7.3 Transmission and reflection at boundaries

Sound waves behave uniquely at boundaries between different media. They can reflect, transmit, or transform, depending on the properties of the materials they encounter. Understanding these interactions is crucial for predicting sound propagation in various environments.

Reflection, transmission, and refraction are key concepts in acoustic wave behavior. Snell's law helps calculate how waves change direction when crossing boundaries. Different boundary types, from rigid walls to fluid interfaces, affect sound waves in distinct ways, influencing their energy and direction.

Acoustic Wave Behavior at Boundaries

Behavior of sound waves at boundaries

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• Incident waves approach boundary at specific angle carrying energy from source medium
• Reflected waves bounce back into original medium changing direction at boundary
• Transmitted waves pass through boundary into second medium potentially changing speed and direction
• Impedance mismatch between media affects reflection and transmission amounts (air-water interface)
• Mode conversion may occur transforming longitudinal waves to transverse waves or vice versa (P-waves to S-waves in seismology)

Reflection and transmission principles

• Reflection coefficient measures ratio of reflected to incident wave amplitudes dependent on impedance mismatch
• Transmission coefficient measures ratio of transmitted to incident wave amplitudes complementary to reflection coefficient
• Energy conservation ensures sum of reflected and transmitted energy equals incident energy
• Normal incidence simplifies calculations with waves perpendicular to boundary (sound hitting flat wall)
• Oblique incidence requires consideration of refraction with waves at angle to boundary (ocean waves approaching beach)

Snell's law for acoustic angles

• Snell's law formula: $\frac{\sin\theta_1}{c_1} = \frac{\sin\theta_2}{c_2}$ where $\theta_1$ is incident angle, $\theta_2$ is transmitted angle, $c_1$ and $c_2$ are sound speeds in respective media
• Angle of reflection equals angle of incidence (billiard ball bouncing off cushion)
• Critical angle marks threshold beyond which total internal reflection occurs (fiber optic communications)
• Refraction changes wave direction due to speed difference between media (light bending in water)

Boundary conditions in sound propagation

• Rigid boundaries create high impedance mismatch leading to near-total reflection (sound reflecting off concrete wall)
• Soft boundaries have low impedance mismatch allowing significant transmission (sound passing through fabric curtain)
• Fluid-fluid interfaces allow both shear and longitudinal waves in both media (water-oil interface)
• Solid-fluid interfaces may cause mode conversion (seismic waves at earth's crust-mantle boundary)
• Layered media produce multiple reflections and transmissions (sound in multi-layer insulation)
• Absorption at boundaries converts acoustic energy to heat (sound-absorbing foam in recording studios)
• Standing waves form due to interference between incident and reflected waves (organ pipes)
• Diffraction bends waves around obstacles or through openings (sound heard around corner)