S4.10.3 Chapter 3

Waves on water

Priya's ripples run from deep water onto a shallow part of the tray. What happens to them?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A Ripples slow down and bunch up as they run onto shallow water.
You picked A. Read on. At the end, the chapter comes back to your guess.
B Ripples speed up and spread out as they run onto shallow water.
You picked B. Read on. At the end, the chapter comes back to your guess.
C Ripples stop where the water gets shallow.
You picked C. Read on. At the end, the chapter comes back to your guess.
D Ripples do not change when the depth changes.
You picked D. Read on. At the end, the chapter comes back to your guess.

By the end you can

Measure the speed of ripples in a tray of water
Explain how water waves slow down and bend in shallow water

Drop a pebble into a pond and rings spread out from the place it fell. Each ring is a line of crests. Scientists study water waves in a shallow tray, seen from above, called a ripple tank.

Seen from above, a line that joins the crests is a wavefront. The wave travels at right angles to its fronts. A tap on the water makes circular wavefronts. A straight bar moved up and down makes straight ones.

Ripples from one tap. The rings are equally far apart. That gap is the wavelength.

The gap between two neighbouring rings, or two neighbouring straight fronts, is one wavelength. To measure it well, measure across several waves and divide. Do not measure a single gap, because the reading error is a bigger share of a small length.

Five fronts have four gaps between them. Four gaps are four wavelengths.

Count the gaps, not the fronts. Five fronts leave four gaps, so the distance from the first front to the last is four wavelengths. A common slip is to divide by five.

To find the speed, use the wave equation from the last chapter: v = fλ. The frequency f is how many taps you make each second. The wavelength λ is the length you measure. Keep λ and v in the same length unit, such as cm and cm/s.

Worked example

Taps at 4 Hz make five fronts with 28 cm from the first to the last. Find the wavelength and the speed.

Five fronts have four gaps. So four wavelengths together are 28 cm.
One wavelength is λ = 28 cm ÷ 4 = 7 cm.
The speed is v = fλ = 4 Hz × 7 cm = 28 cm/s.

Answer: 7 cm and 28 cm/s

Try it

Find the speed of ripples in a tray

You need: a flat tray or a wide dish, with water about 1.5 cm deep, a ruler, a pencil or a stick, a torch, a clock or phone that shows seconds, a phone camera to take a photo from above, if you have one.

Be safe: Keep the tray and the water away from sockets, wires and anything that runs on electricity. Wipe up spills at once, because a wet floor is slippery. Do not shine the torch into anyone's eyes.

Put the tray on a table or the floor, on a dark cloth. Pour in water about 1.5 cm deep. Wait until the water is still.
Lay the ruler flat on the edge of the tray where it shows along the water. Shine the torch low from the side, so the crests show as bright and dark lines.
Tap the water with the tip of the pencil at one end, at a steady beat. Count your taps for 10 s and divide by 10. This is the frequency f in Hz. Try for about 4 taps each second.
Look from straight above, or take a photo. Measure the distance across 5 fronts, which is 4 gaps. Divide by 4. This is the wavelength λ.
Work out v = fλ.

What you will see: The ripples are a few centimetres apart and travel several tens of centimetres each second. Your numbers will differ with the depth of the water and how fast you tap. Measuring four gaps together is far better than measuring one.

If you cannot try it, this is what happens.

Now change the depth. Once the water is shallower than about half a wavelength, water waves travel more slowly where it is shallower. Ripples in a tray 1.5 cm deep are in this range. This is also why ocean waves slow, shorten and rise as they reach a beach.

A flat plate in the tray makes part of the water shallow.

Send straight ripples across the tray from the deep end to the plate. Here is what happens at the edge of the shallow part. The frequency does not change. The source makes the same number of waves each second, and the same number cross the edge each second.

The speed falls in the shallow water. The wave equation v = fλ then tells us the wavelength must fall too, because f stays the same. So the fronts bunch up in the shallow water. The new wavelength is λ = v ÷ f.

Next, let the ripples meet the edge at a slant. Each part of a front slows down as it reaches the shallow part. The part that arrives first slows first, so the front turns. It is like a cart whose wheel meets sand first and swings round.

To describe the turn, draw the normal: the line at 90° to the edge. The ripples travel along arrows at right angles to the fronts. In the slower, shallow water, the ripples turn towards the normal. This bending is refraction.

Ripples cross from deep to shallow water. The fronts are closer together, and the way they travel turns towards the normal. It is drawn with the speed ratio 0.6 of the worked example below. In a real tray the bend is often smaller, because the edge of the plate is not sharp.

Look at the edge. Every front that reaches the edge carries on into the shallow water without a break. So the fronts meet the edge equally far apart on both sides. That joining is why the direction must turn when the wavelength changes.

If the waves run from shallow water into deep water, they speed up and the wavelength grows. They then turn away from the normal. Along the normal itself, waves do not turn. Their wavelength and speed still change.

Worked example

In the deep part of a tray the wavelength is 5 cm. Ripples with a frequency of 4 Hz then cross to the shallow part, where the wavelength is 3 cm. Find the speed in each part.

In the deep part v = fλ = 4 Hz × 5 cm = 20 cm/s.
The frequency stays 4 Hz in the shallow part, because the source does not change.
In the shallow part v = fλ = 4 Hz × 3 cm = 12 cm/s.

Answer: 20 cm/s and 12 cm/s

The ratio of the two speeds is 12 ÷ 20 = 0.6, which is the same as the ratio of the wavelengths, 3 ÷ 5. This works because the frequency is the same in both parts. The picture above is drawn with this ratio of 0.6.

Going further: the sea shows the same slowing on a huge scale. For a wave much longer than the water is deep, the speed is v = √(g × h). Here h is the depth in m, and g is 9.8 m/s². This rule is for tides and tsunamis. It is not for small ripples in a tray.

The speed of a long wave in the sea. 198 m/s is about 710 km/h, nearly as fast as a jet airliner. The shallow sea near a coast slows it to about 36 km/h.

Worked example

A long wave crosses a sea that is 90 m deep. Find its speed.

The rule for a long wave is v = √(g × h).
Put in g = 9.8 m/s² and h = 90 m: g × h = 9.8 × 90 = 882.
The square root is √882 = 29.7.
So the speed is about 30 m/s.

Answer: About 30 m/s

As such a wave nears a coast, the water gets shallower and the wave slows. The energy the wave carries is squeezed into less water, so the wave grows taller. This is why a tsunami is small far out at sea and tall near the shore.

Try it

Ripples over a plate

You need: the tray and the water from before, about 1.5 cm deep, a flat tile or flat piece of board about 1 cm thick, with its top just under the surface, a pencil, a ruler and a torch.

Be safe: Keep the tray away from electrical things, and wipe up spills. Set the tile down gently, because it can crack the tray.

Lay the tile flat at one end of the tray, so that the water over it is about 0.5 cm deep.
Tap the water at the other end at a steady beat. Watch the ripples cross onto the tile.
Look at how far apart the crests are over the tile compared with the deep part. Measure both, across four gaps.
Try to tap so that the ripples meet the edge of the tile at a slant. Watch the way they turn.

What you will see: The crests are closer together over the tile. That is a shorter wavelength at the same frequency, so the waves are slower there. At a slant the ripples turn towards the normal on the tile. The effect is small, so look carefully, or take a photo from above.

If you cannot try it, this is what happens.

Five fronts, with 20 cm across them. Count the gaps between them: one, two, three, four.

A wrong idea many people have

To find the wavelength, divide the distance across the fronts by the number of fronts.

Why it feels right: We count things to share a total. Five fronts look like five waves, so it seems right to divide by five.

The test that breaks it: Look at the picture: 20 cm across four gaps, not five. Divide 20 cm by 5 and you get 4 cm. The real gaps are 20 ÷ 4 = 5 cm.

What is true: Divide by the number of gaps, which is one less than the number of fronts. Six fronts have five gaps.

A wrong idea many people have

The frequency of ripples changes when they pass into shallow water.

Why it feels right: The ripples look different in the shallow part, with the crests closer together. It seems that the number of waves must have changed.

The test that breaks it: Count crests that pass a point in the deep part in 10 s, then in the shallow part in 10 s. The counts are the same. If one more arrived than left, the water at the edge would pile up forever.

What is true: The source sets the frequency, and it does not change when the wave crosses an edge. The speed and the wavelength change. The wavelength is v ÷ f.

A wrong idea many people have

Water waves speed up in shallow water.

Why it feels right: A wave on a beach seems to rush up the sand, and it looks fast as it breaks.

The test that breaks it: The ripples over the tile have crests closer together at the same frequency. With v = fλ, a smaller λ and the same f give a smaller v.

What is true: Waves slow down as the water gets shallower. A breaking wave looks quick because it is rising and tipping over, not because its speed has gone up.

Key words

ripple tank: a shallow tray of water, seen from above, for studying water waves.
wavefront: a line joining the crests of a wave, seen from above. The wave travels at right angles to it.
normal: a line at right angles (90°) to an edge, used to measure angles.
refraction: the turning of a wave when its speed changes at a slanted edge.

Back to your guess

The answer is A: Ripples slow down and bunch up as they run onto shallow water.

Water waves travel more slowly in shallower water. Their frequency stays the same, so the wavelength gets shorter and the crests bunch up. If they meet the shallow part at a slant, they also bend.

Keep it

Copy this into your exercise book.

v = fλ. Measure λ across several gaps: fronts − 1. Shallow water slows waves. f stays the same, so λ shortens and the wave bends towards the normal.

Draw: Straight wavefronts crossing an edge at a slant, closer together in the shallow part, with the normal and two arrows.