L1-07 · Track a wave
A wave is a pattern of ups and downs that travels. Its amplitude is how high a crest rises above the middle line. Its wavelength is the distance from one crest to the next. The rope itself only moves up and down.
Level 1 · Grades 4–6 · about 40 minutes
Big idea. A wave is a pattern of ups and downs that travels. Its amplitude is how high a crest rises above the middle line. Its wavelength is the distance from one crest to the next. The rope itself only moves up and down.
Before this lesson. L0-07, L1-01
Grown-ups. The guide for adults is at the end of this page.
What you will learn¶
I can point to the crests and troughs of a wave and measure its amplitude and wavelength.
I can explain that the rope moves up and down while the wave pattern moves along.
I can tell when two waves are in step and when one is half a cycle behind.
Story: The ribbon that stayed put¶
Maya ties one end of a long jump rope to a fence post. Leo holds the other end and shakes it up and down. Bumps race along the rope toward the post. The high points are called crests, and the low points are called troughs.
Maya ties a red ribbon to the middle of the rope. “Watch the ribbon,” she says. The bumps travel all the way to the post, but the ribbon does not. It just bobs up and down in the same spot. The wave pattern moves along, while each bit of rope only moves up and down.
Maya takes a photo, and they copy the wave onto graph paper. First they draw a middle line where the rope rests. Each crest is 2 squares above the middle line, and each trough is 2 squares below it. So the amplitude is 2 squares, not 4. The crests are 8 squares apart, so the wavelength is 8 squares. In 24 squares of rope, they count 3 crests.
Next Leo shakes harder, but at the same speed. The crests now rise 3 squares above the middle line, so the amplitude is 3 squares. Then he shakes faster. The crests crowd closer together, only 4 squares apart. Faster shaking gives a shorter wavelength.
Maya’s cousin Sam shakes a second rope beside Leo’s, with the same size and speed of shakes. If both hands went up and down together, the two waves would be in step. But whenever Leo’s hand is at the top, Sam’s hand is at the bottom. Sam’s wave is half a cycle behind Leo’s. Each crest on Leo’s rope lines up with a trough on Sam’s rope.
A wave on a grid: its amplitude, its wavelength, a dot that only bobs, and a second wave half a cycle behind.
What the animation shows. A blue wave sits on a grid of squares with a dashed middle line. A marker from the middle line up to a crest says 2 squares, and an arrow from one crest to the next says 8 squares. Then the wave slides to the right while a red dot on the rope only moves up and down. Last, a violet wave appears half a cycle behind, and its crests line up with the blue troughs.
Hands-on: Rope waves¶
About 20 minutes.
You need:
A jump rope or a long toy spring
A partner
A ribbon, sticky note, or clothes peg
Graph paper and a pencil
A shallow tray of water and a pencil (optional)
Steps:
Lay the rope straight along the floor. You hold one end and your partner holds the other.
Clip the ribbon to the middle of the rope.
Shake your end slowly from side to side along the floor, the same amount each time. Watch the ribbon. Does it travel with the bumps, or only move side to side in its spot?
Count how many crests you see on the rope at one moment. Write the number down.
Make bigger shakes at the same speed. Do the bumps get bigger, or closer together? This is a bigger amplitude.
Now make faster shakes of the same size. Count the crests again. Are they closer together, so the wavelength is shorter?
On graph paper, draw a middle line and a wave with crests 2 squares high and 8 squares apart. Label one crest, one trough, the amplitude, and the wavelength.
Optional: tap the water in the tray with a pencil and watch the ripples spread out.
Practice together¶
On graph paper, a wave’s crests are 3 squares above the middle line and its troughs are 3 squares below. The crests are 10 squares apart. What are the amplitude and the wavelength?
Check your answer
The amplitude is 3 squares, measured from the middle line up to a crest, not from trough to crest. The wavelength is 10 squares, the distance from one crest to the next.
Try it yourself¶
1. A wave has crests at 2, 8, and 14 squares along the paper. What is its wavelength? How far apart are a crest and the next trough?
Check your answer
The crests are 6 squares apart, so the wavelength is 6 squares. A trough sits halfway between two crests, so a crest and the next trough are 3 squares apart.
2. Leo says, “The water in a pond wave rushes all the way to the shore.” Think about the ribbon. What would you tell Leo?
Check your answer
The wave pattern travels to the shore, but the water mostly bobs up and down near where it was, like the ribbon on the rope.
Talk about it¶
Why did the ribbon stay in its spot while the bumps raced to the post?
What did you change to make the amplitude bigger? What did you change to make the wavelength shorter?
Two people shake ropes half a cycle apart. Where are the crests of one rope compared with the other?
Check yourself¶
1. A crest is 2 squares above the middle line. A trough is 2 squares below it. What is the amplitude?
A. 4 squares
B. 2 squares
C. 8 squares
Show the answer
B. 2 squares
Amplitude is measured from the middle line to a crest. That is 2 squares. The 4 squares from trough to crest is twice the amplitude.
2. Two waves are half a cycle apart. Where are the crests of one wave?
A. Exactly on the middle line
B. Lined up with the troughs of the other wave
C. On top of the crests of the other wave
Show the answer
B. Lined up with the troughs of the other wave
Half a cycle moves a point from the top of its cycle to the bottom. So each crest of one wave lines up with a trough of the other.
3. Remember Lesson 1. A wave’s crests are 8 squares apart. A trough sits halfway between two crests. What fraction of the wavelength is that? (Review of L1-01)
A. One eighth, which is 1 square
B. One half, which is 4 squares
C. One fourth, which is 2 squares
Show the answer
B. One half, which is 4 squares
Halfway means one half of the whole. One half of 8 squares is 4 squares, because 4 out of 8 is the same as 1 out of 2.
Words to know¶
- amplitude
- How far a crest rises above the middle line. It is also how far a trough dips below it.
- wavelength
- The distance from one crest to the next crest.
- phase
- Where a point is in its up-and-down cycle: at the top, at the bottom, or somewhere between.
For grown-ups: about this lesson
Why it matters. The Grade 4 science standard asks children to model waves in terms of amplitude and wavelength and to see that waves can make objects move NGSS Lead States, 2013. OpenStax defines amplitude as the distance from the resting position to a crest and wavelength as the distance between adjacent identical parts of a wave, such as crests Paul Peter Urone and Roger Hinrichs, 2020. The ribbon shows a second key idea, which OpenStax illustrates with a seagull bobbing in place as waves pass: each bit of rope moves up and down near its own place while the pattern travels Paul Peter Urone and Roger Hinrichs, 2020. Phase is introduced only as “where a point is in its cycle”, with in step and half a cycle behind as the two cases the next lesson needs. The animation draws the wave from a formula on a grid of true squares, so every length on screen can be counted.
Watch for these mix-ups.
Amplitude is the height from trough to crest. (That is twice the amplitude; measure from the middle line to a crest.)
The rope or water travels along with the wave. (Each bit moves up and down near its own spot; the pattern travels.)
Shaking harder makes the wavelength longer. (Bigger shakes give a bigger amplitude; faster shakes on the same rope give a shorter wavelength.)
Half a cycle behind means the wave is half as big. (It has the same size; it is shifted so its crests line up with the other wave’s troughs.)
To make it easier. Skip phase and the second rope. Focus on crests, troughs, and the ribbon that bobs in place.
To make it harder. Time 10 full up-and-down shakes with a stopwatch and work out how long one shake takes (the period). Then try to make a wave with half the wavelength and see how your shaking must change.
Limits. A shaken rope makes a traveling wave that reflects from the post, so real rope waves soon get messy; the drawings show an ideal wave. Water waves are more complex than rope waves: water near the surface moves in small loops, not straight up and down, but the pattern still travels much farther than the water does.
On the computer. Grown-ups can see where this idea leads later in Lab: Interference, written for older learners.
Sources for this guide. NGSS Lead States, 2013Paul Peter Urone and Roger Hinrichs, 2020
Next: L1-08 · Combine waves, then square
- NGSS Lead States. (2013). 4-PS4-1 Waves and Their Applications in Technologies for Information Transfer. Next Generation Science Standards. https://www.nextgenscience.org/pe/4-ps4-1-waves-and-their-applications-technologies-information-transfer
- Paul Peter Urone and Roger Hinrichs. (2020). 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period. OpenStax, Physics. https://openstax.org/books/physics/pages/13-2-wave-properties-speed-amplitude-frequency-and-period