The lever that lets air speak to water

What happens where the medium changes found that a wave meeting a boundary is divided between reflection and transmission by one quantity, the ratio of the two media’s impedances. The layer that makes a reflection vanish cancelled the reflection with a quarter-wave layer, the taper that matches every note removed it with a gradual change, and the mismatch no network can remove found the limit on how broad any such match can be. The absorber whose bandwidth is its thickness and the layer too thin to see applied the same ideas to radar and to seismic echoes.

Each of those matched impedances with a layer or a gradient of material between the two media — something a wave passes through. There is a second way, as old as the lever, which uses no intermediate medium at all: change the ratio of force to motion mechanically. Every land vertebrate carries a device that does this, because every land vertebrate has an ear filled with fluid and a world filled with air, and the boundary between them reflects almost everything.

Air meets water #

The characteristic impedance of a medium for sound is its density times its sound speed. For air at room temperature that is about 413 in the usual units, kilograms per square metre per second; for water it is about 1.48 million, 3,600 times larger. Water is dense and stiff; air is light and soft. A sound wave in air is a large motion with a small pressure, and a sound wave in water carrying the same power is a small motion with a large pressure.

At the boundary the two media must move together and push equally on each other, and a wave arriving from air cannot supply the pressure water needs to move with it without reflecting almost all of its energy. The fraction crossing is $4r/(1+r)^2$ for an impedance ratio $r$, which the figure plots. For air into water it is 0.11 per cent: twenty-nine and a half decibels lost at the surface. A person underwater hears voices from the bank as a faint murmur for exactly this reason, and so, by the symmetry of the formula, does a fish hear little of what goes on in the air.

An inner ear filled with fluid, facing air through a membrane, would face the same loss. The ancestors of land animals, leaving the water, had ears built for water, where no such problem exists: sound in water passes into a fluid-filled ear with no mismatch at all. The middle ear evolved, several times independently in different lineages, to solve the problem the move onto land created.

A lever and a piston #

The solution has two parts, both mechanical advantages.

The eardrum collects the force of the sound over its working area, about fifty-five square millimetres, and the chain of three small bones — hammer, anvil and stirrup — delivers that force to the oval window of the inner ear through the footplate of the stirrup, about three square millimetres. The same force on a seventeenth of the area is seventeen times the pressure. The bones also form a lever: the hammer’s arm is about 1.3 times as long as the anvil’s, so the force at the footplate is 1.3 times the force at the eardrum, and the motion correspondingly smaller. Together the pressure rises about twenty-two times and the velocity falls by the same factor. Force multiplied, and nothing gained found the hydraulic press multiplying force by the ratio of piston areas and dividing the motion by the same ratio, so that the work in equals the work out; the middle ear is a hydraulic press run in reverse, with air on the large piston and fluid on the small one.

Multiplying pressure by $n$ while dividing velocity by $n$ changes how the load looks from the input side. An impedance is a ratio of pressure to velocity, so a load of impedance $Z$ seen through the device appears as $Z/n^2$. With $n = 22$ the fluid of the inner ear looks five hundred times less stiff from the eardrum than it would through a bare membrane, and the ratio the air sees falls from 3,600 to about seven. The electrical transformer does exactly this with voltage and current in place of pressure and velocity, and for the same purpose: a transformer between an amplifier and a loudspeaker makes a low-impedance speaker look like the high impedance the amplifier wants to drive.

How good a match is good enough #

The transmission depends on the pressure gain, and it has a best value.

For an ideal transformer between air and open water the figure plots the fraction crossing against the gain. It rises from a thousandth, peaks at a gain of sixty — the square root of the impedance ratio, where the transformed water looks exactly like air — and falls again beyond, where the water is made to look softer than air and the mismatch returns from the other side. The ear’s gain of twenty-two lands on the rising side, passing forty-three per cent: a twenty-six-decibel improvement on no middle ear at all.

Two things about the curve are worth noticing. The peak is broad: within a factor of two either side of the ideal, the transmission stays above about two-thirds, so the mechanism need not be precise to be effective. And the comparison with open water is itself approximate. The inner ear is not a large body of water into which the wave radiates freely; it is a narrow coiled tube of fluid, partitioned by the membrane that carries the sensory cells, and its input impedance at the oval window, measured in human and animal ears, is several times lower than the plane-wave figure, which moves the ideal gain down towards the ear’s actual value. The plane-wave calculation shows the size of the problem and the principle of its solution; it does not claim the ear achieves only forty-three per cent, and measurements of the middle ear’s transfer suggest it does rather better near its best frequencies.

The band over which it works #

A lever of bones and a membrane have mass and stiffness, and those set the frequencies over which the transformer does its job.

At low frequencies the stiffness of the eardrum and of the air trapped in the middle-ear cavity dominates: the system is hard to move slowly, and the transmission falls. At high frequencies the mass of the bones dominates: it is hard to move quickly. Between them, near a resonance where the two reactances cancel, the transformer works as intended. The figure uses the simplest model with one mass and one stiffness, tuned to a resonance at 1.5 kilohertz, and it reproduces the shape of the measured behaviour: good transmission from under a kilohertz to a few kilohertz, falling on either side. The real middle ear has several resonances and the bones do not move as a simple lever at high frequencies, where their motion becomes more complicated, so the model is a caricature of the shape rather than a prediction of the numbers.

That band is where human hearing is most sensitive, and it is not a coincidence either way. The ear canal itself resonates, as a tube closed at one end, near three kilohertz and adds ten decibels or so there, and the consonants that distinguish one word from another carry most of their energy between one and four kilohertz. The mismatch no network can remove found that any matching network trades bandwidth against the quality of the match; the middle ear makes the trade in favour of a moderate match over the band that speech and most animal calls use.

Two windows, and why only one is pushed #

There is a second, subtler function of the ossicles, visible only when the inner ear’s geometry is taken into account.

The inner ear is a closed space of incompressible fluid with two flexible openings, the oval window, which the stirrup pushes, and the round window, which bulges out as the oval window is pushed in. Fluid can move only if one window moves in while the other moves out, and the sensory membrane between them is driven by the difference in pressure across the two. If sound reached both windows equally and in step, nothing would move. The figure plots the net drive against the phase difference between the pressures at the two windows. For equal pressures the drive vanishes at zero phase and remains small for small differences: at ten degrees it is fifteen decibels down. With the ossicles delivering twenty-two times the pressure to the oval window alone, the drive is large whatever the phase.

That explains the clinical fact that a break in the chain of ossicles — from infection, injury or the bone growth of otosclerosis — costs forty to sixty decibels of hearing, more than the twenty-six decibels of the transformer alone. Without the bones, sound reaches both windows through the air of the middle ear at nearly equal strength and nearly in step, and much of it cancels. Surgery that restores the chain, or replaces the stirrup with a prosthesis, restores most of the lost hearing because it restores both the lever and the imbalance between the windows.

There is a second route into the inner ear that bypasses the transformer entirely, and everyone uses it. When a person speaks, the vibration of the voice travels through the bones of the skull directly to the fluid of the inner ear, shaking the whole cochlea rather than pushing on one window. That bone-conducted sound adds to the sound arriving through the air, and it carries relatively more of the low frequencies, which the skull transmits well. A recording of one’s own voice, heard through the air alone, sounds thin and unfamiliar because the bone-conducted part is missing. Audiologists use the two routes to diagnose hearing loss: a tuning fork held against the skull tests the inner ear directly, one held beside the ear tests the whole chain, and a gap between the two locates the fault in the middle ear. Bone-anchored hearing aids, which vibrate the skull through a small implant, restore hearing to people whose middle ear cannot be repaired, delivering the sound past the boundary the middle ear exists to cross.

The same trick in fish, the other way round #

Fish have the opposite problem in a milder form. A fish’s body is nearly the same impedance as the water around it, so a sound wave passes straight through it, moving the fish and its inner ear together, and only the dense stones of the inner ear, lagging behind the rest because of their inertia, register the passing wave. The fish hears the motion of the water well and its pressure poorly. Many fish carry a gas-filled swim bladder, whose impedance is as far from water’s as air’s is, and which therefore responds to the pressure of a sound by pulsating strongly.

Carp, catfish and their relatives, the largest group of freshwater fishes, connect the swim bladder to the inner ear by a chain of small bones, the Weberian ossicles, derived from parts of the first vertebrae. It is a chain of bones carrying motion from a gas-filled chamber to a fluid-filled ear — the same arrangement as the mammalian middle ear, evolved entirely independently and assembled from different bones. Fish with Weberian ossicles hear higher frequencies and fainter sounds than their relatives without them, and the device works for the same reason: it couples a medium that moves easily to one that does not, through a mechanical ratio.

Whales solved the problem differently when they returned to the water. A toothed whale’s ear is isolated from its skull by air-filled sinuses and receives sound through a channel of fat in the lower jaw whose impedance is close to that of sea water, so that sound enters with little reflection and is guided to the ear. The ancestral air-adapted middle ear was not discarded but reworked, and whale ears are among the most studied examples of an impedance-matching problem solved twice in one lineage.

Transformers everywhere #

Anywhere two media must exchange energy efficiently the same device appears, and it is worth recognising in its other forms. A medical ultrasound probe is a ceramic crystal with an impedance twenty times that of the body; it is fronted with a quarter-wave matching layer of intermediate impedance, the device of the layer that makes a reflection vanish, and coupled to the skin with gel, because a film of air between probe and skin would reflect almost everything, for exactly the reason in the first figure. A loudspeaker horn is a tapered tube that transforms the high-pressure, low-velocity motion at a small diaphragm into the low-pressure, high-velocity motion of the open air, gradually, the device of the taper that matches every note in acoustic form, and it raises a speaker’s efficiency several-fold. The bell of a brass instrument does the same for the air column inside it.

Electrical engineers state the principle as the maximum power transfer theorem: a source delivers the most power to a load whose impedance equals its own, and when the two differ a transformer with the right turns ratio restores the match. The mechanical version is the lever, which the same push, further out introduced as a way of trading force against distance. The middle ear is a lever put to that use, and its turns ratio is a ratio of areas.

What the figures leave out #

The figures reduce the middle ear to a lossless transformer with fixed ratios, plane waves at normal incidence, and one resonance. The eardrum is a cone that does not move as a rigid piston, especially at high frequencies, where different parts of it move in different phases; its effective area is therefore a function of frequency, and the figure of seventeen is a low-frequency value. The ossicles are suspended by ligaments and muscles, and two small muscles can stiffen the chain in response to loud sounds, reducing transmission at low frequencies by up to twenty decibels as a protective reflex. The inner ear’s input impedance depends on frequency in a way set by the travelling wave along the cochlea, which is a subject in itself. None of that changes the central point — that a lever and a ratio of areas turn a thousandth of the power into a substantial fraction — but each changes the numbers.

Still open: how the middle ear came to be #

The middle ear with three bones is a mammalian invention, and it evolved from bones that in the ancestors of mammals formed part of the jaw joint; the transition is recorded in a sequence of fossils in which the bones shrink and detach from the jaw over tens of millions of years. Birds, reptiles and frogs have a single bone, the columella, and achieve a similar transformation with a different mechanism and a narrower band. How the mammalian arrangement’s extended high-frequency response evolved, what the intermediate forms could hear, and why the three-bone chain arose independently more than once within early mammals are questions being worked on with fossils scanned in three dimensions and with models of how each arrangement would have transmitted sound.

The habit worth carrying away is to ask whether a mismatch can be removed without anything in between. A boundary between air and water reflects 99.9 per cent of sound because their impedances differ 3,600-fold, and a lever that multiplies pressure by n while dividing motion by n makes the water look $n^2$ times softer — so three small bones and a ratio of areas turn a thousandth of the power into nearly half, over the band where speech lives. A transformer adds no energy; it only changes what each side sees of the other.