Superposition of Wave Functions - Online Tutor, Practice Problems & Exam Prep

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.