Wave Interference - Online Tutor, Practice Problems & Exam Prep

Wave interference occurs when two or more waves meet in the same medium, leading to temporary disruptions. This interaction can result in constructive interference, where waves combine positively, increasing amplitude, or destructive interference, where waves with opposite signs partially cancel each other out. The resultant amplitude is the sum of the individual amplitudes. Understanding these concepts is crucial for analyzing wave behavior, as they illustrate the principles of superposition and the effects of wave interactions.

Wave Interference & Superposition

Hey guys. So by now we've seen all the different characteristics of a single wave that travels through a medium like a wave on a string. But in some problems, we're going to see two or more waves or pulses that meet along the same medium. So think of two pulses or two waves that meet along the same string. Now these two waves are going to interact with each other. So that's what I want to talk about in this video. This is called wave interference and superposition. Let's check this out.

In our problem here, we have two pulses that are going to approach each other. Imagine that you and your friend have the two ends of the rope and you're both whipping it up once. You're going to create two wave pulses that travel towards each other. In the second example, it's the same idea. We have two pulses, except now one of them is going to be inverted. But the idea is the same. We want to calculate the amplitude of the resultant wave pulse in both of these problems here. So let's go ahead and take a look at this.

Whenever you have two or more waves or pulses that meet along the same medium, they're going to temporarily disrupt each other. The word that you're going to see for this is 'interfere,' so they're going to interfere with each other. So, what happens here is that, if you and your friend whip a string up once, you're going to create two wave pulses and when they start to overlap, they're going to disrupt each other. But this disruption is temporary because afterward, they're both going to basically just keep moving along their merry ways as if nothing ever happened. So that's why this is a temporary disruption.

What actually happens here during this disruption? Well, as they pass through each other as these waves or pulses pass through each other, they're going to combine. The word you're going to see for this is 'superimposed.' So you're going to superimpose these waves, which is just a fancy word for combination. And what you're going to do is form a new wave in which the height is basically just the sum of the heights of the two waves that make it up. So if you have these two wave pulses, as they fully overlap each other, what happens is you're going to create an even bigger wave like this.

So what happens here is that these two waves can only interfere with each other when they have the same frequency. Now, there are two types of interference that I want to talk about. In this first problem, we're going to have two wave pulses, and they're basically both above the x-axis, and this is called constructive interference. Constructive interference happens when you have two waves that meet each other, and the displacements are going to have the same signs. For instance, both of these displacements are going to be positive. So what happens to the amplitude of this new resultant wave? Well, if you have a wave in which the amplitude is 1, and you have another wave in which the amplitude is 1, when they combine, the heights are going to combine. You're going to have h₁ + h₂ or y₁ + y₂. And basically, you're just going to create a new wave in which the new amplitude is just 2. So you just say that the amplitude of this wave pulse is 2.

In the second example, it's the same setup, except now what happens is that one pulse is going to be inverted, and they also don't have the same amplitude. So you notice that one of them has an amplitude of 0.5, the other one has an amplitude of negative one. So now what happens? It's the same idea when you have these two waves that overlap each other. Basically, what happens is that they sort of partially cancel each other out. This is called destructive interference. This is where you have displacements that have opposite signs. So, I like to think of constructive and destructive as: constructive, you're building—constructive sounds like construction, and you're building a bigger wave. Destructive sounds like destruction, and basically, you're destroying the wave. Right?

So what happens here? Well, basically what happens is that the new wave is going to have a height that is h₁ + 2y₁ + y₂, and because these things are opposite signs, you're going to create a new wave in which this thing is going to be slightly downwards like this. So you're going to create a new little wave like this. The amplitude of our new wave is going to be negative 0.5. And then, what happens afterward is again, these wave pulses just separate as if nothing ever happened.

So that's really all there is to it, guys. Let me know if you guys have any questions with this.

Example 1

Everyone, let's see if we can get some practice here so I can kind of visualize this whole phenomenon with wave interference. I have these two wave pulses, a and b, that are traveling towards each other. Imagine you and your friend grab the end of a rope, and you both flick it upwards, and the wave pulses travel towards each other. Now they each travel at a speed of 2 centimeters per second, and, basically, everything on the x and y axis is in centimeters, so we don't have to do any conversions. In other words, both of these things are moving towards each other with a speed of 2 centimeters per second. What we want to do is we want to sort of sketch out, not really calculate anything, what the wave is going to look like at different points throughout this sort of interaction. Alright?

We've actually got a couple of graphs to help us along with that. We're going to measure this, or we're going to draw this out when t equals 1 second and then 2 and then 3 and so on. So let's get started. Basically, what we're going to do here is we're going to take this sort of wave pulse, and we're actually given, you know, where the wave pulse starts and stops in terms of centimeters.

So let's take a look at what happens at t equals one second. Alright? From t equals 0 to 1, one second passes, and if these things move at 2 centimeters per second, then that means that they both are going to move towards each other by 2 centimeters. I'm initially starting here at 6, which means that the blue curve, that blue pulse, is going to travel to the right by 2 centimeters. So, you basically just line up the tip of this pulse, and now you're just going to shift it over by 2 centimeters after one second. The top of this little peak is actually going to happen now at 8, and the bottoms of these two triangles aren't going to be 4 and 8; they're all going to shift over by 2 centimeters to 6 and 10.

What does the red curve look like? Well, the red pulse is going to do the same but except it's actually going to move to the left. So what happens is this peak lines up with 14, but then it's going to move to the left by 2, so it's going to line up with 12. Notice at the top of this peak, it actually isn't 4; it's just 2. So, it's going to end up looking like this: the bottoms of these will look like that, and the rest of the wave pulse is flat.

What happens next? This pulse continues moving to the right. In other words, the top is going to be over at 10 on the x-axis. So the top is going to be at 10, and the bottom is going to be at 8, and this bottom over here is going to be at 12. And then, the rest will be flat. For the red, it will move the same again to the left. So, the tippy top will be over at 10 centimeters, and the bottoms are going to go from 10 to 8, and then 14 to 12. What happens at t equals 2 seconds, these waves will be on top of each other.

Now, remember what happens when waves interfere with each other. When you have waves that are stacked on top of each other, the resultant wave is just adding the two triangles together. So what ends up happening here is these wave pulses stack up, and they add on top of each other, and you'll end up with a strong wave pulse that goes all the way up to 6 centimeters, because it's going to add the 2 and the 4. So the real wave pulse is going to look something like this.

After they pass each other, the blue moves to the right, and the peak goes from 10 to 12. What about the red one? The red is going to move 2 centimeters to the left. So, the top is going to be here at 8. Because these things aren't on top of each other anymore, we don't have to do any addition; they have already passed each other, and the superposition has already happened. So this is what each of these stages look like throughout the motion of these two wave pulses. You can even try it at home if you have a string and create two little wave pulses. You're going to see this weird thing that happens when the pulses hit each other, and they'll add up like this. Anyway, folks, that's it for this one. Thanks for watching.

Here’s what students ask on this topic:

What is wave interference and how does it occur?

Wave interference occurs when two or more waves meet while traveling through the same medium. This interaction leads to a temporary disruption of the waves, which can result in either constructive or destructive interference. Constructive interference happens when the waves combine to form a wave with a larger amplitude, while destructive interference occurs when the waves partially or completely cancel each other out. The principle of superposition explains this phenomenon, stating that the resultant wave at any point is the sum of the displacements of the individual waves at that point.

What is the difference between constructive and destructive interference?

Constructive interference occurs when two waves meet and their displacements are in the same direction, resulting in a wave with a larger amplitude. For example, if two waves with amplitudes of 1 unit each meet, the resultant wave will have an amplitude of 2 units. Destructive interference, on the other hand, happens when the displacements of the waves are in opposite directions, leading to partial or complete cancellation. For instance, if a wave with an amplitude of 1 unit meets a wave with an amplitude of -1 unit, the resultant wave will have an amplitude of 0 units.

How do you calculate the resultant amplitude in wave interference?

The resultant amplitude in wave interference is calculated by summing the individual amplitudes of the waves at each point. For constructive interference, where the displacements are in the same direction, the amplitudes add up directly. For destructive interference, where the displacements are in opposite directions, the amplitudes subtract from each other. Mathematically, this can be expressed as:

$A=A{1}_{}+A{2}_{}$ for constructive interference, and $A=A{1}_{}-A{2}_{}$ for destructive interference.

What is the principle of superposition in wave interference?

The principle of superposition states that when two or more waves overlap in the same medium, the resultant wave at any point is the sum of the displacements of the individual waves at that point. This principle applies to both constructive and destructive interference. In constructive interference, the displacements add up to create a wave with a larger amplitude. In destructive interference, the displacements subtract from each other, potentially leading to partial or complete cancellation of the waves.

Can wave interference occur with different types of waves?

Yes, wave interference can occur with different types of waves, including sound waves, light waves, and water waves. The fundamental principles of interference and superposition apply to all types of waves. For example, in sound waves, constructive interference can lead to louder sounds, while destructive interference can result in quieter sounds or silence. In light waves, constructive interference can produce bright fringes, and destructive interference can create dark fringes in patterns such as those seen in the double-slit experiment.