3.3 The ossicular solution

How do you couple a low-impedance medium (air) to a high-impedance one (cochlear fluid) without losing the energy to reflection? You build a transformer. The middle ear is a mechanical transformer with two compounding tricks.

Photograph of the three middle-ear ossicles: malleus, incus, stapes. — A dissection photograph of the three middle-ear ossicles — malleus, incus, stapes — together about the size of a grain of rice. Anatomist90 · CC BY-SA 3.0 · Wikimedia Commons

Labeled anatomical diagram of the three auditory ossicles. — The ossicular chain in anatomical orientation. The malleus articulates with the eardrum, the incus pivots between malleus and stapes, and the stapes footplate seats in the oval window. Marc Giacone, SVG by Angelito7 · CC BY-SA 3.0 · Wikimedia Commons

Area ratio

The eardrum (tympanic membrane) has an area of about $65\,\text{mm}^2$ . The stapes footplate, which contacts the oval window, has an area of about $3.5\,\text{mm}^2$ . A given pressure acts over the eardrum, producing a force $F = p_\text{TM} \cdot A_\text{TM}$ . That force is transferred (via the ossicular chain) to the stapes, where it acts over a much smaller area: the pressure at the oval window is $p_\text{OW} = F / A_\text{footplate}$ . So

\frac{p_\text{OW}}{p_\text{TM}} = \frac{A_\text{TM}}{A_\text{footplate}} \approx 17.

A 17-fold pressure amplification from area concentration alone.

▶ Derivation: pressure amplification from area concentration

The total force exerted by a pressure $p$ acting over an area $A$ is $F = p A$ . If the same force is transmitted to a smaller area $A'$ (via a rigid mechanical chain), the new pressure is

p' = \frac{F}{A'} = \frac{p A}{A'}.

The pressure amplification is the area ratio: $p'/p = A/A'$ .

For the middle ear, $A_\text{TM}/A_\text{footplate} = 65/3.5 \approx 17$ . Note: this assumes the entire tympanic membrane moves in unison, which is roughly true at low frequencies. At high frequencies the membrane moves in higher-order vibrational modes that reduce the effective area ratio. ∎

Lever ratio

The three ossicles (malleus, incus, stapes) form a lever. The malleus is attached to the eardrum and rotates about a pivot near its head; the incus pivots about the same axis. The geometry is such that the malleus arm is slightly longer than the incus arm, giving a lever ratio of about

\frac{p_\text{out}}{p_\text{in}} \approx 1.3.

This is small but matters.

Combining

The two effects multiply. The total pressure gain from eardrum to oval window is approximately

G_p = \frac{A_\text{TM}}{A_\text{footplate}} \cdot \frac{L_\text{malleus}}{L_\text{incus}} \approx 17 \cdot 1.3 \approx 22.

In decibels, $20\,\log_{10}(22) \approx 27\,\text{dB}$ .

This recovers most — though not all — of the 30 dB lost to impedance mismatch. The net transmission of the middle ear into the cochlea is roughly 60% of the incoming intensity — a remarkable feat for a mechanical assembly the size of a grain of rice.

incoming SPL

60 dB

impedance mismatch loss: -29.6 dB
ossicular pressure gain: +26.9 dB
net SPL at oval window: -2.7 dB