Hey, guys. So up until now, we've seen some pretty basic equations for the wave speed of waves. But in some problems, you're going to have to use something called the wave function to solve problems. So I'm going to introduce you to the wave function in this video and show you exactly what it is. Now if you've never seen it before, it can be kind of scary. So what I'm going to show you is that there's actually only 3 variables you really need to know. Let's check this out here. So the idea here is that the wave function is really just a sinusoidal equation. Remember sinusoidal just means sine or cosine, and it describes the shape of an oscillating wave. When you whip a string up and down, you're producing these sort of sinusoidal graphs that go up and down like this, so they can be described by sine or cosine graphs. Alright? So this is the first time you're going to see something like this in physics, so I want to go a little bit carefully. The equation for this wave function is going to be y of x and t. So you're going to have 2 things inside the parentheses. What it's doing here is it's actually giving you an output. It's giving you the displacement value when you plug into inputs, the position and the time. So you plug in x of t in your equation, and it's going to give you the y value of a particle on that string at a certain position and time. That's what it's telling you. Alright? So depending on the textbook that you're using, you might gonna see this, you might see this written in 2 different ways. So you might see this written as a⋅sinkx±ωt, and in some textbooks you'll see this written as a⋅coskx±ωt. Your professor might also have a preference, so just go ahead and use whatever they're going to use. Now again, both of these are accurate, sine and cosine. They're both sinusoidal. The difference between when you use them is actually where the graph starts. So this blue graph right here, I actually have it in blue and you'll see that the equation or the graph starts at y equals 0. That's when you use the sine graph. You're going to start when y equals 0. And you're going to use this cosine graph when you're starting at the amplitudes. It could be either the upward or the positive or the negative amplitude. So you're going to use the cosine wave function when your wave starts at either plus or minus a. So let's go ahead and talk about the different variables that are inside this equation. We already know that the position in time, x and t. So what about the a, k, and omega? The a is just the amplitude. That's just really the maximum displacement in either direction of the graph, and we've seen that before. The k is something new called the wave number. You don't need to know conceptually what it is. All you need to know is the equation for it, which is 2πλ, the units for this k are going to be in radians per meter. Now the other equation is going to be, or sorry, the other variable is omega, which is the angular frequency, and we've seen that before. The angular frequency is just 2πt, the period. So notice that these two actually kind of look alike. 2πλ, 2πt. The angle of frequency also had another equation which is 2*pi times the frequency. That's really it. Those are your 3 variables. So let's go ahead and take a look at our problem here. So our problem we're going to whip a rope up and down to create a transverse wave. We're told some of the values. We know that the amplitude, this a here, is 0.5. We're also told that the wave speed of the rope is going to be 8 meters per second. Second, that's going to be v. We're also told that the wavelength of the wave that we're producing, so that's going to be lambda equals 0.32. Now we're also told at t equals 0, at the starting point, the end of the rope that we're going to hold is at the maximum upward displacement. So what it looks like is kind of like this. So you're holding this little rope like this at the maximum upward displacement and you're going to flick it up and down to create a transverse wave that's moving to the right like that. Now in the first part of the problem, in part a, we're going to write the wave function for this wave. So we're going to write an equation that's y of x and t. Now to do this we need to figure out whether we're using a sine or a cosine function, and remember that depends on where the graph is going to start. So remember, if we're starting at either the positive or negative amplitude, we're going to use the cosine function. So with this y of x and t, we're going to use a times the cosine of kx+−omegat. The next thing we have to do is actually figure out the sign of this function, whether it's kx plus or kx minus omega t. And to do that, we're just going to use this rule here. The direction of the wave determines the sign of kx plus or minus omega t. The rule is pretty simple. It's the direction is going to be opposite to the sign. So what this means is that your wave moves to the right like in the plus x direction, sign is going to be negative. It's going to be kx minus omega t. And then it's the opposite if you're moving to the left. Minus x, your sign is going to be plus, so it's going to be plus omega t. Alright? So what happens is we're moving to the right, our wave, so that means we're just going to use a minus sign. So it's k x minus omega t. The last thing we need to do is we actually have to figure out the values for a, k, and omega, so we can plug them back into our equation, and that's going to be the full wave function. So we actually already know what the amplitude is. That's what we were given, a equals 0.5. So we need to figure out k next. So this k value, remember, has has an equation. It's going to be 2 pi divided by the lambda, which is going to be 2 pi divided by our lambda value is 0.32. So pretty straightforward. If you go ahead and do this work this out, you're going to get 19.6. So now we have what k is. The last thing we need to do is just figure out what what omega is. So our omega equation, or omega, variable, remember, has 2 equations. We're going to have 2 pi divided by the period, or we have 2*pi times the frequency. Now we're not told in this problem anything about the period. All we know is the amplitude, the speed, and the wavelength. So we don't know the t, so we're not going to use this equation. We're going to use this 1, 2*pi times the frequency. Now unfortunately, we don't know what f is, but we can go figure it out. Remember that the only other place that f shows up is inside of the wave speed equation, v equals lambda f. Now we actually have 2 out of 3 variables. We know the wave speed, and we also have what the lambda is. So we can figure out the frequency. So our frequency is just going to be v over lambda. That's going to be 8 divided by 0.32, and you're going to get to 25 hertz. So that's the frequency of the wave that you're oscillating. Right? So now all we have to do is just plug this back into this f right here, and then we'll figure out omega. So omega is just 2 pi times 25, and if you go ahead and work this out, what you're going to get is a 157.1. So that's what we now plug into this variable here, omega. So now all we're all we're going to do is just pop each one of these variables back into our wave function equation, and then we're done. So our wave function equation is going to be 0.5 times the cosine. Now we have 19.6 for k. We have an x here for the position minus 157.1 times t, and this is our full wave function equation. Notice how when you write out the wave function, if you're just asked to write it out, you're going to plug in values for a, k, and omega, but not for x and t. This is a function here, so you're still going to have these inputs as x and t. Right? So you're still going to have those inputs there. All you have to plug in a, k, and omega. Now the last thing we have to do, that's part a, is just actually evaluate this wave function. We're going to figure out the displacement of a particle at x equals 0.4 and t equals 0.75. So basically what happens is that now that we have this wave function, we're just going to plug in some values and figure out the y value. So y when x is equal to 0.4 and t is equal to 0.75. You're basically just going to plug in these values for x and t inside of your equation. So this is going to be 0.5 times the cosine of 19.6 times 0.4 minus 157.1 times 0.75. So it's a lot of stuff to plug into your calculator. Make sure you do it carefully. And also make sure that your calculator is in radians mode. You cannot forget this. Make sure your calculator is in radians and not degrees, or or you're going to get the wrong answer. What you're going to get here is you're going to get a displacement of negative 0.5. So what this means here is that if your amplitude is 0.5 and negative 0.5, then at this position and time, you're going to have a particle that's right here at the bottom at the negative That's all that means. Alright? So that's it for this one guys. Let me know if you have any questions.
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Wave Functions
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