Superposition of Wave Functions: Study with Video Lessons, Practice Problems & Examples

Wave interference and superposition allow us to calculate the resultant wave from two wave functions, such as sine and cosine. The principle of superposition states that the net wave function is the sum of individual wave functions: y_net = y1+y2. For example, given wave functions, we can find displacement by substituting values for position and time, leading to a straightforward calculation of net displacement.

Superposition of Sinusoidal Wave Functions

Hey, guys. So in a previous video, we saw a more graphical approach to wave interference and superposition. We saw how waves could actually add and combine each other to produce new resultant waves, but we didn't really do any math with that. However, some problems are actually going to give you 2 wave functions, and they're going to ask you to calculate the displacement or the superposition of the resultant wave. The example they're going to work out down here has 2 waves in which given 2 wave functions. We have a sine and a cosine function. We want to calculate the displacement at some point, you know, x and t. So the idea here is we can actually use the principle of superposition to handle these wave functions and that's what I'm going to show you. It's very quick, it's very straightforward as well. Basically, what the principle of superposition says is that we can simply add the two wave functions. So we can just add these 2 wave functions as if they were normal numbers. So the idea here is that we have 2 wave functions, then the net wave function is just going to be y₁+y₂. So it's going to be a₁∙sink₁x±ω₁t+a₂∙sink₂x±ω₂t. So these things don't necessarily have the same amplitude, they don't have to have the same wave number or frequency. The principle of superposition says we could just add them like normal numbers. And by the way, this equation works regardless of whether y1 and y2 are sine or cosine functions. You could have any combination of sine cosine, sine plus sine, sine plus cosine, cosine plus sine, and so on and so forth. So that's really all there is to it, guys. Let's go and take a look at our example here. We have 2 transverse waves. We're given the wave functions, and we want to calculate the displacement of the particle in the string at this position and this time. So what happens is we're given 2 wave functions. So our net wave function is just going to be y1 plus y2 and what this means is that we have 0.3∙sin4x-1.6t+0.7∙cos5x-2t. So that's basically what our new wave function is. You just add the 2 functions together. So if we want to calculate what the displacement is, for a particle at x equals 2 and t equals 0.5, you're just going to plug in these values for the two x's inside of your functions and then the two t's inside of your two functions as well. You just plug everything into your calculator. So basically, all we're going to do is we're going to have y_net=0.3∙sin4∙2-1.6∙0.5+0.7∙cos5∙2-2∙0.5. So just plug and chug. I highly recommend that you solve one of these at a time. Just plug in 1, then add it to the second one, that way you don't make any mistakes on your calculator. And what you're going to get here is when you plug in the sine function, you're going to get 0.24, and that's going to be positive. And when you plug in the one for the cosine function, you're going to get negative 0.64. So that means that the net displacement or the displacement of your particle here because of these two wave functions is just going to be adding those 2 displacements together. So we're going to get 0.24-0.64=-0.4. That's the answer. That's all there is to it, it's very straightforward. So let me know if you guys have any questions about this.

Here’s what students ask on this topic:

What is the principle of superposition in wave functions?

The principle of superposition in wave functions states that the resultant wave function is the sum of the individual wave functions. Mathematically, if you have two wave functions y₁ and y₂, the net wave function y_net is given by:

$y_{\mathrm{net}}=y_1+y_2$

This principle applies regardless of the type of wave functions involved, whether they are sine, cosine, or a combination of both. It allows us to calculate the resultant displacement at any given position and time by simply adding the displacements from the individual wave functions.

How do you calculate the displacement of a particle using superposition of wave functions?

To calculate the displacement of a particle using the superposition of wave functions, follow these steps:

1. Identify the given wave functions y₁ and y₂.
2. Use the principle of superposition to find the net wave function y_net by adding y₁ and y₂:

$y_{\mathrm{net}}=y_1+y_2$

3. Substitute the given values for position (x) and time (t) into the net wave function.
4. Calculate the individual displacements from y₁ and y₂ at the given x and t.
5. Add the individual displacements to find the total displacement.

For example, if y₁ = 0.3sin(4x - 1.6t) and y₂ = 0.7cos(5x - 2t), and you need to find the displacement at x = 2 and t = 0.5, substitute these values into the functions and add the results.

Can the principle of superposition be applied to any type of wave functions?

Yes, the principle of superposition can be applied to any type of wave functions. This includes sine, cosine, or any combination of these functions. The principle states that the net wave function is simply the sum of the individual wave functions, regardless of their specific forms. For example, you can add a sine function to a cosine function, or two sine functions with different amplitudes, wave numbers, or frequencies. The resulting wave function will be the algebraic sum of the individual wave functions.

What are some common mistakes to avoid when calculating the superposition of wave functions?

When calculating the superposition of wave functions, some common mistakes to avoid include:

1. Not correctly substituting the values of position (x) and time (t) into the wave functions.
2. Forgetting to add the individual wave functions algebraically, which means considering the signs (positive or negative) of the displacements.
3. Not using a calculator correctly, especially when dealing with trigonometric functions like sine and cosine.
4. Mixing up the units of measurement for position and time, which can lead to incorrect results.
5. Not simplifying the wave functions properly before adding them, which can cause errors in the final calculation.

To avoid these mistakes, carefully follow the steps of substitution, calculation, and addition, and double-check your work at each stage.

How does wave interference relate to the superposition of wave functions?

Wave interference is a phenomenon that occurs when two or more waves overlap and combine to form a new wave pattern. This is directly related to the superposition of wave functions, as the principle of superposition explains how the individual wave functions add together to create the resultant wave. There are two main types of interference:

1. Constructive interference: When the waves are in phase, their amplitudes add up, resulting in a larger amplitude.
2. Destructive interference: When the waves are out of phase, their amplitudes subtract from each other, resulting in a smaller amplitude or even cancellation.

By using the principle of superposition, we can predict the resultant wave pattern and understand the effects of wave interference.