The physics of the ear

Air, bone, fluid

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.

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.

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 is not decoration. 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. Inside the cochlea the vibration travels along a membrane whose stiffness changes with distance, so each frequency disturbs its own region most, and hair cells there fire nerve impulses to the brain. Position along the cochlea is how the ear codes pitch.

Intensity, and why the scale is logarithmic

Sound carries energy, and the quantity that matters at a detector is how much of it arrives per second on each square metre facing the wave. That is the intensity,

measured in W m⁻², with A the area at right angles to the direction the sound travels. An eardrum of area 60 mm² in a sound of intensity I therefore absorbs a power IA, and that is the calculation an exam builds towards.

Now look at the range the ear covers. The quietest sound a good ear detects has an intensity of 1.0 × 10⁻¹² W m⁻², and a sound a million million times more intense is painful rather than deafening. A linear scale cannot show both ends of that on one axis. Worse, it would misrepresent what the listener reports, because the ear judges by ratio: multiply the intensity by ten and the sound seems one step louder, multiply it by ten again and it seems one more step louder. Equal ratios feel like equal steps, and that is precisely what a logarithm turns into equal intervals.

So sounds are quoted as an intensity level in decibels, measured against the threshold of hearing:

with I₀ = 1.0 × 10⁻¹² W m⁻², the booklet value. Read it once and the numbers stop being arbitrary. Ten times the intensity is +10 dB, a hundred times is +20 dB, and doubling the intensity is close to +3 dB.

The claim at the top of the page dies here. From 40 dB to 80 dB is 40 decibels, so four factors of ten, and the intensity ratio is 10⁴. Ten thousand times, not twice.

Two sounds can also be compared with each other, without either one being referred to the threshold. Their relative intensity level is

which is just the difference of their two levels, and it is what a question means by asking how many decibels louder one sound is than another.

Equal loudness curves

Intensity level is a measurement. Loudness is a judgement, and the two part company as soon as the frequency changes, because the ear is not equally sensitive across its range of about 20 Hz to 20 kHz. An equal loudness curve maps the difference.

Producing one takes a listener and two tones. A reference tone at 1 kHz is set to a chosen intensity level. A test tone at some other frequency is then adjusted until the listener judges the two equally loud, and the level it needed is plotted at that frequency. Repeat across the range, join the points, and the curve shows every combination of frequency and intensity level that sounds equally loud to that listener. Each curve is named by its own level at 1 kHz, and the results are averaged over many listeners.

Three readings come off that family, and questions ask for all three. The ear is most sensitive between about 2 kHz and 5 kHz, where the curves dip lowest, so least intensity is needed there. Sensitivity falls away at both ends, and the bass end is the worse of the two, which is why a very low note needs a far higher intensity level to match a mid tone. And the curves flatten as they rise, so a loud passage is heard with a more even frequency balance than a quiet one.

The lowest curve of all is the threshold of hearing, the minimum intensity a normal ear can detect, quoted at 1 kHz as 1.0 × 10⁻¹² W m⁻². That is where the 0 dB of the decibel scale comes from, so 0 dB does not mean silence; it means the faintest audible sound at the reference frequency.

A meter reading in plain dB weights every frequency equally, so it will call a rumbling low-frequency noise louder than a listener does. The dBA scale fixes that by filtering the signal with a frequency weighting shaped like the ear's own response before the level is computed, discounting the frequencies the ear is poor at. A figure in dBA therefore tracks perceived loudness, and it is the one written into noise-at-work limits.

What damage does to the curves

Hearing is lost in two ways the specification asks about: injury from exposure to excessive noise, and gradual deterioration with age. Both show up as a change in the curves rather than as silence.

The threshold curve lifts, meaning a greater intensity is now needed before anything is heard at all, and it lifts unevenly. The high-frequency end goes first, so the top of the range shrinks well before the middle is affected. Loud noise typically leaves a dip in performance around 4 kHz; ageing raises the whole treble end steadily. The higher equal loudness curves change less, so loud sounds still seem loud while quiet ones vanish.

The practical consequence is worth stating in an answer, because it is what a patient reports. Consonants such as s, f and t carry their information at high frequency and at low intensity, so they are the first things to go, and speech becomes hard to follow in a noisy room long before anyone would say they were deaf.