Used by the air, water, and seawater equations.
Mackenzie’s stated range is 30–40 PSU.
Terminology: \(z=\rho c\) is specific impedance in Pa·s/m. \(Z_c=\rho c/S\) is volume-velocity/tube characteristic impedance in Pa·s/m³; it is not the same quantity.
Choose two presets or select “Custom material” and enter density and sound speed. Their impedances are calculated automatically.
Ideal single quarter-wave transformer: \(z_m=\sqrt{z_1z_2}\), \(\lambda_m=c_m/f\), \(t=\lambda_m/4\). Assumes lossless materials, plane waves, normal incidence, and a layer with the specified longitudinal sound speed.
Do not interchange the symbols: this page uses lowercase \(z\) for the material property. It reserves \(Z_c\) for tube/volume-velocity characteristic impedance. Radiation impedance and complex, frequency-dependent impedance require source geometry, phase, loss, and frequency data beyond \(\rho c\).
Ideal dry-gas model: \(\rho=101325/[287.05(293.15)]=1.204\text{ kg/m³}\), \(c=\sqrt{1.4(287.05)(293.15)}=343.2\text{ m/s}\).
\(z=1.204(343.2)=413.3\text{ Rayl}=0.0004133\text{ MRayl}\). Interpretation: air has very low specific impedance compared with liquids and solids.
The stated water fits give \(\rho\approx998.23\text{ kg/m³}\) and \(c\approx1482.36\text{ m/s}\).
\(z=998.23(1482.36)\approx1.480\times10^6\text{ Rayl}=1.480\text{ MRayl}\). Interpretation: water is roughly 3,580 times the impedance of room-temperature air.
Using \(z_1=413.3\) Rayl and \(z_2=1.480\times10^6\) Rayl: \(r=0.99944\), \(R=r^2=0.99888\), and \(T=1-R=0.00112\).
That is 99.888% reflected and 0.112% transmitted. The positive \(r\) means the reflected pressure is in phase.
Using representative \(z_1=1.500\text{ MRayl}\) and \(z_2=1.628\text{ MRayl}\): \(r=(1.628-1.500)/(1.628+1.500)=0.04092\).
\(R=0.001675=0.168\%\) and \(T=0.998325=99.832\%\). Interpretation: gel removes the much worse probe–air–skin mismatch and couples nearly all modeled energy into skin.
Search the calculator library above to load any row. Values are representative, not certification data; tissue composition, alloy, grain direction, moisture, porosity, frequency, and temperature can materially change them.
| Category | Material | Condition | Density (kg/m³) | c (m/s) | Rayl | MRayl |
|---|
Reference basis: air impedance and units follow the UK National Physical Laboratory acoustics guide; pure-water equations follow the NPL pure-water guide; tissue presets are representative values based on the IT’IS Foundation tissue-properties database; metals use longitudinal-wave data from Evident’s NDT material-velocity table. Concrete, glass, gel, and biological values vary widely and should be measured for design work.
Applications: first-pass checks for ultrasound coupling and transducer matching, sonar and hydrophone interfaces, ducts and horns, building-material inspection, and acoustics education.
Boundary assumptions: homogeneous lossless media, a plane progressive wave, a flat boundary, normal incidence, positive real specific impedances, and no absorption, scattering, shear-mode conversion, surface roughness, or finite-layer interference. For oblique incidence, solids, porous media, radiation loading, or lossy/dispersive media, use angle-, mode-, and frequency-dependent complex models.
Model references: NPL, Speed of sound in pure water; K. V. Mackenzie, “Nine-term equation for sound speed in the oceans” (JASA 70, 1981); Fofonoff & Millard, UNESCO Technical Papers in Marine Science 44 (1983). These values are unsuitable as a substitute for measured data in safety-critical, diagnostic, contractual, or certification work.
Technical author: Starlight Tools Editorial Team. Last reviewed: 19 July 2026. No independent technical review is claimed.
One Rayl is one pascal-second per metre (Pa·s/m), the SI unit of specific acoustic impedance. One MRayl is 1,000,000 Rayl.
The lossless characteristic value \(\rho c\) is often treated as frequency-independent over a useful band. Real materials can be dispersive and lossy, however, so measured complex impedance may vary with frequency.
Gel displaces the air gap between probe and skin. Removing that severe impedance mismatch lets much more ultrasound energy enter the body.
It means the reflected pressure wave is inverted by 180° because the wave travels from higher specific impedance toward lower specific impedance. Reflected energy \(R=r^2\) remains positive.
The simple equations assign all incident energy to reflection or transmission. Absorption, scattering, mode conversion, and lossy layers require additional terms.
Specific impedance \(z=\rho c\) is a material property in Pa·s/m. Tube characteristic impedance \(Z_c=\rho c/S\) also includes duct area and has units Pa·s/m³.
Temperature changes both density and sound speed. At fixed absolute pressure, warmer air generally has lower \(\rho c\); water follows a nonlinear temperature relationship, so use a model within its stated range.
