The Resultant Amplitude of Two Superposed Waves

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  • 0:05 Waves All Around Us
  • 0:45 Wave Superposition
  • 2:46 Constructive Interference
  • 3:08 Destructive Interference
  • 3:33 Lesson Summary
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Lesson Transcript
Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

Have you ever wondered what happens when two waves run into each other? In this lesson we're going to look at that exact scenario and how it affects wave amplitude.

Waves All Around Us

Waves aren't limited to traveling through water; they are everywhere. Every sound you hear consists of sound waves. All the light we can see consists of electromagnetic waves. You are bombarded with waves all day and all night long.

Since waves are traveling all around us all the time, they must at some point come into contact with each other. Do they smash into each other and stop, do they pass through each other without interacting, or do they do something else entirely? In this lesson, we're going to look at what happens when two waves run into each other. Specifically, we're going to focus on how this affects waves' maximum displacement, or amplitude.

Wave Superposition

When two waves come into contact with each other, they don't collide. They pass right through each other without any change in their original speed or direction. However, they do interact as they pass through each other. Only waves of the same type can directly interact with each other. This means a sound wave can interact with another sound wave, or a light wave can interact with another light wave, but a sound wave cannot interact with a light wave.

We call the interaction of waves with one another wave interference. We describe how these waves interact through the principle of superposition, which states that two interfering waves cause a displacement in the medium they're traveling though equal to the sum of the individual waves' displacements.

To try and better understand the principle of superposition, let's look at an example of what we get when two waves interfere with each other.

Red and Green Superposed Waves
wave superposition graph

The resultant wave is the purple wave created by the red and green waves interfering with each other. By using the grid, you can see that the resultant displacement at any given point of the resultant wave is equal to the addition of the displacements of the other two waves at that same point on the x-axis. Essentially, the waves are overlapping to create a bigger or smaller wave, depending on how they interact.

The maximum displacement of a wave is known as its amplitude. A periodic wave's amplitude is located at the highest point in its peaks and lowest point in its troughs. Here we have a positive amplitude and a negative amplitude. In this case, the negative sign is there to show the direction of displacement. One of the amplitudes is in the opposite direction from the other, but the magnitude of displacement is the same.

Displacement vs. Time Graph Showing Amplitude
wave amplitude graph

Since a wave's amplitude is just the maximum displacement of a wave, we can use the principle of superposition to find the amplitude for a resultant wave. The resultant amplitude of the wave we get through the combination of the two interfering waves is equal to the addition of the displacements of those two waves at the same location as the resultant wave's amplitude.

resultant amplitude equation

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Additional Activities

Finding the Amplitude of One Wave Given the Superposition of Waves

Imagine you are a scientist on the look out for other intelligent life in the universe. You run a lab looking for radio wave communications from extraterrestrial life. Your lab looks for radio wave communication by sending a consistent "noise" radio wave at all times and monitoring the amplitude (and possible changes in amplitude) via a monitor. Seeing a change in the amplitude of your known "noise" wave would show that an unknown radio wave has interacted with the noise wave. In the situation described below, use your knowledge of the resultant amplitude of two superimposed waves to discover the amplitude of the alien communication radio wave.


Your lab's consistent "noise" radio wave is modeled by the equation y = cos(x). When monitoring your noise wave, you notice the wave suddenly changes! This is what you've been waiting for - alien communication! The graph below shows the modified wave picked up by the monitor as well as your noise wave.

Noise wave cos(x) in red, modified wave in green

Find the amplitude of the alien communication wave at 0, pi/2, pi/ 3pi/2, 2pi, 5pi/2, 3pi, 7pi/2, and 4pi. Plot these points and sketch the graph of the alien communication radio wave. Can you find an equation for the alien communication wave?


Since the amplitude of superimposed waves is the sum of the amplitudes of the individual waves, we can find the amplitude of the alien wave by subtracting the amplitude of the noise wave from the amplitude of the superimposed wave.

  • At 0, the amplitude is 1 - 1 = 0
  • At pi/2, the amplitude is -2 - 0 = -2
  • At pi, the amplitude is -1 - (-1) = 0
  • At 3pi/2, the amplitude is 2 - 0 = 2
  • At 2pi, the amplitude is 1 - 1 = 0
  • At 5pi/2, the amplitude is -2 - 0 = -2
  • At 3pi, the amplitude is -1 - (-1) = 0
  • At 7pi/2, the amplitude is 2 - 0 = 2
  • At 4pi, the amplitude is 1 - 1 = 0

Plotting these points and sketching the graph results in the purple graph below:

This graph looks like y = -2sin(x). Checking other points of the superimposition verifies this equation.

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