Interface reflection and transmission coefficients

At normal incidence on a plane boundary between two media the split of sound between reflection and transmission is fixed entirely by the two acoustic impedances Z = ρ·c. The pressure reflection coefficient is R = (Z₂ − Z₁)/(Z₂ + Z₁) and the pressure transmission coefficient is T = 2Z₂/(Z₂ + Z₁), where medium 1 is the one the wave arrives in.

Two things surprise people. First, R is negative when the second medium is the softer one (steel into air, steel into a resin-filled defect): the reflected pulse comes back phase-inverted, which is how a flaw echo can be distinguished from a backwall echo on an RF display. Second, the pressure transmission coefficient can exceed 1 — pressure amplitude in the stiffer medium is higher than in the incident medium — without violating energy conservation, because the particle velocity is correspondingly lower.

Energy (intensity) coefficients are the ones to quote for sensitivity budgets: Rᵢ = R² and Tᵢ = 4Z₁Z₂/(Z₁+Z₂)² = 1 − Rᵢ. Tᵢ is also the two-way pressure factor for a pulse that crosses the interface, reflects deeper in, and crosses back — so the dB figure it gives is the real transmission loss you lose from a wedge-to-steel or water-to-steel boundary on every scan.

Typical impedances (MRayl, compression): air 0.0004, water 1.48, perspex/acrylic 3.22, Rexolite 2.45, aluminium 17.1, ferritic steel 46.3, copper 41.6, tungsten 100. This calculation assumes normal incidence and a smooth, clean, unbonded interface; at oblique incidence mode conversion redistributes the energy and these figures no longer apply.

Z = ρ · c
R (pressure) = (Z₂ − Z₁) / (Z₂ + Z₁)
T (pressure) = 2·Z₂ / (Z₂ + Z₁)
R (energy) = R²
T (energy) = 4·Z₁·Z₂ / (Z₁ + Z₂)² = 1 − R²