9.6 Reflection and refraction at flow boundaries

When sound crosses from a region of still air into a region of moving air — or from one flow to another — it refracts. The mechanism is analogous to Snell’s law for an interface between media of different sound speeds, but with an extra complication: the effective sound speed depends on the angle between the wave’s direction and the flow.

Phase matching at the interface

The dispersion relation in a moving medium, from lesson 9.2, is

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

At an interface between region 1 (with mean flow $\mathbf{U}_1$ ) and region 2 (with mean flow $\mathbf{U}_2$ ), three conditions must be matched:

Frequency $\omega$ is continuous (the boundary doesn’t generate new frequencies).
The wavevector component tangential to the interface, $\mathbf{k}_\parallel$ , is continuous (otherwise the wave at one side of the interface wouldn’t match up phase-by-phase with the wave at the other).

From these two and the dispersion relation, the normal wavevector component $k_\perp$ is determined on each side, and the angles of refraction follow.

In the simple case where the flow is parallel to the interface, the formula for the angle of refraction is

\frac{\sin\theta_1}{c_1 + U_1 \cos\theta_1} \;=\; \frac{\sin\theta_2}{c_2 + U_2 \cos\theta_2},

which reduces to Snell’s law when there’s no flow ( $U_1 = U_2 = 0$ ).

What this means

A few concrete consequences:

Refraction by shear layers. Sound from a stationary source crossing a layer of moving air (a jet, a wind shear) bends. The direction of bending depends on whether the flow is “with” or “against” the propagation. This is why jet exhaust focuses its own noise in characteristic patterns rather than emitting omnidirectionally.
Refraction by atmospheric wind shear. A horizontal sound ray in air with a strong wind aloft refracts. If the wind is in the direction of propagation (downwind), the ray bends downward — sound carries well downwind, often surprising the listener. If the wind is against propagation, the ray bends upward into the sky, leaving an acoustic shadow on the ground. This is the standard explanation for why distant sounds are often more audible at night (when temperature inversions promote downward refraction).
Refraction at temperature-driven flow boundaries. Convection cells, thermal layers, the boundary layer at the ground all have associated flow patterns and can refract sound rays.

The acoustic shadow

A more dramatic phenomenon: total internal reflection at a flow boundary. If a sound ray is incident at a steep enough angle on a flow region, and the flow is fast enough, the wave can be totally reflected — unable to enter the flow at all. This creates an acoustic shadow in or behind the flow region.

This is exploited deliberately in some quiet-zone designs (sound-baffle systems using induced air flow as a barrier) and is the natural reason that jet aircraft sound has the characteristic far-field directional pattern it has: the flow itself shadows portions of the radiation.

In hearing

The vocal tract is a non-trivial flow region. When you produce voiced speech, the airflow through the larynx, the constriction at the tongue, the lips — all set up small but real flow patterns that refract the sound being generated. For most speech analysis this can be neglected, but for high-fidelity vocal-tract modelling (speech synthesis, musical-instrument simulation) the flow-coupling matters.

Looking ahead

Chapter 9 has handled three regimes of moving acoustics: Doppler kinematics (lesson 9.1), the wave equation in a moving medium (9.2), supersonic and the Mach cone (9.3), the bookkeeping when multiple actors move (9.4), aerodynamic sound generation (9.5), and now refraction at flow boundaries (9.6).

The wave equation we derived in chapter 4 has carried us this far. In chapter 10, we will see where it finally breaks: the dissipative corrections (viscosity, thermal conduction, molecular relaxation) and the nonlinear corrections (wave steepening, shock formation, the bridge to cavitation). The wave equation is excellent in its regime, but its regime is bounded — and the boundary is what chapter 10 explores.