The ear and the transmission of sound

Air, bone, fluid

Everything the brain will ever learn about a sound leaves the ear as nerve impulses from an organ full of fluid, and the sound itself arrives as a pressure variation in air. Getting from one to the other is a transmission problem, and the structure of the ear is the solution to it.

The ear is a chain of three sections, and each one hands the vibration to the next in a different medium. The outer ear is the visible pinna, which collects sound and funnels it into the ear canal. The canal is a tube closed at the far end by the eardrum, and like any closed pipe it resonates. Its first resonance falls near 3 kHz, which is one reason the ear is at its most sensitive there: a canal about 2.5 cm long fits a quarter of a wavelength between the open air and the drum, so it picks out λ = 4L ≈ 10 cm, and sound of that wavelength in air is a frequency of roughly 3 kHz.

At the end of the canal the eardrum vibrates with the arriving pressure variation. That is the boundary of the middle ear, an air-filled cavity spanned by three small linked bones, the ossicles, which carry the motion across to a much smaller membrane called the oval window.

The pressure step-up in the middle ear

Two things happen on the way across, and both raise the pressure. The ossicles are linked as a lever, so the force at the far end is larger than the force at the near end. More importantly the eardrum has perhaps twenty times the area of the oval window, and the same force spread over a twentieth of the area is twenty times the pressure. Together they lift the pressure by a factor of the order of twenty or thirty.

That pressure increase is what the next stage needs. Beyond the oval window lies the inner ear, and the cochlea inside it is full of fluid. Sound passing straight from air to fluid would meet a huge mismatch and almost all of it would reflect, exactly as an ultrasound pulse reflects at an air gap. The middle ear is the matching stage that gets the energy across.

The mismatch has a name this option meets again. In ultrasound imaging the same problem is written as a difference in acoustic impedance, and the answer there, the coupling gel between probe and skin, does the same job the middle ear does here: it stops the wave meeting an air-to-something boundary that would send it straight back.

The middle ear does not supply energy: in the idealised account, it increases pressure at the expense of displacement amplitude and improves energy transfer from air to the fluid of the cochlea. Describe this as pressure transformation rather than as active power amplification.

Describe the drawing

The middle ear drawn from the lesson's own numbers. On the left the eardrum, on the right the oval window, the two squares drawn in the true ratio of their areas, 55 square millimetres against 3.0. Between them the ossicles are drawn as a lever whose arms are in the ratio 1.5 to 1, so the force arriving as F leaves as 1.5F, and the two arrows are drawn in that ratio too. Beneath, the pressure at each membrane as a bar: the second is 27.5 times the first, which is the lever's 1.5 multiplied by the area ratio of 55 to 3.0. Beneath that, the energy carried across in one cycle at each end, drawn as two bars of exactly the same length, because the larger force moves through a proportionately smaller distance. A large, easily moved membrane concentrating its force onto a small one raises the pressure and adds no energy at all, so the middle ear steps up pressure and is not an amplifier.

Inside the cochlea

Once through the oval window, the vibration travels in the cochlea's fluid and along the basilar membrane that runs the length of its coil. The membrane is not uniform: it is narrow and stiff near the oval window and becomes wider and floppier towards the far end, so each frequency disturbs its own region of the membrane most, high frequencies peaking near the entrance and low frequencies far along.

Describe the drawing

The cochlea drawn uncoiled as a channel tapering from the oval window at the left to the apex at the right, with the basilar membrane running along it as a wedge, narrow at the window and wide at the apex. Three computed response envelopes sit on the membrane: an 8 kHz tone peaking close to the entrance, a 1 kHz tone near the middle and a 200 Hz tone far along by the apex. A row of hair cells lines the membrane, and at each peak the cell beneath is enlarged with an impulse arrow leaving it for the brain: pitch is read off from which place fired, not from any arithmetic on the wave.

Hair cells sit along the membrane, and the ones at the disturbed region are bent by the motion and fire nerve impulses to the brain. Position along the cochlea is therefore how the ear codes pitch: the brain does no frequency arithmetic; it reads an address. A pure tone excites one small stretch of membrane, a chord excites several at once, and the nerve reports which stretches were busy.

That completes the chain the specification asks for, air to bone to fluid to nerve: pinna, canal, eardrum, ossicles, oval window, cochlea, hair cells, brain.