11.4 The ear as an acoustic network

The three elements and the analogy now pay off. The outer and middle ear — canal, eardrum, ossicles, air cavities — is small compared with a wavelength through most of the hearing range, so it is an acoustic circuit: a source (the sound arriving at the ear) driving a network of compliances, inertances, and resistances that ends at the load (the cochlea). This lesson assembles that circuit from the parts built in this chapter and hands it to What is hearing?, where the same components are used to explain what the ear does with them.

A tube as a two-port

A length of duct that is not short compared with a wavelength cannot be a single element, but it is still a two-port: a box with pressure and volume velocity at each end, related by a matrix. For a uniform lossless tube of length $L$ , area $S$ , and characteristic acoustic impedance $Z_0 = \rho_0 c/S$ , the input pressure and flow $(p_1, U_1)$ relate to the output $(p_2, U_2)$ by

\begin{pmatrix} p_1 \\ U_1 \end{pmatrix} = \begin{pmatrix} \cos kL & i Z_0 \sin kL \\[2pt] i\,Z_0^{-1}\sin kL & \cos kL \end{pmatrix} \begin{pmatrix} p_2 \\ U_2 \end{pmatrix}.

where

$p_1, U_1$: pressure and volume velocity at the tube's input Pa, m^3/s
$p_2, U_2$: pressure and volume velocity at the tube's output Pa, m^3/s
$Z_0 = \rho_0 c / S$: characteristic acoustic impedance of the tube Pa·s/m^3
$k = \omega/c$: wavenumber; kL is the electrical length of the tube —

This transfer matrix (or ABCD matrix) is the bridge between the field theory and the circuit theory: it is exact at all frequencies, and cascading tubes just multiplies their matrices. The lumped elements are its low-frequency limit.

▶ The lumped elements are the kL ≪ 1 limit of the tube Derivation

Expand the matrix for a short tube, $kL \ll 1$ , using $\cos kL \approx 1$ and $\sin kL \approx kL$ :

\begin{pmatrix} \cos kL & iZ_0\sin kL \\ iZ_0^{-1}\sin kL & \cos kL \end{pmatrix} \approx \begin{pmatrix} 1 & i\omega\,(\rho_0 L/S) \\ i\omega\,(SL/\rho_0 c^2) & 1 \end{pmatrix}.

Read the off-diagonal terms with the definitions of 11.1. The top-right, $i\omega\,\rho_0 L/S = i\omega M_a$ , is the series inertance of the plug of air in the tube. The bottom-left, $i\omega\,SL/\rho_0 c^2 = i\omega\,(V/\rho_0 c^2) = i\omega C_a$ with $V = SL$ , is the shunt compliance of the air in it. So a short tube is exactly a series mass followed by a compliance to ground — a little L-section of the circuit. A wall of these L-sections is the tube, and in the limit their sum reproduces the wave equation. The lumped elements and the field are the same physics at two ends of $kL$ .

The outer and middle ear as a circuit

Now stack the pieces, ear-canal-inward:

The ear canal ( $\approx 25$ mm, area $\approx 0.4\ \text{cm}^2$ ) is a two-port tube. At low frequency it is a lumped series inertance plus shunt compliance; as $kL \to 1$ near 2–3 kHz its $\cos kL$ term drives a quarter-wave resonance — the ~+10 dB canal gain that shapes the head-related transfer function. The lumped-vs-distributed interactive in 11.1 is precisely this transition, tuned to a canal’s length.
The eardrum and ossicles present a load at the end of the canal. The drum’s tension and the ossicular mass make it, to first approximation, a compliance and a mass in series — a resonant load, tuned near 1 kHz, with resistance from the cochlear termination.
The middle-ear air cavity behind the drum is a shunt compliance $C_a = V_\text{ME}/\rho_0 c^2$ ; the mastoid air cells add more. This is why a change in middle-ear pressure or volume (an effusion, a perforation) shifts the whole network’s admittance — the basis of the tympanogram.
The cochlea terminates the chain: a nearly resistive load (the traveling-wave dissipation of Hearing Ch 4) of high acoustic impedance, which the middle ear’s transformer is built to match to the low impedance of air (Hearing Ch 3).

The middle ear is thus a two-port transformer inserted between a low-impedance source (air, $Z_0 \approx 400\ \text{Pa·s/m}$ specific) and a high-impedance load (cochlear fluid, ~3600× larger), and the area ratio and lever ratio that set its turns ratio are exactly the impedance-match numbers derived in Hearing 3.3. Circuit acoustics is the natural language for why those numbers.

Tympanometry: measuring the input admittance

Everything an audiologist reads from a tympanogram is the input admittance $Y_\text{in} = U/p$ looking into this network from the probe in the canal. Because the canal air sits in parallel with the eardrum (they share the canal pressure), their admittances add,

Y_\text{in} \;=\; Y_\text{canal} \;+\; Y_\text{eardrum},

so the instrument subtracts the canal’s compliance (measured at high static pressure, when the stiffened drum contributes nothing) to isolate the eardrum — the parallel rule of 11.2. Sweeping the static pressure changes the drum’s compliance, tracing the tympanogram’s peak; the whole clinical procedure is a controlled measurement of one node of this circuit (Tools of Audiology 4.1).

⏳ The history — Zwislocki and the acoustic model of the middle ear

By the early 1960s the electrical analogy had matured enough to model the human middle ear as a specific circuit. Jozef Zwislocki built lumped acoustic-analog networks of the ear canal, eardrum, ossicles, and cochlear load whose measured input impedance matched real ears, and used them to design the acoustic couplers and artificial ears still used to calibrate audiometers. The tympanometer on every audiology bench is a direct descendant: it assumes the ear is the circuit this chapter builds, and reports the value of one of its elements. The physics that began with a slab of air and a variational principle ends, four chapters of the sound book and one lumped-element chapter later, as a number on a clinical readout.

Closing the chapter

The lumped-element picture is the low-frequency face of everything in Chapters 4–7. When the object is smaller than a wavelength, the field theory collapses to compliance, inertance, and resistance; those combine by the circuit rules; and the outer and middle ear are their most important application, a small network matching air to cochlea. That is the bridge this chapter was built to carry: the hearing volume inherits these elements and asks what the ear does with them — how it turns this acoustic circuit into the first stage of hearing.