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L1-08 · Combine waves, then square

When two waves meet, their heights add at every point. In step, they make a bigger wave. Half a cycle apart and the same size, they cancel. For light, brightness goes with the amplitude times itself, so the order matters: add first, then square.

Level 1 icon Level 1 · Grades 4–6 · about 40 minutes
Big idea. When two waves meet, their heights add at every point. In step, they make a bigger wave. Half a cycle apart and the same size, they cancel. For light, brightness goes with the amplitude times itself, so the order matters: add first, then square.
Before this lesson. L1-07, L1-02
Grown-ups. The guide for adults is at the end of this page.

What you will learn

Story: When two bumps meet

Maya and Leo stretch a long rope between them. Each one flicks a single bump up toward the other at the same moment. On their graph paper, each bump is 2 squares tall. In the middle, the bumps meet. For an instant, the rope jumps up 4 squares, because 2 plus 2 is 4. Then the bumps pass through each other and keep going.

Next, Maya flicks a bump up and Leo flicks a bump down, each 2 squares. When they meet, the rope looks flat for an instant. An up of 2 and a down of 2 cancel. The bumps did not break. A moment later, they come out the other side.

Back at the table, Leo draws a wave on tracing paper, with crests 2 squares high. Maya draws the same wave on another sheet. They lay one on top of the other, in step. At every point they add the heights: 2 up and 2 up make 4 up, and 2 down and 2 down make 4 down. The new wave has twice the amplitude.

Then Maya slides her sheet half a cycle, 4 squares along. Now each crest sits on a trough. At every point, the two heights are equal and opposite, like up 2 and down 2, so they add to 0. The two waves cancel completely.

Their teacher adds one more rule. For light, brightness goes with the amplitude times itself. Say one light wave has amplitude 1. Then amplitude 1 gives brightness 1, and amplitude 2 gives brightness 4. In step, the brightness is 4 times one wave, not 2 times. Half a cycle apart, amplitude 0 gives brightness 0. The extra light in bright places comes from dark places nearby, where waves cancel.

Wave A, wave B, and their sum, point by point. The bar shows brightness, which is the total amplitude times itself.

What the animation shows. Wave A appears with amplitude 1, and a brightness bar rises to 1. Wave B grows in step with A, and the bottom row, A plus B, becomes twice as tall. At a crest, 1 up plus 1 up is 2 up, and the screen says amplitude 2 gives brightness 4. Then wave B slides until it is half a cycle behind, the sum goes flat, and the screen says amplitude 0 gives brightness 0.

Hands-on: Add two waves on paper

About 20 minutes.

You need:

Steps:

  1. On graph paper, draw a long middle line. Mark a dot on it every 2 squares.

  2. Lay a sheet of tracing paper on top. Draw a wave with crests 2 squares high and 8 squares apart. Start at the middle line on the first dot.

  3. On the second sheet, draw the same wave in a different color.

  4. Lay both sheets on the graph paper, in step. At each dot, add the two heights. Count heights above the line as up and below it as down. Draw the sum on the graph paper.

  5. Slide the top sheet 4 squares along, which is half a cycle. Add the heights at each dot again and draw this second sum.

  6. Make a table with three rows: one wave, two waves in step, and two waves half a cycle apart. For each row, write the amplitude of the sum and that amplitude times itself.

  7. Optional: hold a rope with a partner. Each flick one bump at the same moment, both up, then one up and one down. Watch what happens where they meet.

Practice together

At four dots, wave A has heights 0, 2 up, 0, 2 down. Wave B is in step and has the same heights. What are the heights of the sum? What if wave B is half a cycle behind instead?

Check your answer

In step, the sum is 0, 4 up, 0, 4 down, a wave with amplitude 4. Half a cycle behind, wave B has heights 0, 2 down, 0, 2 up, so the sum is 0 at every dot. The waves cancel.

Try it yourself

1. At one point, wave A is 3 squares up and wave B is 1 square down. What is the height of the sum there?

Check your answer

Up 3 plus down 1 is up 2. The sum is 2 squares up at that point.

2. A light wave has amplitude 3, so its brightness is 3 times 3, which is 9. Two of these waves meet in step. What is the new amplitude? What is the new brightness? How many times as bright as one wave is that?

Check your answer

The amplitudes add to 6. The brightness is 6 times 6, which is 36. That is 4 times as bright as one wave, because 36 is 4 times 9.

Talk about it

Check yourself

1. Two equal light waves meet in step. How bright is the result, compared with one wave?

Show the answer

C. 4 times as bright

The amplitudes add, so the total amplitude is 2 times as big. Brightness goes with amplitude times itself, and 2 times 2 is 4.

2. Two equal waves meet half a cycle apart. What happens?

Show the answer

C. They cancel, so the sum is flat

Half a cycle apart, each crest meets a trough of the same size. Up plus an equal down is 0 at every point, so the waves cancel.

3. Remember Lesson 2. A bag has 3 red counters and 5 blue counters. What is the chance of not drawing red? (Review of L1-02)

Show the answer

B. 5 out of 8

There are 8 counters, and 5 of them are not red. That is 5 out of 8, which is 1 minus 3 out of 8.

Words to know

amplitude
How far a crest rises above the middle line.
cancel
To add up to zero. An up of 2 and a down of 2 cancel.
square
To square a number, multiply it by itself. The square of 3 is 3 times 3, which is 9.
For grown-ups: about this lesson

Why it matters. OpenStax describes superposition: where waves overlap, the resulting displacement is the sum of the individual displacements, giving constructive interference when they are in phase and destructive interference when they are half a cycle out of phase William Moebs, Samuel J. Ling, and Jeff Sanny, 2016. A standard result of wave physics is that for light and sound the intensity (energy carried per second per area) is proportional to the square of the amplitude, so two equal waves in phase give twice the amplitude and four times the intensity of one. The order matters: adding amplitudes first and then squaring is not the same as squaring each and adding. The quantum preview in the Where this leads paragraph follows Feynman’s account of the two-slit experiment, where the chance of arrival is found by adding the amplitudes for the two paths first and then taking the absolute square Feynman, 1966. The animation computes each wave with numpy and draws the sum as exactly the pointwise sum of the two drawn waves.

Watch for these mix-ups.

  • Two waves in step give twice the brightness. (They give twice the amplitude and four times the brightness.)

  • When waves cancel, they are destroyed. (They pass through each other and continue; the flat moment is the sum at that time and place.)

  • Interference makes extra energy at bright spots. (Light is moved, not made: bright places are balanced by dark places nearby.)

  • You can square each wave first and then add. (That gives 1 + 1 = 2 for two equal waves in step and can never give 0; add the amplitudes first.)

To make it easier. Do only the in-step and half-cycle overlays at four dots. Use whole numbers and skip the brightness table, or do just one row of it.

To make it harder. Slide the top sheet a quarter cycle (2 squares) and add again. The sum is a wave in between in size. Measure its amplitude and square it to compare with the other cases.

Limits. “Brightness goes with amplitude squared” is about intensity, the energy carried by the wave. How bright or loud something seems to a person grows more slowly than that. Two ordinary flashlights do not show light and dark stripes, because their waves do not keep a steady step with each other. Seeing interference with light usually uses a single source split into two paths, such as two narrow slits.

On the computer. Grown-ups can see where this idea leads later in Lab: Interference, written for older learners.

Sources for this guide. William Moebs, Samuel J. Ling, and Jeff Sanny, 2016Feynman, 1966

Next: L1-09 · Unknown is not the same as quantum

References
  1. William Moebs, Samuel J. Ling, and Jeff Sanny. (2016). 16.5 Interference of Waves. OpenStax, University Physics Volume 1. https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves
  2. Feynman, R. P. (1966). The Feynman Lectures on Physics. Volume III. https://www.feynmanlectures.caltech.edu/III_toc.html