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.
