Hey guys. So up until now we've been dealing with the velocity equation for transverse waves on strings, which was this guy right here. But sometimes in problems, you're not going to have transverse waves on strings, you're going to have longitudinal waves that are propagating through other types of mediums, like a liquid or a solid or something like that. So I'm going to show you the equation for the speed of longitudinal waves in fluids and solids. What we're gonna see is that it's very similar to this equation right here, just a couple of letters that are different, and then we're gonna check out some examples. Very straightforward. Let's get to it. So basically, your longitudinal wave speed equations break down into two different categories. Whether your wave is traveling through a fluid, like water or, you know, some other liquid, or if they're traveling through a solid, like a metal rod or something like that. If you've ever, like, gone up to a metal pole and you've knocked on it, you've probably heard the sound wave travel up, and sort of echo along the way. So sound waves can travel through fluids and solids. Now your textbooks are gonna do some long derivations for this. I'm just going to give you the equations because your professors won't have you memorize this. So the speed for fluids is going to be βρ, where beta is the bulk modulus of a fluid. This is a number that's always going to be given to you if you need it. Divided by the density of the fluid. So these are both letters that were probably sort of given to you in, or talked about when you did the elasticity chapter. Now in solids, it's gonna look very similar except it's gonna be Young's modulus because we're talking about solids rods. Solids and rods divided by the density of the material. Now remember that all waves regardless of whether they are transverse or longitudinal, always are equal to lambda times frequency. So this relationship still holds for longitudinal waves.

That's it. So that's all there is to it. Let's go ahead and take a look at some examples here. So, we have a liquid, so we know we're going to be dealing with this equation right here. We're told the density of this liquid is 1200. We know it's a longitudinal wave, and we're told that the frequency of these waves is 400 Hz, and we're told that the wavelengths are measured to be 8 meters. What we want to do is we want to calculate the bulk modulus of this liquid here. So basically that's going to be that beta term. So I'm gonna use this equation right here. Let's go ahead and set it up. So I've got my v is equal to the square roots of beta divided by rho is equal to lambda times the frequency. So let's see. I don't have the beta. In fact, that's actually what I'm looking for here, but I do have the density of the fluid. I also have the wavelength and the frequency of that wave. So remember, this is basically just, it's just giving you 4, you know, 4 variables. And I have 3 out of the 4, so I can go ahead and figure this unknown out. Out. This one that's unknown. So I'm just going to square both sides. I'm actually just gonna go ahead and start plugging in some numbers. So I've got the square roots of this bulk modulus divided by 1200 equal to 8 times the frequency which 400 here. So when I square both sides, I'm gonna get beta divided by 1200 is equal to this is gonna be 3200, and I'm gonna square that. And so finally, once you go ahead and plug everything in, you're gonna get that beta is equal to, let's see. I got 1.23 x 10^{10}. And that is the units for that are in pascals.

Let's look at the second example. The second example, we're gonna strike a 60-meter-long brass rod at one end. So let me go ahead and draw this. I've got this little, like, long brass pipe like this, and I've got the distance here. This delta x is equal to 60 meters. So how long does it take for a person on the other end of the rod to hear the sound? What they're actually asking us for is delta t. So the idea here is that you're gonna strike one end of the brass rod, and the sound wave is gonna propagate, it's gonna travel down through that solid and eventually hit the person on the other side. So this person is standing over here like this. So we wanna calculate what's the delta t between you hitting it and them hearing the sound here. Alright? So we're given all the constants that we need, the Young's modulus, and the density and stuff like that. So how do we figure this out? Well, if I'm looking for a delta t and I have velocity and delta x, I'm actually gonna set up an old one-dimensional motion equation. Remember that v is equal to delta x over delta t. Right? So it's just displacement over time. So what I want is I wanna figure out this delta t, so I'm actually just gonna trade places between both these variables. So delta t is equal to delta x divided by the velocity. So I have what the delta x is, and if I want to figure out delta t, I just need to figure out what is the velocity. So how do I do that? Well, now I have the equation to do that, the velocity of a longitudinal wave given Young's modulus and the density here. So that's all I have to do. I just have to basically go over here and figure out what is this what is this v here by using that equation. So remember this v is just gonna be the square roots, and this is gonna be Young's modulus divided by rho. So this is gonna be Eρ. And what you should get is the velocity is gonna be 3200 meters per second. So that's the speed at which a sound wave travels through brass. So this is actually about 10 times faster than the speed of sound through air, and that makes sense. So sound travels much faster through metals and solids because the density of material is much higher there. So this is 3200 meters per second. Now we just plug this back into this equation here. So to finish this off, we have delta t is equal to this is gonna be 60 divided by 32100, and you should get a delta t that's equal to 0.02 seconds. So it takes almost no time for that sound to reach them. Alright so that's it for this one guys, let me know if you have any questions.