### Video Transcript

In this video, we’re talking about
calculating wave motion. In particular, we’re going to see
how to use wave properties to figure out how fast waves move. Or at the same time, if we already
know how fast a wave is moving, then we’ll be able to work backwards and solve for
some of those properties of a wave.

Now, a great place for us to start
is to remind ourselves just what is a wave. A wave is defined as a disturbance
that transfers energy from one point to another. Examples of waves include sound
waves we create when we speak, transferring sound energy from one point to
another. Or if we were holding on to the
free end of a rope and we moved that free end up and down rapidly, then we create a
disturbance in the medium of the rope. And we’re transferring energy down
the rope to the right. And then, of course, like we saw on
our opening screen, there are such things as water waves that wash up on shore from
the ocean, transferring energy in that direction.

Now, it’s possible to create a wave
where this disturbance is very short lived. For example, thinking of holding on
to the free end of this rope, if we just moved our hand up and down rapidly once,
then we would create this single disturbance on the rope. And as it traveled down the rope,
this disturbance would not repeat. But then, if instead of shaking
this rope up and down just once, we kept on shaking it up and down throughout time,
then we would be creating a disturbance along each point of this rope at any instant
in time. So, whether we have a wave that
looks something like this, where there’s just one cycle of the wave, or if it goes
on for many cycles, in either case, we’re still talking about a disturbance that
transfers energy. We still have a wave.

Now, in mentioning wave cycles,
that brings up one of the important fundamental properties of a wave. Here’s the idea with wave
cycles. Let’s say we start out at this
beginning point and we follow along the wave for some distance until we get right to
here. We can see that over this distance,
the wave has gone through one complete cycle. At the outset, it started by rising
up and then it reached some peak value and then descended down to its original
value, then kept going down until it was at a lower value than rose back up until it
was at its starting point.

Now, when we say starting point, we
don’t mean the same starting point in space. But we mean the same starting point
with respect to the wave being a disturbance. If we go back to the very start
point of the wave, we see that from here, the wave was moving upward. And the next point along the wave,
where the wave is moving in the same exact way, is this point here. That’s what we mean when we say a
wave goes through cycles. This is one cycle of the wave. And the distance in space between
these two points is called the wavelength. And the way we often represent
wavelength is using the Greek letter 𝜆.

So, 𝜆, the wavelength, is a
distance often measured in meters. The wavelength tells us how far in
space a wave needs in order to move through one complete cycle. And let’s write that definition
down to help our understanding as we go along. Wavelength is the distance needed
for a wave to move through one complete cycle. Considering the wave we have
on-screen, we can see that this way it goes through one, then two, then three, then
four, and then five complete wavelengths. We could mark out those wavelength
points on our curve like this. So, this wave has gone through five
complete cycles over this distance right here. So, we could call that distance
five times 𝜆, five times the wavelength.

Now, we’ve said that wavelength is
a distance. That means if we were to create
axes and put those axes down on top of our wave, then they could look something like
this. Where on the vertical axis, we’re
showing the height of the wave, say, in units of meters, and on the horizontal axis,
we have the distance the wave has traveled, in meters again. This is the kind of graph we might
expect to see if we we’re investigating a wave’s length.

But there’s another property of
waves which is just as important as wavelength. And to consider what that property
is, instead of using distance in meters on our horizontal axis, let’s replace that
with time in seconds. Now, clearly this is a big
change. We’re not changing from meters to
kilometers or centimeters. In other words, we’re not changing
a distance to a distance. But we’re changing the type of
variable we’re using completely from a distance to a time.

Here’s what this means. Whereas before when we had distance
on our horizontal axis, that meant we could say something like this. We could say at this given distance
from the start point of our wave, the height of our wave above its equilibrium
location is this height right here. That’s the sort of thing we could
figure out if distance was on our horizontal axis. But now that time in seconds is
there, here’s what that means. Now, we say this amount of time
after our wave started moving ahead, the fancy word for that is propagating. After the wave had been propagating
for this amount of time, we could call it Δ𝑡, then based on the wave’s position, we
can see what the wave height is at a given point in time rather than a given point
in space.

The fact that our wave changes in
time, that is that the wave height changes for different time values, shows us that
this wave is in motion. It’s not standing still. But as time passes, the wave
evolves. It moves along. Even though we’ve changed from
distance and meters to time and seconds on our horizontal axis, the concept we used
earlier of wave cycles still applies to this graph. We still move through one complete
cycle of the wave. If we start out, say, at our
original point, and then we move to here. Or likewise, if we start out here
and then follow the wave to here, that’s also one complete cycle.

So, we’re still seeing wave
cycles. But now, instead of seeing how far
apart in distance these cycles are, we’re instead seeing how much time it takes for
each wave cycle to complete itself. In other words, the difference
between this point here and this point here on our wave in the horizontal direction
is no longer a distance in space, but rather we could call it almost a distance in
time. It’s an amount of time, some time
interval. It’s the amount of time that needs
to pass in order for the wave to go through a complete cycle.

And when it comes to waves in
general, this is something we’re interested in. We want to know how many cycles a
given wave will go through per unit of time, say, per second. And that brings us to the second
important property of waves, which is wave frequency. The definition of wave frequency is
that it’s the number of cycles a wave completes in one second of time.

So, going back to our wave on these
axes, if this interval of time was exactly one second, we see we’re going through
one cycle of this wave in one second of time. That would mean that the frequency
of this wave is one cycle per second. We could write that this way. We could say that the frequency of
the wave, we typically represent it using a lowercase 𝑓, is equal to one wave cycle
divided by one second of time. That is, this wave goes through one
cycle every second.

This ratio of units, number of
cycles per number of seconds, can be written another way. A cycle per second is called a
hertz, named after the German physicist Heinrich hertz. And it’s abbreviated capital
Hz. So, we would say that the frequency
of this particular wave, because it goes through one complete wave cycle in one
second, is one hertz. Now, we’ve said that this wave,
because it changes in time, is a wave that’s in motion. And a question arises. Just how fast is this wave
moving? That is, what is the speed of this
wave?

Well, it turns out that we can
combine these two wave properties, wavelength and wave frequency, to find out. We can start thinking of it this
way. If we represent wave speed using an
𝑠, then we know the units of this speed will be, in SI base units, meters per
second. Now, considering these units, if we
think about meters, we know that those are the same units as our wavelength. The wavelength is a distance often
measured in meters. And then, if we think about the
units of one divided by seconds, we could also call it inverse seconds, that’s like
the units of wave frequency, which are number of cycles per second.

From looking at the units of wave
speed then, we can start to see how we might combine wavelength and wave frequency
to get wave speed. Here’s how we do it. If we take the frequency of a wave
and then multiply that by the wave’s length 𝜆, then that’s the same as multiplying
the number of wave cycles that pass a given point over some amount of time by the
distance between cycles of the wave. When we multiply these two values
together then, the number of cycles cancels out and we’re left with a distance, say
in units of meters, divided by a time in units of seconds. And like we saw, wave speed is
measured in those same units. And it does turn out that wave
speed is equal to frequency multiplied by wavelength.

And when we talk about wave speed
𝑠, we mean how fast is this disturbance traveling along the time axis. For example, if we let a bit more
time pass, then our wave will go through this oscillation. And that will represent the wave
covering one more wave length of distance over this amount of time. So indeed, our wave is in motion
and the speed of that motion is determined by the product of frequency and
wavelength. Let’s take a moment now to get some
practice with these ideas through an example exercise.

What is the frequency of the wave
shown in the diagram?

We see in this diagram this
displacement in meters measured against time in seconds. And we see that that displacement
follows a wave like pattern. It goes up then down, then back up
to its original point. And at this point, the cycle begins
again and the wave again moves up and then all the way down and then back up to its
original displacement. Based on this information, we want
to know what is the frequency of the wave.

To figure this out, we can recall
the definition of frequency, that it’s the number of cycles that wave completes in a
time of one second. Looking at our graph, there are a
couple of different ways we can solve for this wave’s frequency. One method involves figuring out
the amount of time it takes for the wave to go through one cycle, on our graph that
looks to be 0.5 seconds, and then calculating frequency based on that.

But another method is to simply
count the number of wave cycles that elapse in one second of time. And we see that that’s equal to two
complete wave cycles. Where here, at the far-right edge
of our horizontal axis, we have one second of time elapsed. So, this wave goes through two
complete cycles in one second of time. In other words, two complete
movements from the wave moving up and then down past its original starting point
then back up to that original displacement. That’s one wave of cycle.

Knowing that this wave finishes two
cycles every one second of time, we can now recall that the unit cycles per second
can be written another way. A cycle per second is equal to
what’s called a hertz, abbreviated Hz. So, the frequency of this wave,
we’ll call it 𝑓, is equal to two cycles per second, or two hertz. That’s the frequency of the wave
shown in the diagram.

Let’s now consider a second
question involving this same wave.

A wave is shown in the diagram. What is the wavelength of the wave
if its speed is 360 meters per second?

Okay, looking at this diagram, we
see a wave with the displacement of that wave plotted against time in seconds. We’re told that this wave has an
overall speed that is motion in the horizontal direction of 360 meters per
second. And based on this, plus the
information in our graph, we want to solve for the wave’s length.

Now, there’s a relationship that
ties together wavelength and wave speed, and it involves a third variable, wave
frequency. That relationship says that wave
speed 𝑠 is equal to wave frequency 𝑓 multiplied by wavelength 𝜆. In our case, though, it’s not wave
speed we want to solve for, but wavelength. So, let’s rearrange this equation
by dividing both sides by wave frequency.

When we do that, the frequency term
on the right-hand side cancels out and we see that the wavelength of the wave is
equal to the wave speed divided by the wave frequency. Since we’re given wave speed, we
want to solve for a wave frequency to ultimately figure out wavelength. Considering wave frequency, let’s
again take a look at our graph. If we start at the origin where the
wave’s displacement is zero, and zero time has elapsed. And then, we follow this wave along
through one complete cycle of the wave, we get to a time value of 0.5 seconds.

We can recall that wave frequency
is equal to the number of cycles a wave goes through as one second of time
elapses. So, as we’re seeing with this
particular wave, our wave frequency is equal to one cycle, since we go through one
cycle of the wave, in 0.5 seconds worth of time. And remember, we figured this out
by tracing out one complete cycle of our wave and seeing how much time it took to do
that.

Now, if we divide one by 0.5, then
that’s equal to two. So, we can write our wave frequency
as two cycles of the wave per second of time. And then, we can recall that one
hertz is defined as one cycle per second. Writing our wave frequency in terms
of this unit then, we would have two cycles per second, which is equal to two
hertz. This, then, is the value that we’ll
insert for 𝑓 in our equation to solve for wavelength 𝜆.

The wavelength of this wave then is
equal to the wave speed, 360 meters per second, divided by the wave frequency, two
hertz. And notice that if we replace the
units of hertz with units of inverse seconds, which is what a hertz is equivalent
to, then we have cancellation of the one-over-seconds unit in our numerator and
denominator. We’re left in the end with units
just of meters, units of distance, which is good since we’re calculating a
wavelength. And when we compute this fraction,
we find that it’s equal to 180 meters. That’s the wavelength of the
wave.

Let’s take a moment now to
summarize what we’ve learned in this lesson on calculating wave motion. Starting off, we saw that a wave is
defined as a disturbance that transfers energy from one point to another. We learned further that waves are
characterized by two primary properties, the wavelength of a wave and the frequency
of a wave. Wavelength is the distance a wave
needs to pass through one complete cycle. And wave frequency is the number of
cycles a wave completes in one second of time. And lastly, we saw that wave speed,
represented by the letter 𝑠, is equal to wave frequency 𝑓 multiplied by wavelength
𝜆.