7.2 Oblique incidence and Snell for sound

When a plane wave hits a boundary at an angle other than normal, the reflected and transmitted waves go off in directions set by the angle of incidence and the speed ratio of the two media. The relationship is Snell’s law — the same law that governs light refracting through glass.

The geometry

Let the boundary be the plane $z = 0$ , with medium 1 above ( $z > 0$ ) and medium 2 below ( $z < 0$ ). The incident wave comes from above with wavevector $\mathbf{k}_i$ making angle $\theta_i$ with the surface normal. After hitting the boundary, two waves emerge:

A reflected wave with wavevector $\mathbf{k}_r$ , angle $\theta_r$ above the boundary.
A transmitted wave with wavevector $\mathbf{k}_t$ , angle $\theta_t$ below the boundary.

All three wavevectors lie in the plane of incidence (the plane containing $\mathbf{k}_i$ and the surface normal) — a consequence of the symmetry.

Snell’s law for sound

The frequency $\omega$ is the same in both media (the boundary oscillates at the wave’s frequency). So the wavenumbers $k_1 = \omega/c_1$ and $k_2 = \omega/c_2$ are different in the two media.

For the boundary conditions to be satisfied at all points along the interface, the tangential component of the wavevector must match across the boundary:

k_1 \sin\theta_i \;=\; k_2 \sin\theta_t,

i.e.

\boxed{\;\;\frac{\sin\theta_i}{c_1} \;=\; \frac{\sin\theta_t}{c_2}.\;\;}

This is Snell’s law for sound — identical in form to the optical Snell’s law (with sound speeds replacing refractive indices via $n = c_\text{vac}/c_\text{medium}$ ).

The reflection is specular: $\theta_r = \theta_i$ (angle of incidence equals angle of reflection), the symmetric analogue.

Refraction examples

Air to water. $c_1 = 343$ m/s, $c_2 = 1480$ m/s. Sound going from air into water bends away from the normal, since $c_2 > c_1$ . A sound coming straight down at $\theta_i = 30°$ refracts to $\theta_t = \arcsin(\sin 30° \cdot 1480/343) = \arcsin(2.16)$ — undefined! Above a critical incidence angle, no refraction is possible and the wave is totally reflected.
Warm to cold air (temperature gradient). $c \propto \sqrt{T}$ . Hotter air → faster sound. A horizontal sound ray in air with a temperature gradient bends away from the warm side and toward the cold side. This is why sound carries farther over a cold pond (downward refraction back to the ground) and worse over hot asphalt (upward refraction away from the listener).
Underwater sound channel (SOFAR). The temperature and pressure profile of the deep ocean creates a minimum in $c$ at about 1000 m depth. Sound rays at intermediate angles refract back toward this minimum from both above and below, becoming trapped in a horizontal waveguide and propagating across entire ocean basins.

Critical angle and total internal reflection

For sound going from a slow medium to a fast medium ( $c_2 > c_1$ ), there is a critical incidence angle

\theta_c \;=\; \arcsin(c_1 / c_2)

beyond which no transmitted wave exists. All energy is reflected. For air-to-water, $\theta_c = \arcsin(343/1480) \approx 13.4°$ — for incidence angles above $13.4°$ from normal, no sound enters the water from the air. (This is why underwater sound cones are narrow when looking up from underneath.)

For sound from water into air, $c_1 > c_2$ and no critical angle exists — but the impedance mismatch still makes the transmission coefficient tiny.

Reflection and transmission coefficients at angle

The amplitude coefficients at oblique incidence are more complex than at normal incidence:

R \;=\; \frac{Z_2 \cos\theta_i - Z_1 \cos\theta_t}{Z_2 \cos\theta_i + Z_1 \cos\theta_t}, \qquad T \;=\; \frac{2 Z_2 \cos\theta_i}{Z_2 \cos\theta_i + Z_1 \cos\theta_t}.

At normal incidence ( $\theta_i = \theta_t = 0$ ), these reduce to the formulas from lesson 7.1.

Geometrical acoustics

For wavelengths short compared to the size of refracting structures, we can trace acoustic “rays” using Snell’s law at each surface — exactly like geometrical optics. The eikonal approximation, ray equations, Fermat’s principle of least time — all carry over. This is the framework underlying:

Underwater acoustics (the SOFAR channel is a ray-tracing problem).
Atmospheric acoustics (sound shadowing, propagation over distances).
Architectural acoustics (image-source method for computing reflections off walls).
Ultrasound imaging (focusing through layered tissue).

Geometrical acoustics is exact in the short-wavelength limit. It fails near caustics (where rays cross), near focal points, and at scales comparable to the wavelength. The next two lessons (Huygens construction and diffraction) cover the wave phenomena that ray theory misses.