(0:00) So today, I will start with a general discussion of wave as an introduction to electromagnetic wave which we will discuss next week. We will start with a very down to earth equation: y equals one third x. And I am going to plot that for you. Here is y; here is x; and that's a straight line through origin: y equals one third x. Suppose now I want this line to move. I want this line to move with a speed of 6 meters per second in the plus x direction. All I’ll have to do now is to replace x in that equation by x minus 6 t. Notice the minus sign. I will go then in the plus 6 direction. The equation then becomes y equals one third times x minus 6 t. So look at it at t equals 1. And at t equals 0, you already have the line. At t equals one, you now have y equals one third x minus 2. That means, here, it will intersect at minus 2, and here it will intersect at plus 6. And the line parallel to the first one. This line is now t equals one and this is t equals 0 and it has moved in this direction with a speed of 6 meters per second.
(1:52) So what is this telling us, that if we ever want something to move with speed v in the plus x direction, all we have to do in our equations is to replace x by x minus v t. And if we want to move in the minus x direction, then we replace x by x plus v t. That's all we have to do.
(2:21) Now I am going to change this to something that is a real wave. I now have y equals 2 times the sine of 3 x. That’s a wave. It is not moving. Not yet. I can make a plot, y as a function of x, and that plot will be like this. This is 0, because the sine is 0, and this is pi divide by three. This is 180 degrees, and this is again 0. This is two pi divide by 3, and this is again zero. And lambda, which we call the wavelength, lambda in this case, is from here to here, and it is 2 pi divide by three, and this is of course, also, from here to there. I want to introduce the symbol k that we often see. We call it the wave number. And k is simply defined as 2 pi divide by lambda. So in our specific case, k is three. This here, is k. you know this number, you can immediately tell what's the wavelength is.
(3:53) Now I want to have this wave move. I want to have a traveling wave. And I want to have it move in 6 miters per second in the plus x direction. So the recipe is not very simple. All I have to do is replace this x to x minus 6 t. So now I get y equals 2 times sine times 3 times x minus 6 t. and if you now look at this curve. this equation and you plot it a little bit later in time than t 0, this is already t0, a little later in time, you will see this, indeed has moved in the plus x direction. And it is moving with a speed of 6 meters per second.
(4:45) So this equation when you look at it. It holds all the characteristic of the oscillation, it holds the amplitude: this 2 is the amplitude, and minus 2 is the amplitude. This information, k, holds the information on the wavelength and this information tells you what the speed is. And the minus sign, which is important, tells you that it goes in the plus x direction, and not in the minus x direction. Can we make such a traveling wave? Yh, we can do that actually quiet easily. Suppose I have here, rotating wheel, rotates with its angular frequency omega, and let it have a radius r, and I gave it 2 units, so that I get the same amplitude that I have here. And I attach to this is a string. I put some tension on the string. So I create a wave as I rotate it. The string is attached here, and as it rotates, the wave is going to propagate into the string with a velocity, let's say, v. So I can generate a traveling wave. The periods of one oscillation, if you are here on the string, you are going up, and you are going down, and you are going up and you are going down. That's all what you are doing when the wave passes by. The period of one whole oscillation, is obviously two pi divide by this omega. The wavelength lambda, that you were creating, here, here, lambda. Well, if you notice the speed which it is traveling, you know that it has been traveling capital T seconds, one oscillation. That's the distance lambda. So this is V times T. This is also v divide by f if f is the frequency in Hertz. And so the frequency, f, is then also given by the speed divide by lambda.
(7:00) And so I could write down this equation now in a some what different form, y equals two times the sine, and now I bring the 3 inside so I get 3 x minus 18 t. This 18 is now that omega, this is omega t. in here, it's all the timing information, omega, period t, everything is in here. Here is all the spatial information, this is k, and here is the information about lambda. So if I know omega, and I know k, then I can also find the velocity, which is omega divide by k. So everything is in here. Omega divide by 3, give me back my 6 meters per second. So while you have the equation, I can ask you any question about the wave. And you should be able then, to answer: wavelength, frequency in hertz, in radius per second, speed, everything.
(8:14) You may ask me now, so why do you discuss this with us? oh well, we are coming up to electro-magnetic wave next week, and in electro-magnetic wave, you are going to see lambda, you are going to see omega, and you are going to see capital t, and you are going to see frequency and you are going to see k's. everything you see here, you are going to see next week, with the exception that y, the displacement y, will not be in centimeters or meters, but it will be in electric field, traveling electric field, volt per meter, or the traveling magnetic field, Tesla. Other than that, all these quantity will return in exactly the same way.
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Now I want to discuss with you, a standing wave first.Standing waves are going to be important.This is a traveling wave, and now comes something even more intuitive, which is a standing wave.Suppose I have a wave traveling in this direction, and I call that y1, and y0 is the amplitude, sine k x minus omega t.Notice now I have all the symbols that we are familiar with.We have the k here, and we have the omega here, and we have the amplitude here.And the minus sign tells me, going in the plus x direction.
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But we have another wave.The wave is exactly identical, in terms of amplitude, in terms of the wavelength, in terms of the frequency, identical, but it's traveling in different direction.So this is y2, which is y0 times the sine of k x plus omega t.This plus sign tells me that it is going in this direction.
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So if it is a string, the net result is the sum of the two.So we will have to add them up.So y is y1 plus y2.So I have to do some trigonometric manipulations, and this is what I leave, I leave you with that. That's high school stuff.After that you will find 2 y0, notice the amplitude has doubled, times the sine of k x times the cosine of omega t.That’s the sum of those 2.And this, this is very very different from a traveling wave.Nowhere will you see a kx minus w (note: stand for omega from now on) t anymore.Kx is here separated under the sine, and wt is separated under the cosine.All the timing information is now separated from the spatial information.
