5.4 Specific acoustic impedance

The combination $\rho_0 c$ appears in every formula we’ve derived this chapter — connecting pressure and velocity, energy and intensity, and (next chapter) the interaction between fields and boundaries. It deserves a name. The specific acoustic impedance of a medium is

For a plane wave in that medium,

Pressure and particle velocity are proportional, with $Z$ as the constant of proportionality. The same $Z$ appears in every plane wave; it is a material property.

Numerical values

| Medium | $\rho_0$ (kg/m³) | $c$ (m/s) | $Z = \rho_0 c$ (Pa·s/m) | |---|---|---|---| | Air at 20°C | 1.20 | 343 | 412 | | Helium at 20°C | 0.18 | 970 | 173 | | Water at 20°C | 998 | 1480 | $1.48 \times 10^6$ | | Steel | 7800 | 5100 | $4 \times 10^7$ | | Aluminium | 2700 | 5100 | $1.4 \times 10^7$ |

Note the enormous variation: water has 3,600 times the impedance of air, steel has 100,000 times. These impedance contrasts are why sound reflects so strongly between air and water (essentially total reflection: most sound from above the surface doesn’t enter the ocean, and vice versa). The middle ear of land animals has a 22:1 ossicular lever specifically to bridge the impedance gap between air and the fluid-filled inner ear; we’ll see that in the Hearing book.

What impedance “is”

Impedance is the ratio of a force-like quantity to a velocity-like quantity. In mechanical systems it appears as force/velocity. In electrical circuits it appears as voltage/current. In acoustics, with pressure playing force and particle velocity playing velocity, it appears as $\tilde p / \tilde v$. The analogy is exact and very productive: every acoustic problem with linear elements can be solved as an electrical-circuit problem with equivalent impedances.

For a plane harmonic wave the impedance is real and equals $\rho_0 c$ — the wave is resistive. For a non-plane wave (spherical, near-field, standing) the impedance has a reactive part (an imaginary component) that stores and returns energy without transmitting it. We’ll meet reactive impedance when we look at sources in chapter 6.

Connections we’ve already made

Using $Z = \rho_0 c$, the formulas from this chapter become:

• Intensity in a plane wave: $\langle I \rangle = P_0^2 / 2Z = \tfrac12 Z |\tilde v'|^2$.
• Energy density: $\langle \mathcal{E} \rangle = P_0^2 / 2 Z c$.
• Pressure–velocity relation: $\tilde p' = Z \tilde v'$.

In each case $Z$ encodes the only material property that matters for what the wave carries. Two different media with the same $\rho_0 c$ (but different $\rho_0$ and $c$ individually) carry plane waves identically — same intensity for same pressure, same velocity for same pressure, same energy density.

Why this matters at boundaries

The reason impedance is so central is that every reflection at a boundary is governed by the impedance mismatch. Two media joined at a surface, with impedances $Z_1$ and $Z_2$, partially transmit and partially reflect an incident wave. The reflection coefficient for a plane wave at normal incidence is

For $Z_2 \gg Z_1$ (e.g. sound in air hitting water), $R \approx 1$ — almost full reflection. For $Z_2 = Z_1$, $R = 0$ — no reflection (impedance-matched). For $Z_2 \ll Z_1$ (water-to-air, looking up from underneath), $R \approx -1$ — full reflection with phase inversion.

This formula governs:

• Why echoes happen at walls (acoustic impedance of air vs. concrete).
• Why ultrasonic transducers need impedance-matching layers (gel between transducer and skin).
• Why the ear canal terminates at the eardrum’s impedance, not free space.
• Why a stethoscope works — converting tiny air-side pressure changes into liquid-side audible ones.

We will derive this formula and many of its consequences in chapter 7. For now: $Z = \rho_0 c$ is the single number that summarises what plane waves do in a medium. Memorise it.