2.4 The ear canal

After the pinna, the sound enters the external auditory meatus — the ear canal — which is a roughly cylindrical tube about 25 mm long that ends at the eardrum. Its resonant behavior is well-approximated as a closed tube (closed at the eardrum, open at the entrance). The lowest resonance of such a tube is at $\lambda = 4L$ , so

f_\text{res} = \frac{c}{4L} = \frac{343\,\text{m/s}}{4 \times 0.025\,\text{m}} = 3430\,\text{Hz}.

▶ Derivation: the quarter-wave resonance of a closed tube

Consider a cylindrical tube of length $L$ , closed at one end and open at the other. The closed end forces a pressure antinode (the wall cannot move, so pressure can vary freely there but velocity must be zero). The open end forces a pressure node (pressure must equal atmospheric, which is uniform).

A standing wave $p(x, t) = A\sin(kx)\cos(\omega t)$ has a node at $x = 0$ and an antinode at the first place where $\sin(kx) = \pm 1$ , namely $kx = \pi/2$ , so $x = \pi/(2k) = \lambda/4$ .

If the closed end is at $x = L$ and the open end at $x = 0$ , the longest standing wave that fits is $L = \lambda/4$ , i.e., $\lambda = 4L$ . The frequency is $f = c/\lambda = c/(4L)$ .

For $L = 25$ mm and $c = 343$ m/s, $f = 3430$ Hz. ∎

The interactive below lets you play with the geometry. Toggle each end between open and closed, vary the length, and read off the resonant frequencies. The preset: human ear canal button sets the configuration to closed at the right end (the eardrum) and open at the left (the canal opening), at 25 mm long — the standard configuration that produces the 3 kHz resonance. The animated standing wave is the pressure field $p(x, t)$ inside the tube; the dashed envelope is its amplitude $|p(x)|$ — nodes where it touches zero, antinodes where it peaks.

left end:

right end:

length L

25 mm

mode:

f₁: 3.43 kHz
f₂: 10.29 kHz
f₃: 17.15 kHz
f₄: 24.01 kHz

mixed ends → odd-harmonic series (n = 1, 3, 5, …)

The ear canal’s first resonance falls at roughly 3 kHz, with a gain of about 10 dB at that frequency.

This is exquisite engineering, accidental or otherwise: the canal’s resonance falls right in the middle of the speech frequency range. Human speech communication is partly a story of the human ear having been pre-amplified for human speech, by the geometry of the canal’s resonance.

Try the open-open and closed-closed configurations in the interactive too. These produce the full integer harmonic series ( $f_n = n \cdot c/(2L)$ ) rather than the odd-only one ( $f_n = (2n - 1) \cdot c/(4L)$ ) of the mixed boundary. The difference between the two boundary-condition families — even-integer vs odd-only multiples — runs through every tube-resonance problem in physics: organ pipes (open-open or closed-open), flutes (open-open), clarinets (closed-open, hence their dominant odd harmonics), and your own vocal tract during vowel production (closed at the larynx, open at the lips, again odd-only).