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.