Sound - Physics for Pilot Assessments | Pilot Assessment Training

Sound is the one wave you can check against your own experience all day, and it is also the one the knowledge tests calculate with most often. By the end of this lesson you should be able to say what is physically travelling when a sound reaches you, explain why a sound cannot cross a vacuum, quote the speed of sound in air and say what changes it, connect amplitude and frequency to loudness and pitch, and handle an echo calculation without losing the factor of two that half the candidates lose.

What is actually travelling

Something has to vibrate. A loudspeaker cone moves out and in, a guitar string flicks back and forth, a vocal fold flaps — and each time the surface moves outwards it pushes the air immediately in front of it, crowding those air particles a little closer together. That crowded region does not stay put: the squeezed particles push on their neighbours, crowding them in turn, and the region of crowding travels outwards through the air. When the surface moves back it leaves the air in front of it momentarily thinner than normal, and that thinned region follows the crowded one outwards in the same way. What travels from the source to your ear is that pattern of crowded and thinned air — compressions and rarefactions, the vocabulary from the wave anatomy lesson — arriving in the order they were made.

So sound is a longitudinal wave: the air particles move backwards and forwards along the same line the sound is travelling, not across it. They do not make the journey themselves. Each particle shifts a fraction of a millimetre and returns, exactly as the rope in the earlier lesson did, and the air by your ear when somebody speaks to you across a room is the same air that was there before — a fact easily checked by noticing that you feel no wind. What arrives at your eardrum is the pressure variation, which pushes it in and out at the frequency of the source, and what your ear then does with that is biology rather than physics.

It needs something to travel through

Because sound is nothing more than particles passing a push along, it cannot exist where there are no particles. This is the classic demonstration: an electric bell hung inside a sealed glass jar, ringing away audibly, with the air slowly pumped out. As the air thins the ringing fades, and with a good vacuum it becomes inaudible even though you can plainly see the hammer still striking the bell. Let the air back in and the sound returns. Nothing about the bell changed; what changed is that there was no longer any material to carry the wave. A vacuum is silent, and space, being close to a vacuum, is silent too.

Note the contrast, because it is the reason the final lesson of this module exists. You could still see the bell throughout, so whatever light is, it crossed the vacuum that stopped the sound. Any wave that needs a medium is called mechanical, and sound, water waves and waves on a rope are all mechanical. That is a genuine physical divide between two families of wave and not a technicality.

How fast sound goes

In air at ordinary temperatures sound travels at about 340 metres per second. Treat that as the working figure — the true value is around 331 metres per second at the freezing point of water and about 343 at 20 degrees Celsius, so 340 is a rounding chosen because it divides nicely and is close enough for every item in the bank. It is fast on human timescales and slow on the scale of distances: about a kilometre every three seconds, which is why you can count the gap between a lightning flash and its thunder and get a rough distance in kilometres by dividing the seconds by three.

The speed depends on the material, and the pattern runs the opposite way to most people's guess: sound goes fastest in solids, slower in liquids, and slowest in gases. Roughly 340 metres per second in air, about 1500 in water, and about 5000 in steel. The mechanism is the one from the states of matter lesson in the earlier module. In a gas the particles are far apart, so a crowded region has to wait while each particle crosses a relatively large gap before it can pass the push on. In a liquid the particles are already touching, and in a solid they are not only touching but bound to one another, so a shove is handed on almost immediately. Tighter coupling means a faster wave, and the density of the material and the stiffness of the bonds between its particles are what settle the figure.

Temperature matters too, within a gas. Warm the air and its particles are already moving faster and colliding more often, so they pass a disturbance along sooner and the speed of sound rises — by roughly 0.6 metres per second for each degree Celsius, which is worth knowing as a direction rather than as a number to calculate with. Cold air carries sound more slowly than warm air. What does not affect the speed is the sound itself: a loud sound and a quiet one travel at the same speed, and so do a high note and a low one, which is the only reason music arrives as music rather than as its parts in order of pitch.

Example 1 — how far away is the storm

You see a lightning flash, and the thunder arrives 6 seconds later. Taking the speed of sound as 340 metres per second, how far away did the lightning strike?

The light of the flash reaches you essentially instantly — it is so much faster than sound that its travel time can be ignored — so the 6 seconds is the time the sound alone took.
Distance is speed × time, from the mechanics module: $d = 340 \times 6 = 2{,}040\ \text{m}$
So the strike was about 2 kilometres away — and note there is no halving here, because the sound made a one-way trip from the strike to you. Halving belongs to echoes, which come later in this lesson.
Check against the rule of thumb from earlier: seconds divided by three gives kilometres, and $\frac{6}{3} = 2$ kilometres. It agrees, because a kilometre takes sound about three seconds.

Loudness and pitch

Two properties of the wave map onto the two things your ear notices. The amplitude sets the loudness: a bigger movement of the loudspeaker cone crowds the air more strongly, so the pressure swings further from normal at each compression, more energy arrives each second, and the sound is louder. The frequency sets the pitch: more compressions arriving per second is heard as a higher note. They are independent, which is why a volume control changes the loudness of a note without changing which note it is, and they are the pair that items most often swap over.

Human hearing runs from roughly 20 hertz at the bottom to roughly 20 000 hertz — 20 kilohertz — at the top, with the upper limit falling steadily with age. Below 20 hertz the vibrations are felt rather than heard and are called infrasound; above 20 kilohertz they are called ultrasound, and the prefix ultra- there means beyond, not powerful. Ultrasound is ordinary sound in every physical respect, obeying the same wave equation at the same speed; the only special thing about it is that human ears do not respond to it.

Echoes, and the factor of two

Sound reflects off a hard surface, and a reflected sound heard as a separate event is an echo. Everything about echo calculations follows from one geometric fact: the sound has to get to the reflecting surface and come back, so the distance it covers is twice the distance to the surface. The time you measure is the time for the whole round trip, and dividing that time by the speed of sound gives you the whole round trip, not the distance to the wall. That is the halving step candidates forget.

The listener is in the same place as the source, so the sound covers the distance twice. Divide the total path by two to get the distance to the wall.

Distance to a reflecting surface from an echo

d = \frac{v t}{2}

$d$: the distance to the reflecting surface, in metres (m)
$v$: the speed of sound in the medium, in metres per second (m/s)
$t$: the time from making the sound to hearing the echo, in seconds (s)
$2$: the round trip — out to the surface and back again

It is safer to do the calculation in two visible steps than to use that formula in one go: find the total distance travelled first, from speed × time, then halve it. Doing it that way makes the halving a deliberate act rather than something to be remembered, and it also lets you work the problem backwards, where you double the distance first and then divide by the speed.

Example 3 — working backwards to a time

A wall is 510 metres away. How long after a sound is made will its echo be heard? Take the speed of sound as 340 metres per second.

The sound must cover the distance twice, so the total path is 2 × 510 = 1020 metres.
Rearrange distance = speed × time for the time by dividing both sides by the speed, which leaves $\text{time} = \frac{\text{distance}}{\text{speed}}$ .
So the time is $\frac{1020}{340} = 3$ seconds.
Check it forwards: 340 × 3 = 1020 metres of travel, which is 510 metres out and 510 metres back. Correct.
Note that doubling comes first here and halving came first in Example 2. Which way round depends on whether you are given the time or the distance, which is why it is worth thinking about the round trip rather than memorising a formula.

Example 4 — the wave equation applied to sound

A tuning fork sounds a note of frequency 170 hertz in air. What is the wavelength of the sound? And what would the wavelength of the same note be in water, where sound travels at 1500 metres per second?

In air, use $\text{wavelength} = \frac{\text{speed}}{\text{frequency}}$ from the previous lesson: $\lambda = \frac{340}{170} = 2\ \text{m}$
In water the frequency is unchanged, because the frequency is set by the fork and the crests must leave the boundary as often as they arrive: still 170 hertz.
So in water the wavelength is $\frac{1500}{170}$ , which is about 8.8 metres.
The note is the same note — the pitch you hear depends on the frequency, not on the wavelength — but it is stretched out over more than four times the distance, because the wave is travelling more than four times as fast.

Why it matters in aviation

The fact that the speed of sound depends on temperature has a direct consequence for fast aircraft. Air gets colder with height through most of the region airliners cruise in, and colder air passes a disturbance along more slowly, so the speed of sound up there is lower than it is near the ground. The quantity that matters to a fast aeroplane is its speed compared with the local speed of sound, because the effects that eventually limit it come from air accelerating towards that speed as it flows over the wing, not from the airspeed on its own.

So the same true airspeed represents a larger fraction of the speed of sound at high level than it does low down. That is why a high-level cruise is flown to a Mach number rather than to an airspeed: holding an airspeed while climbing into colder air would let the ratio creep upwards, whereas holding the ratio keeps the aircraft a known margin away from the limit. It also explains why the outside air temperature gauge is a performance instrument, and why a temperature well above or below the expected value changes the numbers a crew is working with.

What the test asks

The factor of two in an echo item is the single most reliable trap in the sound bank. Speed multiplied by time gives the whole journey, and the whole journey is twice the distance being asked for, so an answer that used the time without either halving the distance or doubling it first is wrong — and it will be sitting there among the options, because it is exactly double or exactly half the right answer. The second trap is sound in a vacuum: an option claiming that sound travels through space slowly, or faster than in air because there is nothing in the way, when the correct statement is that it does not travel at all. The third is swapping loudness and pitch, usually as an option saying that increasing the amplitude raises the pitch. And expect one item on the ordering of speeds through solids, liquids and gases, where the intuitive but wrong answer is that sound goes fastest through air because air is the thing we normally hear it in.