Guys, in the last couple of videos, we took a look at the equation for wave speed for waves on strings, which is really just this guy right here. Well, in some problems, they're going to give you the initial setup of a problem. For example, we have an oscillating blade that's creating transverse waves on a stretched string. What they're going to do in different parts of the problems is they're going to change some of the variables. For example, in part a, we're going to quadruple the tension. In part b, we're going to double the frequency. They're going to ask you to calculate what happens as a result of this change. So, we're going to calculate the new wave speed and the new wavelength. Now in these problems, it's often difficult to understand which variables affect others inside of this equation here. So, I'm going to show you how this works. I'm going to give you some very simple rules for solving these problems. Let's go ahead and check this out here. We're going to start with part a. Our setup here tells us that we have an oscillating blade which is really just a little blade that's attached to the string that's on a motor that's vibrating up and down. It's creating waves at a frequency of 35 hertz. So, it's basically spinning up and down at a frequency of 35. We're told the tension on this string here is 98. We're also told that the mass density of this is just 2. And initially, we calculate the wave speed, so this whole entire wave is moving to the right with 7 meters per second. And we also thought that the wavelength, this lambda here, is equal to 0.2. Now you can just trust me on that or you can actually plug in all the values inside of this equation and you'll actually get those numbers. So, in part a, we're going to quadruple the tension on the string and then we want to calculate the new wave speed. So, basically what they're saying here is I'm going to calculate this new tension, which I'm going to call . It's just going to be quadruple what the initial tension was. So it's going to be , which is going to equal 392, and now we want to calculate the new wave speed, so I'm going to call this . And that brings us to the first rule. The first rule to solve these problems is that the velocity, remember, depends on the mass, the tension, the length. It depends on the physical properties of the medium. So any time you change any one of these values, tension, mass, or length, you're always going to change the velocity. So remember to change any one of these values here, like the tension or the mass and the length, it's going to directly impact the speed of the wave that's on the string. So, what happens is to calculate our new velocity, all we have to do is just use this same equation here, but now we just have to use our updated or our new value for tension. So we're just going to use the square roots of . Now is the only one that's changed, now we just have 392, and our value hasn't, it's just 2, right? The mass and length hasn't changed. So, what you end up getting here is you end up getting 14 meters per second, which is exactly twice what the initial wave speed was. I'm going to call this here. And that's because we quadrupled the value that's inside this numerator, so effectively you are doubling the wave speed. Alright? So that's the first rule. Let's take a look now at the second part of the problem. Now we're going to double the frequency of the oscillator. So what happens here is remember that we have this little spinning blade that's oscillating, vibrating up and down, and all that happens is now we're going to increase that and we're going to double it. So what happens is we're going to move this vibrating blade and it's going to go faster. It's going to create these little bunched-up waves like this. We want to do the same thing here. We want to calculate the new wave speed and also the new wavelength. So now what happens is we have our new frequency is going to be twice the original frequency, so I'm going to call this original frequency 35 hertz. So, we just have , which equals 70, and now we want to calculate the new wave speed. I'm going to call this and I'm going to call this . So now what happens is we're going to take a look at our second rule. Oursecond rule says that the frequency of a wave depends only on the oscillator frequency. So, what happens here is that this oscillator frequency is directly going to impact what this is. So I'm going to call this , which is really what this is, and these things are equal to each other. So whatever this is, that's going to be that's inside your problems. Now remember that the frequency of the wave speed depends only on the physical properties of the medium, tension, mass, and length. So because depends only on their oscillator frequency, changing it does not affect the velocity. Changing the oscillator frequency only affects and . So, what happens is any change that you make to this is going to directly impact . It's not going to impact the string. So, what happens as a result? Well, basically, if has to increase or decrease, then that means that your wavelength is now going to have to decrease or increase proportionally because this has to remain the same. That's what the rule is. So, what this means here is that by changing the oscillator frequency, your wave speed actually doesn't change at all. So is just the initial wave speed, which is going to be 7 meters per second. And that's the answer. Your wave speed does not change at all. So, what happens to the wavelength as a result? We can actually just go ahead and calculate this by using this equation here, . If you solve for , what you're going to get is , the initial divided by the new frequency, what you're going to get here is the 7 meters per second divided by the new frequency of 70. You're going to get a wavelength that's 0.1 meters. So, this actually makes some sense. If you're doubling the frequency of the oscillator, you're going to create these more bunched-up waves like this, and that just means that the wavelength is going to decrease because now you're going to create more bunched-up waves like this. The wave speed remains the same. Your frequency increases, but your wavelength is going to decrease as a result. Now it's 0.1 whereas initially it was 0.2. Alright? So that's how these problems work. Let me know if you guys have any questions.