https://youtu.be/mvhS06VP1oE
PHYS 1101: Lecture Twenty-Five, Part Four
Okay, that’s my question for you to lead into the second fundamental kind of motion that we need to think carefully about for this wave motion. And that is the motion of the crest of the wave. It’s related to the motion of this red spot that goes up and down, but it’s different, has some different characteristics.
Here’s a main concept bullet for you. Here’s the language we use to describe this. The motion of this disturbance, that’s the crest in this wave, that’s the displacement of this red spot from equilibrium, before the source is turned on, the red spot, like all other segments of the string, want to be horizontal here. This is their equilibrium position. But when this source disturbs them from that equilibrium position, that disturbance you would describe as being this displacement from equilibrium of that object, so it’s really this disturbance that is what’s propagating along. That’s the nature of this wave, and that’s the language that we use. It’s not the red spot that’s moving to the right, but rather, it’s the disturbance. The displacement of the string from its equilibrium position, that’s what’s moving to the right.
Okay. Here’s an important feature of a wave. A definition for you. What we call the wavelength, specifically will mean it’s the characteristic repeat length of the wave. If I have this periodic wave, because I have a source, that’s going up and down periodically, that’s going to give rise to this wave that has a periodic repeat pattern to it. The distance between the crests is the characteristic repeat length of this wave. I could have drawn this line between two crests. I could have drawn the line between two troughs. It’s the same length. Whatever characteristic, if I start at one spot of the disturbance and I go through until I get to the same location again, that distance is the wavelength. So we’re looking at a snapshot in time, we’ve taken a picture of this wave, and we’re measuring that characteristic length.
Okay. This variable, Lambda, let me spell it out for you; it’s another Greek symbol. It will have units then of length, of meters, in SI units. It’s the distance between crests. Okay. Now I want you to go back and watch the movie and do a careful comparison. If you pause the movie at one snapshot, focus on the location of one of these crests. At that same snapshot, picture where the next crest is. Now, turn the movie on and let it play. As it’s playing, how long does it take for this crest, this spot, to move forward one wavelength? How long does it take to move one wavelength?
Compare that time to the period of the source. The time it takes the plunger to go up and down once. That’s the variable, capital T, that we discussed above. Is the time it takes for the crest to move one wavelength less than, equal to, or greater than the source period? It will be, obviously, much less than, or, looks approximately equal to, or it will be obviously much greater than to distinguish between these three choices. Go back and compare that and see how it compares. See how it relates.
Once you’ve thought about that, you’re prepared to appreciate what we’re going to define as the velocity of our wave. The velocity is going to be the velocity of a crest. Okay. It’s the real meters per second that the motion of this crest undergoes. If it’s moving to the right, it’s a velocity vector to the right with a magnitude of, say, one meter per second. Every second that goes by, this crest advances by one meter.
To get you to take your time and really digest what this means, the velocity of the wave, I want you to imagine a wave that you see at a ball game. Likely every one of you has been to a ball game of some kind where a wave starts at one end of the coliseum or the stadium, a group of people stand up, put their hands in the air, and then people next to them stand up or put their hands in the air, and then the people next to them put their hands into the air, and as you watch this from the other side of the stadium, it looks really neat. You see this wave that sweeps around the perimeter of the stadium. Eventually, the wave gets to you and you participate. When it’s your turn, you stand up, put your hands in the air and come back down.
Picture that wave. Then I want you to just estimate, what’s a reasonable velocity, or, let’s just think about magnitude here. So the speed of the wave of people in a crowd in a stadium. Remember speed has to be a distance per second that this disturbance travels. In other words, if you picture watching a wave in a stadium, every second that goes by, does the wave move forward by a third of a meter? Remember, that’s about a foot. Does it move by roughly three yards every second? Or does the wave sweep by and travel 30 meters, 30 yards, about a third of a football field, every second? Which is the reasonable scale for the velocity of a wave, for a people wave in a stadium?
Okay. Here’s an important connection between these different features I’ve pointed out. It’s a connection, but in some way, it’s a nice way to cleanly think of how these different features separate and how they’re related. Let’s think about the source first. There’s always some source that generates this wave. What are the properties that are caused or due solely to the source? Only the source sets these properties. What are they? One of the first things is the period, the time it takes to repeat one cycle has to be caused by the source and the source only.
If the source goes up and down every second, so does every object in this wave. That’s the characteristic time for this wave. The amplitude is always something uniquely set by the source. When you go back and watch that movie, notice that if the amplitude went between ten and two centimeters and back and forth, so did every spot on that string, go up and down by that same amount. Those are the unique features that are caused by the source. The time characteristic of the wave and the amplitude has nothing to do with the properties of the medium so to speak.
The velocity of the wave. This is the complement to that. It has nothing to do with the source properties. It is just how the material responds to being disturbed. It’s completely set by the properties of the medium. In other words, if you think about this wave that sweeps through a crowd at a stadium, what’s setting the pace that this disturbance moves with, is really how fast people react to their neighbor once this wave approaches them.
My guess is, if you did a scientific study at the beginning of the ball game, when people are fresh, they just have gotten there, the wave probably travels, and it’s probably reasonably quick. People respond quickly to seeing their neighbor stand up and sit back down, so they get up quick and stand back down. I suspect, as the game goes on, and more and more beverages are consumed, people’s response starts to slow down. I bet the wave starts to propagate slower near the end of the ball game compared to the beginning. That response of how one neighbor reacts to another has to do with the medium that that wave is propagating in.
It’s the combination of these two that leads to, or has the consequence of, this wavelength, or this visual wavelength. Here’s the relationship. The wavelength that you end up with, here’s my wave, it’s traveling, let’s say to the right, with the speed v wave. I know that it takes the time of one period for one spot on the wave to go up and down. It also takes one period for the crest of the wave to move over one wavelength. That combination tells me, then, that this distance, the distance of one wavelength, the distance from crest to crest, if the velocity is such that I travel that distance in this time, then that wavelength, that distance, is defined as the velocity of the wave times the period.
You remember a velocity, it’s distance divided by time, and you multiply by time to then figure out the distance. Another way to write it is that the velocity of the wave is equal to any distance divided by the time interval over which that distance is traveled. If I make the distance one wavelength, the time it takes to travel that distance is the period. One cycle time. These two equations are equivalent; I’ve just algebraically rearranged them.
This is the one I prefer to remember only because it shows the cause and effect more clearly. The wavelength is the result of, it’s caused by, a value that’s characteristic of the medium times a property that has to do with the source. The wavelength is the velocity of the wave times the period. So let me try to draw a nice picture of it for you here again. Here’s this wavelength, is the velocity of the wave times the period. I’m going to grab a green highlighter, and I’m going to emphasize the variables or the features that are a property or a result of the medium.
The velocity of the wave depends on the nature of the interaction between all these segments in my medium. The segments of my string. The tension of the string, how strong this interaction is between the two, how massive they are, how quick each segment responds or accelerates given the force exerted on it. How much a person’s been drinking in this crowd as this wave in the crowd sweeps by. Property of the medium.
In blue, let me highlight features that are set by the source. The amplitude is an important feature. Remember when I reset this animation here, this plunger was sitting here and the string is nice and straight. This is the zero location. The plunger then oscillates up and down relative to that equilibrium, as does every segment of the string. The amplitude then, the same thing from the previous chapter that we worked with, capital A, has to be equal to half of the total distance between the peak and the valley. A is half the peak to valley.
The motion of each segment oscillates between a y coordinate of +A and -A. That is a feature has to do with the source. The time that it takes for a segment to go up and down, the period. The period of the plunger going up and down, that’s also a characteristic that’s set by the source. The period, you remember, I can relate in terms of an angular frequency or I can relate in terms of the frequency in Hertz, cycles per second. If this is a feature of the source, so then is Omega, and so then is the cycles per second, the Hertz. All of these are features of the source.
So notice in my equation, the wavelength that results is a combination of these two effects. Property of the medium, property of the source. Let me go ahead and highlight Lambda in yellow so all three of these terms are highlighted, or really the whole equation is to emphasize, because it’s a very important connection between these features.
Okay, the kind of wave we’ve been looking at was a wave on a string, and when you slowed down and really watched the motion of the wave crest, in comparison to the motion of a segment in the string, I hope that you appreciated that the segment in the string went up and down, that was transverse, or perpendicular to the direction that the wave moved in. The wave moved to the right, the crests were moving to the right, yet the segment in the string went oscillating up and down, up and down, transverse wave then. If the motion of a particle in the medium is perpendicular to the direction that the wave travels, it’s a transverse wave.
If you’ve ever played with a slinky, though, you realize that you can have other kinds of waves. A slinky is a great example of what’s called a longitudinal wave. If you were to take your slinky and paint one of the rings pink, what you would discover is that if you plucked a slinky here and you sent this compression pulse that then moved down the slinky, it’s going to move to the right, if you in slow motion watch the motion of this one ring in the slinky, you would see that it actually moved backwards and forwards as the wave went by. Its motion, it’s oscillatory, but it’s moving in general, parallel and antiparallel to the wave direction. Not perpendicular like a transverse wave.
Okay. We’re going to spend the rest of the lecture working on waves on a string. And in order to do that, I need to define carefully for you how you calculate these properties of the string that impact this wave. Another way to say that is what sets the velocity of the wave? What are the properties of this medium, of this string that set that? There are really two. For any wave on a string, the velocity of that wave is given by this mathematical equation highlighted in yellow. It’s the square root of two important features of the string.
In the numerator, I have to put the tension of the string in units of Newtons. 20 Newtons, 10 Newtons, what’s the tension in the string? If you were to pluck the string, how plucky is it? That’s another measure of the tension. The more tension you have, the more important or the stronger is this action-reaction force between neighboring segments and so the bigger the velocity of the wave is, the faster that’s going to propagate. Bigger tension, bigger wave speed.
But, inertia plays an important role, because ultimately it’s the acceleration of each segment that leads to an overall motion, and we know that acceleration depends in an important way, on the inertia. For a string, that has to be how massive that adjacent segment is. You capture that effect of the inertia by a quantity that we call the linear density. It’s the mass of that segment divided by the length of that segment. Mass over length.
If I take any length of this string and if I know the mass of that total length, the ratio of those numbers gives me the linear density and that’s a unique number, characteristic for that type of string. If you are to cut the length in half, you would have half the mass, so the ratio ends up giving you the same number. By using this linear density, it’s a number that’s just a unique property of that type of string. Just how massive that string is.
Okay. This is how you calculate the velocity of the wave. This is what sets it. Okay. Here are the remaining quiz questions for the lecture, and I’m asking you to just put some of these ideas together to just consider these answers. What you’re looking at here is a snapshot, so a freeze-frame of two waves. It’s a red wave or a wave on a red string and a different wave on a blue string. Both of those waves are traveling to the right with a speed of 12 meters per second, meaning the crest of each of them moves to the right with 12 meters per second.
Question 17: how do the wavelengths of the two waves compare? Question 18, which wave, if either, has the higher frequency? Let me emphasize what you can pick out from this graph, is the wavelength. That’s the distance between, or the characteristic repeat length of the wave, but the frequency, the characteristic time of this wave, is related through the velocity. Think about the connection between the velocity, Lambda, and the period, and therefore the frequency to answer question 18.
Question 19 now has you putting a few of these things together. Imagine you attach a particle to each of these waves. I’m going to attach it here for wave two and let’s attach it here, it doesn’t really matter where, for wave number one. As time goes on and these waves are traveling to the right, we have learned that a particle on each of these strings is going to be going straight up and down. Question 19 is asking us about that simple harmonic motion of the particle going up and down, not of the wave crest moving to the right. What’s the maximum speed of this particle as it goes up and down? And how do they compare between waves two and wave one? In essence, which particle would have the greater maximum speed?
Let’s go through that analysis for one of these. Let’s put a particle on wave two. Here’s wave two. I’m sticking a post-it on it. Here’s some information that I have. In addition to the velocity of the wave being 12 meters per second, I know that the wavelength, the repeat distance for the red wave, is 2 meters. The other thing I can read off from this graph, this is showing me the maximum displacement here, the positive value and the negative value, so I know the amplitude. It looks to me from my graph that my amplitude for this simple harmonic motion up and down is 0.5 meters.
If this particle is going up and down like a mass on a spring, then my maximum speed is at y=0, it’s at my equilibrium position, and further I know that it has a value of A times Omega. Well, what is A? Well, I worked that out. It’s a half a meter. What is Omega? Well, Omega is the angular frequency, it’s certainly related to the period of the motion. 2π over the period is also 2π times the frequency if I need that relationship. Well, this showing me a snapshot in time, so I don’t really have time information, although I do indirectly, because I’m told the velocity of the wave.
Remember, Lambda is equal to v wave times the period. I can then rearrange that to say that the period has to be Lambda divided by v wave. I have numbers for each of those, so I can calculate that my period is 2 meters divided by my 12 meters per second wave. That’s one-sixth of a second, so I can then go back and plug that in for Omega. It’s 2π divided by one sixth of a second. Omega is 37.7 radians per second. V max, I can now go in and calculate, which is A times Omega as being 0.5 meters times 37.7 radians per second and that ends up being about 18.8 meters per second. So there’s an example of calculating one specifically.