Complex Waves as Superpositions of Sinusoidal Waves

Instructor: Damien Howard

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

When sinusoidal waves come into contact with each other, they create a resultant wave. In this lesson, we'll learn how the properties of the interacting sinusoidal waves can create complex waves.

Making Waves

Let's try a little experiment that you can do yourself at home. All you need is something filled with water, like a bucket or sink. Put the tip of your finger in the water, and you'll see that it makes ripples: little waves.

Our mini-experiment will be putting two fingers in, one from each hand. What do you think will happen when the ripples hit each other? You can see that the ripples pass through each other, seemingly without affecting one another. Well, that's how it looks to our eyes, but it's not quite the case. During the brief moment when the ripples are touching there is an effect, and then they go back to normal once they pass through each other.

In this lesson, we're going to learn how waves interacting with each other can affect one another. We will look at sinusoidal waves, and see how their interaction can create complex waves.

Sinusoidal Waves

Before we start looking at how sinusoidal waves interact with each other, we need to learn what it means for a wave to be sinusoidal. Essentially, a sinusoidal wave is any wave whose graph takes the form of a sine wave.


Sinusoidal Wave
sinusoidal wave


A wave function like the one in the image above has several properties, but for this lesson we're going to focus on three of them. The first is the wave's amplitude, which is the height of the wave's peaks and troughs. The second is the wave's phase, which is the distance from the origin of the graph to the first time it crosses the axis. Finally, the third is the wave's frequency, which is the number of waves that pass a point over some period of time.


Amplitude, Phase, and Frequency
wave elements


Superposition

Now that we know what a sinusoidal wave is, what happens when two or more of them interact with each other? When waves come into contact with each other, they temporarily create a new resultant wave.

The displacement of the resultant wave at any point on the graph will be determined by the displacements of the waves that created it. In general, a displacement is a measurement of the change in an object's starting position. For example, the displacement of a water wave is the change in height in the water from where the water was before the wave traveled through it.

A resultant wave's displacement is always equal to the sum of the displacements of the waves that created it. We call this the principle of superposition, and the act of the wave's interacting we call interference. To understand how all of this works, let's look at an image of superposed waves.


Interfering Green and Blue Waves
wave superposition


If you compare the displacement of the resultant red wave to the interfering green and blue waves, you can see that it is the sum of the two. By changing the amplitude of the initial waves, we can change the displacement of the resultant wave.


Blue Wave with Increased Amplitude
displacement with changing amplitude


Complex Waves

So far, our two interfering waves have lined up perfectly with each other. This is not always the case, and when they don't line up we can end up with more complex waves forming.

Varying Phase

Now, the question is what would cause the initial waves to not line up perfectly? Having aspects of the initial waves other than amplitude differ can cause the initial waves to stop lining up. One such example of this is when we have waves with varying phases.

Before we get into the superposition of waves with varying phases, we need to understand something called magnitude. To find the magnitude of a displacement we need to find its size while ignoring its direction. Since our displacement in this case is in one dimension, vertical in the graphs, we denote direction by using a negative sign. If the displacement has a negative sign, it means it is in the opposite direction to a displacement with a positive sign. To find the displacement's magnitude, its size without the direction, we take its absolute value.

When the only difference between the interfering waves is their amplitude, the magnitude of the resultant wave's displacement at any point on the graph is always greater than or equal to the magnitude of the displacement of any initial wave. Once we start changing their phase, this is no longer guaranteed. When the interfering waves' peaks and troughs no longer line up, you can find places where the magnitude of the resultant wave is less than the magnitude of one of the interfering waves.


Green Wave with a Phase Shift
displacement with varying phase


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