Lesson Video: Modeling Longitudinal Waves

In this video we learn what longitudinal waves are, how they compare to transverse waves, and how to model them mathematically.

12:19

Video Transcript

In this video, we’re going to learn about modeling longitudinal waves. We’ll see what these waves look like compared to transverse waves. And we’ll see mathematically how to represent them.

To get started, imagine that as a deep-sea explorer, you’ve recently made the discovery of your career. Way towards the bottom of Mariana’s Trench, the deepest part of the Pacific Ocean, you’ve discovered a new species of fish. As the discoverer of this species of fish, you have the right to give it its name. And you call it the howler fish because of its almost unbelievable capacity for generating sound. Proud of your discovery, you dream of bringing the howler fish to millions of aquarium goers worldwide.

The only problem is the fish generates such an amazing amount of sound you’re worried it might damage the glass in the aquarium tank where it resides. To figure out the minimum safe distance for the howler fish to be away from the glass wall, it will be helpful to know something about modeling longitudinal waves.

Longitudinal waves can be compared to another type of wave we’ve seen before. Imagine that you take a rope and you tie one end of it to a pole. And then holding on to the free end of the rope, you start to shake it up and down. When you do, this up-and-down motion of your arm starts to generate these pulses in the rope. And the pulses don’t stand still, they move from left to right down the rope. Because the amplitude of this wave is perpendicular to the direction that the wave travels in, this type of wave we’ve seen is called a transverse wave. We’re able to recognize it because if we put 𝐴 and 𝑉 next to one another, there will be a right angle between them.

A longitudinal wave is different. For a longitudinal wave, the wave amplitude and the wave velocity are in the same direction. They’re parallel. So, what would make a wave like this where the oscillations are in the same direction as the wave motion? Maybe you’ve played with a slinky before. A slinky is a series of thin coils of metal that wrap on one another. Let’s imagine that you get a nice long slinky and you lay it out flat on a table top. And then, holding one end of the slinky, you start moving it quickly back and forth, forward and backward. So, unlike with the rope, where we moved it up and down, we’re moving the slinky along the direction of its length and then pulling back.

After we start moving the slinky like this, let’s say we take a snapshot when we’re pushing the slinky in, that is when our hand is compressing the slinky overall. At that instant in time, if we look along the slinky, we see that there are sections of the slinky where the coils are very close together. And in-between those, there’re sections where the coils are further apart. What we’re seeing is a wave that’s traveling along the length of the slinky. It’s not moving up and down like the rope did, but it’s moving left to right.

And here’s something interesting. Recall as we look back at our transverse wave that we have peaks and troughs to the wave, high points and low points. Our longitudinal wave has those same high and low points. They just look a little different. For our longitudinal wave, the high points, the peaks, happen where the coils are scrunched tightly together, where they’re compressed. In fact, whenever the coils are bunched closely together, this is called a compression. That’s a pressure high point along the wave.

And then, in-between the compressions, at the spots where the coils are stretched out far from one another, these are like the troughs in our transverse wave. And they’re called rarefactions. One way to help remember this term is that at a rarefaction, the material that the wave is moving through is rare, or it’s sparse.

With this longitudinal wave, we’ve shown just one snapshot in time. So, we might wonder what happens if we let time move forward. If we do that, just like with our transverse wave, the wave overall moves from left to right. So, our compression points would move to the right, and our rarefaction points would move that same way too. And so, the areas along our slinky that used to be compressions are now rarefactions and vice versa.

One interesting fact about this motion is that the overall position of our slinky doesn’t change as the wave is moving through it. That means that our slinky is the medium for that wave but that it’s not moved ultimately by the wave. This happens for transverse waves as well. If we imagine a small segment of our rope, that segment is disturbed by the wave as the wave passes through, but eventually the segment returns to its equilibrium location.

So, our wave requires a material medium to move through. And while it’s in that medium, the medium shifts. But after the wave is gone, the medium returns to its equilibrium location. When we talk about longitudinal waves, maybe the best example of longitudinal waves are sound waves. When we talk, we create compressions and rarefactions in the air in front of us. And this wave travels through the air to our listener’s ear.

Imagine that we model the direction of our voice as contained within a cone. As we talk to our friend, we create compressions, areas of high air molecule density, and also rarefactions in-between them where the air molecules are less dense. When we studied transverse waves, we saw that a function that describes the wave’s height at any position and at any time is the wave amplitude multiplied by the sine of the wavenumber times its position minus angular frequency of the wave times time.

This function, it turns out, works not just for transverse waves, but also for longitudinal waves. As we look at this expression though, this may raise a question. What’s the amplitude 𝐴 of a longitudinal wave? For a transverse wave, understanding the amplitude wasn’t too difficult. It was the maximum height of the wave above equilibrium. With our longitudinal wave though, we don’t have displacement perpendicular to the direction of our wave. That displacement is in the same direction as the wave.

If we wanted to measure the amplitude of this soundwave the same way we measure the amplitude of the transverse wave, we might look at one of our compression points, realizing that that corresponds to a peak on our transverse wave, and one of the rarefaction points, realizing that corresponds to a trough, and then say that the amplitude of our longitudinal wave is half the distance between these two points. When it comes to sound waves though, there’s another way we can do it.

With our diagram as is, we’ve ignored any air molecules that appear outside of the cone where we’ve said the voice energy travels. But if we were to draw a few of those air molecules in, it might look something like this. We can say that this air is undisturbed air. It’s unperturbed by any sound waves moving through it. And related to the density of this air, it has a certain pressure.

Let’s say we call that pressure of the undisturbed air 𝑃 sub air. And then, looking at the high-pressure area of one of our compressions, we call the air pressure there 𝑃 sub 𝑐. We can measure the amplitude of this sound wave we’ve created by taking the difference between the pressure at a compression point and the pressure of the air as it would be undisturbed.

So, sometimes when we see a mathematical description for a wave, we’ll see the amplitude given in units of pressure rather than in units of distance. When the amplitude is in units of pressure, that’s a clue that we’re speaking of a longitudinal wave. Along these lines, sound waves are sometimes themselves called pressure waves, or compression waves.

Speaking of compression, when large plates underneath Earth’s surface contact and press against one another, another type of longitudinal wave is created, called seismic P-waves. These waves are so powerful that rather than moving air molecules back and forth in the atmosphere, they move earth and rock underneath Earth’s surface back and forth. This is an earthquake! Knowing the two main types of longitudinal waves, let’s get a bit of practice mathematically describing these waves through an example.

Consider a sound wave modeled by the equation 𝑠 as a function of 𝑥 and 𝑡 equals 4.00 nanometers times the cos of 3.66𝑥 minus 1256𝑡, where 𝑥 is measured in meters and 𝑡 is measured in seconds. What is the maximum displacement of the sound wave? What is the wavelength? What is the frequency of the sound wave? What is the speed of the sound wave?

We’ll label the maximum displacement of the wave capital 𝐴. The wavelength we’ll call 𝜆. The frequency is 𝑓. And the speed is 𝑣. We know that in general for a sound wave, the function that describes its height as a function of position and time is wave amplitude 𝐴 times the sine of wavenumber times position minus angular frequency times time.

If we look at the wave function we’re given, we see that instead of sine we have a cosine term. This doesn’t change the basic description of our function though. All cosine, instead of sine, means is a phase shift of 90 degrees. It’s like shifting the wave one-quarter of a wave cycle.

So, looking at our given function 𝑠 and comparing it to the general equation for a sound wave, we can say that the amplitude of our wave is equal to 4.00 nanometers, that the wavenumber 𝑘 is equal to 3.66 inverse meters, and that the angular frequency 𝜔 is 1256 inverse seconds. This tells us the answer to our first question, what the amplitude of our wave is. It’s 4.00 nanometers.

Next, to solve for the wavelength of this wave, we recall there’s a relationship between 𝜆 and wavenumber 𝑘. 𝑘 is equal to two 𝜋 divided by 𝜆. So, we can say that 𝜆 is two 𝜋 over 𝑘, or two 𝜋 over 3.66 meters. To three significant figures, this is 1.72 meters. That’s the wavelength of this wave.

Next, to solve for wave frequency, we can recall the relationship between 𝑓 and 𝜔, the angular frequency. 𝜔 is equal to two 𝜋 times 𝑓. So, 𝑓 equals 𝜔 over two 𝜋, or 1256 over two 𝜋 hertz. This rounds to 200 hertz. So, this sound wave moves through 200 cycles every second.

Finally, we want to solve for the wave speed 𝑣. We can recall that wave speed is equal to the product of wavelength and wave frequency. And since we’ve solved for the wave’s wavelength and frequency earlier, we can plug those values in now. In terms of the exact values for 𝜆 and 𝑓, two 𝜋 over 3.66 meters for 𝜆 and 1256 over two 𝜋 hertz for 𝑓. When we multiply these terms together, we see the factors of two 𝜋 cancel out, and our velocity can be written as 1256 over 3.66 meters per second. To three significant figures, this is 343 meters per second. That’s the speed of this sound wave.

Now let’s summarize what we’ve learned about modeling longitudinal waves. We’ve seen that, unlike transverse waves, longitudinal waves displace matter in the same direction as the wave moves. Another way of saying that is the wave amplitude is in the same direction as the wave velocity.

On the other hand, like transverse waves, we’ve seen that longitudinal waves are modeled by the equation the wave height as a function of its position and time is equal to its amplitude times the sign of its wavenumber times position minus angular frequency times time. And we saw that for longitudinal waves, the wave amplitude might be measured as a pressure difference rather than as a distance. And finally, we saw that two common examples of longitudinal waves are sound waves and seismic P-waves.

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