Phase Constant - Video Tutorials & Practice Problems
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Phase Constant of a Wave Function
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Hey guys. So up until now, all of the wave functions waves that we've seen so far have either started at Y equals zero like these graphs over here or they started at the amplitudes like these graphs over here. Now, occasionally you're gonna run into a situation or run into a problem where you'll have a graph that doesn't start at either one of those special points. And the example, we're going to work out down here, we have a wave with an amplitude of 4 m. It oscillates between four and negative four, but the starting position at T equals zero X equals zero is actually a displacement of three. So it starts here, not at zero and not at the amplitudes. And in this problem, we're going to have to write a wave function for this wave. So I'm going to show you how to write these wave functions by using something called a phase constants. And in textbooks, this can be pretty confusing, but I'm going to show you a step by step process for how to deal with these kinds of problems. Let's go ahead and check this out here. So remember that we could use either signs or cosigns to describe these graphs here. And it really just depended on where our starting locations were. If we started at Y equals zero, we used the sign and if we started at Y equals either one of the amplitudes, then we use the cosine graph. So then how do we use these, or how do we write a wave function for something that doesn't start at those sort of special positions? Well, it turns out that we can actually write any wave function by using either sine or cosine. We can actually describe this graph here by using signs or cosines. All we have to do is we just to add a constant inside of our parenthesis here. So we have a sign or a cosine KX plus or minus omega T plus or minus this Greek letter which is called Phi and this phi here represents or is called a phase constant. And what this basically does is it shifts the graph to the left or to the right from a normal, from its normal starting point. So basically the only difference between these graphs here is where they start. So we can take this graph and we can actually shift it to the left or to the right. And we'd be able to sort of line it up and make it look like these other graphs over here or basically any point in between. And that's what that phase constant does as it shifts. So I'm going to get back to this in just a second here. I'm going to go ahead and start out our problem. So we're told that this, um we're told that some of the characteristics of the properties of this wave, the wave number is uh is 10 radiance per meter. So that's K it's 10, we have the angular frequency that's omega which is going to be 62.8. We also have the amplitude which is four. So here are all our values. And this first part here, we want to write the wave function using a sine function. Remember we can use because these graphs are sinusoidal, we can use either sines or cosines. We want to use a sine function here. So before I do that, I want to actually draw out what an unsifted normal sine graph looks like it's going to look like this, you're gonna go up and down and up and down like this and you're going to start at zero. So this is my unsifted wave function, this is gonna be a sine KX. And because we're traveling to the right, this is going to be minus omega T and because this isn't shifted right, there's no shift here. We don't have to include a phase constant like this. Now what happens our graph? The one that's in blue here actually is shifted. If you look at the first hill, the first little crest, it actually happens to the left of where the normal sine graph starts or where that normal sine graph has its first crest. So that means that we have to write this as a sign, this is gonna be KX minus omega T and this is plus or minus this phi constant. And this shift here between the first hill of, of, of, of the graph and our normal sort of unsifted sine graph that's going to be what that phase constant represents and this is what we want to calculate here. All right. So that brings us to the list of steps. The first thing you're going to do is you're going to write out your YX of T equation with, and you're going to include that P, the constant, that phase constant here. So we have this YX and T and you have a times the sign. Now we have KX plus or minus Omega T, we actually know that it's going to be minus because again, this wave moves to the right and now we have plus or minus this Phi and I'm gonna use Phi s because we're using the sign like this. All right. So the next thing we have to do is we actually have to determine the sign of that Phi or it could be plus or minus. How do we do that? Well, actually turns out that it's the same rule as when we figured out whether this KX plus or minus Omega T was plus or minus. So the rule is if the graph is shifted to the right of its normal starting position, then the phi is going to be negative. And if it's shifted to the left, it's going to be positive. So it's basically the same exact rule here. So that means that our graph or our phase constant is going to be positive because we have been shifting to the left here from a normal starting position. All right. So let's go ahead and now take a look at the third step, which is we're going to plug in all the given values that we have. So Y when X is equal to zero and T is equal to zero is going to be the amplitude, which is four times the S the K is 10. And when we plug in zero, this just goes away minus 62.8. That's omega times zero again, that just goes away plus R five constant here. All right. Now, the fourth step is we just have to go ahead and solve 45. So what happens is here is the problem tells us that we plug all this stuff in. You're what you're gonna get at the end is just equal to 3 m. That's the displacement here. So we end up with a pretty simple expression. What we end up with here is that four times the sign of our phase constant is equal to three. So now we just to do some division, we have the sign of our phase constant is equal to 3/4. And when you go ahead and take the arc sign, you're gonna do the inverse sign, you're gonna do the inverse sign of 3/4. And what you should get if your calculators have plugged in radiance mode is you should get 0.85 radiance. So that basically this 0.85 now represents the shift to the left from the normal starting graph. All right. So the last thing we have to do is just now go ahead and write our complete wave function now that we figured out this phase constant. So our complete wave function is gonna be uh for part A, this is gonna be Y of X and Z equals this is gonna be four times the sign of 10 X. Remember we just keep the XS when we're just writing it out minus 62.8 T and now we have plus 0.85 and you could put the rad or not. So it's just plus 0.85. And this is it, this is your full wave function using a sine function. All right. So that's the answer. Now, we're going to do the same exact thing except we're gonna have to write a wave function by using a cosine function and we're going to see something interesting. So in part B now we have to do is we want to write our Y of S and T equation except now we want to use a cosine. So this is going to be a times the cosine of KX plus or minus. Actually, it's gonna be minus omega T and then we have plus or minus. I'm gonna call this phi C for cosine. So what does a normal cosine graph look like? Remember a normal cosine graph starts here at the amplitude and it's gonna go down and up like this. So now this is our unsifted cosine graph. So this is uh our Y equals cosine. So this is gonna be a cosign of KX minus omega T without the phase constant. That's what a normal cosine graph looks like. So if you take a look here, what happens is that our first hill actually is shifted to the right from what a normal cosine graph looks like. And so with our Phi C here, if we're shifted to the right actually is going to be positive. So in order to, to, so um so basically, we figured out where we figured out that this is actually going to be a negative cosine or negative Phi C, that's what our phase constant is going to be. All right. So we just do the first two steps. Now, we just have to plug in all the given values we're going to end up with the same exact expression that we had over here. So our Y of X and T is just going to be four times the cosine and you're going to end up with canceling out just these KX minus omega T because X and T are zero. So you end up just basically with a minus phi C, you have to keep that minus sign there. And again, our initial position is going to be three. So you end up with a little expression like this, you end up with cosine of negative Phi C is equal to um oops is yeah, negative five C is equal to 3/4. And when you solve for this, what you're gonna get is P five C is equal to negative 0.72 radiance. So here we got a negative answer and that makes sense because the graph was shifted like this. So now all we have to do is just now write our complete expression yxt is just gonna be a cosine or I'm sorry, this is gonna be four times the cosine four cosine of 10 X minus 62.8 T minus 0.72 radiance. So what we've done here is we've actually written the same exact wave. Our blue wave right here can be written with a sign or cosine graph as we've seen. So what happens here is that when you do this, when you write using a sign or a cosine your phase constant that you calculate is going to be different. And that's OK. And it's because you're starting from a sign or a cosine and you have to shift in different directions to get to the same graph. All right. So that's how you deal with these kinds of problems. Let me know if you have any questions.
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Problem
Problem
A wave traveling to the right has an Amplitude of 15 cm, wavelength of 40 cm, and oscillates 8 times per second. At t = 0, the displacement of a particle at x = 0 along this wave is +15 cm. Write the wave function, including the phase constant, using a sine function.