Acoustic Impedance Calculator - Z = rho c for Any Medium
Use this acoustic impedance calculator to compute Z = rho c from density and speed of sound, with material presets and a reflection coefficient.
Acoustic Impedance Calculator
Results
What Is Acoustic Impedance Calculator?
An acoustic impedance calculator returns the characteristic Z of a medium from its density and the speed of sound in it, using Z = rho * c, with results in Pa.s/m, Rayl, and MRayl and an optional pressure reflection coefficient between two media.
- • Acoustics and engineering homework: Confirming textbook values for the characteristic Z of air, water, tissue, and metals in Pa.s/m and Rayl.
- • Medical ultrasound physics: Estimating the pressure reflection coefficient at tissue boundaries such as fat-muscle or muscle-bone.
- • Audio transducer design: Picking a matching layer so the impedance of a transducer and the surrounding fluid stay close to reduce reflected power.
- • Underwater acoustics and sonar: Comparing seawater Z against hull material Z to predict how much acoustic energy crosses the hull.
The Z of a medium decides how much of a sound wave's energy crosses an interface and how much is reflected. A much higher Z acts almost like a wall to the wave; similar Z values transmit almost everything.
Characteristic Z is a property of the bulk medium and does not change with frequency for most gases, liquids, and homogeneous solids.
If you already know the frequency and wavelength of the wave but not the speed of sound in the medium, the wave speed calculator returns v = f * lambda and its rearrangements in one panel.
How Acoustic Impedance Calculator Works
The calculator reads the density and the speed of sound in the medium, computes Z = rho * c in Pa.s/m, converts it to Rayl and MRayl for cross-checking against textbook tables, and, when a second medium is provided, returns the pressure reflection coefficient r = (Z2 - Z1)/(Z2 + Z1) and the transmission coefficient 1 - r^2.
- rho (density): Mass per unit volume in kg/m^3. Air at 20 C is 1.204, water at 20 C is 998.2, soft tissue averages 1060.
- c (speed of sound): Phase speed of longitudinal sound waves in m/s. Air at 20 C is 343, water at 20 C is 1482, longitudinal steel is 5960.
- rho2, c2 (second medium): Density and speed of sound in a second medium, used only when both values are positive. The calculator then returns Z2 and the reflection coefficient against the first medium.
Pa.s/m, Rayl, and MRayl are the same family of units. One Rayl equals one Pa.s/m, one MRayl is one million Rayl. The calculator returns all three so you can read the answer in the units your textbook uses.
Worked example: dry air at 20 C
rho = 1.204 kg/m^3, c = 343 m/s, no second medium.
Z = 1.204 * 343 = 413.0 Pa.s/m. In Rayl that is 413, and in MRayl it is 0.000413.
Z = 413 Pa.s/m = 413 Rayl = 0.000413 MRayl.
This is the textbook value for the characteristic Z of air at 20 C, used as a reference for air-coupled ultrasound and speaker design.
Worked example: air-to-water reflection at 20 C
Medium 1: rho = 1.204, c = 343 (air). Medium 2: rho = 998.2, c = 1482 (water).
Z1 = 413 Pa.s/m. Z2 = 1,479,332 Pa.s/m. r = (1,479,332 - 413)/(1,479,332 + 413) = 0.99944. T = 1 - 0.99944^2 = 0.001114.
r = 0.9994 (pressure), T = 0.0011 (intensity fraction transmitted).
Almost all of the pressure wave is reflected at an air-water boundary, which is why a coupling gel is needed between a probe and skin in medical ultrasound.
According to Wikipedia - Acoustic impedance, the characteristic specific acoustic impedance Z_0 = rho * c and the more general specific acoustic impedance z = p / u both have units of pascal-seconds per metre (Pa.s/m) and are called the rayl, while the volume-velocity acoustic impedance Z = p / U has units of pascal-seconds per cubic metre (Pa.s/m^3) and is the unit most often called the acoustic ohm.
When the wave is described as a sinusoidal harmonic instead of by Z = rho * c, the harmonic wave equation calculator returns the displacement y(x,t) at any position and time.
Key Concepts Explained
Four concepts are worth keeping next to the result panel: characteristic versus specific impedance, Z = rho * c, the pressure reflection coefficient, and the Rayl versus MRayl units.
Characteristic Z = rho * c
Z is the product of density rho and sound speed c, both bulk properties of the medium. It has units of pascal-seconds per metre (Pa.s/m), which is the same as the Rayl. It equals the specific acoustic impedance z = p / u that a plane wave sees in an unbounded medium.
Specific z = p / u
z is the local ratio of acoustic pressure to particle velocity at a single point. It depends on position and frequency, while Z = rho * c does not.
Pressure reflection coefficient r = (Z2 - Z1)/(Z2 + Z1)
r is the fraction of the pressure wave that bounces back at a flat boundary. Identical mediums give 0, an open end gives -1, a rigid wall gives +1.
Rayl and MRayl in practice
Air sits in the hundreds of Rayl, water and soft tissue in the 1-2 MRayl range, and structural metals in the tens of MRayl.
These four concepts describe how much of a sound wave crosses a flat interface and how much reflects, which is the practical question behind impedance matching.
For the steady-flow pressure-versus-velocity counterpart of the acoustic pressure-versus-particle-velocity relation, the Bernoulli equation calculator returns the pressure drop across a pipe or venturi from flow speed and height.
How to Use This Calculator
Five steps are enough to read the characteristic Z of any medium and, optionally, the reflection coefficient at a second medium.
- 1 Pick a material preset: Use the preset selector to load standard density and speed of sound for air, water, soft tissue, steel, aluminum, glass, or helium. Pick Custom to enter both by hand.
- 2 Enter density rho: Type the mass per unit volume of the first medium in kg/m^3. The preset already fills this, but you can override it for a specific temperature.
- 3 Enter speed of sound c: Type the longitudinal sound speed in the medium in m/s. For air at 20 C, 343 m/s is a clean reference; for water at 20 C, use 1482 m/s.
- 4 Optionally enter a second medium: Type rho2 and c2 for the second medium if you want a reflection coefficient. Leave either field at 0 to hide the reflection and transmission rows.
- 5 Read the Z, reflection, and transmission: The result panel shows Z in Pa.s/m, Rayl, and MRayl, plus r and 1 - r^2 when a second medium is present.
For air to water at 20 C, pick Air (20 C) as the first preset (Z1 = 413 Rayl) and Water (20 C) as the second (Z2 = 1.48 MRayl), then read r = 0.9994 in the result panel.
If the same medium is excited by a single mechanical oscillator instead of a traveling wave, the vibration natural frequency calculator returns the natural frequency in hertz from the mass and stiffness.
Benefits of Using This Calculator
These benefits are the workflow improvements when Z = rho * c, the Rayl and MRayl conversion, and the reflection coefficient are no longer done by hand.
- • Single-panel density and speed of sound: Enter rho and c once, and the result panel returns Z in three units at the same time, instead of doing the multiplication and the Rayl or MRayl conversion separately.
- • Seven material presets: Air at two temperatures, helium, water, soft tissue, steel, aluminum, and glass load standard density and speed of sound values that match published tables.
- • Reflection coefficient at any interface: Type rho2 and c2 and the calculator returns r = (Z2 - Z1)/(Z2 + Z1) and the transmission fraction, the two numbers that decide how much acoustic energy crosses a boundary.
- • Three impedance units in one result panel: Switching between Pa.s/m, Rayl, and MRayl is no longer needed; the calculator echoes all three at once for cross-checking against textbook tables.
- • Real-time updates on input change: Edit any density or speed field and the impedance, reflection, and transmission values refresh in the result panel without a recalculate click.
When the reflection coefficient has to be reported as an amplitude ratio in dB for a cable, antenna, or acoustic matching layer, the decibel calculator converts a linear ratio to decibels in one step.
Factors That Affect Your Results
Four factors decide the number you read, plus two limitations when Z = rho * c is used as a model for a real material.
Temperature changes both rho and c
Air at 0 C is roughly 331 m/s and 1.293 kg/m^3, while air at 20 C is 343 m/s and 1.204 kg/m^3, giving a noticeably different Z.
Soft tissue composition shifts Z by a few percent
Fat, muscle, and bone all sit near 1.5-1.7 MRayl but not at exactly the same value. The 1060 kg/m^3 and 1540 m/s values used here are textbook averages from medical ultrasound references.
Material is not always isotropic
Single-crystal solids and fiber-reinforced composites have different Z along different axes. The longitudinal values used here assume an isotropic solid, the standard textbook case.
High-frequency absorption is not in Z
Characteristic Z describes lossless propagation. Real materials absorb sound, especially at high frequency, and that loss shows up as an attenuation coefficient, not in Z.
- • Z = rho * c assumes a homogeneous, lossless medium and a plane wave. Near boundaries, in small enclosures, or at very high frequency the wave is no longer a plane wave and the local specific impedance z = p / u differs from the bulk Z.
- • The reflection coefficient formula r = (Z2 - Z1)/(Z2 + Z1) is exact only for a plane wave hitting a flat boundary larger than the wavelength. At curved or rough interfaces, scattering changes the reflected fraction.
According to Engineering Toolbox - Speed of Sound in Solids, longitudinal sound speed in steel is 5960 m/s and in aluminum is 6420 m/s at room temperature, which sets Z for those metals in the tens of MRayl.
According to NDE-Ed.org - Ultrasound Physics, medical ultrasound imaging assumes an average speed of sound of 1540 m/s in soft tissue, which sets the textbook characteristic Z for soft tissue at about 1.6 MRayl.
Frequently Asked Questions
Q: What is acoustic impedance?
A: Acoustic impedance is the opposition a medium presents to a sound wave. The characteristic Z = rho * c is the product of the medium's mass density and the speed of sound in it, with units of pascal-seconds per metre (Pa.s/m), which is the same as one Rayl. This calculator returns that characteristic Z, not the volume-velocity acoustic impedance p / U that has the Pa.s/m^3 unit.
Q: How do you calculate acoustic impedance?
A: Multiply the medium density in kg/m^3 by the speed of sound in the medium in m/s. For dry air at 20 C, 1.204 * 343 = 413 Pa.s/m, which is also 413 Rayl or 0.000413 MRayl. For water at 20 C, 998.2 * 1482 is about 1,479,000 Pa.s/m, or 1.48 MRayl.
Q: What is a Rayl and how does it relate to MRayl?
A: One Rayl equals 1 Pa.s/m, the same unit used for the specific acoustic impedance z = p / u and the characteristic Z = rho * c. A megarayl, written MRayl, equals one million Rayl. Gases sit in the hundreds of Rayl, liquids and soft tissue in the 1-2 MRayl range, and metals in the tens of MRayl.
Q: What is the reflection coefficient for sound at an interface?
A: When a sound wave crosses from medium 1 into medium 2, the pressure reflection coefficient is r = (Z2 - Z1) / (Z2 + Z1). An identical pair of mediums gives r = 0, while a much larger Z2 gives r near +1 and a much smaller Z2 gives r near -1.
Q: Why does acoustic impedance matter for medical ultrasound?
A: An ultrasound image is built from reflections at tissue boundaries. Each boundary reflects a fraction of the wave equal to r = (Z2 - Z1)/(Z2 + Z1), so the contrast between fat, muscle, and bone comes from the small differences in their characteristic Z, all of which sit near 1.5-1.7 MRayl.
Q: What is the difference between characteristic and specific acoustic impedance?
A: Characteristic Z = rho * c describes a uniform medium and equals the specific impedance z = p / u in a plane wave, with units of Pa.s/m (Rayl). The general specific z varies with position and frequency in a real wave, while the volume-velocity acoustic impedance p / U has units of Pa.s/m^3 and is not what this calculator returns.