4.2 Refraction of sound in different media

Fundamentals of Sound Refraction

Refraction of sound waves

When a sound wave passes from one medium into another, its speed changes. Because the speed changes but the frequency stays the same, the wave bends at the boundary between the two media. This bending is refraction.

The direction and amount of bending depend on two things:

• The angle of incidence (the angle at which the sound wave hits the boundary)
• The ratio of sound speeds in the two media

Several properties of a medium determine how fast sound travels through it: density, elasticity (how stiff the material is), and temperature. Any variation in these properties, whether at a sharp boundary or across a gradual gradient, can cause refraction.

Sound behavior across media

Different media transmit sound at very different speeds, and that speed difference is what drives refraction at every interface.

• Air is the slowest common medium for sound (~343 m/s at 20°C). Air is highly compressible, and its sound speed shifts noticeably with temperature and humidity.
• Water conducts sound much faster (~1480 m/s). It's far less compressible than air, and its sound speed varies with temperature, pressure, and salinity.
• Solids transmit sound fastest of all (steel is ~5960 m/s). Solids are the least compressible and can support both longitudinal and transverse (shear) waves, unlike fluids.

At any interface between two media, some sound energy reflects and some transmits through. How much energy crosses depends on the acoustic impedance mismatch between the two materials. A large mismatch (like air-to-water) means most energy reflects; a small mismatch allows more transmission.

Attenuation also differs: air absorbs sound energy quickly over distance, water allows propagation over very long distances, and solids vary widely depending on the material.

Calculations with Snell's Law

Snell's Law relates the angles and speeds on each side of a boundary:

$\frac{\sin\theta_1}{c_1} = \frac{\sin\theta_2}{c_2}$

where $\theta_1$ is the angle of incidence, $\theta_2$ is the angle of refraction, $c_1$ is the sound speed in the first medium, and $c_2$ is the sound speed in the second medium. All angles are measured from the normal (the line perpendicular to the boundary).

Using Snell's Law step by step:

1. Identify the sound speeds in both media ($c_1$ and $c_2$).
2. Measure or determine the angle of incidence $\theta_1$ from the normal.
3. Plug into the equation and solve for $\sin\theta_2$: $\sin\theta_2 = \sin\theta_1 \times \frac{c_2}{c_1}$
4. Take the inverse sine to find $\theta_2$.

Which way does the wave bend?

• Entering a faster medium ($c_2 > c_1$): the wave bends away from the normal ($\theta_2 > \theta_1$).
• Entering a slower medium ($c_2 < c_1$): the wave bends toward the normal ($\theta_2 < \theta_1$).

Critical angle and total internal reflection

When sound travels from a slower medium into a faster one ($c_2 > c_1$), there's a special incident angle called the critical angle:

$\theta_c = \arcsin\left(\frac{c_1}{c_2}\right)$

At the critical angle, the refracted wave skims along the boundary at 90° to the normal. Beyond this angle, no sound transmits into the second medium at all. Instead, all the energy reflects back into the first medium. This is total internal reflection.

Applications of sound refraction

Refraction shows up across a wide range of fields:

• Underwater acoustics: SONAR systems rely on understanding how sound bends through water layers of varying temperature and salinity. Deep ocean sound channels (the SOFAR channel) trap sound by continuous refraction, allowing signals to travel thousands of kilometers. Marine animals exploit these same channels for long-distance communication.
• Atmospheric propagation: On a calm night, a temperature inversion (warm air above cool air) bends sound waves back toward the ground, making distant sounds surprisingly audible. Wind gradients similarly refract sound, bending it downwind and creating "shadow zones" upwind.
• Seismic exploration: Geologists send sound waves into the earth and analyze how they refract through rock layers to map subsurface geology and locate oil or mineral deposits.
• Medical ultrasound: Imaging systems must account for refraction as sound passes through tissues with different speeds (fat, muscle, bone) to produce accurate images.
• Acoustic lenses: Specially shaped materials refract sound to focus or redirect it, similar to how optical lenses work with light.
• Non-destructive testing: Engineers send ultrasound through manufactured parts and use refraction patterns to detect internal flaws without cutting the material open.