7.3 Transmission and reflection at boundaries

Acoustic Wave Behavior at Boundaries

When a sound wave traveling through one medium hits a boundary with a different medium, it doesn't just keep going. Some of the wave's energy reflects back, some transmits through, and depending on the angle and materials involved, the wave may change direction or even change type. Predicting how much energy reflects versus transmits comes down to the acoustic impedance mismatch between the two media.

Behavior of sound waves at boundaries

Three things can happen when a sound wave reaches a boundary:

• Incident wave: the original wave approaching the boundary, carrying energy through the source medium
• Reflected wave: the portion that bounces back into the original medium, reversing direction at the boundary
• Transmitted wave: the portion that crosses into the second medium, potentially at a different speed and direction

The relative amounts of reflection and transmission depend on the impedance mismatch between the two media. A large mismatch (like air meeting water, where water's impedance is roughly 3,500 times greater) means most energy reflects. A small mismatch means most energy passes through.

Mode conversion can also occur at boundaries: a longitudinal (pressure) wave can partially convert into a transverse (shear) wave, or vice versa. This is common in seismology, where P-waves hitting a rock layer boundary generate both reflected P-waves and converted S-waves.

Reflection and transmission principles

Two coefficients quantify what happens at a boundary:

• Reflection coefficient (R): the ratio of reflected wave amplitude to incident wave amplitude. Determined by the impedance mismatch between the two media.
• Transmission coefficient (T): the ratio of transmitted wave amplitude to incident wave amplitude.

Energy conservation constrains these values: the reflected energy plus the transmitted energy must equal the incident energy. (Note that amplitude coefficients and energy/intensity coefficients are related but not identical, since intensity depends on amplitude squared and on the medium's impedance.)

Normal incidence is the simplest case, where the wave hits the boundary straight on (perpendicular). At normal incidence for pressure amplitudes:

$R = \frac{Z_2 - Z_1}{Z_2 + Z_1}$

$T = \frac{2Z_2}{Z_2 + Z_1}$

where $Z_1$ and $Z_2$ are the characteristic acoustic impedances of the first and second media.

Oblique incidence is more complex because the transmitted wave refracts (changes direction), and you need Snell's law to find the transmitted angle before calculating coefficients.

Snell's law for acoustic angles

When a wave hits a boundary at an angle, Snell's law relates the incident and transmitted angles to the sound speeds in each medium:

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

where $\theta_1$ is the angle of incidence (measured from the normal to the boundary), $\theta_2$ is the angle of transmission, and $c_1$, $c_2$ are the sound speeds in the first and second media.

A few key results follow from this:

• Angle of reflection equals angle of incidence. The reflected wave always bounces back symmetrically, just like a billiard ball off a cushion.
• Refraction bends the transmitted wave. If $c_2 > c_1$, the wave bends away from the normal (toward the boundary surface). If $c_2 < c_1$, it bends toward the normal.
• Critical angle. When $c_2 > c_1$, there's a critical angle $\theta_c = \arcsin(c_1 / c_2)$ beyond which no wave transmits into the second medium. All energy reflects back. This is total internal reflection, the same principle used in fiber optic cables and also relevant in underwater acoustics where sound can get trapped in certain ocean layers.

Boundary conditions in sound propagation

Different boundary types produce very different acoustic behavior:

Rigid boundaries (like a thick concrete wall) have extremely high impedance compared to air, so nearly all sound energy reflects. The reflected pressure wave arrives in phase with the incident wave, creating a pressure maximum at the surface.

Soft (pressure-release) boundaries have very low impedance relative to the source medium. An example is the water-air interface as seen from underwater: sound in water hitting the surface reflects with a phase inversion, creating a pressure minimum at the boundary.

Fluid-fluid interfaces (like a water-oil boundary) transmit and reflect longitudinal waves according to the impedance mismatch. Both media support pressure waves, so the analysis follows directly from the reflection and transmission coefficients.

Solid-fluid interfaces are more complex because solids support both longitudinal and shear waves while fluids only support longitudinal waves. An incident longitudinal wave can generate both wave types in the solid, which is mode conversion. Seismic waves at the Earth's crust-mantle boundary are a classic example.

Layered media produce multiple reflections and transmissions at each layer boundary. Sound bouncing back and forth within layers can interfere constructively or destructively, which is the principle behind multi-layer acoustic insulation design.

Two additional phenomena worth noting:

• Absorption at boundaries converts some acoustic energy into heat. Sound-absorbing foam in recording studios works this way, reducing reflections by dissipating energy.
• Standing waves form when incident and reflected waves of the same frequency interfere in a confined space. Organ pipes and room resonances are common examples. The pattern of nodes (zero displacement) and antinodes (maximum displacement) depends on the boundary conditions at each end.