Acoustic Impedance Calculator, z = ρ·c, Z = ρ·c/A, R & T at a boundary

The acoustic impedance calculator solves the three core relations of linear acoustics for any medium or interface: the characteristic specific acoustic impedance z = ρ · c (intensive property of the medium, measured in rayls = Pa·s/m); the geometry-aware acoustic impedance Z = ρ · c / A for a duct or aperture (with A the cross-sectional area, in acoustic ohms = Pa·s/m³); and the normal-incidence amplitude reflection and transmission coefficients R = (Z₂ − Z₁)/(Z₂ + Z₁) and T = 2Z₂/(Z₂ + Z₁) at a plane boundary between two media of different impedance.

The page below explains the formulas in detail, gives the standard reference values for gases, liquids, solids and biological tissues, derives the boundary reflection/transmission formulas from first principles, and shows worked numerical examples that can be checked against the calculator's output.

How to use the acoustic impedance calculator

1. Pick a mode: Specific impedance z = ρ·c, Acoustic impedance Z = ρ·c/A, or Boundary R & T (normal incidence).
2. In the impedance modes, pick which variable to solve for. The natural choice is z (you give ρ and c); the calculator also back-solves for ρ = z/c and c = z/ρ, useful for sanity-checking a measurement.
3. Enter the density ρ in kg/m³ and the speed of sound c in m/s. Click any preset chip to load verified values for common materials.
4. In the geometry mode, also enter the cross-sectional area A in m². The geometry-aware impedance is what matters for ducts, horns, and Helmholtz resonators.
5. In the boundary mode, enter the impedances Z₁ and Z₂ (in rayls) of the two media on either side of a plane interface. The calculator returns R and T plus the power/intensity coefficients R² and 1 − R².
6. Click Calculate. The result shows the solved variable, the formula, and a results panel with the supporting quantities. The result panel uses exponential notation for very large or very small numbers (e.g. 1.48e+6 for water, 413.3 for air).

What is acoustic impedance?

Acoustic impedance is the opposition a system presents to an acoustic flow driven by an acoustic pressure. The name emphasises the analogy with electrical impedance: pressure is analogous to voltage, volume flow to current, and acoustic impedance is the ratio of the two. Two closely related quantities appear throughout the literature:

• Specific acoustic impedance z (lowercase) is the ratio of acoustic pressure to acoustic particle velocity: z = p / v. It is an intensive property of the medium. The SI unit is Pa·s/m; the customary MKS unit is the rayl (named after Lord Rayleigh).
• Acoustic impedance Z (uppercase) is the ratio of acoustic pressure to acoustic volume flow rate: Z = p / Q = z / A. It depends on both the medium and the geometry (the cross-sectional area A of the aperture, duct, or pipe). The SI unit is Pa·s/m³, also called the acoustic ohm.

For a plane progressive wave in a lossless fluid, z reduces to a real number: z = ρ · c. This is the characteristic specific acoustic impedance (often written z₀). In the more general case (standing waves, reactive terminations, lossy media) z becomes a complex number with resistive (energy-dissipating) and reactive (energy-storing) parts.

The formula z = ρ · c

For a plane progressive acoustic wave in a uniform lossless medium, combining Newton's second law with the adiabatic compressibility gives that pressure and particle velocity are in phase, with constant ratio:

z = ρ · c

where ρ is the volumetric mass density (kg/m³), c is the speed of sound in the medium (m/s), and z is the characteristic specific acoustic impedance (rayls = Pa·s/m). Rearranged: ρ = z / c and c = z / ρ. For solids the wave speed depends on type: longitudinal waves use c_L = √((K + 4G/3)/ρ) where K is the bulk modulus and G is the shear modulus, while transverse (shear) waves use c_T = √(G/ρ). Liquids and gases cannot support shear (G = 0), so only longitudinal waves propagate and c = √(K/ρ). The calculator accepts a single c value, so for solid media use the longitudinal wave speed (the more common engineering value).

Geometry-aware acoustic impedance Z = ρ · c / A

The acoustic impedance of a duct, horn, or aperture of cross-sectional area A is:

Z = ρ · c / A

with units Pa·s/m³ (acoustic ohms). This is the impedance that enters the Helmholtz resonator formula, the loudspeaker-enclosure interaction, and the lumped-element models used in horn design. As the area shrinks, Z grows, a smaller opening presents more opposition to the same pressure-driven flow, which is why a tweeter horn with a small throat has a higher input impedance than an equivalent woofer with a large cone.

Reference values of ρ·c for common materials

The table below lists the specific acoustic impedance z = ρ·c for materials commonly encountered in acoustics, ultrasonics, and audio engineering. Values are at standard conditions (20 °C and 1 atm for gases; 20 °C for liquids and solids).

The single most striking pattern in the table is the five-orders-of-magnitude gap between gases (~10² rayls) and condensed matter (~10⁶ to 10⁷ rayls). This is why almost no acoustic energy crosses an air-water boundary, why whales cannot hear boats in the air above them, and why underwater SONAR uses frequencies 1,000× lower than airborne radar to achieve comparable range.

Reflection and transmission at a plane boundary

When a plane acoustic wave hits a flat boundary between two media of different characteristic impedances z₁ and z₂, part of the wave is reflected and part is transmitted. For normal incidence the amplitude coefficients are:

R = (z₂ − z₁) / (z₂ + z₁) (amplitude reflection coefficient) T = 2 z₂ / (z₂ + z₁) (amplitude transmission coefficient)

The corresponding power / intensity coefficients are R² for the reflected share and 1 − R² for the transmitted share. For a lossless boundary these sum to exactly 1. Three special cases:

• Matched impedances (z₁ = z₂): R = 0 and T = 1. The wave passes through with no reflection. This is the design goal of every acoustic coupling layer.
• Rigid termination (z₂ → ∞, e.g. air against a concrete wall): R = +1 and T = 0. The wave is fully reflected with no phase change.
• Pressure-release termination (z₂ → 0, e.g. sound in water at the air interface): R = −1 and T = 0. The wave is fully reflected with a phase inversion.

For air (z₁ ≈ 413) against water (z₂ ≈ 1.48 × 10⁶): R ≈ 0.99944, so R² ≈ 0.99889, about 99.89% of acoustic intensity is reflected, and only 0.11% enters the water. This is why SCUBA divers cannot hear someone shouting at them from the surface, and why sonar transducers need a couplant gel to displace air between the probe and the skin.

Why z is dimensionless in some texts and rayls in others

Specific acoustic impedance has dimensions of pressure × time / length, the same as mechanical impedance (force / velocity). In SI units this is Pa·s/m, and the MKS rayl is the same dimension by definition (1 rayl = 1 Pa·s/m). Modern acoustical standards (ISO 80000-8:2020) prefer Pa·s/m. The geometry-aware form Z has units of Pa·s/m³ (acoustic ohms), sometimes seen as "rayl per square metre" (Rayl/m²) in older MKS-system texts.

Acoustic impedance in medical ultrasound

Diagnostic ultrasound relies on the partial reflection of sound at tissue boundaries to build an image. The echo amplitude at any boundary is set by the impedance mismatch: a small mismatch (fat-muscle, z ≈ 1.41 vs 1.70 MRayl) gives a faint echo, while a large mismatch (soft tissue-bone, z ≈ 1.70 vs 7.55 MRayl) gives a bright echo and almost total reflection. This is why ultrasound cannot see through bone or air-filled lungs (the air-tissue mismatch reflects 99.99% of the beam). Blood (1.66 MRayl) and soft tissue (1.70 MRayl) are within 3% of each other, which is why ultrasound can image blood vessels and soft organs in a single pass.

Acoustic impedance matching and quarter-wave transformers

When two media have very different impedances (e.g. air and water, or piezoelectric transducer and skin), a matching layer is inserted between them to reduce the reflection coefficient. The optimum single-layer impedance is the geometric mean, z_match = √(z₁ · z₂). For air-water that gives z_match ≈ 24,800 rayls, exactly the impedance of typical acoustic gels and rubbery polymers, which is why those materials are used as ultrasound couplants. A quarter-wave transformer, a layer of thickness λ/4 with impedance √(z₁ · z₂), gives zero reflection over a band around that frequency. The same trick is used in optics with anti-reflection coatings.

Standing waves, organ pipes, and room modes

The acoustic impedance determines how a wave reflects from the end of a pipe. An open end acts approximately as a pressure-release boundary (R = −1, phase inversion); a closed end acts as a rigid boundary (R = +1, no phase change). These reflections set the boundary conditions for the standing-wave modes: an open-closed pipe (like a clarinet) has fundamental wavelength 4L; an open-open pipe (like a flute) has 2L. Higher z = ρ·c means more energy flux for the same particle velocity, which is why an organ pipe radiates much louder in helium (high c, modest z) than in a heavy gas like SF₆.

Common pitfalls

• Confusing ρ·c with the speed of sound c. c tells you how fast a wave moves; z tells you how strongly the medium resists flow. Both depend on the medium but measure different things.
• Using the speed of sound in air for water. Air: c ≈ 343 m/s. Water: c ≈ 1482 m/s. Steel: c ≈ 5960 m/s. Confusing them gives wildly wrong impedance values, off by a factor of 4 to 17.
• Forgetting to divide by A. z is the specific (intensive) impedance; Z = z/A is the geometry-aware (extensive) impedance. The distinction matters for ducts, horns, and Helmholtz resonators, but not for free-field waves.
• Applying the normal-incidence formulas at an angle. R = (z₂ − z₁)/(z₂ + z₁) and T = 2z₂/(z₂ + z₁) are valid for normal incidence only. At oblique incidence cos θ factors appear, and above a critical angle total internal reflection can occur.
• Reporting intensity reflection as R instead of R². The amplitude reflection coefficient is what the formula returns. To get power / intensity reflection, square it.

Frequently Asked Questions

What is the difference between acoustic impedance z and characteristic impedance z₀? For a plane progressive wave in a uniform lossless medium, z equals the real number ρ · c. This is the characteristic specific acoustic impedance, often denoted z₀. In the more general case, standing waves, ducts, boundary layers, z becomes a complex number z = r + i x where r is the resistive part (in-phase with particle velocity, dissipates energy) and x is the reactive part (out-of-phase, stores energy). The characteristic impedance z₀ is the special case where the impedance is purely real.

What are the SI units and customary units of acoustic impedance? Specific acoustic impedance z has SI units of Pa·s/m (pascal-second per metre). The customary unit is the rayl (named after Lord Rayleigh), with 1 rayl = 1 Pa·s/m exactly. Acoustic impedance Z (which includes the geometry, Z = z/A) has SI units of Pa·s/m³, also called the acoustic ohm.

How do I compute the reflection coefficient at an air-water interface? At 20 °C, the specific acoustic impedance of air is z₁ ≈ 413.3 rayls (1.2041 × 343.21) and of fresh water is z₂ ≈ 1.48 × 10⁶ rayls (998.2 × 1482). The amplitude reflection coefficient is R = (z₂ − z₁)/(z₂ + z₁) ≈ 0.99944. Squaring gives the power reflection coefficient R² ≈ 0.99889, so about 99.89% of acoustic intensity is reflected and only 0.11% enters the water. This is why sonar and underwater acoustic communication need a coupling layer to displace the air.

How do I convert between intensity and pressure reflection coefficients? The amplitude pressure reflection coefficient R relates pressure amplitudes: p_reflected = R · p_incident. Because intensity is proportional to pressure squared (I = p² / z), the intensity (or power) reflection coefficient is R². same air-water case, R² ≈ 0.9989, so 99.89% of incident intensity is reflected. If a problem statement gives a percent reflected as a percentage, that is the intensity reflection coefficient and you should take its square root to recover the amplitude coefficient.

What is the acoustic impedance of air at body temperature (37 °C)? Air at 37 °C has density ρ ≈ 1.137 kg/m³ and speed of sound c ≈ 353 m/s. The specific acoustic impedance is z = 1.137 × 353 ≈ 401.4 rayls, slightly lower than the 20 °C value (≈ 413 rayls) because density falls faster than c rises as temperature increases. Both are within 5% of the standard 413 rayls.

How does the acoustic impedance of biological tissues affect ultrasound imaging? Diagnostic ultrasound relies on the small impedance differences between soft tissues to produce images. Blood (z ≈ 1.66 MRayl), soft tissue (muscle, liver, z ≈ 1.70 MRayl), and fat (z ≈ 1.41 MRayl) are all within a few percent of each other, which is why ultrasound passes through the abdomen reasonably well. Bone (z ≈ 7.55 MRayl) and air-filled lung (z ≈ 413 rayls) are extreme mismatches that reflect almost all the beam, which is why ultrasound cannot see through bone or aerated lung. Image contrast is set by R = (z₂ − z₁)/(z₂ + z₁) evaluated at every tissue boundary the beam crosses.

Why does the speed of sound in a solid depend on whether it is a longitudinal or shear wave? For an isotropic solid, the longitudinal wave speed is c_L = √((K + 4G/3)/ρ) and the shear (transverse) wave speed is c_T = √(G/ρ), where K is the bulk modulus and G is the shear modulus. Liquids and gases cannot support shear, so c_T = 0 and only longitudinal waves propagate; c = √(K/ρ). The calculator's solid-material presets use the longitudinal wave speed, so the resulting z = ρ · c is the longitudinal characteristic impedance. For shear-wave impedance, enter c_T instead of c_L.

References

• Kinsler, L. E., Frey, A. R., Coppens, A. B., Sanders, J. V. (2000). Fundamentals of Acoustics, 4th ed. Wiley. ISBN 0-471-84789-5.
• Attenborough, K., Postema, M. (2008). A pocket-sized introduction to acoustics. Zenodo. doi:10.5281/zenodo.7504060.
• Rossing, T. D., Fletcher, N. H. (2004). Principles of Vibration and Sound, 2nd ed. Springer. ISBN 978-1-4757-3822-3.
• Fletcher, N. H., Rossing, T. D. (1998). The Physics of Musical Instruments, 2nd ed. Springer. ISBN 978-0-387-21603-4.
• ISO 80000-8:2020. Quantities and units, Part 8: Acoustics.

Worked example

Take air at 20 °C: density 1.204 kg/m³ and sound speed 343 m/s, so the specific acoustic impedance is Z = 1.204 x 343 = 413 rayl. Water has a density of 998 kg/m³ and a sound speed of 1,483 m/s, giving Z = 1.48 x 10^6 rayl. The ratio between the two is about 3,580:1, which is why almost all the sound energy reflects at an air-water boundary instead of crossing it. Enter those densities and speeds into the calculator and you get the same figures, along with the reflection coefficient for any pair of media.