11.2 The acoustic–electrical analogy

The three elements of 11.1 obey laws with the same algebraic form as the three elements of an electrical circuit. That shared form is a working tool, not a coincidence to be admired and set aside. Once the correspondence is fixed, every result of linear circuit theory (series and parallel combination, Kirchhoff’s laws, resonance, transfer functions, Bode plots) carries over to acoustics unchanged, and a small acoustic system can be drawn as a circuit and solved by inspection.

The dictionary

Match the across variable to voltage and the through variable to current:

Pressure $p$ ↔ voltage $V$ (the effort that drives flow)
Volume velocity $U$ ↔ current $I$ (the flow that responds)

With that single choice, the element laws line up term for term:

Acoustic compliance $U = C_a\,\dot p$ ↔ capacitor $I = C\,\dot V$ . $C_a$ is a capacitance.
Acoustic inertance $p = M_a\,\dot U$ ↔ inductor $V = L\,\dot I$ . $M_a$ is an inductance.
Acoustic resistance $p = R_a\,U$ ↔ resistor $V = R\,I$ . $R_a$ is a resistance.

where

$p \leftrightarrow V$: acoustic pressure plays the role of voltage (the across / effort variable)
$U \leftrightarrow I$: volume velocity plays the role of current (the through / flow variable)
$C_a \leftrightarrow C$: compliance behaves as a capacitance m^3/Pa
$M_a \leftrightarrow L$: inertance behaves as an inductance kg/m^4
$R_a \leftrightarrow R$: acoustic resistance behaves as a resistance Pa·s/m^3

This particular choice — pressure as voltage — is the impedance analogy. (There is a dual, the mobility analogy, that maps pressure to current instead; it makes mechanical mass become a capacitor and is handy for electromechanical transducers, but we use the impedance analogy throughout because it keeps acoustic impedance $Z_a = p/U$ equal to electrical impedance $V/I$ .)

Impedance and the single-frequency shortcut

Drive everything at one frequency, $p, U \propto e^{i\omega t}$ , and every time derivative becomes multiplication by $i\omega$ (Foundations 3.3). The element laws collapse to algebraic impedances $Z_a = p/U$ :

Z_{C} = \frac{1}{i\omega C_a}, \qquad Z_{M} = i\omega M_a, \qquad Z_{R} = R_a.

The compliance’s impedance falls with frequency (a soft spring passes fast wiggles easily); the inertance’s rises with frequency (mass resists rapid acceleration); the resistance is flat. The imaginary sign tells the phase: $Z_M = i\omega M_a$ leads (pressure ahead of flow, as with an inductor), $Z_C = 1/(i\omega C_a) = -i/(\omega C_a)$ lags. These are exactly the RLC reactances of Foundations 3.3, relabelled.

Combining elements: series and parallel

Because the analogy is exact, elements combine by the ordinary circuit rules — you only have to know which physical arrangement is “series” and which is “parallel”:

Series — elements that carry the same volume velocity, one after another along a single flow path (air pushed through a neck and then into a cavity). Series impedances add: $Z = Z_1 + Z_2 + \cdots$ .
Parallel — elements that share the same pressure across them, offering alternative paths for the flow (a vent alongside an eardrum, both seeing the ear-canal pressure). Parallel admittances add: $Y = Y_1 + Y_2 + \cdots$ , with $Y = 1/Z$ .

The parallel rule is exactly the tympanometry bookkeeping of Tools of Audiology 4.1: the probe sees the ear-canal air in parallel with the eardrum, so their admittances add, $Y_\text{total} = Y_\text{canal} + Y_\text{eardrum}$ — which is why the canal contribution can be subtracted off.

▶ A mass and a compliance in series resonate Derivation

Put an inertance $M_a$ in series with a compliance $C_a$ — a neck feeding a cavity. The series impedance is

Z(\omega) \;=\; i\omega M_a + \frac{1}{i\omega C_a} \;=\; i\!\left(\omega M_a - \frac{1}{\omega C_a}\right).

The two reactances have opposite sign, so at one special frequency they cancel and $Z \to 0$ (with only the small resistance left): the circuit passes volume velocity freely. That is resonance, at

\omega_0 M_a = \frac{1}{\omega_0 C_a} \quad\Longrightarrow\quad \omega_0 = \frac{1}{\sqrt{M_a C_a}}.

It is the acoustic copy of the LC resonance $\omega_0 = 1/\sqrt{LC}$ , and — substituting $M_a = \rho_0\ell/S$ and $C_a = V/\rho_0 c^2$ — it is the Helmholtz resonator of the next lesson. The mass of the neck and the springiness of the cavity trade energy back and forth, exactly as in the mass-on-a-spring of Chapter 2.

What the analogy buys

Once a small acoustic system is a circuit, its behaviour is a transfer function you can read off:

Resonances and antiresonances are the frequencies where series or parallel combinations vanish or blow up — the pitch of a bottle, the notch a vent cuts in a hearing-aid response.
Filtering falls out for free: a compliance to ground is a lowpass shunt; an inertance in series rolls off the highs. A hearing aid’s tubing, vent, and receiver form a filter network engineered exactly this way.
Impedance matching — the entire raison d’être of the middle ear (Hearing Ch 3) — is a two-port transformer inserted between a high-impedance load (cochlear fluid) and a low-impedance source (air), and it is designed with these same rules.

⏳ The history — From telephones to the acoustic circuit

The electrical analogy grew out of the telephone industry’s need to design earpieces, microphones, and horns quantitatively. Arthur Kennelly and Kurô Nukiyama measured the “motional impedance” of telephone receivers around 1919, treating the acoustic load as a circuit element seen through the coil. Through the 1920s and 30s, A. G. Webster, Warren Mason, and Harry Olson at Bell Labs and RCA turned the correspondence into a full discipline: Olson’s Dynamical Analogies (1943) tabulated the mechanical, acoustical, and electrical duals and let engineers draw a loudspeaker or a resonator as a schematic and compute its response. The framework is why a modern hearing aid or cochlear-implant microphone can be designed on a circuit simulator before any air moves (Beranek 1954).

The next lesson works the most important two-element circuit in full — the Helmholtz resonator — and 11.4 assembles the ear canal and middle ear into a small network of these parts.