Characteristic Acoustic Impedance from Density and Sound Speed

The Acoustic Impedance Formula

For a progressive plane wave in a lossless, linear, homogeneous medium, the ratio of acoustic pressure $p$ to particle velocity $u$ is a material constant:

Here $\rho$ is the equilibrium mass density and $c$ is the phase velocity of the wave. $Z$ is intensive: it does not depend on the size of the sample or the cross-sectional area of the beam.

Dimensional analysis

Since $1\ \text{Pa} = 1\ \text{kg}/(\text{m}\cdot\text{s}^2)$, this is identical to $\text{Pa}\cdot\text{s/m}$:

Equivalent forms

Substituting $c = \sqrt{K/\rho}$, where $K$ is the bulk modulus of a fluid, gives

For an ideal gas, $c = \sqrt{\gamma P/\rho}$ with adiabatic index $\gamma$ and static pressure $P$, so

Check for air ($\gamma = 1.4$, $P = 101\,325\ \text{Pa}$, $\rho = 1.2\ \text{kg/m}^3$): $Z = \sqrt{1.4 \times 101\,325 \times 1.2} \approx 412.6\ \text{rayl}$, in agreement with the direct product $1.2 \times 343 = 411.6\ \text{rayl}$ to within rounding of the inputs. For water ($K \approx 2.2\ \text{GPa}$): $Z = \sqrt{2.2\times10^{9} \times 1000} \approx 1.48\times10^{6}\ \text{rayl}$.

Sound speed is not particle velocity

The speed $c$ is how fast the wavefront travels. The particle velocity $u$ is the small oscillatory velocity of the medium itself. In air, 1 Pa rms of sound pressure produces $u = p/Z = 1/411.6 \approx 2.4\ \text{mm/s}$, while the wavefront moves at about $343\ \text{m/s}$. The same 1 Pa in water gives $u \approx 0.68\ \mu\text{m/s}$. That difference is the physical meaning of impedance: a high-$Z$ medium needs more pressure to produce the same particle motion.

Which "Acoustic Impedance" Is This?

Four quantities share the name. This calculator computes only the first.

The three acoustic forms are related by the area $S$: $Z_a = z/S$ and $Z_m = zS$. Electrical speaker impedance is a different quantity entirely and is outside the scope of this tool.

Units and Exact Conversion Factors

Input units:

• Density: $1\ \text{g/cm}^3 = 1000\ \text{kg/m}^3$
• Sound speed: $1\ \text{km/s} = 1000\ \text{m/s}$

A convenient shortcut follows from these two factors. A density in $\text{g/cm}^3$ multiplied by a speed in $\text{km/s}$ gives $Z$ directly in MRayl, because $1000 \times 1000 = 10^{6}$.

How to Calculate Acoustic Impedance: Worked Examples

Step-by-step method

1. Choose a density and a sound speed that describe the same medium, wave mode, temperature, and pressure.
2. Convert both to base units: $\text{kg/m}^3$ and $\text{m/s}$.
3. Multiply: $Z = \rho c$ in rayl.
4. Divide by $10^{6}$ for MRayl, or by $10$ for CGS rayl.

Example 1: Air at the calculator default

In CGS units this is $41.16\ \text{CGS rayl}$.

Example 2: Fresh water

Example 3: Unit switching preserves the physical value

Entering $\rho = 1.2\ \text{kg/m}^3$ and then changing the density unit to $\text{g/cm}^3$ rewrites the field as $0.0012$. Changing the speed unit to $\text{km/s}$ rewrites $343$ as $0.343$. The physical inputs are unchanged, so the result stays at $411.6\ \text{rayl}$:

Example 3b: Cortical bone using the MRayl shortcut

With $\rho = 1.9\ \text{g/cm}^3$ and $c = 4.08\ \text{km/s}$ (longitudinal):

The equivalent base-unit calculation is $1900 \times 4080 = 7\,752\,000\ \text{rayl}$.

Example 4: Soft tissue

With $\rho \approx 1060\ \text{kg/m}^3$ and $c \approx 1540\ \text{m/s}$:

Example 5: Temperature matters

Air at $0\,^\circ\text{C}$ and 1 atm has $\rho \approx 1.275\ \text{kg/m}^3$ and $c \approx 331.3\ \text{m/s}$, giving $Z \approx 422\ \text{rayl}$. At $20\,^\circ\text{C}$, $\rho \approx 1.204\ \text{kg/m}^3$ and $c \approx 343.4\ \text{m/s}$, giving $Z \approx 413.5\ \text{rayl}$. The calculator does not correct for temperature or pressure; supply values for your actual conditions.

Reference Table: Typical Acoustic Properties

Values are representative room-temperature figures for longitudinal waves. Real materials vary with composition, temperature, and grade, so use datasheet values for design work.

The span from air to steel covers about five orders of magnitude, which is why the comparison plot uses a logarithmic axis.

Why Impedance Mismatch Matters: Reflection and Matching

This tool computes the intrinsic $Z$ of one medium. It does not compute interface coefficients, but you can use the results to estimate them. At normal incidence between media with impedances $Z_1$ and $Z_2$, the intensity reflection coefficient is

and the intensity transmission coefficient is $T = 1 - R = \dfrac{4 Z_1 Z_2}{(Z_1 + Z_2)^{2}}$.

Two practical consequences follow.

• Medical ultrasound: A transducer face placed on dry skin traps a thin air film, and nearly all the energy reflects. Water-based coupling gel has $Z$ close to that of tissue ($\approx 1.5\ \text{MRayl}$), which removes the air gap and restores transmission.
• NDT and transducer design: Piezoceramic elements have impedances well above that of water or tissue. Matching layers with intermediate impedance, commonly near $\sqrt{Z_1 Z_2}$ for a single quarter-wave layer, reduce the mismatch loss and broaden the usable bandwidth.

Inputs, Presets, and Validation

• Density $\rho$: valid range $0.000001$ to $100000$ in the selected unit. Default $1.2\ \text{kg/m}^3$, step $0.1$.
• Sound speed $c$: valid range $0.000001$ to $100000$ in the selected unit. Default $343\ \text{m/s}$, step $1$.
• Sliders: logarithmic, with exponents from $-6$ to $5$ in steps of $0.01$. Each step scales the value by about $2.3\%$. The number boxes accept direct typing.
• Presets: Air ($1.2\ \text{kg/m}^3$, $343\ \text{m/s}$, giving $411.6\ \text{rayl}$) and Fresh Water ($1000\ \text{kg/m}^3$, $1480\ \text{m/s}$, giving $1.48\ \text{MRayl}$).
• Invalid input: values $\le 0$ or $> 100000$ trigger a message such as "Enter Density from 0.000001 to 100000 in the selected unit." Empty or non-finite entries show an em dash (—) as the result and disable the copy button.

Because the limits apply in the selected unit, the reachable result range depends on the unit pair:

Comparison Plot and Workspace Tools

The interactive SVG plot places the result on a logarithmic impedance axis spanning orders of magnitude $10^{p}$. Fixed benchmark markers show air ($411.6\ \text{rayl}$) and water ($1.48\ \text{MRayl}$). An orange circle marks your medium and carries a label of the form "Your medium: X rayl". The plot supports pan, zoom, and viewport reset.

Two utilities are built in:

• Copy: places the density, speed, selected units, result in rayl and MRayl, and the plane-wave assumption on the clipboard.
• Print / PDF: produces a clean browser print layout that keeps the full content even when panels are collapsed.

Computation and Display

The calculator runs entirely in your browser as vanilla ES modules (js/math.js). There is no telemetry and no external API call, and it works offline once loaded. Arithmetic uses IEEE 754 binary64 floating point. Results are displayed to at most 6 significant digits, switching to exponential notation for values $\le 10^{-6}$ or $\ge 10^{9}$.

Scope and Limitations

• Plane-wave, lossless model. The result assumes a progressive, non-dispersive plane wave in a homogeneous, lossless medium.
• Matching wave mode and state. Density and sound speed must refer to the same wave mode and the same thermodynamic state. In solids, longitudinal and shear waves have different speeds and therefore different impedances; use the speed of the mode you are analyzing. The calculator does not infer or correct for temperature or pressure.
• No frequency dependence. Complex frequency-dependent impedance, attenuation, dispersion, pipe or cavity resonance, and reactive effects are not modeled.
• No interface calculation. The tool returns the intrinsic $Z$ of one medium. Use the reflection formula above, with two results, for a normal-incidence estimate.
• Not for electrical or duct impedance. Speaker impedance in ohms and resonator input impedance are different quantities.

Frequently Asked Questions