Acoustic Impedance

What acoustic impedance measures

Push on a column of air and it barely resists — it is light and springy. Push the same way on a column of water and it pushes back hard. Acoustic impedance quantifies exactly that resistance: for a given acoustic pressure, how little the medium moves. Formally, the specific (characteristic) acoustic impedance is the ratio of acoustic pressure p to the particle velocity u in a travelling plane wave:

Z = p / u = ρ · c

where ρ is the medium's density and c is the speed of sound in it. The unit is the rayl (named after Lord Rayleigh): 1 rayl = 1 Pa·s/m = 1 kg/(m²·s). A medium that is both dense and stiff (so sound travels fast through it) has high impedance. Air, being light and slow, has very low impedance; steel has impedance roughly 100,000 times larger.

The crucial physical point is that impedance is what a boundary compares. A single uniform medium never reflects sound. Reflection happens only where Z changes — at an interface between two media, or where density or stiffness jumps. The amount that reflects depends not on either impedance alone but on their ratio.

Reflection and transmission at a boundary

At any boundary, two conditions must hold for all time: the acoustic pressure must be continuous (otherwise there would be infinite force on a massless interface), and the normal particle velocity must be continuous (the media stay in contact). A single transmitted wave generally cannot satisfy both — so a reflected wave appears to balance the books. Solving the two continuity equations for normal incidence gives the pressure reflection coefficient:

R = (Z₂ − Z₁) / (Z₂ + Z₁)        (amplitude of reflected pressure / incident)
T = 2·Z₂ / (Z₂ + Z₁)             (amplitude of transmitted pressure / incident)

Energy is what most applications care about. The intensity reflection and transmission coefficients are:

R_I = R² = [(Z₂ − Z₁)/(Z₂ + Z₁)]²
T_I = 1 − R_I = 4·Z₁·Z₂ / (Z₁ + Z₂)²

Read these and the whole subject falls out. If Z₁ = Z₂ the impedances match: R = 0, nothing reflects, everything transmits. If Z₂ ≫ Z₁ (going into a much harder medium), R → +1 — almost total reflection with the reflected pressure in phase. If Z₂ ≪ Z₁ (going into a much softer medium, like sound in water hitting air), R → −1 — again near-total reflection, but with a 180° phase flip. Either large mismatch reflects nearly all the energy; only the ratio Z₂/Z₁ being close to 1 lets sound through.

Note a counter-intuitive consequence: a soft medium reflects as strongly as a hard one. A diver tapping a pipe underwater is almost inaudible above the surface not because air "blocks" sound, but because the air–water impedance ratio is enormous in both directions.

Real impedances and reflection numbers

Here are characteristic impedances for common media at room temperature, with the resulting reflection when sound passes from air into each (and a few interfaces that matter in practice).

Medium	Density ρ (kg/m³)	Sound speed c (m/s)	Impedance Z (rayls)
Air (20 °C)	1.20	343	415
Water	1000	1480	1.48 × 10⁶
Soft tissue (avg)	1060	1540	1.63 × 10⁶
Fat	950	1450	1.38 × 10⁶
Bone (cortical)	1900	4080	7.8 × 10⁶
PZT ceramic (transducer)	7500	4000	~30 × 10⁶
Steel	7800	5900	4.6 × 10⁷

Now the reflections at common interfaces, computed from R_I = [(Z₂ − Z₁)/(Z₂ + Z₁)]²:

Interface	Impedance ratio Z₂/Z₁	Energy reflected R_I	Energy transmitted
Air → Water	~3,570	99.89%	0.11%
Air → Soft tissue	~3,930	99.90%	0.10%
Soft tissue → Bone	~4.8	~32%	~68%
Fat → Soft tissue	~1.18	~0.7%	~99.3%
Soft tissue → Air pocket (lung/gas)	~0.00025	~99.9%	~0.1%
Transducer (PZT) → Soft tissue	~0.054	~80%	~20%

The last row is the engineering problem ultrasound has to solve: a bare ceramic transducer pressed on tissue would lose 80% of its energy to reflection at its own face. The fix is impedance matching.

Impedance matching

To get a wave across a big mismatch, you bridge it with an intermediate layer. The optimal single-layer match has impedance equal to the geometric mean of the two media and a thickness of a quarter wavelength (measured inside the layer):

Z_match = √(Z₁ · Z₂)        thickness d = λ_layer / 4

The quarter-wave layer works because the wave reflected from its back face travels an extra half wavelength and returns 180° out of phase with the front-face reflection, cancelling it. At that one design frequency the layer is effectively invisible and transmission is total. For the PZT-to-tissue problem, Z_match = √(30 × 10⁶ · 1.6 × 10⁶) ≈ 6.9 × 10⁶ rayls, achievable with a loaded epoxy. Many transducers stack two matching layers for a broader bandwidth.

The everyday version of impedance matching is ultrasound gel. The real culprit in imaging is not the transducer face but the thin air gap between probe and skin: a tissue→air interface reflects ~99.9% of the wave. Gel (Z close to water) fills the gap and removes the air, so the wave reaches the body almost undiminished. The same principle explains horn loudspeakers (a flaring horn gradually matches the high impedance of a small driver to the low impedance of open air) and the middle-ear ossicles (which match low-impedance air to the high-impedance fluid of the cochlea, recovering most of the ~30 dB you would otherwise lose).

Computing reflection and matching

// Characteristic impedance Z = rho * c, in rayls
function impedance(rho, c) { return rho * c; }

const Z = {
  air:    impedance(1.20, 343),    // ~415
  water:  impedance(1000, 1480),   // ~1.48e6
  tissue: impedance(1060, 1540),   // ~1.63e6
  bone:   impedance(1900, 4080),   // ~7.8e6
  pzt:    30e6,
};

// Pressure reflection coefficient at normal incidence
function reflectionCoeff(Z1, Z2) { return (Z2 - Z1) / (Z2 + Z1); }

// Fraction of INTENSITY (energy) reflected and transmitted
function energySplit(Z1, Z2) {
  const R = reflectionCoeff(Z1, Z2);
  const Ri = R * R;
  return { reflected: Ri, transmitted: 1 - Ri };
}

console.log(energySplit(Z.air, Z.water));    // { reflected: 0.9989, transmitted: 0.0011 }
console.log(energySplit(Z.tissue, Z.bone));  // { reflected: ~0.32, transmitted: ~0.68 }

// Optimal single quarter-wave matching layer
function matchingLayer(Z1, Z2, freq, c_layer) {
  const Zm = Math.sqrt(Z1 * Z2);           // geometric mean impedance
  const thickness = c_layer / freq / 4;    // lambda/4 inside the layer
  return { Zm, thickness };
}

// Match PZT (30 MRayl) to tissue (1.63 MRayl) at 5 MHz, layer sound speed 2500 m/s
const layer = matchingLayer(Z.pzt, Z.tissue, 5e6, 2500);
console.log(layer.Zm.toExponential(2));      // ~6.99e6 rayls
console.log((layer.thickness * 1e6).toFixed(1) + ' um'); // 125.0 um quarter-wave thickness

// Transmission Loss in decibels across a single interface
function transmissionLossDB(Z1, Z2) {
  const { transmitted } = energySplit(Z1, Z2);
  return -10 * Math.log10(transmitted);
}
console.log(transmissionLossDB(Z.air, Z.water).toFixed(1) + ' dB'); // ~29.6 dB lost

Where acoustic impedance shows up

• Medical ultrasound. Every bright line in an ultrasound image is an echo from an impedance change; gel and quarter-wave matching layers maximize the wave that gets in and back. Gas-filled lung and bowel are nearly opaque because tissue→gas reflects ~99.9%.
• Nondestructive testing. Sonar-like probes find cracks and voids in metal because a flaw is an impedance discontinuity that reflects a detectable echo.
• Architectural acoustics. Anechoic chambers use graded-impedance foam wedges so the wave meets no sharp boundary and is absorbed rather than reflected.
• Loudspeakers and brass instruments. Horns and flared bells are impedance transformers matching a small, high-impedance driver or reed to the low impedance of open air.
• Hearing. The middle-ear bones are a biological impedance-matching network between air and cochlear fluid; damage to them causes conductive hearing loss.
• Underwater and seismic. Sonar and seismic surveys read reflections off impedance layers in the ocean and crust; oil exploration maps "acoustic impedance contrasts" between rock strata.

Common mistakes

• Thinking only hard media reflect. A soft medium (low Z₂) reflects just as strongly as a hard one — the sign of R flips, but |R| → 1 in both extremes. Sound trying to leave water for air bounces back almost completely.
• Confusing pressure and intensity coefficients. R = (Z₂−Z₁)/(Z₂+Z₁) is for pressure amplitude; energy goes as R², and transmission is 1 − R², not 1 − R.
• Using the arithmetic mean for matching. The ideal matching impedance is the geometric mean √(Z₁Z₂), not (Z₁+Z₂)/2. The arithmetic mean does not cancel the reflections.
• Forgetting the matching layer is frequency-tuned. A quarter-wave layer is perfect only at the frequency where its thickness equals λ/4. Away from it, transmission degrades — hence multilayer broadband designs.
• Ignoring angle. The simple R formula is for normal incidence. At oblique incidence you need the angle-dependent form and must watch for total internal reflection and mode conversion in solids.
• Mixing up specific vs. acoustic impedance. Specific acoustic impedance (p/u, in rayls) differs from acoustic impedance defined as pressure over volume velocity (which depends on cross-sectional area). Keep the definitions straight when working with ducts and horns.