the two arms by $\sigma =\pm 1$. Their positions are $y_{\sigma }=\sigma r{e}_r(\Omega {\tau }_{\sigma }^{\ast })$. To radiating far-field accuracy,The retarded time equation therefore gives, with ${\mu }_r=\Omega r/c_0$,For the first arm, $\sigma =1$, this is the requested phase expansion. The more explicit geometric error also makes the dimensional meaning of the printed $O(1/R)$ term clear.

Apply the Taylor theorem to the rotating line force. Since $d{e}_{\varphi }/d\alpha =-e_r$, its first two orders, expressed in a common reference frame, areAlso $v_{\sigma }=\sigma \Omega r{e}_{\varphi }(\Omega {\tau }_{\sigma }^{\ast })$, so the radial Mach number isThe absolute value causes no change of sign in this subsonic limit. Multiplying these two expansions before summing is essential: both the shifted force and the moving-surface retarded Jacobian contribute at the same order.

Denote the integrated numerator, including that Jacobian, by $\mathcal{B}$. Pairing the two blades cancels all terms odd in $\sigma$, including the first Doppler correction to the axial load. HenceThe factor $a/3$ comes from $a^{-2}{\int }_0^a{r}^{2}dr$. Only the time-dependent term radiates at order $R^{-1}$. Applying $-{\partial }_{x_i}$ to the integral now gives ${\rho }_{\mathrm{rad}}^{\prime }=(4\pi c_0^{3}R{)}^{-1}n\cdot {\partial }_t\mathcal{B}$, and thusThere are also nonradiating terms of order $R^{-2}$. Multiply by $c_0^2$ for the acoustic pressure. The stated coefficient uses the rotation convention fixed above; reversing the rotation reverses the corresponding signed load and phase convention.

This is a compact rotating two-blade loading source acting as an acoustic quadrupole. The compact total rotating force cancels, leaving the first spatial moment of the loading. Its two factors of the observer's projection into the rotor plane produce ${\sin }^2\theta$: there is no leading sound on the rotation axis and the density amplitude is maximal in the rotor plane. The configuration repeats after half a rotation, explaining frequency $2\Omega$. These statements concern amplitude; the corresponding acoustic intensity has a ${\sin }^4\theta$ factor.

If $F=0$, the displayed first-order contribution vanishes as well. For completeness, expanding the axial Jacobian to its next even order givesThe first remaining axial-load radiation is thenThus the constant axial total force does not produce the lower-order term, even though its moving spatial distribution can radiate at a higher order.