Compressions, rarefactions and the wave equation

In Discover you found the vibration behind every sound, and followed it to your ear. Good. But a sentence such as “the pattern of pushes travels through the air” is a promise, not an explanation.

This layer keeps the promise. You will see exactly what is squashed and what is stretched, learn why a sound wave is a completely different shape of wave from a wave on a rope, get the one equation that ties speed, frequency and wavelength together, find out why steel carries sound seventeen times faster than air, and take the ear apart properly.

By the end you should be able to take any sound and say three things about it with numbers: how fast it is travelling, how often it is vibrating, and how long each wave is.

Air is not empty. Every cubic centimetre around you holds an enormous number of molecules, flying about, bumping into each other billions of times a second. On average they are evenly spread, and the pressure is the same everywhere.

Now put a loudspeaker cone in the middle of it and push the cone forward. The molecules right in front have nowhere to go instantly, so they crowd together. That crowded patch has slightly higher pressure than normal. We call it a compression.

Pull the cone back and it leaves a little extra room behind it. The molecules there spread out. That thinned patch has slightly lower pressure than normal. We call it a rarefaction.

Now move the cone in and out steadily, hundreds of times a second. You make compression, rarefaction, compression, rarefaction, and each one shoves the next patch of air, so the whole striped pattern marches outwards. That marching pattern of high and low pressure is the sound wave.

Waves come in two shapes, and sound belongs firmly to one of them.

Stretch a long rope across the floor and flick one end sideways. A hump runs down the rope to the far end. But look at any one point of the rope: it moves up and down, at right angles to the direction the hump is travelling. That is a transverse wave. Water ripples and light behave this way.

Now stretch a slinky spring along the floor and, instead of flicking it sideways, push one end sharply forwards and back along its own length. A bunched-up patch of coils races down the spring. Watch one coil: it slides forwards and back, in the same line as the travelling patch, and stays roughly in place. Between the bunched patches the coils are stretched apart.

That is a longitudinal wave, and a slinky is doing exactly what air does when a drum sounds. Bunched coils are compressions; stretched coils are rarefactions.

Freeze a sound wave in mid-air and measure the distance from one compression to the next. That distance is the wavelength, written with the Greek letter lambda, λ.

Now unfreeze it and stand still while it goes past. In one second, f complete waves sweep over you, where f is the frequency in hertz. Each of those waves is λ metres long. So in that one second, the front of the wave must have moved f × λ metres.

That is the speed. And so we have the single most useful equation in the whole of wave physics:

v = f × λ

Speed equals frequency times wavelength. It is not a law about sound in particular; it is simply what the words mean, and it is true for every wave there is.

“343 metres per second” is not one of those numbers that is simply true, like the number of sides of a triangle. It is the answer to a specific question: how fast does sound travel in dry air at 20 degrees Celsius?

Change the material and the answer changes enormously. Change the temperature and it changes a little. Change the loudness, the pitch or the wavelength and it does not change at all, which is a surprise worth taking seriously.

Two properties of a material decide its answer:

• Stiffness: how strongly the particles resist being squashed and how hard they spring back. More stiffness means a faster hand-off, so faster sound.
• Density: how much mass has to be shifted each time. More density means more sluggishness, so slower sound.

Speed rises with stiffness and falls with density. In practice stiffness wins by a mile going from a gas to a solid, which is why steel beats air even though steel is thousands of times denser.

Now temperature. Warm air is air whose molecules are moving faster, and molecules that already move faster carry a push to their neighbours sooner. So warm air carries sound faster than cold air.

The school formula is a straight line and is accurate enough for anything you will meet:

v = 331.3 + 0.606 × (temperature in °C)

Every extra degree adds about 0.6 metres per second. So:

• At 0 °C, a cold Shimla morning: 331.3 m/s.
• At 20 °C, a pleasant room: 343.4 m/s, which we round to 343 throughout this topic.
• At 40 °C, a May afternoon in Nagpur: 355.5 m/s.

That is a change of about 7 per cent across the year, small enough to ignore in most problems and large enough to be real. Notice what is not in the formula: pressure. At ordinary pressures, sound travels at the same speed at sea level and on a hill. Only the temperature matters.

Your ear can hear a pin drop and can also survive a thunderclap. Between those two the energy arriving differs by a factor of about a million million: 1,000,000,000,000.

No ordinary scale can show that. If a whisper were one millimetre on a ruler, a thunderclap would be a thousand kilometres along it.

So we squash the scale. The decibel scale counts multiplications rather than additions:

• +10 dB means the sound is carrying ten times as much energy, and sounds about twice as loud to you.
• +20 dB means a hundred times the energy.
• +30 dB means a thousand times.

Zero decibels is not silence. It is the quietest sound a healthy young ear can just detect, and it is the point the whole scale is measured from.

In Discover you found three ways to raise the pitch of a string: make it shorter, tighter or thinner. Now we can say why, and put numbers on the first one.

A string clamped at both ends can only vibrate in whole patterns that fit exactly between the clamps. The simplest one has the string bulging in the middle and still at both ends, and that pattern makes the note you hear. Make the string half as long and the same wave has half the distance to cover on each swing, so it completes twice as many swings a second. Half the length is exactly double the frequency, a jump musicians call an octave.

Tension and thickness work through how sharply the string springs back. Tighter means a stronger restoring pull, so a faster spring-back, so a higher note: frequency rises with the square root of the tension, so you need four times the tension to double the pitch. Thicker means more mass to move, so a slower spring-back and a lower note.

The ear has a hard engineering problem to solve, and the solution is worth admiring.

Sound arrives in air, which is easy to move. It must end up shaking liquid in the inner ear, which is much harder to move. Shout at a swimming pool and almost all the sound bounces off the surface; only about a thousandth of it gets in. If your ear worked that way you would be nearly deaf.

The middle ear is the fix. Three small bones take the movement of the fairly large eardrum and concentrate it onto the much smaller oval window, while acting as a lever at the same time. Area concentration multiplied by lever action gives a pressure gain of roughly twenty times.

That is exactly enough to push the liquid properly. The middle ear is a matching device, the same idea as the gears on a bicycle: it trades a big, weak movement for a small, strong one.

Human hearing runs from about 20 Hz to 20,000 Hz. That is a range of a thousand to one, which musicians count as about ten octaves, since each octave is a doubling and ten doublings multiply by 1,024.

It is a generous window, but it is only a window. Sound exists above and below it, made by ordinary vibrations, obeying the same rules, and simply not detected by us.

• Above 20,000 Hz is ultrasound. Bats shout up to about 120,000 Hz and listen for the echoes. Dolphins go higher still. A hospital scanner uses millions of hertz to draw a picture of a baby.
• Below 20 Hz is infrasound. Elephants call at around 14 Hz and hear each other across kilometres. Volcanoes, earthquakes and large storms make it, and networks of infrasound microphones listen worldwide for distant explosions.

And the window narrows as you age. Nearly everyone loses the top end, from about 20 kHz in childhood to perhaps 12–15 kHz by middle age. It is normal, it is gradual, and loud noise makes it happen sooner and go further.