Superposition of Wave Functions - Video Tutorials & Practice Problems
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Superposition of Sinusoidal Wave Functions
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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. But some problems are actually gonna give you to wave functions. And they're gonna ask you to calculate the displacement or the superposition of the resultant wave. The example they're gonna work out down here has waves in which we're given two wave functions, we have a sign sign and a cosine function. And we want to calculate the displacement at some, 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 gonna show you. It's very quick. Uh It's very straightforward as well. Basically with the principle of superposition says is that we can simply add the two way functions. So we can just add these two wave functions as if they were kind of just normal numbers. So the idea here is that we have two wave functions then the net wave function is just gonna be Y one plus Y two. So it's gonna be a, one times the sign of K one X MA plus or minus Omega T. Um I'm gonna call this Omega one plus A two sign of K two X plus or minus Omega two T. So these things don't necessarily have to 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 Y one and Y two are sine or cosine functions, you could have any combination of sine cos we have sin plus sine, sine plus cosine, cosine plus sine, 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 two 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 two wave functions. So our net wave function is just gonna be Y one plus Y two. And what this means is that we have 0.3 times the sine of four X minus 1.6 T plus 0.7 times the cosine of five X minus two T. So that's basically what our new wave function is. You just add the two functions together. So if we want to calculate what the displacement is uh for a particle at X equals two and T equals 0.5. You're just gonna plug in these values in for the two X's inside of your functions and then the two Ts inside of your two functions as well. You just plug and chug everything into your calculator. So basically, um all we're gonna do is we're gonna have Y nets is equal to 0.3 times. The sign, this is gonna be four times two minus 1.6 times 0.5 plus 0.7 times the cosine of five times two minus two times 0.5. So basically, you just plug and chug, I highly recommend that you just solve one of these at a time. So just plug in one and then add it to the second one. That way you don't make any mistakes in your calculator. And what you're gonna get here is when you plug in the sine function, you're gonna get 0.24 and that's gonna be positive. And when you plug in the one for the cosine function, you're gonna get negative 0.64. So that means that the net displacement or the displacement of your particle here because of these two wave functions, it's just gonna be adding those two displacements together. So we're gonna get zero points 24 minus 0.64. And this is Sequels negative 0.4. That's the answer. That's all there is to it very straightforward. So let me know if you guys have any questions for this.
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