Guys, in the last couple of videos, I introduced you to waves and I also showed you the wave speed equation, \( v = \lambda f \), which applies to all kinds of waves whether they are transverse or longitudinal. Now in some problems, we're going to be dealing with transverse waves on strings. You're pulling some string with some tension, you're whipping it up and down. But this equation isn't going to be enough to solve our problems. So we're going to need another equation to solve, and that's what I'm going to show you in this video. I'm going to show you a special equation that we use to calculate the velocity for waves, specifically well on strings. So let's go ahead and check this out here. Let's take a look at a problem. We got a tension of 100 Newtons and we're flicking it up and down to create a transverse wave. Basically, you're doing this. You've got some little string like this. It's attached to the wall at some point and you're just basically whipping it up and down to create a transverse wave. So let's go ahead and take a look here. The string has a mass of 0.5 kilograms. So first, the tension is 100. Our mass is 0.5, and the length of the string \( l \) is going to be 1.2 meters. So we want to figure out what the frequency of this wave is.

You're flicking this thing up and down. What's the frequency? But we're told that the wavelength is 15 centimeters. So the last piece of information we know is that \( \lambda \) is equal to 0.15 once you do the conversion here. So what's \( f \)? Well, the only equation we've seen so far is the wave speed equation, \( v = \lambda f \). So let's start there. We've got \( v = \lambda f \). Now I'm going to solve for \( f \). So I've got this \( f \) here is equal to \( v \) divided by \( \lambda \). So if I want to figure out the frequency, I just need both of these things. Now the problem is I actually am told what the wavelength is, but unfortunately I don't know what this wave speed is. What is \( v \)? And given the other variables that I have here there's no way to calculate \( v \). So it turns out that this equation isn't enough to solve this problem, and so we're going to need another equation. That's the point of this video. So \( v = \lambda f \), that applies to all waves, but wave speed is also determined by the physical properties of the medium itself.

What does that mean? It means that the velocity for waves on a string depends on the physical properties of the string. Those are the things that you can measure. Things like tension and mass and length. So this new equation here that relates the speed with these physical properties is going to be the square root of the tension divided by this Greek letter \( \mu \). This Greek letter \( \mu \) is just the mass divided by the length of the string. Most textbooks refer to this as the mass density of the string. So we have \( v = \lambda f \), but specifically for waves on strings, we also have this other equation. Since some problems, we're going to have to use both. So let's go ahead and get back to our problem here. We want to figure out the wave speed. Now we're just going to use this new equation here. So we have \( v = \sqrt{\frac{\text{tension}}{\mu}} \). Now we're not given \( \mu \) directly. Right? We are given the tension, so we can go ahead and pop that in. We're not given the \( \mu \), but we actually do have mass and length so we can figure it out. So one thing we can do here is we can actually just go ahead and rewrite this entire expression. So we're going to have the tension on top and remember, \( \mu \) is really just mass divided by the length. So really we're just going to have this fraction dividing by another fraction here. This is going to be \( m \) over \( l \). Now I'm just going to plug in all the numbers. So this is going to be the square roots of 100 divided by and and this is going to be 0.5 divided by 1.2. So if you go to work this out, you're going to get a speed of 15.4 meters per second. Are we done yet? Not quite because remember, this is just the \( v \). This is just the wave speed. We're supposed to find the frequency. So we gotta pop this thing back into this equation over here and then we'll solve. So we got now this wave speed here is 15.4. Now we're going to divide it by the wavelength, which is 0.15. When you go ahead and work this out here, you're going to get 103 Hertz. So that's how to calculate the frequency. Sometimes you just have to use a combination of these equations.

Alright, guys. So that's it for this one. Let me know if you have any questions.