9.2 The wave equation in a moving medium

Consider air moving with steady uniform velocity $\mathbf{U}$ in a fixed (laboratory) frame. A sound wave propagates through it. What is the equation governing the sound, as observed from the lab frame?

The transformation

In a frame moving with the fluid, the medium is at rest and the acoustic wave equation is the standard one:

\frac{1}{c^2}\, \frac{\partial^2 p'}{\partial t^2} \;=\; \nabla^2 p'.

To rewrite this in the lab frame, replace the time derivative by the convective derivative

\frac{D}{Dt} \;\equiv\; \frac{\partial}{\partial t} \;+\; \mathbf{U} \cdot \nabla,

which is the rate of change following the fluid. The convected wave equation is

\boxed{\;\;\frac{1}{c^2}\!\left(\frac{\partial}{\partial t} + \mathbf{U} \cdot \nabla\right)^{\!2} p' \;=\; \nabla^2 p'.\;\;}

When $|\mathbf{U}| = 0$ , this reduces to the standard wave equation. When $|\mathbf{U}|$ is non-zero, the propagation becomes anisotropic: sound travels faster downstream than upstream.

Plane waves in flow

Substitute the plane wave ansatz $p' = P_0\, e^{i(\omega t - \mathbf{k} \cdot \mathbf{r})}$ :

\frac{1}{c^2} (\omega - \mathbf{U} \cdot \mathbf{k})^2 \;=\; |\mathbf{k}|^2.

Take square roots:

\omega - \mathbf{U} \cdot \mathbf{k} \;=\; \pm c\, |\mathbf{k}|.

The dispersion relation now depends on the angle between $\mathbf{k}$ and the flow $\mathbf{U}$ . Define $\theta$ as the angle between them:

\omega \;=\; \mathbf{U} \cdot \mathbf{k} \pm c\, |\mathbf{k}| \;=\; |\mathbf{k}|\, (U \cos\theta \pm c).

The phase velocity in the direction of $\mathbf{k}$ is

v_\text{phase} \;=\; \frac{\omega}{|\mathbf{k}|} \;=\; U \cos\theta \pm c.

Downstream ( $\theta = 0$ ): phase velocity $U + c$ . Sound propagates faster than in still air. Upstream ( $\theta = \pi$ ): phase velocity $-U + c = c - U$ . Slower than still air. Cross-stream ( $\theta = \pi/2$ ): phase velocity is $c$ , but the wave is swept sideways by the flow.

The wind shadow

For $U < c$ (subsonic flow), sound can propagate upstream, just slowly. The geometry of wavefronts is asymmetric — wavefronts emitted by a stationary source in flow appear compressed downstream and stretched upstream.

For $U > c$ (supersonic flow), sound cannot propagate upstream. The wave equation has no real solution travelling against the flow. Any disturbance is swept downstream, forming a Mach cone (next lesson).

This is why a supersonic aircraft is “invisible” — acoustically — to anyone in front of it until the shock wave passes overhead.

Refraction by flow gradients

If the flow velocity varies with position (e.g., wind faster aloft than near the ground), sound rays refract. This is the wind-shear analogue of the temperature-gradient refraction discussed in lesson 7.2. Combined, temperature and wind gradients explain most of the long-range propagation patterns observed in atmospheric acoustics — why thunder claps that propagate downwind sound very different from those that propagate upwind, why summer evenings sometimes carry sound much farther than summer afternoons.

Sound in a duct with flow

A common practical case: sound propagating along the axis of a duct with steady mean flow $U$ along that axis. For axial propagation,

A wave going downstream has phase velocity $c + U$ .
A wave going upstream has phase velocity $c - U$ .
The wavelength of a fixed-frequency wave is longer downstream than upstream.
The frequencies of tube modes shift slightly relative to the no-flow case.

These shifts are the basis of ultrasonic flow meters: launch sound pulses both directions and measure the time-of-flight difference, which is proportional to $U$ . The technique is used in HVAC, fuel flow, and medical blood-flow measurement.

The Mach cone awaits

For $U > c$ , the wave equation in flow loses one of its real-valued plane-wave solutions and the propagation geometry becomes degenerate — concentrated on a Mach cone. The next lesson works out this transition.