Velocity of Longitudinal Waves - Online Tutor, Practice Problems & Exam Prep

Longitudinal waves travel through fluids and solids, with their speed determined by the equations: βρ for fluids, where β is the bulk modulus, and Eρ for solids, where E is Young's modulus. The relationship between wave speed, wavelength, and frequency remains consistent: v = λf. Understanding these principles is crucial for analyzing sound propagation in different materials.

Speed of Longitudinal Waves (Fluids & Solids)

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 10¹⁰. 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.

A metal bar $$has a length of 1.5 m and a density of 6400 kg/m³. Sound waves take 3.9 × 10^-4 s to travel along the length of the bar. What is Young's modulus for this metal?

• A. 2,311 Pa
• B. 9.47 × 10¹⁰ Pa
• C. 2.46 × 10⁷ Pa
• D. 3.744 Pa

Example 1

Welcome back, everyone. In this problem, we're focused on a container full of oxygen gas. 32 grams of oxygen occupies a volume of 0.0224 m³. The speed of sound in this container is 317 meters per second. We want to figure out the bulk modulus for oxygen gas. So, in other words, we have a mass of 0.032 kilograms. I'm just going to convert that to kilograms. A volume that it occupies, which is 0.0224 m³. We're told the speed, which is a little v, of sound in this container is 317 m/s. And, ultimately, we want to figure out the bulk modulus, which is represented by the capital letter beta here. So that's really what we're looking for. Remember, in these types of problems here, we're dealing with waves of longitudinal, of waves, or sorry, speed of longitudinal waves. There are 2 forms. One with beta over rho and one with Young's modulus over rho. One is for liquids. This actually sort of should say fluids because in this case, we're dealing with a gas, but it behaves like a liquid, and then one for solids. So we're really just looking at this equation over here, our velocity equation. Alright. So let's sort of put all this stuff together, and we're just going to use our velocity equation for longitudinal waves in fluids, which is going to be the bulk modulus divided by rho. We're looking for this bulk modulus here, so we just need those other two variables. I need to figure out the density, and I also need the speed of sound. Now we actually have what the speed of sound is. We were just given that, where it's just 317 meters per second. What we really need to do is find the density here because all we have is mass and volume. So, that leads us to, how do we find density? How do we figure out this rho here inside of this equation? Well, we're really just going to come over here and remember the definition of density is always just mass divided by volume. Now remember, mass over volume has to be in kilograms per meters cubed, so you have to have things in the right units. So that's why we converted this thing to 0.032. So this really just becomes 0.032, that's in kilograms, divided by a volume of 0.0224, and that's going to be in meters cubed. Now, if you go try to figure this out, what you're going to get actually here is you are going to get 1.43, and that's going to be kilograms per meters cubed. So this is the density that you're going to plug into this equation over here.

All right. So now I'm just going to go ahead and rearrange. So first, we're going to have to square both sides to get rid of the square roots. So this is just going to be beta over rho. And then lastly, all we have to do is just multiply the rho over. So in other words, beta is equal to v² times rho. So in other words, it's 317, that's the speed of sound in this container squared, times our density that we figured out was 1.43. Now go ahead and work this out. What you're going to get is you are going to get a bulk modulus of 1.44 x 10⁵ pascals. So this is our final answer. And looking through our answer choices, it looks like that's going to be answer choice c. So that's the correct answer. Hopefully, you got that one right. Let me know if you have any questions. If you didn't, thanks for watching.

Here’s what students ask on this topic:

What is the equation for the speed of longitudinal waves in fluids?

The speed of longitudinal waves in fluids is given by the equation:

$\sqrt{\frac{\beta }{\rho }}$

where $\beta$ is the bulk modulus of the fluid and $\rho$ is the density of the fluid. The bulk modulus is a measure of the fluid's resistance to compression, and the density is the mass per unit volume of the fluid. This equation is crucial for understanding how sound waves propagate through liquids.

How do you calculate the speed of longitudinal waves in solids?

The speed of longitudinal waves in solids is calculated using the equation:

$\sqrt{\frac{E}{\rho }}$

where $E$ is Young's modulus of the solid and $\rho$ is the density of the solid. Young's modulus measures the stiffness of the material, and the density is the mass per unit volume. This equation helps in understanding how sound waves travel through solid materials like metals.

What is the relationship between wave speed, wavelength, and frequency for longitudinal waves?

The relationship between wave speed ($v$), wavelength ($\lambda$), and frequency ($f$) for longitudinal waves is given by the equation:

$v=\lambda f$

This means that the speed of the wave is equal to the product of its wavelength and frequency. This relationship holds true for both transverse and longitudinal waves, making it a fundamental concept in wave mechanics.

How does the density of a material affect the speed of longitudinal waves?

The density of a material significantly affects the speed of longitudinal waves. According to the equations for wave speed in fluids and solids:

$\sqrt{\frac{\beta }{\rho }}$ for fluids and $\sqrt{\frac{E}{\rho }}$ for solids,

the speed of the wave is inversely proportional to the square root of the density ($\rho$). This means that as the density increases, the speed of the wave decreases. Conversely, a lower density results in a higher wave speed. This is because denser materials have more mass per unit volume, which requires more energy to propagate the wave.

What is the bulk modulus and how does it relate to the speed of longitudinal waves in fluids?

The bulk modulus ($\beta$) is a measure of a fluid's resistance to compression. It is defined as the ratio of the change in pressure to the relative change in volume. The bulk modulus is crucial in determining the speed of longitudinal waves in fluids, as given by the equation:

$\sqrt{\frac{\beta }{\rho }}$

where $\beta$ is the bulk modulus and $\rho$ is the density of the fluid. A higher bulk modulus indicates that the fluid is less compressible, leading to a higher speed of sound through the fluid.