Calculate depth/distance from time and wave velocity

Follow one sonar pulse

The boat sends a pulse of sound straight down. So far the sound has travelled 0 m.

Drag along the track, or press play, and watch how far the sound has travelled at each moment.

Rearranging s = v × t

Cover the quantity you want. For an echo, s is the there-and-back distance, not the depth.

Tap the quantity you want to find. The triangle shows you the formula.

Predict, then check

Same echo time, different material. Commit before you look.

A sonar echo takes 0.7 s to come back when the sound travels in water. Now imagine the same 0.7 s echo time, but with the sound travelling in air. Compared with the water case, where is the reflecting object?

What you need to know

• An echo is sound that bounces off a surface and comes back to you, so you hear it again.
• Speed is distance travelled for every unit of time: speed = distance ÷ time, so distance = speed × time.
• In s = v × t, s is distance travelled (m), v is speed (m/s) and t is time taken (s).
• Have a goA sound travels 60 m in 3 s. What is its speed, with the unit?

  20 m/s

  Speed = distance ÷ time = 60 ÷ 3, and metres over seconds gives m/s. Multiplying instead would find a distance, not a speed.

• Units must match: metres with seconds gives m/s, kilometres with hours gives km/h. So 0.7 km becomes 700 m first.
• Have a goA pulse travels 0.5 km in 2 s. Work out its speed in m/s.

  250 m/s

  Convert first: 0.5 km × 1000 = 500 m, then 500 ÷ 2 = 250. Dividing 0.5 by 2 would give a speed in km/s, not m/s.

• Wave speed depends on the medium: sound is about 340 m/s in air, but 1,500 m/s in water in the sonar example.
• The longer an echo takes to return, the further away the object is.
• Sonar uses this underwater: the speed of sound in water is known, so echo time gives a distance.
• Sound goes there and back, so the distance to the object is half the distance the sound travelled.
• Have a goJo times an echo from a wall in air: 1.5 s. 'Sound goes 340 m/s, so the wall is 340 × 1.5 = 510 m away!' Jo is very sure. How far away is the wall really?

  255 m. Jo found the there-and-back distance (510 m).

  Jo stopped one step early. The sound went to the wall and back, so the wall is only half of the distance the sound travelled.

• Sonar in four steps: pulse down and reflected, echo time measured, speed × time gives there-and-back distance, depth is half.

The big picture

An echo makes the sound travel to an object and back. Use distance = speed × time to find the whole trip, then halve it to get the depth or distance to the object.

⚠ Watch out

Using speed × echo time as the depth. That is the distance the sound travelled down and back, so the depth is half of it.

Memory hook

Echo = there AND back. Speed × time gives the whole trip, then halve it for the way there.

Check yourself

Why do you halve speed × echo time to get a depth? Then try it: water, 1,500 m/s, echo after 1.2 s. How deep? (Speed × time is 1,800 m, there and back.)