What wave impedance is and why it matters
The characteristic (acoustic) impedance of a medium, Z, measures how much pressure a sound wave produces for a given particle velocity. It is a property of the material: dense, stiff materials such as water or steel have a much higher impedance than air. When a wave travels from one medium into another, the size of the impedance step decides how much of the wave passes through and how much bounces back at the boundary.
This calculator computes the impedance of the first medium from its density and wave speed, then compares it with the impedance you enter for a second medium to give the pressure reflection and transmission coefficients. It applies to plane waves meeting a flat boundary head-on, and is useful for ultrasound and sonar basics, acoustic insulation questions, and understanding why sound in air barely enters water.
Formulas and definitions
For plane waves at normal incidence the relationships are:
- Z1 = ρ × c, with density ρ in kg/m³ and wave speed c in m/s. The unit is the rayl (kg/(m²·s), equivalently Pa·s/m).
- Reflection coefficient R = (Z2 − Z1) / (Z2 + Z1), the ratio of reflected to incident pressure amplitude.
- Transmission coefficient τ = 2 Z2 / (Z2 + Z1), the ratio of transmitted to incident pressure amplitude; note τ = 1 + R.
- Z2 is the impedance of the second medium, entered directly in the same units as Z1.
- The boundary note labels |R| below 0.1 a good match, below 0.4 a moderate mismatch, and 0.4 or more a strong reflection boundary.
Worked example
Take air with ρ = 1.225 kg/m³ and c = 343 m/s, and a second medium with Z2 = 1,000. Then Z1 = 1.225 × 343 = 420.175.
R = (1,000 − 420.175) / (1,000 + 420.175) = 579.825 / 1,420.175 ≈ 0.4083. The transmission coefficient is 2 × 1,000 / 1,420.175 ≈ 1.4083, which equals 1 + 0.4083. Because |R| is at least 0.4, the note reads Strong reflection boundary. These match the calculator's defaults.
A pressure transmission coefficient above 1 can look odd, but it does not mean energy is created. Intensity depends on pressure squared divided by impedance, and the fraction of intensity transmitted is 4 Z1 Z2 / (Z1 + Z2)² = 1 − R² ≈ 0.833 in this example.
Common mistakes and how to interpret the result
- Confusing pressure and intensity coefficients. R and τ here describe pressure amplitude. The energy reflected is R² and the energy transmitted is 1 − R², and τ can exceed 1.
- Mixing units for Z2. Z2 must be in the same units as ρ × c (rayl). Water is roughly 1.5 million rayl, so entering 1,000 as in the default is only a placeholder.
- Ignoring angle. The formulas hold only at normal incidence. Oblique waves need Snell's law and angle-dependent coefficients.
- Forgetting that R is signed. A negative R means the reflected pressure wave is inverted, which happens when the second medium has lower impedance than the first.