Hey, guys. So in the last couple of videos, we saw the equations for the wave function, which are sine and cosine functions. Well, in some problems, you're going to be asked to calculate two different kinds of velocities. In the example that we're going to work out here, we have a wave function given to us. In part a, we're going to calculate the velocity of the wave, which is really just the propagation velocity, and we've seen that before. In part b, we're going to calculate something else entirely, which is called the transverse velocity. Now, these things sound very similar. The velocity of a transverse wave versus the transverse velocity of waves. They sound very similar, but they're actually very different ideas. In this video, I am going to show you the differences between these two velocities and how we calculate the transverse velocity of waves. Let's go ahead and check this out here. So we're going to start with what we know. Remember that the velocity of a transverse wave had another name, we call it the propagation velocity. Basically, this was just the velocity of the wave pattern that is moving left and right. So if you take a string and whip it up and down, as we have in our diagram over here, you're going to create a wave, and this wave pattern overall moves to the right like this, and that's what the propagation velocity is. Now, this is different from the transverse velocity of waves. They sound similar, but they're actually very different. Because what's happening here is that as you're whipping the string up and down, the particles that are on the string are also moving. They're moving up and down perpendicular to the direction of the wave. So that's what this transverse velocity means. Transverse just means perpendicular. It's the perpendicular velocity of the particles that are on the string or on the wave, and those particles move up and down. So that's the difference between these two velocities. The propagation velocity is the overall wave pattern moving to the left or right. The transverse velocity is going to be the velocity of the particles that are moving up and down. Alright? So before I actually get into the equation, I want to go ahead and start our problem now that we know the differences between these two. So in part a, we want to calculate the velocity of the wave, which is really just the propagation velocity, and we've seen how to do this before. If we have our wave function equation, we've actually seen this exact wave function before. Remember that this has the form \(a \times \cos(kx - \omega t)\). So we have the velocity of the wave, the propagation velocity, is really just going to be given by this equation right here, \(\lambda/ k\). Remember that was our shortcut equation. So we have \(v = \omega / k\). So our omega value is just 6, our k value is 0.4. If you go ahead and work this out, you're going to get 15 meters per second. So overall, this whole entire wave is moving to the right at 15 meters per second. What about the transverse velocity? That's what we're going to calculate in part b. Well, now I'm going to show you the equations for this. Remember that different textbooks will use different sorts of equations for the wave function. Some will use a sine and then some will use a cosine. I am actually going to give you both of the transverse velocity equations, basically based on which one your textbook actually uses. So if we're using a sine, your transverse velocity equation is also going to be a function of x and c, and it's going to look like this: \( \pm \omega a \times \cos(kx \pm \omega t)\). So we picked up another value of \(\omega\) outside of our cosine equation and we also have this \(\pm\) here. This \(\pm\) is actually related to this \(\pm\) that's inside of our \(kx \mp \omega t\). So basically, what happens is that the rule is that they have the same signs. If you have a plus sign here, you're going to have a plus sign outside of your equation. If you have a minus sign inside of your wave function equation, then you're going to pick up a minus sign when you take the transverse velocity. Alright? So, let's take a look at the other equation for cosine. It's going to look a little bit different. So this equation is going to look like this. It's going to be \(\mp \omega a\). Now we're going to have \(\sin(kx \pm \omega t)\). So the idea here is that this is a minus over a plus, which actually means that it's going to be the opposite sign of whatever you have inside of here. So if you have \(kx+\omega t\), then you're going to pick up a minus sign outside of your equation for the transverse velocity. If you have a minus sign, then it's going to be the opposites. And also, the other thing to realize is that the sine turns into a cosine and the cosine turns into a sine, so the equation of the trig function is always going to flip. Alright, so that's just a little bit about the transverse velocity equations. So let's go ahead and actually use them. We want to calculate the transverse velocity of a particle that's at some position and time. So we need to figure out which one of the equations we're going to use. So \(v_{\text{tx}}\), which one are we going to use? Well, we're starting off with a wave function that is of the form \(a \cos\), so we have a cosine here of \(kx\) and we have \(- \omega t\). So we're going to actually use this equation here to start off with, which means our equation is going to have this form right here. So the first thing we have to do is figure out whether we have a plus or a minus sign that's outside of our equation. Remember the rule. If we have a minus sign that's here, then what happens is we're going to have a plus sign inside of our transverse velocity equation. That's exactly what happens here. So we have a plus sign here. Next we have \(\omega\) and \(a\). So our \(\omega\) remember is just going to be 6, our \(a\) is going to be 3. Now we have the sine and this is going to be our \(k\), this is going to be 0.4, that doesn't change, \(- 6t\). Right? So what's goes inside the parenthesis doesn't actually change. So this is actually the format that we're going to use. So all we have to do now is just plug in the values. So \(v_{\text{tx}}\), so now we're going to plug in when \(x\) is equal to 0.75, and when \(t\) is equal to 0.2. So all you have to do now is just plug in the values. So we're going to have \(18 \times \sin(0.4 \times 0.75 - 6 \times 0 .2)\). So that's what happens when I plug in all the numbers. Remember to keep your calculator in radians, and what you'll get is \( -14.1 \text{ meters per second}\). So this is the perpendicular velocity of a particle at that specific time. It's going downwards at 14.1 meters per second. That's all that means. Alright. So that's how we use the wave function. The last thing I want to mention here is that the propagation velocity that will be calculated in part a is going to be constant at all points on the wave, whereas the transverse velocity is actually going to change with the position and the time. Remember, it depends on \(x\) and \(t\), so if you change those values, it's going to change the transverse velocity. So this just means that some particles in the wave could be going up and others are going down depending on where you look along the wave. Alright. So let's finish things off and talk about the last part here, which is calculating the maximum transverse velocity of the particles. So remember the idea was that when we had the wave function, this \(y\) equation here, the maximum displacement that we could possibly have was really just the amplitude, either the positive or negative one. It was basically just whatever is outside of your trig function, either sine or cosine. It's the same idea for the maximum transverse velocity. It's just going to be whatever is outside your sine and cosine, so that's just going to be \(\omega a\). So we want to calculate \(v_{\text{t max}}\) here, so this is just going to be \(\omega \times a\), and we actually know both of those values. This is just going to be \(6 \times 3\), so our maximum transverse velocity is 18 meters per second. Alright? So that's how you work out these problems. Let me know if you guys have any questions.