How can you measure the distance to a cliff with a shout and a stopwatch?

Sound comes back

When a sound wave strikes a hard, smooth surface, a good part of it is reflected. The surface does not absorb the energy but returns it, and that return is the echo.

The ear only separates the original from its echo if more than about a tenth of a second passes between them. Below that the brain hears one lengthened sound, which is why you get an echo from a distant cliff and not from a nearby wall.

A tenth of a second at 340 metres per second is 34 metres of travel. Since the journey is there and back, the surface has to be at least about 17 metres away before you hear a separate echo.

Calculating a distance

The formula is distance equals speed times time divided by two. That division by two is the whole point, and forgetting it is the commonest mistake in every echo calculation.

The reason is simple: the stopwatch measures the time for the sound to reach the surface and return. The distance you want is only the first half of that journey.

The same calculation drives a ship's sonar measuring the depth of the sea, and medical ultrasound measuring distances inside the body. Only the speed of sound in the medium changes: about 1,500 metres per second in water and in tissue, against 340 in air.

Acoustics: when an echo is a problem

Hard surfaces — concrete, glass, tiles — reflect sound well. In an empty hall the reflections come back again and again from every wall, mix with the speech and leave it smeared and hard to follow.

Soft, porous materials — curtains, carpets, upholstery, an audience of people — absorb the energy instead of returning it. The wave enters the fibres, wears itself out against them, and ends up as heat.

This is why a concert hall is designed carefully: part of it reflects so the sound reaches the back rows, and part absorbs so nothing is smeared. The same reasoning decides where absorbing panels go in a noisy classroom.

Worked examples

1. A shout returns from a cliff after 5 seconds. How far away is the cliff?

   1. In that time the sound went there and came back
   2. The total distance is 340 times 5, which is 1,700 metres
   3. The distance to the cliff is half of that

   Answer: 850 metres

2. A sonar sends a pulse and receives the return after 0.4 seconds. How deep is the sea, if sound travels at 1,500 metres per second in water?

   1. Total distance is 1,500 times 0.4, which is 600 metres
   2. Divide by two for the round trip

   Answer: 300 metres

3. Why does a classroom start to echo once the chairs and the pupils are taken out of it?

   1. People and fabric absorb sound
   2. Without them only hard walls are left
   3. Every reflection returns into the room instead of being absorbed

   Answer: Because the absorbing surfaces were removed and only reflecting ones remain

Common mistakes

Forgetting to divide by two

The measured time covers a round trip. Without the division the answer comes out twice too large, and this is the error that recurs in nearly every test on the topic.

Using 340 metres per second in water as well

The speed of sound depends on the medium. In water it is about 1,500 metres per second, so a sonar calculation using the air value gives a depth four and a half times too small.

Expecting an echo from a nearby wall

The ear needs more than about a tenth of a second to separate two sounds. A wall three metres away returns the sound in far less than that, and the brain hears a single sound.

More in Waves and Sound

• If a wave travels towards the shore, why does a cork not travel with it?
• What happens to the wavelength when the frequency doubles and the speed stays the same?
• Why does a ringing bell fall silent when the air is pumped out of the jar?
• Is a high-pitched sound necessarily a loud one?
• Why is a guitar string loud over a wooden box and almost silent without one?