Wave propagation: incident (blue), reflected (orange), transmitted (green)
Standing wave pattern (amplitude envelope)
Acoustic impedance Z = ρ·c (measured in rayls, or Pa·s/m) is the product of a material's density ρ and the speed of sound c within it. When a sound wave encounters a boundary between two materials with different impedances, it partially reflects and partially transmits according to the pressure reflection coefficient R = (Z₂ − Z₁)/(Z₂ + Z₁) and the transmission coefficient T = 2Z₂/(Z₁ + Z₂). The power reflection coefficient is R², meaning that a large impedance mismatch — for example, from air (Z = 415 rayl) to water (Z = 1.48 MRayl) — reflects almost all the sound energy (R² ≈ 0.9989, a 30 dB transmission loss). This principle is fundamental to ultrasound medical imaging and sonar design.
The simulation visualises incident (blue), reflected (orange), and transmitted (green) waves moving in real time across a boundary at 55% of the canvas width. A second panel shows the standing wave envelope on the incident side, which forms when the reflected wave interferes with the incident wave. You can select from five real materials (air, water, soft tissue, bone, steel), adjust frequency (0.1–10 kHz), and control wave amplitude, while the stats panel computes R, T, transmission loss in dB, standing wave ratio (SWR), and energy conservation.
What is acoustic impedance and why does it matter?
Acoustic impedance Z = ρ·c determines how much opposition a material offers to a propagating sound wave. When two materials have different impedances, sound cannot cross the boundary without partial reflection. The mismatch determines the fraction of energy transmitted: for small mismatches (like water to soft tissue, Z₁ = 1.48 MRayl, Z₂ = 1.63 MRayl) nearly all energy crosses; for large mismatches (air to tissue) almost none does, which is why ultrasound gel is applied to skin before medical scans.
What does the reflection coefficient R represent?
R = (Z₂ − Z₁)/(Z₂ + Z₁) is the pressure reflection coefficient, ranging from −1 to +1. Its square R² is the fraction of incident power that is reflected. When Z₂ > Z₁ (harder medium), R is positive (no phase inversion); when Z₂ < Z₁ (softer medium), R is negative (180° phase shift on reflection). For a perfect impedance match (Z₁ = Z₂), R = 0 and 100% of sound is transmitted.
What is the standing wave ratio (SWR)?
The SWR = (1 + |R|)/(1 − |R|) measures the ratio between the maximum and minimum acoustic pressure amplitudes on the incident side, where the incident and reflected waves interfere. An SWR of 1 means no reflection (perfect match). An SWR approaching infinity means total reflection (no transmission). In the simulation, air–steel gives SWR ≈ ∞ while water–soft tissue gives SWR ≈ 1.06, showing near-perfect matching.
Air has Z ≈ 415 Pa·s/m while soft tissue has Z ≈ 1.63 MRayl — a mismatch of roughly 3,500:1. Inserting this into the power reflection formula gives R² ≈ 0.9989, meaning only about 0.1% of sound energy enters the body. Ultrasound gel (with Z close to tissue) eliminates the air gap at the skin surface, raising transmission to nearly 100% and making medical imaging possible.
In sonar and non-destructive testing, transducers are matched to the load medium using quarter-wavelength matching layers with impedance Z_match = √(Z₁·Z₂). A layer exactly one quarter-wavelength thick creates destructive interference for reflected waves, minimising the overall reflection. Multiple matching layers are stacked in modern piezoelectric transducers to achieve broadband transmission across large impedance mismatches.
The pressure transmission coefficient T = 2Z₂/(Z₁+Z₂) gives the amplitude ratio of transmitted to incident pressure. However, energy (intensity) transmission is T_energy = (Z₁/Z₂)·T² = 1 − R². When Z₂ > Z₁, the transmitted pressure can exceed the incident pressure (T > 1) even though energy is conserved, because the higher-impedance medium carries the same power at lower particle velocity but higher pressure.
Transmission loss (TL) = −20·log₁₀(|T|) in decibels measures how much weaker the transmitted signal is compared to the incident signal, in logarithmic scale. A TL of 0 dB means no loss; 6 dB means the transmitted amplitude is halved; 20 dB means a 10-fold amplitude reduction. Air-to-steel gives TL ≈ 33 dB, while water-to-soft tissue gives TL ≈ 0.4 dB.
In the simple planar wave model used here, the reflection and transmission coefficients at a single boundary are frequency-independent. However, frequency matters greatly in layered systems (where resonance and anti-resonance between layers produce frequency-dependent transmission), in lossy materials (where higher frequencies are absorbed more strongly), and in biological tissue where frequency determines both resolution and penetration depth in ultrasound imaging.
Air: 415 Pa·s/m; water: 1.48 MRayl; soft tissue: 1.63 MRayl; fat: 1.38 MRayl; bone: 6–8 MRayl; steel: 47 MRayl; lead: 24 MRayl; rubber: 0.9–1.8 MRayl. The large mismatch between bone and soft tissue (R² ≈ 0.68) is why ultrasound cannot image through bone — most energy is reflected at the cortical bone surface.
On the incident side, the total acoustic field is the superposition of the incident wave and the reflected wave travelling in opposite directions. When |R| > 0, they interfere to form a standing wave pattern with spatial maxima (antinodes) at λ/2 intervals. The envelope plot shows this amplitude distribution — wider envelope means stronger partial standing wave. When R = 0 (perfect match), the envelope is flat, confirming no standing wave forms.