welcome to Unit 12 on traveling waves in this unit. We're going to look at the concept of a wave moving through space and time s. So, for example, we might see something like Triangle Wave moving to the right. And if this is, uh, the way that some time we'll say at T equals zero seconds, then some time later, we'll see it over here. This will call at T equals just tea sometime later. Okay? Eso here. We're looking at a displacement in the Y direction and this is along the X. So, looking at this, let's learn some of the vocabulary here. First of all, the front side of the of the wave is called the leading edge. And you might guess that the backside is then called the trailing edge. Okay, So leading edge and trailing edge Now, the distance from the very beginning of the waves of the very front of the leading edge, the very back of the trailing edge. This distance is called the wavelength, which we do know as Lambda Wavelength and the time that it takes for the entire wave to pass a particular point. So if the time it takes further the leading edge to hit it and then the trailing edge to pass. It is our period t. Okay, so we have a lambda and we have a t describing this traveling wave scenario. Now, if we were to imagine or look at the motion of an individual point on the X axis here, what would it look like as a function of time? We'd say, Well, the displacement we'll call Displacement of position X as a function of time, we know that it's going to start by shifting up, and then it's going to shift back down. So say we look at this point here at some time in it's going to start to go up and then it'll come back down. So there's a plot of our displacement as a function of time noticed that it kind of looks like the wave itself. If we had a more complicated wave, for example, say we had a wave that looks something like this again. This is why versus X here and we're looking at this exposition. Actually, Let's look at this one right here. Then we know AT T equals zero is going to start going down and then it's going to come up. Then it will come back down again. Notice how that it's kind of mirrored from what the y versus X looks, where it looks like we're looking at again just the displacement of a particular position X as a function of time. Okay, eso thinking If we could describe the displacement of each point along the X axis as a function of time, we would be able to describe mathematically all of these waves put together. Um, one more question needs to be answered, though, which is how fast is this wave moving forward? Now we have lambda, which is in meters, and we have period, which is in seconds. So it makes sense that we would say the wave speed V is equal to Lambda divided by T. Or if we wanted to use frequency, we could say Lambda Times F. So this is an important relationship. This is the wave speed. It tells us how fast the wave is moving forward, and it will be helpful later on here. As we develop this mathematical model that will describe the displacement of every position as a function of time in order to do this. We're going to consider Sinus oil waves. Okay, so these airwaves that look like a sine or cosine function Okay, so there is a continuous wave here, traveling through the medium. So again, looking at why as a function of X, But again, we could draw displacement of a particular point. And we know that it would look very similar as we just saw. So if we want to have a function that is the displacement of a point X at a time T what is it going to look like? Well, let's consider. First of all, AT T equals zero AT T equals zero. We see that the displacement for a particular position is gonna look a lot like a sign plot. So we're going to say d of X zero seconds. It is going to be equal to We've got some amplitude, just like we had before when we were considering ocelot or emotion and then looks like a sine function. So I'm gonna say sign and notice that because it's a sign function every time we go a distance Lambda forward. So this is our total wavelength Lambda here. Every time we go a distance slammed of forward. We want to repeat and be back to the same position. So in order to do that, we're going to say is two pi over Lambda Times X Now that would satisfy everything. But of course, we have to remember the phase constant that we recently learned about. So we're gonna throw in a phase Constant here is well, just in case we need it. Okay, so now we have it for at T equals zero seconds. What? What about sometime later? Well, sometime later, Remember, this is moving forward with the velocity V. So the distance it's going to move forward is going to be BT. So that's going to be the distance you travel forward. Remember, Because V is equal to X over t. That means X is equal to b times T. So if we want to shift a plot to the right direction toe, that means that we need to go take our variable X and turn it into X minus. VT Remember the minus VT when we apply it to the independent variable is how we move, how we shift a graph to the right. So let's go ahead and plug that in and see what we find. We have d of X T sequel to a sign of two pi over Lambda Times X minus VT Plus, if I Okay, well, we know that the okay is equal to Lambda over tea. And so if I were to put that in, I would end up with D of X T is equal to a sign of two pi over Lambda Times X minus. We have Lambda over tea times, Little tea time plus fi. Now notice here when I distribute this through the land as well cancel and I'll end up with two pi over landed times X minus two pi over tea time Big tee times little t remember two pi over Big T is something we've seen before. It's called Omega. It's the angular frequency and we'll also have this two pi over Lambda thing here. So this is angular frequency. We're gonna go ahead and define that two pi over Lambda is equal to a variable K, which we call the wave number. Okay, so wave number is equal to two pi over the wavelength. All right, so d of X t then becomes a rather simple looking formula we have a sign of K X minus Omega times, Time plus fi. And so this is for a wave moving to the right. Now, if we had wanted it to move to the left, remember, all we would have had to do is plug in X plus V T instead of X minus VT. So to move it to the left, we would say D of X is equal to a sign of K X plus omega t plus if I so that's moving left. But these new variables we can also rewrite our wave speed. Remember, we had the is equal to Lambda Times f eso lambda. It's going to be equal to two pi over K half is equal to omega over two pi, so we'd end up with V is equal to Omega. Okay, cool. Uh, so now we have a different way to express in terms of variables that will immediately show up inside our equation. Now, I should mention here Ah, lot of times people will rewrite. This is Lambda equals the over F and correspondingly, you can change this the reason being that the is going to come from the properties of the material as we'll see in the next section. And F is going to come from the properties of the source in physics one or two where we consider optics. This will be a particular importance for a couple of concepts. Um, but here we're just going to need to remember that V comes from property of the material. F comes from the properties of the source. So the source determines frequency material termine svi and together they determine wavelength.