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[Assistant] And you find it'll work and you can do this in your head. I certainly don't want to repeat all of that.


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[Assistant] The reason it's going to work is that every time you take a derivative of x, you bring down an alpha, and still keep e to the alpha t. So, when you put that guess into this equation, you're going to get alpha squared plus gamma


[User] Please transcribe this text

[Assistant] Times a e to the alpha t equal to zero.


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[Assistant] And if this is going to vanish, it's not because of A and it's not because of e to the alpha t, it's because this guy in brackets is going to vanish.


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[Assistant] So if that's 0, you all recognize this to be a quadratic equation.


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[Assistant] These are the roots. Now, here you have to look at, so anyway, what does this mean?


[User] Please transcribe this text

[Assistant] You can find a solution to this equation that looks like a to the αt. A can be any number you like.


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[Assistant] No restriction is placed on a, but α cannot be any number you like. α has to be the solution of this quadratic equation, and there are two solutions. So, you can call them α plus and α minus.


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[Assistant] That's when I branched off into two different cases. Case one, you can see, look, when you do quadratic equations,


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[Assistant] The weight going to fall out is depending on whether the stuff inside the bracket is positive or negative.


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[Assistant] So let's take the easier case when it's positive. That means γ over 2 is bigger than ω0. γ, you remember, is the force of friction.


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[Assistant] Not only non-zero, but bigger than this amount controlled by the natural frequency. In that case, let's not go into details, but you can all see


[User] Please transcribe this text

[Assistant] Uh, you put whatever value you have for γ, the square root of this is some number, the whole thing is going to be two real choices you get. One will be minus γ over 2 plus whatever is inside the square root.


[User] Please transcribe this text

[Assistant] Other would be minus gamma over 2 minus what you have in the square root.


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[Assistant] The only thing I want you to notice is that both these roots are negative.


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[Assistant] This guy is obviously negative because it's a minus γ over 2 minus something else. That's definitely negative. You might be a little worried here that this is minus γ over 2 and this is plus something.


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[Assistant] But the plus something is smaller than the minus something because what's inside the square root is


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[Assistant] Square, if you do the square root of that, you will get exactly γ over 2. What you are taking away from it, this number.


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[Assistant] Therefore, this square root is bigger, is smaller than this number, so the overall thing will still come out negative.


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[Assistant] So why am I so eager that it should come out negative? Why am I eager that the root should be negative? Yes?


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[Assistant] Uh, no, even if I had positive, if I had two derivatives, it will work out.


[User] Please transcribe this text

[Assistant] Won't converge to what? This is not a cosine function.


[User] Please transcribe this text

[Assistant] Yeah, that's our main problem. When you write an exponential function,


[User] Please transcribe this text

[Assistant] Usually people write exponentials like this for population growth. Everyone says our problems are increasing exponentially. They always think of exponential as rising.


[User] Please transcribe this text

[Assistant] Of course, it's also possible that if α were a negative number,


[User] Please transcribe this text

[Assistant] Times the absolute value of α, just to show you it's a falling exponential. And we have a stomach only for falling exponentials. You've got to think. If I pull this mass and I let it go,


[User] Please transcribe this text

[Assistant] I expect x to go down from whatever value I had, not to keep growing indefinitely. That doesn't make any physical sense.


[User] Please transcribe this text

[Assistant] And one of the great things about the union of mathematics and physics is the equation


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[Assistant] Automatically it works out that way. It gives you an answer that makes sense.


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[Assistant] Okay, so what you get then is a solution that looks like this. One number a times e to the one of the roots, uh, plus another number b times e to the


[User] Please transcribe this text

[Assistant] I wrote them, they're both e to the αt, but I write it as minus the absolute value so you can all see their falling exponentials. You understand if a number is negative, like negative 5, that can be written as the minus 1 times the absolute value of the number.


[User] Please transcribe this text

[Assistant] So it is e to the αt, but I want to write it in this form, so it's falling.


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[Assistant] So this is a case where if you pull the mass and you let it go,


[User] Please transcribe this text

[Assistant] It will just relax, eventually coming to x equal to 0, which is its normal location, equilibrium location.


[User] Please transcribe this text

[Assistant] Now, how about the numbers a and b? Right now they are completely arbitrary.


[User] Please transcribe this text

[Assistant] In order to find A and B, we need to be given more information other than simply the mass and the spring constant. They're all sitting here and the friction coefficient, but that's not enough to tell you what it does on a given day.


[User] Please transcribe this text

[Assistant] It wants to know, well, yes it's the mass, it's that spring, but did you pull it or did you release it in this way or that way?


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[Assistant] That's what will control the answer. They are called the initial conditions. One initial condition is, what was the value of x at time t equal to 0? That's asking you how much you pulled it when you let it go.


[User] Please transcribe this text

[Assistant] The other initial condition is, what is the velocity, namely the derivative of x at time 0?


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[Assistant] Most of the time what you will find is you pull the mass and you release it, then it's got a non-zero x and a zero velocity.


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[Assistant] But it doesn't have to be that way because I could have pulled it and released it at t equal to 0. It's starting to move inwards, and that's when you jump in and you set your clocks to 0.


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[Assistant] Well, at that instant, x has a value and the velocity also has a non-zero value. They don't all have to be 0, but they are some two numbers you can pick at will.


[User] Please transcribe this text

[Assistant] You can see that if I put t equal to 0, I'm going to get


[User] Please transcribe this text

[Assistant] Times z to the 0, which is 1. I'm going to get a plus b is x0.


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[Assistant] That's one condition on A and B. Now take a derivative, namely find the velocity, then put t equal to 0.


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[Assistant] And set that equal to some pre-assigned number, you'll get a second equation for a and b, and we all know we can solve those two equations.


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[Assistant] That's how you fit these three parameters to the solution tailor-made for that particular situation.


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[Assistant] I can pick these numbers, x0 and v0, but why not keep going? Maybe the acceleration at time 0 is some number. Why is that not up to me to decide?


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[Assistant] Can anyone tell me why that's not negotiable? Yes?


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[Assistant] Can I pick the acceleration of a body at will? Yes?


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[Assistant] Yeah, I'm not sure I heard the exact answer, but it sounds close enough to what I had in mind, which is Newton will tell you what the acceleration of a body is.


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[Assistant] Namely, mass times acceleration is the force, and the force, once it's given in any given problem, which is given by where the body is,


[User] Please transcribe this text

[Assistant] And what the velocity is in this example, you have no choice on what the acceleration is. The initial position and velocity are not given by Newton's law.


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[Assistant] Newton's law says if the force is so-and-so, the acceleration is so-and-so, but he doesn't tell you what should be the velocity at a given time, what should be the position at a given time. That's why there are two free numbers. That's why you have the answer with two free numbers that you can pick to bring it in accord with these two numbers.


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[Assistant] Okay, this solution to the equation with x equal to 0, from now on I'm going to call xc, where c stands for the complementary function.


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[Assistant] Now, there's another thing that could have happened here,


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[Assistant] Another alternative. You can think about what I'm saying.


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[Assistant] The other alternative is when this γ over 2 is smaller than this ω0. Then, of course, you have the square root of a negative number. So, then you go back to the state where you're finding the root, and you say α equals minus γ over 2 plus or minus square root.


[User] Please transcribe this text

[Assistant] Of gamma over 2 square minus omega naught square, but that number is negative.


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[Assistant] That can be written as a negative sign times the square root of this number.


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[Assistant] And the square root of minus 1 will come out as an i. So either you will get two real roots.


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[Assistant] This is a mathematical result, nothing to do with physics. A quadratic equation with real coefficients either will have two real roots or it will have two roots which are complex conjugates of each other.


[User] Please transcribe this text

[Assistant] You learned the notion of complex conjugation, which is the change i to minus i. You can see the α plus and α minus have the same real part, minus γ over 2, and have opposite imaginary parts. So, in this example, α plus is the complex conjugate of α


[User] Please transcribe this text

[Assistant] And I've told you this relationship works both ways, that for α- is a complex conjugate of α+.


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[Assistant] I'm going to call this fellow in brackets as ω prime.


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[Assistant] Then there's a second solution. The real part is exactly the same. The imaginary part is opposite,


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[Assistant] Omega prime is a new frequency connected to the natural frequency and the frictional force. That's why we've got three Omegas. We've got to keep that in mind.


[User] Please transcribe this text

[Assistant] If I have a driving force, I reserve the symbol ω for the frequency of the driving force. Here is yet another quantity with units of frequency, which I'm calling ω prime. So, what does this guy look like? Well, first of all, it's full of complex numbers and we don't like it.


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[Assistant] We know the oscillator should have a real coordinate, so you ask, I have not picked a and b yet, how shall I pick them so x is real?


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[Assistant] Well, forget A and B for a minute and look at this number and this number. Do you guys realize that they're complex conjugates of each other? Because i has gone to minus i.


[User] Please transcribe this text

[Assistant] So, for example, if A and B were equal numbers, this would be a real number, because number plus its conjugate, you kill the imaginary part, because they have opposite imaginary parts. So, A and B real is a good answer, but it's too restricted.


[User] Please transcribe this text

[Assistant] Instead of a, looks like absolute value of a times e to the i phi.


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[Assistant] B should look like absolute value of A times e to the minus iΠ. In other words, B should be the complex conjugate of A.


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[Assistant] That means the same absolute value but opposite phase. I remind you, in the complex plane,


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[Assistant] If a number z is sitting there, the complex conjugate is sitting here, they have opposite angles.


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[Assistant] A and B both being fully real means this phi is zero.


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[Assistant] But you see, Φ doesn't really have to be 0. There is too much of a restriction.


[User] Please transcribe this text

[Assistant] Phi can be whatever you like, provided the phi that A has, the phase that A has, is the same as the phase that B has. That means the angle in the complex plane of the number A and the number B have to be opposite of each other.


[User] Please transcribe this text

[Assistant] You guys should know what it is. I've told you many times this is unforgivable if you don't know what this is. e to the iθ plus e to the minus iθ over 2 is cosθ. Since you don't have the over 2, you can write it as 2a e to the minus


[User] Please transcribe this text

[Assistant] Okay, so this is the cosine of whatever is in here, ω't


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[Assistant] In view of what's to come, instead of calling this phi, I'm going to call it phi 0.


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[Assistant] Uh, because another phi is going to come in and I want to separate those two.


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[Assistant] So 2 times a is a new number. I'm going to call that c. And this is the bottom line.


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[Assistant] So we've done all this yesterday. I've given you the notes, but I want you to understand.


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[Assistant] Where are we going with all of this? If you pull a mass and let it go,


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[Assistant] There are two things that can happen. I really wish I had time to bring you these different masses and show you.


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[Assistant] If you have no friction, everyone knows what's going to happen. You pull the mass and let it go, it oscillates forever.


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[Assistant] In real life, there is always some friction. So, let's turn on a little friction. What will happen is what's very familiar and expected.


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[Assistant] Go to 4, go to 3, go to 2, rock back and forth, and slowly it may wiggle around, but you won't be able to see it. That's called a damped oscillation. That's really what's described by this solution.


[User] Please transcribe this text

[Assistant] Here it looks like a cosine of something, but the number in front of it itself is falling with time.


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[Assistant] And the picture for that will look like the following. If we just draw an exponentially falling function as your


[User] Please transcribe this text

[Assistant] And you multiply that by an oscillating cosine, it will do this. This is a damped oscillation.


[User] Please transcribe this text

[Assistant] But another extreme case which you may not think about is, if you crank up the friction more till γ over 2 is bigger than ω 0, then what happens is, if you pull the mass and let it go, it doesn't even go to the other side of x equal to 0.


[User] Please transcribe this text

[Assistant] It just slowly gets to zero in an infinite amount of time. You understand? If you pull it from here, you let it go, no friction is this for eternity.


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[Assistant] Small amount of friction will slowly damp the oscillation.


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[Assistant] Too much of friction, you come back slowly and you come to rest.


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[Assistant] These are the different cases, and the mathematics tells you the different cases. So that's what we like about the union of mathematics and physics. Once you wrote down the basic law of Newton, you have no choice as to where it takes you. You go where it takes you. These are the different


[User] Please transcribe this text

[Assistant] Okay, so this is what I did earlier on in the day, but now let's go back.


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[Assistant] So let's go back to the more general problem I had, which is a right-hand side which is not 0. So let's write the right-hand side. It's not 0.


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[Assistant] This is called a driven oscillator. Driven oscillators are present all the time. For example, if you have a swing, swing goes back and forth.


[User] Please transcribe this text

[Assistant] You may choose to push the swing back and forth. If you've got a kid in the park, you're applying a force, which is an oscillating force.


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[Assistant] Whereas in the absence of external forces, all oscillations will eventually die down. Do you guys understand that?


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[Assistant] Why? Why do all oscillations die down in the absence of external forces?


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[Assistant] Well, in this case it's not air resistance but friction. But what friction does is that in the presence of friction you cannot prove a law of conservation of energy. We saw that. Whenever friction acts, the energy, total mechanical energy goes down.


[User] Please transcribe this text

[Assistant] So, kinetic plus potential will decrease with time, so the oscillations get damp.


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[Assistant] But if you are pushing it from outside, you can provide a constant source of energy, so you can maintain a steady oscillation, and that's the problem we're looking at. That is this problem.


[User] Please transcribe this text

[Assistant] This problem, well I said, I don't want to solve this equation, I'm going to manufacture another problem. z double dot plus γ times z dot plus ω 0 square z is f


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[Assistant] So, let's be very clear. This is the real physics problem that occurs all the time.


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[Assistant] This is clearly not of our world, because in our world you don't have a force, e to the iωt. All forces in the real world are real.


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[Assistant] So this is a complex force. If you break down the e to the iωt,


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[Assistant] You all know it's cosωt, which I like, plus i sinωt, which I don't like.


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[Assistant] So I'm solving a problem in which on top of the force I'm really applying, there's an imaginary force put on top of it.


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[Assistant] So, what is the strategy? Why are we taking the problem and making it worse than it is? Because we have a strategy. The strategy is the following.


[User] Please transcribe this text

[Assistant] If you have a right-hand side cosωt, ask yourself what you're trying to do. You're looking for some function x of t which will obey this equation. You can try various functions. Maybe x is a cosine.


[User] Please transcribe this text

[Assistant] Sorry, no derivative will be a cosine, one derivative will be a sine. So, you cannot make it into a cosine.


[User] Please transcribe this text

[Assistant] If you took a sine, the middle guy will look like a cosine. You can try to balance that against this. The end two terms look like cosines. So neither a sine nor a cosine works. Now it turns out something in between will work. But something that always works is the following. If the right-hand side, instead of being cosωt, was e to the iωt, you can guess the answer immediately.


[User] Please transcribe this text

[Assistant] Namely, z itself will behave like e to the iωt, because then if you take two derivatives,


[User] Please transcribe this text

[Assistant] That's going to look like e to the iωt times some numbers. That's going to look like e to the iωt. That's going to look like e to the iωt. So, the beauty of the exponential is that when you take derivatives, you keep getting the exponential.


[User] Please transcribe this text

[Assistant] Okay, but now the problem is, this is not the problem. No one told you to solve this problem. But our strategy is the following.


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[Assistant] The right-hand side has got a real force and imaginary force. The real force will produce the x that I want. The imaginary force, purely imaginary, i times nωt, will produce a purely imaginary answer.


[User] Please transcribe this text

[Assistant] So at the end of the day, I will find the answer and keep only the real part. That's the plan. That's going to be the strategy.


[User] Please transcribe this text

[Assistant] This z is going to be the x that I want, and we'll have a part hanging around y that I don't want, but y is being driven by the sine ωt, and x is being driven by the cos ωt. But now let's solve it by inspection.


[User] Please transcribe this text

[Assistant] Now it's very easy to take derivatives of exponential. Just pull down iω every time you see a derivative.


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[Assistant] So z double dot will become iω squared, which is minus ω squared.


[User] Please transcribe this text

[Assistant] Z dot will become I omega, there's a gamma there.


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[Assistant] And ω 0 squared is just multiplying by a number, so this is what I get on the left-hand side, and I want it to equal this on the right-hand side.


[User] Please transcribe this text

[Assistant] And you're almost there, because the time dependence of the two sides are identical, so you can cancel it.


[User] Please transcribe this text

[Assistant] And what you find is yes, I can satisfy the equation that's given to me, provide the number z0, the number in front of it, obeys this condition.


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[Assistant] That's trivially solved, because we all know how to get the solution to that one. z0 is obtained by dividing both sides by everything that multiplies z0, which happens to be ω0 squared minus ω squared plus iω


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[Assistant] So the beautiful property of the exponential function is that you take what is a differential equation and turn it into an algebraic equation.


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[Assistant] Because the act of taking derivatives generally is a nasty operation. Sines become cosines, x cubed becomes 3x squared, but exponential remains exponential. That's the property you're trying to exploit here. So now we've got z


[User] Please transcribe this text

[Assistant] The function, the complex function looking for is


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[Assistant] E to the i omega t divided by this quantity downstairs.


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[Assistant] This quantity downstairs I'm going to call I of ω. I of ω is a complex number.


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[Assistant] It's got a real part and an imaginary part. The real part is ω 0 square minus ω square. The imaginary part is ω γ.


[User] Please transcribe this text

[Assistant] It's not a fixed complex number. At every frequency of the driving force ω, it's a different complex number. But if you're sitting at one frequency of the driving force, it's a particular number. Everyone should know how to work it out.


[User] Please transcribe this text

[Assistant] Omega 0 is root of k over m. Omega is given to you. That's a real number. Gamma is given to you. That's an imaginary part. If you actually put in numbers, it might look like 6


[User] Please transcribe this text

[Assistant] The thing I call impedance, or I, is just a complex number. But it varies from frequency to frequency. You can see that.


[User] Please transcribe this text

[Assistant] Okay, now we got z of t. If the problem given to us was z, we're done. We don't have to do any more work. Here is the answer.


[User] Please transcribe this text

[Assistant] Well, eat the iωt is cosωt plus i sine ωt, so drop the i sine ωt. But that would not be right. Why is that not right?


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[Assistant] Anyone know why that's not the way to get the real part yet? Yep. Yes?


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[Assistant] Yes, this number downstairs is not a real number. It's a complex number.


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[Assistant] So if you have a complex number on the top divided by a complex number on the bottom, the real part is not simply the real part of the top. You've got to write the whole thing so you can clearly separate the real and imaginary parts. And that's what I'm going to do next.


[User] Please transcribe this text

[Assistant] Every complex number we know can be written as an absolute value times e to the


[User] Please transcribe this text

[Assistant] Right? Every z looks like mod z e to the iθ, because then it's mod z cosθ, which is x, plus i times mod z sinθ, which is iy.


[User] Please transcribe this text

[Assistant] So, every complex number can be written that way, and what will be the magnitude of this guy? Well, by the Pythagoras theorem, it'll be the square of the real part plus the square of the imaginary part under root. Every complex number that you have,


[User] Please transcribe this text

[Assistant] We'll have an absolute value which is that squared with that squared under root, and it'll have a phase Φ obeying the condition tan Φ is the imaginary part divided by the real part.


[User] Please transcribe this text

[Assistant] I want you to realize that in a given problem, with a given mass, given friction, given frequency, both these are known.


[User] Please transcribe this text

[Assistant] There's simply a magnitude of I, and the phase Φ can be found by this process. Now, write it like that.


[User] Please transcribe this text

[Assistant] So, I got e to the iΦ divided by absolute value of i, e to the iΦ, and I got e to the iωt on the top. I'm almost there.


[User] Please transcribe this text

[Assistant] I'm almost there, because e to the i Φ in the bottom can be written as e to the minus i Φ on the top.


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[Assistant] So that this whole thing becomes F 0 over m divided by the absolute value of i e to the iωt minus phi.


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[Assistant] Now it's easy to take the real part because this is cosωt minus Φ plus i sinωt minus Φ. These are all real numbers.


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[Assistant] So, they'll just multiply the whole thing, but the real to imaginary separation now comes from just taking the cosine part of this exponential.


[User] Please transcribe this text

[Assistant] And that is our final answer. That's the answer we've been looking for, which is x of t is equal to f 0 over m divided by the absolute value of i, cosine ωt minus phi.


[User] Please transcribe this text

[Assistant] We went there because the answer is not a cosine and it's not a sine.


[User] Please transcribe this text

[Assistant] So cosωt-φ, if you know your trigonometry.


[User] Please transcribe this text

[Assistant] So it's got some cosωt, it's got some sinωt, and that's why it's hard to guess that answer. But if you go through complex numbers, it comes out very easily.


[User] Please transcribe this text

[Assistant] So the next thing is to analyze the properties of the answer you got.


[User] Please transcribe this text

[Assistant] Notice that even though it's oscillating like a cosine, it's not in step with the applied force.


[User] Please transcribe this text

[Assistant] The applied force is cosωt. Its maximum occurs at 0, 2π, 4π, etc. This guy's maximum occurs somewhat later. The first maximum occurs when ωt is equal to 5. So, it's said to be lagging the function. You can imagine two functions oscillating. One is behind the other one. It reaches the maximum slightly later. So, the first property is, in a driven oscillator, the x will not be in step with the


[User] Please transcribe this text

[Assistant] We'll be lagging, or it could be leading, depends on what phi is.


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[Assistant] Then what about the amplitude of oscillation? That's the thing that's very interesting. How big is the oscillation?


[User] Please transcribe this text

[Assistant] It looks like f naught over m divided by the absolute value of i. Let me write the absolute value of i for you.


[User] Please transcribe this text

[Assistant] So the amplitude of oscillation depends on the frequency.


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[Assistant] You've got to draw the graph. As a function of frequency, what does the 1 over this look like? Pardon me?


[User] Please transcribe this text

[Assistant] Yes, but a-b² and b-a², that's why I've not been paying attention.


[User] Please transcribe this text

[Assistant] Okay. Yeah, so I'm pretty careless on what I put in, but this is one occasion where I sort of know it's okay, because I'm going to square this. I don't have to remember who is ω


[User] Please transcribe this text

[Assistant] I have to be very careful where I don't square things, like here. But if I'm squaring it, it doesn't matter. Okay, so go back here and plot something divided by this denominator and ask, what is it at various frequencies? When ω is 0, that goes away, that goes away.


[User] Please transcribe this text

[Assistant] And omega 0 squared is k over m, and I claim what you will find initially is this will become


[User] Please transcribe this text

[Assistant] F naught over k. That'll be the value of this whole amplitude at 0 frequency.


[User] Please transcribe this text

[Assistant] I hope that makes a lot of sense to you. Zero frequency means the force is not even oscillating. It's a constant force. What x are you going to get?


[User] Please transcribe this text

[Assistant] You're going to get the force divided by the force constant. That's the x to which you will pull the spring.


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[Assistant] If you put all the ω's and masses back into proper place, you will find it's f0 over k.


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[Assistant] Then what happens is, as omega increases, this denominator, one of these guys in denominator is starting to vanish.


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[Assistant] If γ were a very, very tiny number, when ω0 equals ω, you will be dividing by a very small number and the answer will have a huge rise.


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[Assistant] Then you can see when ω goes to infinity, very large ω, you're dividing by ω squared, it'll fall off like this.


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[Assistant] Now, one of the homework problems is to find out exactly where it peaks. If γ is very small, you don't have to do much work to know it peaks at ω, equal to ω0, but otherwise it's near that. That's a phenomenon that's called resonance. Resonance is the following. You are applying a force,


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[Assistant] You go on varying the frequency and response to your force, the system oscillates. The amplitude of oscillation is the largest when the applied frequency is very near the natural frequency.


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[Assistant] If we take the limit of a very small gamma, this maximum is very close to omega 0. That's called resonance.


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[Assistant] If gamma's really tiny, you can get graphs where it's incredibly peak.


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[Assistant] So that's when systems start vibrating a lot. That's why when your car goes over certain bumps,


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[Assistant] They're hitting your car at a certain frequency, and if that frequency is anywhere near the natural frequency of vibration of your suspension system in the car, the car begins to resonate.


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[Assistant] So there's this famous bridge in Tacoma that was getting hit by these winds. They were coming at some frequency and by bad luck they coincided with the natural frequency of the bridge. So there are these famous movies made for high school in which the bridge oscillates more and more and more than


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[Assistant] That's another example of resonance. People always say, yeah, I can resonate with what the person is saying.


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[Assistant] The person is saying whatever the hell he wants. Some people think it's interesting, some people think it's boring. It's the same thing here. The force is acting at a certain frequency.


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[Assistant] If your natural frequency is close to that, your response is high. If it's low, you don't care.


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[Assistant] Same thing, that's mechanical systems selecting a frequency. In fact, this is how radios work. If you have a radio, if there's only one radio station,


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[Assistant] You build an electrical circuits in which the currents like to go back and forth or oscillate at a certain frequency, radio stations should send signals at that frequency, your response will be very high.


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[Assistant] What they do is, each one sends a signal at a different frequency.


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[Assistant] You take your radio and there are dials inside.


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[Assistant] Well, I forgot. Do you know what a dial is?


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[Assistant] You are nowadays pushing some other button. There was a time when you had to turn a dial. When you turn a dial, you're changing the properties of the electrical circuit and changing its natural frequency.


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[Assistant] You move the natural frequency until it lines up with the frequency of the radio station, with one particular station.


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[Assistant] Then the response from that station is very high, and that's the sound you hear. Other stations are also present, but you are not near their maximum. They are sitting somewhere here and somewhere here.


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[Assistant] So you want the station frequency to come under this resonance peak.


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[Assistant] So if you've got a bad radio, it'll pick up sound from many stations because it's not very refined. If the peak is broad, you will get sound from many stations. You want to build a system which is very finely sensitive to the frequency.


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[Assistant] And that requires γ to be very small. If γ is very, very small, you can see if γ is going to 0,


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[Assistant] One over very tiny is very big, and you can get enormous spikes in response, and that's what people need.


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[Assistant] Okay, the last thing, you need to do this problem set. By the way, I know this problem set is very nasty. Towards the end, it got tough, you got email.


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[Assistant] I agree with you that this whole course is somewhat more difficult than it has always been.


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[Assistant] Okay, I'm aware of that fact, and I will bear this fact in mind when you go into your final, because for some reason, I wanted to use problems not in our book, but technical reasons, so I got them from here and there or made them up. They generally tend to be more challenging and more time consuming.


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[Assistant] So if you thought this is a tough course, well it's supposed to be a tough course, but maybe not this hard or not this time consuming.


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[Assistant] So I will remember that in the end at the, when I make the final exam or look at your grade. I will have to compare you not simply to your predecessors, but I'll have to give you some, cut you some slack and then compare you to previous generations.


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[Assistant] This use of complex numbers is something from which I will not, however, back down.


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[Assistant] It turns out that yes, you can do all of this without using complex numbers. Why? Because in the real world, everything has got real numbers. So you can, in fact,


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[Assistant] Okay, so I'm going to give you a problem where you cannot do it in a meaningful way.


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[Assistant] The power of complex numbers is they turn differential equations into algebraic equations. All you have to do is find the roots of a quadratic equation.


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[Assistant] Whereas what he started out was to find the solution of a differential equation.


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[Assistant] That advantage you will lose if you stop using complex numbers. And when you do quantum mechanics in second term, complex numbers are in the equations of motion. The analog of F equals ma, there's an i sitting there. So, you cannot escape the i.


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[Assistant] It's there. So I'm saying, depending on how much exposure you have, you have to do more or less work, but you just got to get used to complex numbers. You got to get used to the absolute value, the phase angle, how to manipulate them.


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[Assistant] If you already know it, good. If you don't know it, you have to do more work than other people. I'm not going to let off on it, yes.


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[Assistant] No. Anything you need to know with differential equations is what I've done for you in the class today. That's it. That's the only part you know. And with complex numbers, you turned it into


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[Assistant] So, you have to go over these steps and work them out.


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[Assistant] That's the part I cannot do for you. That's something you have to do.


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[Assistant] Alright, what is finally missing is that x of t, if I wrote for you what I wrote above,


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[Assistant] Which is x0. I don't want to write the x0 again. It's f0 over mi. Maybe I'll write the whole thing. F0 over m divided by absolute value of i times cosine


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[Assistant] It turns out this is not the full answer to the question, what happens when I push a mass? Let's understand why.


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[Assistant] Suppose I go to this mass and spring system. I grab it at t equal to 0 and start shaking it. Well, at t equal to 0,


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[Assistant] x was 0, because I grabbed it at t equal to 0. Well, at t equal to 0, if I go to this solution, it's not giving me 0. It's saying it's f


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[Assistant] So what happened to the freedom I had to assign the velocity and position arbitrarily? Well, it turns out that I can still do


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[Assistant] That's because this solution has got a part that's actually missing.


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[Assistant] And the part that's missing is this complementary solution, Ae to the minus γt over 2, cosine of ω't minus φ0.


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[Assistant] You can always add the solution to this solution. Why is that?


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[Assistant] Because if x obeyed a certain equation, if I put that in, I'm saying adding this part will not change the fact that it obeys the equation.


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[Assistant] Let's ask ourselves why. Go back to the top line there.


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[Assistant] F, x double dot plus gamma x dot plus omega naught squared x.


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[Assistant] Will be equal to F 0 over m cosωt, if you put this in.


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[Assistant] Now, what if you keep adding this to the solution? Will you mess it up? Imagine adding this to the left-hand side, but I claim on the left-hand side there'll be no extra terms because this guy double dot plus gamma times this guy dot plus ω r squared times this guy is 0.


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[Assistant] This is the function which you say is annihilated by the operations on the left hand side. It's like saying the following. If you want a function whose derivative


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[Assistant] Why do you add the constant? Why doesn't it screw up everything? Because the derivative of a constant is 0.


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[Assistant] And yet, the combination of derivatives appearing in the equation act on this to give you 0.


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[Assistant] That's why you can add it without screwing up what you've done so far. So the way you do these problems is, you first find this function that depends on the applied force, and you add to this solution, you get for free, even in the absence of an applied force.


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[Assistant] This is called a complementary function, and this is called a particular solution. This particular solution depends on the force, and this is the function. I don't know why it's called complementary. Maybe it complements the answer, or it's a freebie and you don't pay for it with any force, but it's called complementary. But you've got to add the two to get the final answer.


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[Assistant] So, you will still have these numbers a and Φ0 hanging around, and how do you select them? You select them if on top of it somebody tells you that this driven oscillator had a certain coordinate and certain velocity at t equal to 0. So, you'll evaluate the whole expression at


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[Assistant] By putting t equals 0 everywhere, you'll get one equation involving a and phi 0.


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[Assistant] And you'll take the derivative for the velocity, set that equal to some given number. You've got a second equation involving a and point 0. You'll juggle the two equations and solve for the two numbers, a and point 0.


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[Assistant] So that's what I tried to do yesterday, but this is not the first time it's happened to me that I tried to go fast. In the end I lost time because I was not happy with the way it was explained. And I usually come back and say sorry and then do it all over again.


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[Assistant] But there's only so much I can do. The rest is up to you guys. You have to do the work and you have to do more problems.


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[Assistant] Okay, so now we are moving off to another topic. Another topic. Somebody was sighing with relief. It's not very different.


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[Assistant] Okay, this is, these are waves. But I think it's going to be, look, I think this probably is the hardest point of what we're doing in this term, at least mathematically. Everything else we do, heat, hot water, cold water, that kind of stuff you've done in high school, that's all in your future. It's going to be a lot easier.


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[Assistant] Okay, waves. So, everyone has a good feeling for waves. You go to a lake, you know, if you're a hitman for the mafia, you're dropping things in with some weight. They go down. What's the first thing they do?


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[Assistant] Once a body is falling into the water, it's carving out some body of water and diving in, so the water next to it dives in to fill that hole.


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[Assistant] But then the water even next to it drives in to fill the hole left behind by these guys.


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[Assistant] For you, it looks like a bunch of ripples traveling outwards from the center.


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[Assistant] So, this is the classic example of a wave. If you've got one example, you can latch onto it every time I say wave.


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[Assistant] So what's the wave in this problem is that if I draw a plot of the height of the water versus the place where I drop the rock, maybe I should do a top view here.


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[Assistant] This is the top view after a while you got these ripples emanating from this point.


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[Assistant] Keep your eye next to the water, you will find these ups and downs going outwards.


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[Assistant] So the wave is some displacement of a medium. That's what people say.


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[Assistant] That's not a perfect definition because we know electromagnetic waves have no medium. They travel in vacuum. But I think for this purpose, in this course, you should imagine a wave as what happens when you excite a medium.


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[Assistant] So here's what, here's a way to think about the wave. You all understand the harmonic oscillator pretty well, right? Mass and spring system in equilibrium will sit there forever.


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[Assistant] If you give it a kick, it'll start vibrating around its equilibrium position. There you have to keep track of only one variable, x, which is the location of the mass.


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[Assistant] Of a medium, that means at every point in space there is something that's ready to oscillate.


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[Assistant] So, you start a system with one degree of freedom, a system with infinite number of degrees of freedom, because the height of the water at each point is an independent variable. It can jump up and down with time. And that's what is oscillating.


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[Assistant] So, if you don't do anything to the water, it'll stay at that height. But if you fiddle with it, you drop something, you hit it with something, it'll start vibrating. But accept this vibration not of one variable, but of a variable in space and time.


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[Assistant] x was a function of time, and the name we use for the wave, if something is oscillating, such as the water.


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[Assistant] So if I want to measure from some point x, I'm going to use the symbol psi of x to denote the height of the water from its normal position at the point x at time t.


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[Assistant] So x is not a dynamical variable here. x labels the point. Ψ is the dynamical variable. Ψ is the thing jumping up and down.


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[Assistant] Okay, now there are two kinds of waves. There's the longitudinal wave and the transverse wave. And we define them as follows. The wave travels out with some velocity. I'm going to use the symbol b for the velocity of the wave. That's something for which all of you have good intuitive understanding, so there's no need to mess it up right now.


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[Assistant] You see when the waves get to the shore, you take the distance divided by time, that's the velocity.


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[Assistant] In sound, you know, you light up a firecracker, after a while you hear it, you can see how long it took and how far you are, and you can find the velocity. That's the velocity of sound in air.


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[Assistant] Now, these velocities are all variable. Unlike the velocity of light that doesn't seem to depend on anything, all other velocities.


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[Assistant] Depending on various conditions, the velocity of sound depends on the temperature, for example.


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[Assistant] Or you can take the following thing. You can take an iron rod.


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[Assistant] Which you know is made up of a lot of atoms, and you take a hammer and you hit it here.


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[Assistant] You send, you basically compress these atoms and they compress the atoms next to them, and the shock waves travel through this rod. That's also called sound. And that travels at a speed which is much faster than the speed of sound in air.


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[Assistant] And there is some kind of a mountain here, and you're over here, and the train is coming here.


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[Assistant] It turns out, if the train lets out a sound, the sound will travel around the bend and you will hear it.


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[Assistant] But actually the pounding of the train on the tracks will set up shock waves in the track that you can hear by putting your ear to the track.


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[Assistant] You hear it by depending on the sound coming through the air, because the waves going through the solid travel much faster.


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[Assistant] So most of the things, please don't do this at home. Uh, this should be done by a trained person. But I certainly did it as a kid. Listen for trains before they come around the bend and you can actually hear it track.


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[Assistant] Or in water, if there are explosions in water, you can hear them faster if you're under the water than if you're outside the water, because the waves travel faster in water than in air.


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[Assistant] Or thunder and lightning is the most famous example. Something happens in the sky, the light gets to you first and the sound gets to you. They're traveling at different velocities.


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[Assistant] Okay, so now let's talk about longitudinal and transverse waves. When I talk to you, I think you sort of know what happens. The air in this room is at some


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[Assistant] And the pressure is constant, and the pressure is pushing your rear drum from the outside.


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[Assistant] Going through your, uh, eustachian canal, pushing the drum from the inside, and the two pressures balance, and you don't feel anything.


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[Assistant] If I talk, uh, the vocal cords move back and forth, and they increase and decrease the pressure as I talk. If I push out, I increase the pressure. If I go backwards, I decompress the air and reduce the pressure. Those shock waves propagate your ear.


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[Assistant] But the eardrum is getting the pressure different from the outside, but the inside pressure is fixed.


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[Assistant] So what happens is when I decompress, the eardrum comes out and it goes in. It goes back and forth and the eardrum moves.


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[Assistant] Then behind the eardrum are the little bones, and then the bones go, and there is some tube, and there's, no, the doctors take over from this point. So, some loose hair is shaking in this fluid, then it picks up some nervous things and goes to your


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[Assistant] Then you figure out, you know, you at least know what the person's saying. You're not sure why he's talking in this particular fashion. But you can at least understand and translate.


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[Assistant] That's a long chain of transducers of energy from sound to the stuff in your ear into your brain.


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[Assistant] I draw the line of the eardrum. I don't want to go on the other side of the eardrum because it's a very shaky term. But the part, this part I know very well. It's a sound wave.


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[Assistant] And it's a longitudinal wave for the following reason.


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[Assistant] It's also going from me to you. So the motion is in line with the velocity.


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[Assistant] The motion of the medium is in the same direction as the motion of the wave.


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[Assistant] A transverse wave is what you find in water if you are on a string. Take a string, clamp it with that, and you give it a little flick like that.


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[Assistant] This bump after a while will end up here. So, the rope is moving transverse, but the velocity of the signal is to the right.


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[Assistant] So do not confuse the motion of the medium with the motion of the signal.


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[Assistant] Okay, the rope at any given point is either at rest or going up and down, but the signal is going from left to right.


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[Assistant] So the medium doesn't have to actually move in the direction of the wave.


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[Assistant] And each one tells the person next to the next person some secret, and the secret is eventually transmitted all the way to


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[Assistant] What has traveled is a secret. The actual person in the first place has not gone over physically to the end.


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[Assistant] So what happens is, you're sitting there, not doing anything, suddenly you hear something, you pass it on to the next person, then you've gone back to doing nothing, then the signal travels. That's what will happen in the rope here. This section of the rope is not doing anything.


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[Assistant] When this pulse hits it, it'll jump up and down for a while, then it'll quiet down and somebody to the right will start moving. Okay, so there are the examples of longitudinal and transverse waves.


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[Assistant] Now, I want to consider one concrete example of a wave, so you people have a good feeling of how to study waves in physics.


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[Assistant] So there are many, many waves, water waves, electromagnetic waves, elastic waves. I'm going to do one. I think to do one thing in some depth will give you a feeling for how we analyze waves. That's going to be waves on a string. So you have to imagine that I have a


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[Assistant] That's being clamped at two ends. And somebody plucks the string.


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[Assistant] So into that shape and lets it go, we know the string is going to vibrate back and forth.


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[Assistant] Let's measure the coordinate x, which is the point on the string.


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[Assistant] And psi is the displacement of the string at that point.


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[Assistant] You guys got that? This is really the x-axis. This is not my string. That is my string.


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[Assistant] Each point is labeled by an x value that it would have the string relaxed to the normal position. We are asking, what is the displacement of the string at the point x at time t?


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[Assistant] So psi is our new dynamical variable, namely the variable for which we like to write the equations governing its motion. And we want to ask, how does psi vary with time? What will it do? If I start it out, what will it do?


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[Assistant] Do you understand the parallel to the mass and spring system? The mass and spring system, you got a mass and you got a spring. You pull the mass out to some new location, you let it go, and you want to know what it does. The answer was cosωt.


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[Assistant] Now at every point x, I have some string.


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[Assistant] And I displace all those degrees of freedom and I let them go, and they're going to do something, and I want to know how they're all varying.


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[Assistant] So previously we had only one coordinate, which I call x, which varied with time. Now at every point x, I have a coordinate, ψ, that varies with time.


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[Assistant] And our job is to find the equation satisfied by ψ.


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[Assistant] So let us take a string and the string has to be under some tension. That means you hang some weights on the end or you tighten it with some screws. There's a tension so that the string is willing to, dying to break apart if you cut it. So t is the tension on the string. Without the tension none of this will work.


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[Assistant] The fate of a siney, tiny segment of length dx.


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[Assistant] An essential parameter here is called μ, which is the mass per unit length.


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[Assistant] Put the string on a weighing machine, you find the mass and you divide by the length.


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[Assistant] The textbook may use f for tension because t is the time period also, so it depends on what symbol they use in the book. I'm not completely sure. I'm going to call it t, because it's been called t in the past.


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[Assistant] I pull this thing and I want to know what the whole string will do.


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[Assistant] Who is going to tell me now, what authority will decide the behavior of this string? What do we have to appeal to?


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[Assistant] Very good. So he said, find the force that will return it to the original position. And what was the it that you mentioned? Pardon me?


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[Assistant] Right, but uhm, we are going to apply. Yes, any other answers before I go? Yes?


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[Assistant] Yeah, it will look like that. That is correct. But you, all I want you to understand is there are no new laws.


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[Assistant] I'm not going to say, well, we studied masses in spring. Today is time to study strings, and I'm going to tell you a new law of motion. There's only one law of motion.


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[Assistant] That's F equals ma. The whole purpose of my exercise now is to show you that really does control. That's why it's a super law.


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[Assistant] Everything mechanical follows from that law, and I want to show you F equals ma is all I have, and I've got to use it properly. That's going to tell me what the string will do.


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[Assistant] The string is of course a long and extended and complicated object. I isolate a portion of it.


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[Assistant] The total force on it, and I'm going to equate it to mass times acceleration.


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[Assistant] This force on the little segment is the tension pulling this way, the same tension pulling that way. The tension on the string doesn't change from point to point.


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[Assistant] But the angle at which it is pulling is not necessarily the same. Here there is some angle, θ, plus Δθ, and here there is some angle, θ.


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[Assistant] This force is at an angle, say, θ below the horizontal. This is pulling above θ plus Δθ.


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[Assistant] The tangent to the string at this end and that end are not quite the same, and that's the key. Because if they were exactly the same,


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[Assistant] The same tension on the two ends, if you take any object like this and apply exactly the same force.


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[Assistant] They will cancel, because they're opposite forces, and F is 0, ma will be 0, and this thing won't move.


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[Assistant] The reason it moves is because even though the magnitude of the force is the same, the angle at which it's acting is not the same.


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[Assistant] So I'm going to find out the vertical component of the force here and take the difference from there.


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[Assistant] And the force on the other side will be t times sine theta. Theta is the angle made by the string.


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[Assistant] From the horizontal, and that angle itself is changing over this tiny length.


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[Assistant] Next approximation, that's the force. That's going to be mass times acceleration.


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[Assistant] The mass of this little segment is the mass per unit length times the length of this segment.


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[Assistant] Now, what is the acceleration in the language of calculus? What is the acceleration here?


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[Assistant] Ah, he said d2x over dt squared, but the d by dt squared is right. It's not d2 of x.


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[Assistant] So, the acceleration is the second derivative, and I write partial derivative because ψ can vary with x or it can vary with time, and I want to take the second time derivative.


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[Assistant] Now come to the left-hand side and you assume the angles involved are very small.


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[Assistant] The angles involved are very small. I remind you guys of the following identity,


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[Assistant] It begins as θ plus dot, dot, dot. Do you remember the series I wrote for you?


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[Assistant] So, tanθ is 1 minus θ² over 2, and we have decided that we are going to create θ so small that θ² we're not interested in.


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[Assistant] Okay, so the angle of approximation, which is sine θ over cosine θ, is now θ over 1, which is θ.


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[Assistant] So, sinθ is sinθ approximately equal to θ, cosθ approximately equal to 1, tanθ maybe approximately replaced by θ.


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[Assistant] So, first nice break. We don't have to work with sine. We just want t of θ plus Δθ minus that, which is Δθ.


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[Assistant] Now, I got a dx here, so what I'm going to do is divide by a dx and multiply it by a dx.


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[Assistant] You know from the calculus, if you move horizontally by dx and you move vertically by dψ, dψ dx is the tan of the angle. That's one of the first things you learned in calculus.


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[Assistant] Now, some of you may find the derivation hard.


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[Assistant] Forget about the details. You're not responsible for that. But some of you may say,


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[Assistant] I don't ever want you to tell me something which you cannot prove, unless it's an axiom or a law of nature. Well, this is not a new law of nature, so I have to tell you where I got this.


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[Assistant] I got this by applying f equals ma to a tiny portion of the string.


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[Assistant] And you can ask yourself, when you pull a string up why does it come down? It comes down because there's a tension on the string.


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[Assistant] And the tension here, and the tension here have vertical components that don't quite match. You see the string is flattening out like this.


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[Assistant] The vertical downward component of this is bigger than the vertical upward component of this force. If you resolve it in the vertical horizontal part, the downward part is bigger than the upward part in this picture. Therefore, there's a net force bringing you down.


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[Assistant] So, what happens is because the derivative here and here are not the same, because the derivative is the slope. So, the whole thing depends on the rate of change of the rate of change, and that's why you get this. But the mass of the tiny segment also depends on dx. So, dx cancels in the end. You drop it, and you get this equation.


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[Assistant] So what I want you to think of is dθ dx is really d by dx


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[Assistant] This is very important. What you have to know is that ψ, the displacement of the string at the point labeled x horizontally from the left end, at a time t, varies with time according to this equation.


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[Assistant] This is a very famous equation. It turns out you take any elastic medium and you disturb it a little bit from equilibrium, always seems to obey equations of this type.


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[Assistant] So, our job from now till sometime is to analyze the equation and see what it means. What are the consequences of this wave equation?


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[Assistant] And even though I called it B, it has units of velocity, I have no right to argue that it really is the velocity of the wave. We'll have to find out.


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[Assistant] It got dimensions of velocity, you can check by putting the units in.


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[Assistant] Let me write down a solution to this equation.


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[Assistant] So this is like saying, I got some equation, second derivative of x.


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[Assistant] But now I got second derivative with respect to space, second derivative with respect to time, and I know that this guy is going to oscillate up and down.


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[Assistant] If you imagine a string, if you take a string or a water wave which is traveling back and forth, the ripples are traveling, if you took a snapshot of the wave at a given time, it'll be going up and down in space.


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[Assistant] Or if you stood at one point in the water and have the wave go past you, the water will go up and down in time at a given space.


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[Assistant] Understand, if the wave's coming towards the shore, you stand in one place, you don't have to move. The wave will go past you, and where you are, the water will go up and down with time.


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[Assistant] Or, if you took a picture of it at one instant, the wave will go up and down in space at a given time.


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[Assistant] So waves oscillate in space and time, and you have to guess the answer.


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[Assistant] Now, if I had more time, I can tell you guys how to dead use the answer. It's not very hard, but I just don't have that kind of time, so I'm going to just write down a solution.


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[Assistant] In fact, there are many, many solutions to the equation. It's very general. So, I'm going to write a particular solution.


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[Assistant] Looks like this. There's some number at A, which is the amplitude.


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[Assistant] And it looks like cosine some number called k times x minus ωt.


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[Assistant] Unfortunately, this k is not a force constant. It is some new symbol.


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[Assistant] And omega is another symbol whose meaning we will deduce shortly.


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[Assistant] At least you can see that I just cannot write a wave like x minus t or something, because cosine should have an argument with just no dimensions. So if I have an x, I should multiply it by some number that's got units of inverse x.


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[Assistant] And if I have a time, I should multiply by something that looks like a frequency. From dimensional analysis, I look for a solution of this form to these equations.


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[Assistant] Once again, whether you solve, this by the way is called a partial differential equation, because it involves partial derivatives.


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[Assistant] All I'm going to be doing, I'm going to be content with showing you that this solution, when I stick it into this equation, actually works.


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[Assistant] Let us first verify that it works. Then we'll come, uh, next Monday and analyze in detail what does this really do. We'll come to that in a moment. Let's verify that it works.


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[Assistant] So, let's take dψdx. What happens when you take dψdx? When you take a partial derivative in x, it says forget about time, treat x as the only variable, then you'll get


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[Assistant] Kx-ωt times the derivative of what's inside the argument, which happens to be k.


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[Assistant] Now, if you differentiate the second time, d2ψdx², you do the whole thing again, and then you'll get


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[Assistant] Now if you take d 2 psi dt square.


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[Assistant] If you take 1 over v squared times that, you will get this.


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[Assistant] What does the wave equation say? It says the second derivative in x should be 1 over v squared times the second derivative in time. And for my solution, that's the second derivative in x.


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[Assistant] That's a second derivative time. I want them to match.


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[Assistant] Well, they're almost there. a matches a, cos matches the cos, but ω and k are not independent numbers. ω over b should be equal to k.


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[Assistant] This wave equation for waves on a string.


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[Assistant] We'll have solutions that look like this where a can be any number you like.


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[Assistant] K and ω, between the two of them, there is one degree of freedom. You can pick K at will and pick ω so that ω equal to Kv, or pick ω freely and choose K so that K is ω over v. But there is a relation between the two of them, which you can write in any number of ways.


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[Assistant] So, let us put that knowledge back into the solution. Let us say ω is in fact kV, so that we write it explicitly in terms of the k which is completely free. So, ψ now looks like A cos


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[Assistant] Minus, now here's where you guys got to do a little mental algebra here. If I pull the k out, can you see that t will be multiplied by ω over k? But ω over k is v.


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[Assistant] And that is a very pretty result, and I just, I will conclude by telling you what that is telling. It says the ψ of x,t, which depends on x and depends on


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[Assistant] Depends on x and t through this combination x minus bt. In other words, if x minus bt has a certain value, psi has a certain value.


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[Assistant] If I change x, and if I change t, keeping x minus bd constant, the function doesn't change, right? Because it's function only of this combination.


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[Assistant] I will argue that that really means it's the wave that's traveling at a velocity v.


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[Assistant] Let's take a function e to the minus x squared. You must have seen this function in statistics. It looks like this. It likes to have x equal to 0.


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[Assistant] Take the function where x is replaced by x


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[Assistant] It has the maximum. When this guy on top vanishes, that is, say, when x equal to vt.


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[Assistant] So, if you took the function e to the x minus vt squared, the minus sign, at t equal to 0 it'll be here, at other time it'll be here, but at every instant the bump will be moving to the right, you can see with the velocity v, because the peak is at a location vt.


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[Assistant] So in a wave, the whole profile moves at the velocity, and therefore this symbol we use here is in fact the velocity of the wave.


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[Assistant] It just looks like cosine kx, and we all know what cosine kx looks like. It's got a maximum at x equal to 0.


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[Assistant] If you wait a little longer, where is this maximum? A little later. If you increase x a little bit, if you increase time a little bit,


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[Assistant] You've got to increase x a little bit so that these two still cancel each other.


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[Assistant] Originally x was at 0 and t was at 0, you're at a maximum.


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[Assistant] After some time delta t, if you go to the point delta x so that these two numbers cancel, you're still riding the maximum.


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[Assistant] But that means the Δx over Δt is v. That means if you move to the


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[Assistant] After a time, Δt, you will ride the crest. That means the crest is moving at the velocity v.


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[Assistant] So, I think to summarize what I've shown you so far is that when you take a string and you have a tension on it, you pull it and let it go, it vibrates according to this equation, that's the first statement.


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[Assistant] This is a solution. I'm not telling you it's the most general one or the only one or anything. I want us to at least have one concrete solution.


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[Assistant] And I've not told you yet how to visualize the function, but I have told you whatever the function is.


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[Assistant] It is drifting to the right at a speed that we call b.


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[Assistant] Because any function of x minus vt has a property that if it did something at x equal to 0, it does the same thing at x equal to vt. That means the whole shape is bodily transported to the right. So, this really signifies a wave traveling to the right. If you wanted a wave traveling to the left, you should put a plus sign here.


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[Assistant] Six o'clock in the morning? Ooh, it's far too early for me, I'm afraid.


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[Assistant] Okay, so let me get some details. What's your name?


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[Assistant] Okay, so let me get some details. What's your name?


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[Assistant] That's Keiko Yuchini. Okay. And you're female. And your nationality?


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[Assistant] Right, 21C Willow College. And how long are you planning on staying with Homestay?


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[Assistant] Okay, do you have any preference for a family with children or without children?


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[Assistant] Alright, now what about you? Are you a vegetarian or do you have any special food requirements?


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[Assistant] Now the location. Are you familiar with the public transport system?


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[Assistant] Now, let me go check on the computer and see who I've got.


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[Assistant] Listen, leave it with me. I'll check my records and I'll give you details this afternoon.


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[Assistant] Dreams of finding lost treasure almost came true recently. A new machine called the Revealer has been invented and it has been used to detect gold which has been buried in the ground. The machine was used in a cave near the seashore where, it is said, pirates used to hide gold.


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[Assistant] The pirates would often bury gold in the cave and then fail to collect it. Armed with the new machine, a search party went into the cave hoping to find buried treasure.


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[Assistant] The party dug a hole two feet deep.


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[Assistant] They finally found a small gold coin, which was almost worthless.


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[Assistant] The party then searched the whole cave thoroughly.


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[Assistant] Yo, this is Radical Redemption, Asia Tour. September starts now, yesterday Wuhan. Right now we're in creamfields in, uh, Taiwan, Taipei. And next week, uh, two China gigs again. Looks fucking awesome. Let's go.


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[Assistant] So right now we're on our way to China. We're doing two shows, uhm, one on Friday, one on Saturday. Right now it's Thursday. And then we'll be back on Sunday, so it's a quick in and out to China. Let's see what kind of madness it will bring this time.


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[Assistant] So yesterday we did a show at Mishland and right now we're in the car, uh, three hour drive to Ningbo, right? To S86 Club.


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[Assistant] Insane stuff in their gloves. Like the LED and the CO2 blower stuff and sound system. Pumping.


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[Assistant] Like crazy, yeaah. Also the booth monitors.


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[Assistant] Enter the room and take your seats and please leave all the refreshments and snacks outside. You are not allowed to bring drinks or snacks in this room.


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[Assistant] I'd like to ask all the participants to enter the room and take


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[Assistant] Professor Batry is a columnar who will be introducing the speakers.


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[Assistant] And it looks at public and private things as well as architecture at large. She is a scholar of


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[Assistant] Thank you very much. Now I'd like to introduce Professor Kim Jaehee from Sungkyunkwan University who will make his presentation.


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[Assistant] As, uh, Professor Yuki said, I would also like to refer to Gilbert de Sibondon, a philosopher of France, as well as, uh, Bernard Stiegler, another philosopher that continued on with the same thought.


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[Assistant] I am using many philosophical terms and I want to save time.


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[Assistant] I will be reading mostly from my paper.


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[Assistant] Since you have downloaded my paper, then you can look at it while I read. So I will now proceed.


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[Assistant] You are not ready? Have you not downloaded the material yet?


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[Assistant] And machinize humans at the same time while opening up a discussion of the post-human.


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[Assistant] They fear that superior non-human beings such as strong artificial intelligence and artificial super-intelligence will someday come to dominate humans.


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[Assistant] This fear of post-humanization is especially pronounced when it comes to the question of alienation that surrounds the topic of labor.


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[Assistant] The transition from a human society to a post-human society transforms not only the relationship between humans and machines, but also the concept of labor itself. Should we not then be looking at the possibility of post-labor condition in which humans and machines do not confront and struggle against each other,


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[Assistant] Identifies the locus of a lenient opposition between labor and capital around the ownership of means of production. Simon Donne focuses on the physio, what psychological discontinuity between the human labor and the technical object being worked with.


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[Assistant] And we're left to either organize or assist the operations of machines. They oil machines, dispose waste, replace parts. In other words, they become the servants taking care of machines or managers adjusting the relationship between machines.


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[Assistant] Transitioning from a social paradigm of labor to one of technical activity requires a fundamental change in the way we understand life, as being oriented by substance to being oriented by relations. In this sense,


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[Assistant] Believe that human society could evolve from biological groupings based on division of labor to a trans individual collective based on creative and technical activities. Capital technology and critical literacy.


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[Assistant] However, Bernard Stigler criticizes Simon Don's view as being overly optimistic and overlooking the relationship between technology and capital. Unlike Simon Don,


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[Assistant] Very much like Deleuzian individuals in the control society.


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[Assistant] I am no longer a unique individual with one body, but a mere data bank which provides and screens in order to operate the computer network. I search data, but am also a part of it. I am nothing but the owner and consumer of data.


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[Assistant] I as individual can no longer make a we a group of unique individual eyes, but rather becomes an unspecified one, which has lost their singularities. One is the loss and liquidation of individuality.


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[Assistant] The one composed of de-individualized individuals deletes to the becoming anthropod of human society. That is, humans have become something between an ant covered with prosthetics and a spider eating itself in the midst of a network. A society has become merely a multi-agent system akin to that of insects.


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[Assistant] The pre-individual nature becomes technological industrial with humanity symbolic mental and motor functions now externalized.


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[Assistant] From fragmented workers to the stressed top executives, motivation declines, instrumentalization is accelerated and labor is depreciated.


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[Assistant] Symbolic misery is the same for a worker at a remote job site or a company executive. The proletariat of the hyper-industrial age have lost their knowledge and executive ability to control machines and in turn have become the tools of them. The labor is deprived not only of a savoir faire but also savoir vivre.


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[Assistant] The appropriate form of post-labor probably lies between Simondon's planning, planning, and Stegler's prescription.


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[Assistant] In other words, Simondon's argument that the paradigm of less should be shifted from labor to technical activities is still valid for envisioning post-labor in a post-human society. But the critical interventions of Stiegler are a necessary condition for Simondonian post-labor to be realized.


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[Assistant] First of all, we need to break out of the labor capital frame of mind, which only regards technical activities as means to improve productivity and limits the relationship with technical objects as that of ownership.


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[Assistant] There is a need to reestablish, uh, there is a need to establish socio-cultural and political economic conditions that allow anyone to freely choose, make, invent and repair their own technical objects and to relate to the world in a different manner. If the transition from labor to post-labor is to contribute to the enhancement of


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[Assistant] Leisure as an opposition to labor is not sufficient in our thinking of post-labor. The technical activity of invention is not captured.


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[Assistant] We are moving from a human oriented society based on labor to a post human society based on technical activities.


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[Assistant] The post human society is preconditioned on mutually collaborative coexistence of humans and technical objects. Humans and machines should no longer compete over labor, but find a way to coexist in the realm of technical activities. The dominate, dominated relationship between humans as actors of labor and machines and tools for labor


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[Assistant] If post humanity were to be characterized as superhumanity, it should not be a reinforcement of human capacity to dominate nonhumans, but trans individuality based on intellectual and emotional communication and cooperation with nonhumans.


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[Assistant] Post-labor should be an embodiment of such post-humanity.


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[Assistant] Yes, thank you very much.


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[Assistant] Uh, I'd like to ask a more specific question in the same vein. Among the things that were said, you said that


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[Assistant] According to the law of principle and market, the technology continues to be developed. And then how can you...


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[Assistant] How can you not comply with the market rule or design outside the market? That's a very difficult question. You're asking how we can get away from the capitalist system itself. I don't think I'm really


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[Assistant] able to answer that question. What I talked about today is the possible forms of post-labor, and I define that as technical activity.


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[Assistant] Technical activity has free creative and inventive attributes because we can assemble, make and experiment.


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[Assistant] In the past, technical activities were focused on labour in the industrial society. If we can break away from the paradigm, I think we can.


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[Assistant] Uh, arrive at the true nature of technical activity at the same time.


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[Assistant] Labor and leisure was thought as a dual thing. It was either or. But then there are things that are not work.


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[Assistant] There could be an activity that can be active and that produces something, but still not be labor nor leisure. It's not about being bound to productivity activity, but it's not, it's also not about doing nothing.


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[Assistant] And I think that is where we can find the answer. We can refer to this as technical activity or something else, but this concept is a path that we must find in the post-labor era.


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[Assistant] Then, as a condition for post-labor, we are


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[Assistant] Mentioning technology, but there must be political and economic system that allows this to happen.


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[Assistant] There were social prejudices, in other words, there are hierarchical orders, uh, for example, there was an order between manual workers and intelligent workers, white-collars, and


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[Assistant] It's not the idealist one that I'm talking about, it's a, if you look at


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[Assistant] The technical workers of engineers, uh, working nowadays, they are still bound by the laws of capitalism and, uh, many of the technical activities that we're talking about, it is only possible with, uh, funding. So it's difficult to, uh,


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[Assistant] I said this, but as a philosopher, I believe that we need to set a certain philosophical principle and the desirable way forward.


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[Assistant] If I can continue on, uh, what you first said.


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[Assistant] It's something that I fully covered during the presentation and I find it to be the same. So you, uh, reiterated, uh,


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[Assistant] What I said, you will summarize what I said. I wasn't trying to say that Stiegler was completely pessimistic. He thought that


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[Assistant] It could be poison and medicine.


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[Assistant] So there was a duality of technology, but how we approach technology, how we perceive technology will determine the direction. Simon Don overlooked the


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[Assistant] Also agree that he wasn't pessimistic, but he was more neutral.


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[Assistant] It was, uh, technology has become more poisonous, but how can you make technology, uh, function more like medicine? That was what Stiegler focused on. Simon Donne.


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[Assistant] I was a thought leader during the turn of the century when we were just moving to


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[Assistant] Uh, the information society and I thought, uh, the thoughts, uh, of Simon Don were worth revisiting.


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[Assistant] Technical activity is, like you said, much better than labor.


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[Assistant] Make sure that it's technology that reorganizes relationships.


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[Assistant] At a time, people would use their body to do things and if we can reduce the scope of these kind of activities, we can place more focus on technology and that will allow


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[Assistant] The relationship between human and nature to change rather than dominating or be dominating, we can focus on


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[Assistant] Making this sort of inventive or adjusting activities and making that the essential nature of people. So.


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[Assistant] When we can reduce the scope of labor, artistic and technical activities, as well as activities that can either be defined as labor or leisure,


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[Assistant] It can also be negotiated and that is what I wanted to focus.


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[Assistant] Furthermore, uh, can I pose a question?


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[Assistant] I'd like to ask this question to Mr. Yamakawa. Since I'm a philosopher, I would like to


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[Assistant] Clarify the concept. My basic position is that AI or robots


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[Assistant] Have already emerged, and we need to coexist with these new emerging things. And that's where the posthuman society is headed towards.


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[Assistant] But I'm curious about one thing. I think you already mentioned this, but when you create AI, why do you emulate natural intelligence? Artificial intelligence is, was invented to


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[Assistant] Be contrasted with natural intelligence, but robots are soft programs that represent artificial intelligence.


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[Assistant] And I don't think it's even possible because natural intelligence and artificial intelligence are essentially different from what I know. Research on natural intelligence is just starting. We really don't even know how the human brain works. But,


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[Assistant] Brain. Automation and autonomy I think were intermixed. Automation is a concept that's different from autonomy.


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[Assistant] To place, but is complete automation possible? And can we?


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[Assistant] Create their own rules in how it operates and behaves. Do you think that will be possible?


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[Assistant] Or do you think that automation becomes more perfected?


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[Assistant] And the function improves, uh, is that what you're talking about when you're referring to autonomy?


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[Assistant] In fact, artificial intelligence actually emerged in the 70s, not now. At that time, it was based on a rationalistic model, not about emulating the human brain. But that didn't work, so it went away for a while. Recently, it has revived.


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[Assistant] Because artificial intelligence was combined with brain science, so they have adopted an empirical experience of bottom-up material.


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[Assistant] AI is actually following what the philosophers have been doing. At first it was a rationalistic model and then it became an empirical model and nowadays it becomes a phenomenical model.


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[Assistant] Calculations, uh, uh, the calculation that people find hard to do is very easy for machines, but what comes naturally to human beings like walking is very difficult for machines. Another thing is, uh,


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[Assistant] The design principle, it seems to be a little bit different. During the industrial age, design was about, uhm, adapting


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[Assistant] People to machines, but now in the information society it goes the other way around. So a mnemonic technology is not only applied to that area, but in other areas as well. The other issue is the issue of autonomy. I think you're a little bit confused here. The autonomy in machines and


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[Assistant] The self-consciousness in strong AI are two different things. So that's why I asked about AGI.


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[Assistant] This is what AGI is about, uh, deck card interesting set.


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[Assistant] Many interesting things. Uh, first of all, he said that animals were machines and the human body is also a machine, but humans are not machines because they not only have bodies, but also have spirits.


[User] Please transcribe this text

[Assistant] And that is something that machines cannot do. He decided to example machines cannot use language or signs. Secondly, machines


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[Assistant] Is, uh, completely different. What makes, what really defines humans would probably be sex. Uh, is that possible?


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[Assistant] Sometimes when you are so precise.


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[Assistant] It becomes impractical. What's more important is that people die, humans.


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[Assistant] are not based on silicon, it's based on carbon and


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[Assistant] But they also have emotions. If you have no emotions but just reason, then we call that kind of a person a psychopath.


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[Assistant] What you're saying is that until now people had to program machines, but will there be a time when machines are able to do this by themselves?


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[Assistant] Until now, machines just emulated the human body. Proclaim machines or, uh, uhm,


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[Assistant] Equipments were stronger than the human body, but now are we able to simulate the human brain, not just the body? The amount of data interchanged between humans is surpassed by the amount of information interchanged between machines.


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[Assistant] A self-driving car because of a problem because of autonomy. But what's interesting here is that this project, uh, three or four hundred years ago was created to serve, uh, humans.


[User] Please transcribe this text

[Assistant] Even if machines become autonomous, they...


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[Assistant] I don't believe they will ever have self-consciousness.


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[Assistant] As they now, on desperate moments in Paris-Roubaix, 33 kilometres out from the finish, 54 seconds the gap, and the man driving at the front, the world champion, Peter Sagan.


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[Assistant] It is a difficult decision to make, but right now they just want to see that group brought back because they're actually some, uh, four fifty seconds behind them.


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[Assistant] A minute 10, 70 seconds. It's a painful gain, but he's gaining second by second, kilometer by kilometer. 24 left, 15 miles to race. It's about 35 minutes at the speed he's going.


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[Assistant] 1100 metres, a minute 12, the gap now.


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[Assistant] Yes he was. He hasn't had the time today, he's always had somebody with him.


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[Assistant] He milled Taliyah, no big surprises, except to swap them back around once again. Vladimir banned coming in, that's what we saw earlier. They didn't ban Vladimir in game three, and Xiaohu picked it up, and he destroyed everyone.


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[Assistant] But again, Kingzone are very clearly saying we just don't want to fall behind early and especially not in the bot lane. So now they have a playmaker. I'm looking at things like Galio mid lane or the Sion again. Things where you can roam from B to D side down and punish everyone.


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[Assistant] And this change is important again because Kingston are playing more for the early game. They need a jungler that can really match what Skarner is doing and Peanut is now on the Ola.


[User] Please transcribe this text

[Assistant] Fiora Illaoi type champion, cause I'm not expecting him to go AP top.