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The Doppler effect is a change between emitted and received frequency caused by motion of the source or receiver. An approaching siren can send more wave crests per second to a microphone than the source emits per second; a receding siren sends fewer. The source need not change its own oscillation. The Doppler shift is $\Delta f=f_o-f_s$, where $f_s>0$ is emitted frequency and $f_o>0$ is received frequency, both measured in hertz.

For acoustic waves, motion must be specified relative to the material carrying the wave. The speed of sound is measured in that material's rest reference frame. Moving the source changes the spacing of successive wavefronts; moving the receiver changes how quickly it crosses those wavefronts. These two mechanisms give different exact factors in the classical Doppler effect for sound, even for the same closing relative velocity.

This makes a microphone a motion probe: a stable tone supplies a repeating clock, and changes in arrival spacing reveal the source's motion toward or away from the receiver. An acoustic Doppler measurement can test the prediction directly. A passing source also illustrates why geometry matters: its speed can stay constant while the received frequency changes continuously, as in Doppler frequency along a trajectory.

The following sound model assumes linear acoustics in uniform stationary material with isotropy, constant speed of sound and negligible wave dispersion. Source and receiver speeds are below the speed of sound, and there is a single direct propagation path. Air motion requires transforming source and receiver velocities into the air's rest reference frame; a nonuniform flow requires a more general propagation model. At the sonic threshold the simple periodic-arrival picture fails; moving-point acoustic wavefronts and the Mach cone describe the relevant geometry, while finite shock waves require further physics. Light in vacuum instead requires the relativistic Doppler effect: the acoustic formula cannot be converted into its exact optical counterpart just by replacing the speed of sound with the speed of light.

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 obeysThe 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.^[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 $c=340\mathrm{m}{\mathrm{s}}^{-1}$ and a source moving through it at $U=680\mathrm{m}{\mathrm{s}}^{-1}$. 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 $M\downarrow 1$ and $0$ as $M\to \infty$.

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 $-M{c}_0{\mathbf{e}}_x$ 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 ${\partial }_{s}F=0$ onThe partial derivative condition gives $x=-(U^2-c^2)s/U$. Substituting into $F=0$ gives $y^2=c^2(U^2-c^2)s^2/U^2$, henceFor $M=2$, the Mach cone section has slope magnitude $1/\sqrt{3}$, 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.

A sonic flow has Mach number exactly one: the local fluid speed relative to the body equals the local sound speed. In uniform gas flowing at $U=c$ past a resting body, an upstream-directed acoustic wave has zero body-frame velocity. A sonic point is a location where this equality holds. In the moving-point acoustic wavefronts construction, old spheres touch the source at the origin; the Mach angle tends to $90^{\circ }$ as the supersonic flow approaches this threshold.