Reflection and Transmission in Solids

Reflection and transmission of elastic waves at interfaces between solids with different acoustic impedance.

Conceptual context

When an elastic wave travels through a solid and encounters an interface where medium properties change, part of its energy bounces back and part continues into the second material. This partition is not accidental: it depends exclusively on the acoustic impedances of both media. Impedance summarizes in a single number the medium's opposition to wave passage, and the impedance contrast governs how much amplitude and energy are reflected or transmitted.

This phenomenon has direct consequences in engineering (ultrasonic inspection, coupling layer design), in geophysics (detection of seismic discontinuities) and in medicine (ultrasound imaging). Understanding reflection and transmission is prerequisite to any analysis of wave signals crossing more than one medium.

🟢 Essential level

The central idea is that a wave cannot cross an interface without adapting to the new medium conditions. If the second material has the same impedance as the first, the wave passes without noticing the boundary. If impedances are very different, most of the wave bounces back.

$Z_1$ and $Z_2$ are the two numbers being compared. Each is calculated as the product of medium density and wave velocity in that medium; the same product can be read as $Z$ when referring to a generic medium impedance. A dense and stiff medium (like steel) has high impedance. A light and flexible medium (like air) has low impedance.

$R$ indicates what fraction of the amplitude bounces back. If impedances match, $R$ is zero: no reflection. If they differ greatly, $R$ approaches unity in absolute value: nearly all the wave bounces back.

$T$ indicates what fraction of the amplitude passes to the second medium. For normal incidence, reflected and transmitted amplitudes add up to the incident amplitude in terms of displacement continuity.

A confusing point is that the transmitted amplitude can be larger than the incident one without violating any law. This happens when the second medium has lower impedance. There is no contradiction because transmitted energy depends on the product of squared amplitude and impedance, and that product always respects conservation.

At this level it suffices to retain that impedance contrast produces reflection, and absence of contrast produces total transmission.

🔵 Formal level

Basic formulas

The boundary condition at the interface requires continuity of displacement and normal stress. Imposing these two conditions on plane waves at normal incidence yields the reflection and transmission coefficients in amplitude:

Amplitude reflection coefficient

Amplitude transmission coefficient

The relationship between both coefficients satisfies the continuity identity. Summing contributions verifies that the total amplitude at the interface (incident plus reflected on side 1, transmitted on side 2) is consistent.

Real student questions

How can the transmitted amplitude be greater than the incident one? Because energy does not depend only on amplitude, but on the product of squared amplitude and impedance. In a low-impedance medium, amplitude grows, but energy per unit amplitude is less.

Does reflection always cause energy loss in the second medium? It is not a loss, it is a partition. Total energy is conserved: what is not transmitted is reflected. There is no dissipation at an ideal interface.

Final synthesis

Reflection and transmission in solids are a direct consequence of acoustic impedance contrast at an interface. The impedance ratio completely determines the coefficients for normal incidence. Energy conservation requires that the sum of energy coefficients equals unity. The didactic key is not to confuse amplitude with energy: a transmitted amplitude greater than one does not violate physics, because transported power also depends on the impedance of the receiving medium.