How Sound Travels and How We Hear It

Clap your hands and a person across the room hears it about a hundredth of a second later. Nothing crossed the room in that time but a disturbance: no air from your hands reached their ears. What did arrive was a pattern, a few millimeters of squeezed air followed by a few millimeters of stretched air, passed along by trillions of molecules that each moved less than the width of a hair.

Everything about sound, from the pitch of a violin to the rumble of thunder and the way your ear tells them apart, comes from that one idea: a push handed from one bit of air to the next. Here is a loudspeaker doing it. Press play to hear the tone, and drag the cone's frequency to watch the pattern in the air squeeze closer together as the pitch rises:

Two things are moving in that picture, and they are easy to confuse. The red and blue bands race off to the right, one wavelength for every swing of the cone. The orange parcels of air go nowhere: each rocks forward and back about a fixed spot and ends where it started. The bands are the sound. The air only carries them, the way a crowd carries a stadium wave without anyone leaving their seat.

This page builds up how that works, piece by piece. First what passes the push along and what sets its speed, then what pitch and loudness are in this picture, how sounds add up into music and voices, and finally how a few square centimeters of tissue in your head turn the bands back into the experience of hearing. Most figures make a sound when you press their play button; nothing plays until you do, and the volume is kept low.

A push that travels#

Start with a simpler material than air: a row of beads joined by springs. Each bead stands for a slab of air, with mass, and each spring for the air's tendency to push back when squeezed. Press Push to shove the piston in and back once, and keep an eye on the orange bead in the middle:

The squeeze, shown as red springs, runs down the row at a steady speed, and behind it the springs relax. The orange bead lurches forward when the squeeze reaches it and settles back when the stretch behind follows; the trace below records that single wobble. Once the pulse has passed, every bead is back where it began.

Now change the materials. With stiff springs and light beads, push again: the pulse shoots down the row. With soft springs and heavy beads it crawls. A bead can only start moving once its spring has been squeezed enough to overcome its inertia, so the speed depends on the ratio of the two:

For air the same formula holds with the air's resistance to squeezing, its bulk modulus, in place of the springs, and its density in place of the beads. But a row of beads hides the real question. Air has no springs. What pushes back when you squeeze it?

Molecules doing the pushing#

Air is mostly empty space with molecules flying through it at the speed of a rifle bullet, about 500 m/s at room temperature, bouncing off each other several billion times a second. Here are a few hundred discs standing in for them, each moving at random. Press Push now to move the piston in and out once, and watch the strip plot of how crowded each slice of the box is. The piston also pushes by itself every few seconds:

A crowded band, a region where discs are packed a little more tightly, breaks away from the piston and crosses the box. The discs right in front of the piston get hit harder and rebound faster; they catch up with their neighbors and hit them a little harder; those hit the next layer. The push is handed forward one collision at a time, which is why the band moves at about the speed of the discs themselves, while each orange disc keeps wandering in its own random way and ends up nowhere in particular.

That is what the springs in the bead model stood for. A gas pushes back when squeezed because squeezing it packs more molecules against each surface, so more of them bounce off it every second. Pressure is the rain of those impacts, and sound is a ripple in that rain.

What sets the speed#

If the push is passed along by molecules hitting molecules, it cannot outrun them. Heat a gas and its molecules move faster, so the push should travel faster too. A light gas, whose molecules move faster at the same temperature, should carry sound faster than a heavy one. Drag the temperature and switch gases:

Air at room temperature carries sound at 343 m/s, about a kilometer in three seconds, which is why you can count the seconds between lightning and thunder and divide by three for the distance in kilometers. Its molecules average about 500 m/s, so sound runs at roughly two thirds of their speed in every gas, at every temperature. Helium, whose atoms are seven times lighter than an air molecule, carries sound almost three times faster; heavy carbon dioxide carries it slower.

The formula behind the curves reads the same story in symbols:

Here $T$ is the absolute temperature, $M$ the molar mass, $R$ the gas constant, and $\gamma$ a number near 1.4 for air that accounts for the gas heating up a little as each squeeze passes [1]. Pressure appears nowhere. That should be surprising, since the squeezing force in our bead model was the pressure.

Climb through the standard atmosphere and watch the three curves. Drag the altitude :

From the ground to 11 km the pressure falls to a fifth, the temperature falls by 71 degrees, and the speed of sound drops from 340 to 295 m/s, tracking the temperature. Above 11 km the temperature stops falling. The pressure keeps dropping, by half again at 16 km, yet the speed of sound stays flat. That flat stretch is the clean experiment: pressure changes by a factor of three, the speed of sound by nothing.

The reason is the one in the quiz. Thinner air pushes back more weakly when squeezed, but there is proportionally less of it to set moving, so the ratio of stiffness to density, and with it the speed, does not change. What the molecules carry the push with is their own speed, and that depends only on temperature.

A wave of pressure#

Back to a steady tone. In the first figure, the pressure curve and the displacement curve were shifted against each other by a quarter of a wavelength. That shift is the most confusing thing about sound waves, so it is worth a figure of its own. Drag the probe along the air and compare what the parcels are doing with the pressure there:

Put the probe where the frozen pressure curve peaks. The parcels on the left of the probe are moving right and the parcels on its right are moving left: they are closing in, so the air between them is being squeezed. Where the displacement is largest the parcels on both sides are pushed the same way by the same amount, so nothing is squeezed and the pressure is normal. Pressure tracks how quickly displacement changes from one parcel to the next, not how big the displacement is.

Because the air moves along the same line the wave travels, this is called a longitudinal wave, unlike a wave on a rope, where the rope moves sideways. A microphone, and your eardrum, respond to the pressure, so from now on the red pressure curve is the one we will plot.

Pitch is how often#

The loudspeaker's cone moved back and forth 220 times a second, and the air delivered 220 squeezes a second to anything in its path. That rate is the frequency, and our ears hear it as pitch. Because every squeeze travels at the same 343 m/s, a higher frequency packs them closer together in space. Sweep the frequency from the lowest note of a pipe organ to the edge of human hearing, and press play to hear it:

At 440 Hz, the A above middle C, the squeezes arrive every 2.3 ms and sit 78 cm apart in the air. The lowest organ pipes make waves over 10 m long, longer than most rooms; the highest sounds a young adult can hear, around 17 000 Hz, make waves 2 cm long, shorter than a fingertip. That thousandfold range in size is why low notes bend around corners and through walls while high notes cast sound shadows: an obstacle only blocks a wave that is small compared with the obstacle.

Sweeping the slider also shows something about the ear. The step from 110 to 220 Hz and the step from 1760 to 3520 Hz sound like the same jump in pitch, although one is 110 Hz and the other sixteen times larger. We hear ratios of frequency, not differences. Play two notes on a keyboard and compare:

Every C on the keyboard has exactly twice the frequency of the C below it: C3 is 130.8 Hz, C4 261.6 Hz, C5 523.3 Hz. Two notes an octave apart sound like the same note, higher. Between the Cs, the twelve keys of an octave split the doubling into twelve equal ratios of about 1.059 each. On the frequency axis under the keys, those equal musical steps get wider and wider in hertz; to the ear they are all the same size. The ear works in ratios, and it will again.

Loudness is how hard#

Frequency is how often the air is squeezed. How hard it is squeezed, the size of the pressure swing, is the amplitude, and we hear it as loudness. The numbers here are startling. Normal air pressure is about 101 000 pascals. Drag the sound level and see how much a sound changes it, and how far it moves the air:

The faintest sound a young ear can detect at 1 kHz swings the pressure by about 20 micropascals, a five-billionth of an atmosphere, and moves the air back and forth by about ten trillionths of a meter, a tenth of the width of an atom. At the threshold of pain the swing is about 60 pascals, still less than a tenth of a percent of the air's pressure. Between the two lies a factor of three million in pressure, and the ear handles all of it.

Numbers that span six orders of magnitude are awkward, so sound levels are quoted in decibels1Strictly, decibels of sound pressure level, dB SPL. A decibel is only a ratio, so every decibel scale names its reference; this one is 20 µPa.: twenty times the base-ten logarithm of the pressure swing over 20 µPa. Every 20 dB is ten times the pressure; every 6 dB is roughly double. The scale happens to match hearing well, because the ear is logarithmic too. Compare the ladder of everyday sounds on a decibel scale with the same sounds on a plain pressure scale, and listen to a tone stepping down in 10 dB stairs:

On the pressure scale a whisper, a conversation, and a vacuum cleaner are all indistinguishable from silence; only the loudest sounds register. Yet the ear tells a whisper from a conversation without effort. Experiments that asked listeners to judge "twice as loud" found that a sound must rise by about 10 dB, three times the pressure, to be heard as twice as loud [2]. The stairs in the tone drop by 10 dB each; most people hear each step as roughly halving the loudness.

Distance is the other thing that makes sounds quieter. A small source sends its sound out in every direction, so the same energy spreads over a growing sphere. Drag the listener away from the source:

Each time the distance doubles, the same energy covers four times the area, so the intensity drops to a quarter and the level by 6 dB: from 1 m to 2 m to 4 m to 8 m the meter reads 94, 88, 82, 76 dB. That is the inverse square law. Indoors, walls send the sound back and the law fails after a few meters. Outdoors it holds until the air itself begins to absorb the sound, which it does far more for high frequencies than low: distant thunder is a rumble because its crackle has been filtered out on the way.

Sounds add#

When two sounds reach the same spot, the air there gets two sets of pushes at once and simply adds them. The pressure at each instant is the sum of the two pressures. That sounds trivial, but it has striking consequences. Shift the second wave against the first with the phase , and listen to the sum:

In step, the waves reinforce and the sum is twice as tall. At half a cycle apart each squeeze of one wave lands on a stretch of the other, and the air does not move at all: two sounds make silence. This is interference, and noise-cancelling headphones run on it. A microphone outside hears the noise and the speaker inside plays it flipped upside down, so what reaches the eardrum is nearly flat. It works best for steady low drones like an engine's, whose waves are long and predictable.

Two tones that are close in pitch but not equal are more interesting, because their phase difference drifts. Set the second tone a few hertz away from the first, 440 Hz, and listen:

The two tones slide into step, reinforce, slide out of step, cancel, and come back, so the sum throbs. These swells are beats, and they come at the difference of the two frequencies: 444 and 440 Hz beat four times a second. Piano tuners and guitarists listen for them: when two strings that should match beat slowly, they are almost in tune, and the beating stops when they are exactly equal. Push the difference past about 15 Hz and the beats are too fast to follow; the pair sounds rough, and further apart it separates into two notes.

The color of a sound#

A piano and a clarinet can play the same A, equally loud, and you still tell them apart at once. The difference, their timbre, lives in the shape of the wave. The pure sines used so far sound plain, like a tuning fork; instruments make waves with corners and wiggles. Early in the 1800s Joseph Fourier showed that any repeating shape can be built by adding sine waves whose frequencies are whole-number multiples of the repetition rate.2Mathematicians spent most of the following century settling exactly which shapes this works for. Every sound a microphone can record is among them. Here is a tone built from ten of them. Drag the tips of the stems to set the strength of each, and listen:

The bottom of the figure is the tone's spectrum: one stem per harmonic, its height the strength of that sine. The top is the wave they add up to. The fundamental alone is a pure sine at 220 Hz. Add overtones at 440, 660, 880 Hz and up in the proportions 1, ½, ⅓, ¼, and the sum becomes a sawtooth, the bright buzz of a bowed string. Keep only the odd ones and it becomes a square wave, close to a clarinet's hollow low notes. The pitch never moves: the pattern still repeats 220 times a second.

Now shift the phases, sliding each harmonic a different distance along in time. The wave changes shape beyond recognition, yet sounds almost the same. The ear behaves like the spectrum, not the waveform: it splits a sound into its frequencies and largely ignores how they line up. Hermann von Helmholtz made this case in 1863 [3]; the last section shows the part of the ear that does the splitting.

Where do an instrument's harmonics come from? A guitar string is pinned at both ends, so those two points stay still whatever it does. Only ripples that fit a whole number of half-wavelengths between the ends reinforce themselves on every round trip. Step through the shapes that fit:

Each shape is a standing wave: the string swings in place, with nodes that never move and antinodes that swing the most. On a guitar's 65 cm A string, ripples travel at about 143 m/s, so the first shape, half a wavelength long, vibrates at 143 ÷ (2 × 0.65) = 110 Hz, and shape number at that many times 110 Hz. A string's overtones are exact harmonics, which is why it has a clear pitch. A drumhead's shapes are not whole-number multiples, so most drums thud rather than sing.

A real pluck starts many shapes at once. Just before you let go, the string is two straight lines meeting at your finger, and that triangle is one particular mix of the shapes. Drag the plucking point along the string, then pluck it:

Pluck near the bridge and high harmonics are strong: the thin twang of picking close to it. Pluck at the middle and every even harmonic vanishes, because every even shape has a node there, and pulling on a point that a shape keeps still cannot start it. Pluck a third of the way along and every third harmonic goes. The sound here is built the same way: each shape starts in the amount the triangle contains and then fades, the high ones faster, which is why a plucked note mellows as it dies away.

Pipes, resonance, and the voice#

Wind instruments use the same trick with air. In a pipe the air itself sloshes back and forth, and the ends set the rules. At an open end the air moves freely, but its pressure is held at the room's, so the pressure cannot swing there. At a closed end the air cannot move at all. Compare the two kinds of pipe, both 60 cm long:

In the open pipe, like a flute, every harmonic fits, as on a string. Close one end, as a clarinet's reed does, and only patterns with still air at the closed end and free air at the open end fit: a quarter wavelength, three quarters, five quarters. The even harmonics vanish and the lowest note drops an octave. That is why a clarinet, about as long as a flute, plays almost an octave lower, and why its low notes sound hollow, like the square wave.

Strings, pipes, and solid objects all have frequencies they prefer. Tap a wine glass and it rings at its own. Now push it with sound instead: a tone whose pressure pushes the rim back and forth. Sweep the frequency of the push past the glass's own, here 660 Hz, and watch how the rim responds:

Well below its own frequency the glass follows the push, in step and only a little. Well above, it cannot keep up and barely moves, half a cycle behind. At 660 Hz each push arrives just as the rim is already moving that way, and the swings grow until friction drains energy as fast as the pushes add it. This is resonance. The sharpness Q sets how tall and narrow the peak is: something that loses little energy each cycle rings long after a tap, and responds hugely, but only near its own frequency.

Your voice uses resonance too. The vocal folds in your larynx snap open and shut about 100 to 250 times a second, making a buzz rich in harmonics. Above them sits the vocal tract, a tube about 17 cm long, closed at the folds and open at the lips. Like the clarinet it resonates at a quarter wavelength and its odd multiples, about 500, 1,500, and 2,500 Hz for a relaxed "uh", boosting the harmonics near them [4]. Your tongue and lips move these resonances. Drag the point on the vowel chart, or change the pitch of the voice:

The peaks of the purple curve are the formants, and the first two decide the vowel. In Peterson and Barney's measurements of 76 speakers [5], "ee" has a low first formant and a high second, "ah" a high first, and "oo" both low. Now slide the pitch from . The harmonics move, the formants stay put, and the vowel stays the same. That is how you understand a child and a bass saying the same word.

Echoes and moving sources#

Sound that meets a hard wall bounces off it, as light does off a mirror. Clap in front of a cliff or a big building and the clap comes back. Drag the wall nearer or further, and clap:

The echo takes as long as the sound needs to reach the wall and come back: twice the distance divided by 343 m/s. The reflection spreads as if from a mirror image of you behind the wall. An echo that returns within a few tens of milliseconds fuses with the original into one sound. For speech, reflections up to about 50 ms late make a voice fuller rather than doubled [6]. Past about 17 m the echo is a tenth of a second late: a separate clap. Bats time their echoes this way to hunt in the dark.

A siren rushing past drops in pitch the instant it passes: a high note coming, a low note going. The siren never changes its note. It moves while it makes it, and each wavefront spreads from the spot where the source was when it sent that front out. Speed the source up:

Ahead of the source the fronts crowd together, so more of them reach you each second and the pitch is higher. Behind, they spread out and the pitch drops. This is the Doppler effect: a source approaching at speed v is heard at f·c/(c − v), and receding at f·c/(c + v). A horn at highway speed, Mach 0.09, falls about three semitones as it passes. At the speed of sound the fronts pile into one shock at the nose; faster still they trail inside a cone with sin θ = 1/M. When that cone sweeps over the ground, people hear a sonic boom: a double crack from the shocks at the nose and the tail.

The ear#

All of this ends at the ear. The outer ear funnels sound into the ear canal, a tube about 2.5 cm long sealed at its far end by the eardrum. Behind the eardrum, in a small air-filled cavity, three tiny bones, the ossicles, carry the vibration to the cochlea, a coiled tube of fluid buried in the skull, where it becomes nerve signals. Point at the parts to name them, and play a tone to set them moving:

The motion is enlarged enormously here. At the loudness of conversation the eardrum moves by a few billionths of a meter. Why the bones at all? Because the cochlea is full of fluid, and sound in air bounces off water almost completely. Air is light and gives way easily; water is dense and stiff, so a pressure that sets air moving barely moves water. Of the energy in a sound wave striking a water surface, only about a tenth of a percent gets in. The middle ear has two tricks to do better. Change them:

The eardrum has about 17 times the area of the stirrup's footplate, so the force it collects is concentrated on a patch 17 times smaller, and the pressure rises 17-fold. The hammer and anvil add a lever of about 1.3. Together they raise the pressure about 22 times, some 27 dB [7], which in this simple model lets around 40% of the energy into the fluid instead of 0.1%. A perfect match would need about 60 times. When the middle ear fails, from fluid behind the eardrum in a child's ear infection or bones fused by disease, hearing drops by tens of decibels.

Inside the cochlea, running its whole length, is the basilar membrane, a strip of tissue that carries about 3,500 rows of hair cells. Unroll the cochlea and pick a frequency:

The membrane is narrow and stiff at the base, by the stirrup, and about five times wider and far floppier at the apex. So every stretch of it is tuned to a different frequency, from 20 kHz at the base to 20 Hz at the apex. Georg von Békésy watched the result in cochleas from cadavers [8]: a tone sends a ripple along the membrane that grows until it reaches the place tuned to it, then dies out abruptly.3In a living ear the outer hair cells push on the membrane in time with the sound, making the peaks far taller and sharper than in Békésy's cadavers. The figures here show the passive version. Donald Greenwood fitted the map from place to frequency [9]. Here is the membrane unrolled, with two tones:

Each tone sets its own stretch of membrane moving, and the hair cells there fire. The cochlea does in fluid and tissue what the spectrum did on paper: it splits a sound into its frequencies, which is why you hear the harmonics and not the shape of the wave. Tones close in pitch stir overlapping stretches and blur into beats or roughness; tones far apart are heard as two. And because every sound passes the base on its way in, the hair cells there take the most wear: hearing loss from age and noise begins with the highest pitches.

Clap your hands once more. The squeeze leaves your palms at 343 m/s, handed from molecule to molecule while each one only rocks in place. It spreads and fades by 6 dB for every doubling of distance, bounces off the walls, and reaches your ear a few milliseconds later as a pressure wiggle of a fraction of a pascal. The eardrum follows it, three bones turn it into a push on fluid, and a ripple runs along the cochlea. A clap holds every frequency at once, so the whole membrane stirs, base first, and within a few milliseconds your brain knows what made the sound and where.

References#

1. L. E. Kinsler, A. R. Frey, A. B. Coppens, and J. V. Sanders, Fundamentals of Acoustics, 4th ed. (Wiley, 2000).
2. S. S. Stevens, “The measurement of loudness”, Journal of the Acoustical Society of America 27(5), 815–829 (1955).
3. H. von Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music, translated by A. J. Ellis, 2nd English ed. (Longmans, Green, 1885); first German edition 1863.
4. G. Fant, Acoustic Theory of Speech Production (Mouton, The Hague, 1960).
5. G. E. Peterson and H. L. Barney, “Control methods used in a study of the vowels”, Journal of the Acoustical Society of America 24(2), 175–184 (1952).
6. H. Haas, “The influence of a single echo on the audibility of speech”, Journal of the Audio Engineering Society 20(2), 146–159 (1972); translated from Acustica 1, 49–58 (1951).
7. J. O. Pickles, An Introduction to the Physiology of Hearing, 4th ed. (Emerald, 2012).
8. G. von Békésy, Experiments in Hearing, translated and edited by E. G. Wever (McGraw-Hill, 1960).
9. D. D. Greenwood, “A cochlear frequency-position function for several species: 29 years later”, Journal of the Acoustical Society of America 87(6), 2592–2605 (1990).