Lumped-element acoustics
When the wavelength dwarfs the object, air behaves as a circuit.
Everything so far has been a field theory: pressure and velocity defined at every point, tied together by the wave equation, resolved into plane waves and modes. That machinery is exact, but it is overkill whenever the thing you care about — a cavity, a short tube, a small port — is much smaller than a wavelength. In that regime the pressure is essentially uniform across the object, its internal spatial structure disappears, and the entire field collapses to a handful of numbers: a compliance, an inertance, a resistance. The partial differential equation becomes an ordinary circuit.
This is not an approximation of convenience; it is the native language of small acoustic systems, and it is exactly the regime the ear lives in. A human ear canal is about 25 mm long — a quarter-wavelength only near 3 kHz, and far shorter than a wavelength across most of hearing — so at low and mid frequencies the canal, the eardrum, and the middle-ear cavities behave as lumped compliances and masses wired together. The impedance-transformer action of the middle ear, the shape of a tympanogram, the tuning of a vent or an earmould: all of it is circuit acoustics. This chapter builds those circuit elements from the wave physics of the previous chapters, then hands them forward to the hearing volume.
- 11.1 The long-wavelength limit: from field to circuit — the criterion ; pressure and volume velocity as the “voltage” and “current” of acoustics; the three lumped elements — compliance, inertance, resistance — derived from a compressed cavity, an accelerated plug of air, and viscous loss.
- 11.2 The acoustic–electrical analogy — pressure ↔ voltage, volume velocity ↔ current; the impedance of each element; how to wire them in series and parallel and read off a resonance.
- 11.3 The Helmholtz resonator — a neck’s inertance and a cavity’s compliance in series: the acoustic mass-on-a-spring, its resonant frequency, its damping, and why a blown bottle sounds the note it does.
- 11.4 The ear as an acoustic network — the two-port of a uniform tube; the ear canal, eardrum, and middle-ear cavity as a small acoustic circuit; tympanometry as a measurement of input admittance.
The chapter leans on the complex-impedance picture from Foundations 3.3 and the driven oscillator of Chapter 2; it is the low-frequency counterpart to the full field theory of Chapters 4–7, and the bridge into What is hearing?.