| 00:00 | Okay, so we were talking about previously off camera off recording how to |
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| 00:08 | real data which uh coming out from instruments, which do not record displacement |
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| 00:16 | though it shows displacement right here. is the equation, we call it |
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| 00:20 | wave equation and it records displacement, that's not what our instruments record. |
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| 00:27 | record our instruments record some combination uh and velocity and acceleration and strain and |
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| 00:35 | so that you know whenever you look data on your workstation screen, you |
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| 00:41 | wiggles. But there is never any on never, there's never a scale |
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| 00:47 | your on your workstation. All that is the relative size of those various |
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| 00:57 | . Because we really don't know. So let's see how we can apply |
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| 01:05 | recognition of of uh physical reality to fact that this theoretical discussion was in |
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| 01:13 | of displacing. Take this equation and differentiate both sides by with time. |
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| 01:24 | sides of time. Okay, so we have three derivatives of of uh |
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| 01:30 | with respect to time. Take one those three derivatives and and uh take |
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| 01:36 | of those three derivatives and apply it to the displacement. And you get |
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| 01:41 | velocity right? So after you've done you have the secondary to respect the |
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| 01:47 | of the velocity. I'm sorry, it into presentation mode. Oh and |
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| 02:00 | I should actually uh sit some screen . So okay, so let me |
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| 02:06 | that again, we're going to differentiate sides of this is with respect to |
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| 02:13 | to time. As a matter of , I'm gonna do it right now |
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| 02:18 | front of you but I have to it this way back out of |
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| 02:21 | So I'm gonna take here uh turn like so yeah. And then I'm |
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| 02:38 | to um um take another derivative Excuse me, I'm gonna I'm gonna |
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| 03:02 | okay, so I'm gonna take another on the left side and on the |
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| 03:14 | side. Okay? And then I'm take this thing that I've just applied |
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| 03:29 | I'm gonna move it inside here right . And I'm gonna do the same |
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| 03:37 | over here. I'm gonna move it boots. 2nd. What I want |
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| 03:40 | do is move this over here. the same thing over here. |
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| 04:03 | now I'm gonna take this derivative one I just apply. I don't know |
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| 04:10 | it's so difficult and I'm gonna move inside here and then I'm gonna recognize |
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| 04:30 | story. Video is simply uh uh gonna call it uppercase V. So |
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| 04:41 | is the the particle velocity, the of the displacement with respect to |
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| 04:48 | Okay? And I've got the same over here, separate different going up |
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| 05:05 | M. Component. Okay, now see this is an equation that looks |
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| 05:11 | like the original equation but now it's equation for particle boss, same |
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| 05:17 | same material properties, everything is is same. So uh we can add |
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| 05:23 | together, right? We can um those two equations together, like so |
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| 05:57 | a similar thing over here, add right sides together. We're gonna do |
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| 06:07 | for the M. Component. And so now we have the same |
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| 06:15 | for a combination of displacement and particle velocity, not wave velocity, |
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| 06:21 | velocity. And then we can add functions on here. And so then |
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| 06:28 | beginning to be more like the uh out of our of our instruments. |
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| 06:35 | a combination of all these defamation And so that's why uh we can |
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| 06:44 | the simple equation like we start off here um to our data is because |
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| 06:55 | all that does is it changes. waiver waves are still propagating in the |
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| 07:02 | way. So that's an excellent Here's an interesting point. And electro |
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| 07:10 | , they do similar stuff. They have electromagnetic sources and they have electromagnetic |
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| 07:16 | and the waves travel through the earth a very similar way, but not |
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| 07:21 | the same because it's highly attentive. of our sensitive waves are weekly attended |
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| 07:27 | it's highly dispersing. Uh Whereas our waves are weakly dispersing Stephanie. Do |
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| 07:34 | know what I mean? By It means that the different frequencies travel |
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| 07:38 | different philosophies. So uh aside from two differences, electromagnetic waves are very |
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| 07:46 | like uh sound waves because of the generation. They don't travel as |
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| 07:52 | So they're not as good for exploration most contexts but in some context they're |
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| 07:58 | they're crucial. Um We don't talk that in this course because this is |
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| 08:03 | cycling ways and but the electrostatic magnetic are very similar with this difference when |
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| 08:11 | measure an electromagnetic wave, they know what they're measuring. Their measuring |
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| 08:18 | And so uh when an electromagnetic uh makes uh looks at his data on |
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| 08:27 | graph. He know he's got a on there, he knows it's volts |
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| 08:31 | he knows the source was in volts now. Now our instruments are also |
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| 08:37 | , but uh but we don't know translation between votes and displacement and |
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| 08:44 | So we don't know that. So data doesn't have any scale on theirs |
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| 08:49 | a definite scale. And so they do stuff like inversions. They can |
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| 08:54 | they can directly invert their data for properties of the uh medium in |
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| 09:02 | Whereas we have, what we call is not really inversion, it's really |
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| 09:07 | uh integration. Uh They they can is is mathematical inversion. So they |
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| 09:16 | do stuff like migration. They do . They do a mathematical inversion. |
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| 09:23 | uh um those are all good It's important to remember that our wiggles |
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| 09:31 | mean anything by themselves, but only relationship to other wiggles. Uh and |
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| 09:38 | arrival time. Uh it compared with I think you captured all of that |
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| 09:55 | your recording and on your zoom. now I'm going to get out of |
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| 10:05 | without saving the changes. So final looks good. So I want to |
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| 10:31 | this money, Which okay, so you still are you seeing this on |
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| 11:01 | own? So the next topic is waves. So um Stephanie, oh |
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| 11:11 | at the end of this afternoon you an exam and I will email you |
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| 11:17 | an exam, just a quiz you tuning in um before Wednesday but also |
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| 11:29 | gonna have a homework assignment which is write another um uh written question which |
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| 11:37 | can hand into me next friday. it'll be concerning what we talked about |
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| 11:44 | afternoon and maybe this morning if you . And if you want more than |
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| 11:47 | question you can do that. And you're gonna grade, you're you're gonna |
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| 11:54 | in the papers handing the quizzes, of your hand in the quizzes to |
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| 12:01 | time before midnight on Wednesday you send to me and I'll have them graded |
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| 12:10 | uh friday and we'll talk about it . Okay, so body waves uh |
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| 12:19 | we have we now have a waiver and we have p wave solutions with |
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| 12:24 | types and how uh we're gonna talk today, many types of solutions and |
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| 12:31 | uh plane waves come naturally to this . And these can be some together |
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| 12:38 | many ways to form a crucial, idea that they can form any solution |
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| 12:43 | summing together plane waves. But if have a localized source, then the |
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| 12:49 | waves radiate outwards from that source and no plane waves at all. Right |
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| 12:53 | occurred waste. And so that makes a difference. And then finally we |
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| 12:59 | to here, which is in the of this course waves and raise. |
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| 13:07 | gonna talk about wave fronts and and and you'll see the differences and you |
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| 13:19 | how both concepts are useful. In seismic exploration, we always have |
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| 13:29 | of receivers and the receivers received the at different times. Obviously the near |
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| 13:39 | receive it soon and the far ones it later. And the difference is |
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| 13:44 | move Out. And we're gonna analyze . We're gonna talk about the content |
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| 13:51 | wave interference that we talked about And then of course, uh the |
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| 14:01 | equation admits shear wave solutions. And gonna be in our data, although |
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| 14:07 | do our best to keep them out our p wave data. Uh still |
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| 14:11 | leak in sometimes. And then, know, what I said about one |
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| 14:15 | signal is um, another man's Maybe it's a good idea to actually |
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| 14:20 | at those share waves instead of trying get rid of them. And then |
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| 14:25 | converted weights. You probably think, know, how converted waves come |
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| 14:30 | You definitely will know it by the of this afternoon. And then how |
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| 14:37 | been a coordinated discussion what we call convolutional model of wave fabrication. It's |
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| 14:46 | way of intuitively thinking about wave And then uh maybe we'll get a |
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| 14:53 | and we'll get around to talking about seismic images. So we start with |
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| 14:59 | p waves. So let's uh start and we'll talk about the scale a |
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| 15:11 | equation um um where the unknown function the pressure and it's very in three |
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| 15:18 | in time. The equation looks like . Okay, so let's um |
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| 15:28 | Let's think first about uh motion only in the X- three directions. So |
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| 15:34 | term is zero and this term is and we have only this. So |
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| 15:40 | gonna do what we did before we gonna guess and then verify. So |
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| 15:45 | we're gonna guess uh this solution and we're gonna verify. And how do |
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| 15:55 | verify while we stick it into the ? So the left side of the |
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| 15:58 | is uh this because we take the derivative of this thing here with respect |
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| 16:06 | time and it brings down in front the it means one times one times |
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| 16:11 | original function. And on the right of the equation, we start off |
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| 16:18 | this, we uh we plug in all right, our guest here and |
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| 16:27 | from these two uh g derivatives, get minus one over v square. |
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| 16:33 | see the coefficient of Z up here minus 1/3. So do that twice |
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| 16:41 | we get minus 1/3 squared. That from this derivative out in front. |
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| 16:46 | got the v squared. That comes the equation. These two re squared |
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| 16:50 | cancel each other. These two minuses each other and we get left over |
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| 16:55 | this which is the same as Yeah. So we verified that this |
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| 17:01 | works however, it's not a good . And so so I want you |
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| 17:09 | tell me, Stephanie, tell me this is not a good solution. |
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| 17:22 | I think this is a difficult So you talk you tell me why |
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| 17:29 | is not a good solution. Here's answer. You can't have something like |
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| 17:48 | in the new in the power and exponent. It's got to be dimension |
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| 17:55 | . If it's not to mention this there's a big difference here is time |
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| 18:00 | units of seconds or or or um or days or what whenever you have |
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| 18:10 | an exponent, it's got to be list. So mm and there's another |
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| 18:23 | which I didn't even think about unless have pressure. And on the right |
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| 18:26 | have dimensional. So we're gonna put here a multiplication content but we still |
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| 18:31 | this problem here in the um um . So we gotta fix it up |
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| 18:41 | . All exponents must be dimensional. we have to multiply this by um |
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| 18:50 | a factor which has dimensions one over . And so we're gonna call that |
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| 18:55 | . That's gonna turn out to be angular frequency of the way. And |
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| 19:04 | this um constantly that we just put previously that might depend on on omega |
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| 19:12 | so uh let's check this out. go through the same kind of of |
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| 19:17 | left side and right side and so . And we get, we conclude |
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| 19:22 | uh the right side is equal to left side, It still works. |
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| 19:26 | so is there still a problem? it's not the kind of solution we |
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| 19:35 | , it's unstable at large. Um time goes to infinity And that large |
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| 19:44 | goes to zero. So the way solve that is to make it a |
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| 19:57 | by putting in I the square root -1 right there. So from uh |
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| 20:13 | , if you remember your your junior , maybe a senior level course in |
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| 20:21 | , we learned that the exponential to times a factor here is equal to |
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| 20:29 | of that same factor plus i times sine of that same factor. And |
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| 20:36 | so that's a fundamental result of complex and you might want to uh review |
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| 20:45 | uh when I was your age, always bought a text book from the |
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| 20:53 | at the beginning of every semester and I kept it. So the modern |
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| 20:58 | do it that way, you buy and keep them. Yeah. |
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| 21:05 | for example, in this course there's textbook uh Sheriff and Delfin is sort |
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| 21:11 | a reference book and it's a good , but it's not a textbook. |
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| 21:16 | text is basically the set of pdf , which I uh put on the |
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| 21:21 | . So uh do you have something to that uh from your previous coursework |
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| 21:29 | an undergraduate? Yeah, but the didn't use a textbook. Thank |
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| 21:56 | Thanks. I'll tell you an embarrassing for my youth. I was about |
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| 22:04 | years old and I was at a and I met one of the great |
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| 22:14 | from the previous generation. You might the name Sir Harold Jeffreys, the |
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| 22:19 | of Jeffrey's, they forget. So was the most highly respected geophysicist, |
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| 22:27 | part of the 20th century call him Dean of Geophysics. He was a |
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| 22:37 | at Cambridge, I think. And he was widely respected and made fundamental |
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| 22:44 | . But he um, wow, didn't keep up with the ties late |
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| 22:57 | his life, maybe after he Plate tectonics came along. And he |
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| 23:04 | believe it because he had not believed in continental drift. And he uh |
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| 23:13 | thank your comics And he lived a old age. He lived about 95 |
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| 23:22 | his dying day. He was fighting the stupidity of those. So I |
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| 23:33 | him when I was a young man he was an old man and he |
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| 23:37 | not in a mood to accept any from me about plate tectonics. So |
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| 23:43 | didn't even try, but I did to shake the great man's hand. |
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| 23:48 | uh um and then I made a for a social faux pas, Lady |
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| 24:00 | was standing right there beside uh Harold she was not quite as old as |
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| 24:07 | was 10 years younger. And so turned to her and I said, |
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| 24:11 | , Lady Jeffries, what do you ? And she forgive me very kindly |
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| 24:19 | my ignorance. Just an american She thought uh I am a |
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| 24:29 | And so I should have known about famous textbook by Jeffrey's and Jeffrey's, |
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| 24:34 | was the standard textbook and mathematical physics the UK. I should have |
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| 24:40 | but I didn't. And uh she I am a mathematician. Anyway, |
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| 24:50 | I immediately withdraw with you from the not wishing to expose my ignorance |
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| 24:58 | And Sir Harold died a few years that. Still fighting the good fight |
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| 25:05 | . Okay, so in your are your notes somewhere from your complex |
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| 25:12 | course, you will have um known identity between this expert e to the |
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| 25:21 | to the i phi it's exactly equal cosine of five plus i sine of |
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| 25:28 | . And so this is a good uh to point out that uh |
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| 25:40 | uh military solutions have both uh even odd parts and even parts of the |
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| 25:50 | closer, closer minus time. Of the minus offset and the ipod, |
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| 25:57 | parts have opposite signs. So the solution has both even an odd and |
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| 26:04 | we could do is we could work everything using these two parts of it |
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| 26:11 | . Um but it's much easier to to do it together in this form |
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| 26:17 | uh figure out that at the end the process when we get to predicting |
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| 26:22 | actual uh an actual observable. We're have both even in odd parts in |
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| 26:31 | solution. And we will combine these parts of the exponential function in an |
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| 26:36 | way. So sexually result is It won't have any imaginary component like |
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| 26:42 | when we finally get to evaluating spectral . So now what we're gonna do |
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| 26:49 | we're gonna put in i here and that it works in the same logic |
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| 26:54 | we did it all we do is an I squared right here and then |
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| 26:58 | squared right here and everything works. now the question is there's still a |
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| 27:04 | , is this a physically reasonable careful of her after having disposed of |
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| 27:11 | elementary mathematical issue. Is it physically ? Well, um right side is |
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| 27:25 | with the left side. Especially, what we just talked about. So |
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| 27:32 | a real number. This is a number. Don't worry about this. |
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| 27:36 | may be confident that when we use initial conditions and boundary conditions to solve |
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| 27:42 | particular problem, any particular problems resulting will be real. So just relax |
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| 27:48 | that issue. No, this solution a one day solution going only in |
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| 28:00 | vertical direction. But is it going or down? It's going in the |
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| 28:05 | of positive Z. Since the phase constant at larger times. If Z |
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| 28:13 | also larger. so if t is t is getting larger, uh this |
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| 28:20 | will be the same if Z also larger in the right proportion. And |
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| 28:25 | what that means is that the phase constant at larger, uh at larger |
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| 28:32 | of Z plus Z, larger times going in the plus C direction. |
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| 28:38 | furthermore that that's not the only thing want to have, we want to |
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| 28:41 | having waves that move in the direction minus. So we just put in |
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| 28:46 | plus or minus and you can verify this is going so I suppose you're |
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| 28:52 | little bit surprised to see how many we have to make to this to |
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| 28:57 | it into a good solution. Now got a good solution for plane |
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| 29:08 | So now it's is it the only ? Well, you think that's an |
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| 29:14 | family solution, so we can put here any frequency we want and we |
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| 29:19 | have a different coefficient for every one them. So that's a family of |
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| 29:26 | . As a matter of fact. in here, we have assumed we've |
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| 29:31 | of implied as we've been working through that the velocity is a constant. |
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| 29:36 | what if velocity is a function of ? It still works, you can |
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| 29:43 | through it and prove to yourself that no point in proving that, at |
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| 29:48 | point in verifying this, guess. we ever assume that the velocity is |
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| 29:54 | of uh of time, it's actually of frequency no doctor about this |
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| 30:12 | Since the wave equation is linear in some two more solutions with different values |
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| 30:20 | frequency still works. That's also a that some, why do we call |
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| 30:28 | plain ways? Well, it's because had no variation in the X and |
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| 30:35 | directions only in the Z direction. it's like a plane. So that's |
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| 30:39 | it's still hungry. No. Um is why they're so important. Fourier |
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| 30:52 | that any physical function of time and may be represented as a sum of |
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| 30:56 | waves and this is uh this is to Wikipedia so you can look up |
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| 31:02 | you. Um So because of four work, it means that by finding |
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| 31:15 | family a plane wave solutions that we found. We've already found all |
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| 31:19 | All we have to do is find the coefficients and I add all the |
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| 31:23 | together. These coefficients are called the and they're found by fitting initial conditions |
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| 31:30 | boundary conditions. Now notice that this oscillates on forever and no matter how |
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| 31:39 | time gets still oscillating back and forth and Cozzens not getting any bigger. |
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| 31:47 | getting any smaller. And the same with position Z. Uh we don't |
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| 31:56 | want that. We want to have solution which is localized in uh for |
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| 32:00 | , if you have an impulsive we want to have a solution which |
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| 32:04 | zero uh before the source goes off then dies away after, you |
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| 32:14 | something dies away gradually after the source finishes however fourier during jesus that any |
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| 32:26 | signal can be decomposed into a spectacular waves which combine constructively at short times |
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| 32:33 | destructively of all times. So you , it's, it's okay, you |
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| 32:41 | take these waves which are not localized time and we can construct a solution |
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| 32:48 | is localized in time and localized in . Now that's all one day vertically |
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| 32:56 | . So now let's talk about uh in any direction. And so the |
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| 33:04 | has a solution like this with infinite of frequencies and all they have is |
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| 33:13 | affect your product here instead of Uh you said K three times |
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| 33:20 | We have K vector dot x, . And the relationship between this K |
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| 33:26 | this omega is given right. And you can verify that this expression still |
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| 33:32 | in three dimensions. Uh so The wave equation still satisfies the wave |
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| 33:39 | this verification does require that we have relationship, remember here that it's about |
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| 33:48 | length of the way back to the , not in its direction, socialization |
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| 34:07 | we have developed, it is good any frequency in any direction. All |
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| 34:18 | that was for the scalar wave So now let's consider the factual |
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| 34:26 | The rich, I'm gonna make a assumption that uh solution looks like this |
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| 34:35 | a vector, uh constant out It's not a constant. But when |
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| 34:41 | say constant, I mean it doesn't with time or space, it might |
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| 34:44 | with frequency. And so we verify this works same kind of logic as |
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| 34:53 | . Now again for simplicity, we're consider one D. Propagation. But |
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| 35:00 | from using the vector wave equation. so uh constant here is gonna be |
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| 35:06 | vector pointing in the Z. This is a unit vector in the |
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| 35:12 | . And so uh our guess is it's uh plane wave looks like |
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| 35:19 | Yeah at a given place two different . The difference in phase at two |
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| 35:30 | times is given by this. So is the phase of time to and |
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| 35:35 | given positions. The finest the phase time, one at the same |
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| 35:43 | And so that's uh the Z terms out. And you get only Omega |
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| 35:51 | T 2 -51. Let's assume that two times correspond to two successive |
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| 35:58 | So you're sitting at at uh placing Earth. You see a wave uh |
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| 36:05 | see this wave go by and when axl um in the wave You mark |
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| 36:14 | time. And then uh the next comes that is two successive peaks. |
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| 36:21 | we call that a difference in phase two pi and so the difference in |
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| 36:27 | is given by to pay over uh . And that's the definition of the |
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| 36:33 | frequency and one over the frequency is period. And do the opposite |
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| 36:41 | Look at two different places at the time. Go through the same kind |
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| 36:48 | manipulation. And you find that for successive peaks, uh the wavelength separating |
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| 36:55 | two peaks is given by the velocity the cycling frequency. Uh or in |
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| 37:01 | of angular frequency extra fence or two . So now we can see why |
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| 37:07 | call the velocity the velocity of the . We just found out that this |
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| 37:12 | the definition of the wavelength and this the definition of the period. So |
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| 37:20 | wavelength divided by the period. That's of the definition of the wave |
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| 37:28 | So finally we understand why we identify as the wave. Oh, |
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| 37:40 | Uh Is this true carefully and why it false? Mhm. We just |
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| 38:11 | about this. It's false because these unstable. It's not a plane |
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| 38:17 | It just at time equals infinity. thing gets to be infinity and uh |
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| 38:25 | equals infinity. This gets to be . So what you need is the |
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| 38:29 | hear you don't have the eye. now is um is this one |
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| 38:38 | Well, this one couldn't be true this one doesn't have the eye |
|
| 38:42 | Right. And now we got the here. So now is this one |
|
| 38:57 | ? I think you said yes. uh that is so all we have |
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| 39:01 | a bunch of terms here. Lots them. And they differ in their |
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| 39:11 | . We know it's only true if link of this thing of this vector |
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| 39:23 | . Is related to the velocity in right way. So that's not that's |
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| 39:30 | assured here. So I don't say it's true with the caveat who was |
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| 39:36 | a warning? Maybe not true unless have the right magnet. Yeah, |
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| 39:45 | uh so here's a question about the . It's just one true. I |
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| 40:02 | warn you. These are trick Remember that the condition between K and |
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| 40:22 | is a scalar, this is affecting . So this is false because this |
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| 40:30 | not what we approved and to convince of that, just go back to |
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| 40:34 | scene. So this relationship between period , correct? Yeah, that's |
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| 40:52 | Yeah, that one's true. now the only problem with all that |
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| 40:58 | about plain ways, we never ever any plane waves in our data. |
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| 41:09 | basically the reason is because our waves from sources and the sources are localized |
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| 41:16 | . And so the waves spread out the sources and they're always curves. |
|
| 41:22 | they're never ever playing life. So way we deal with us, remember |
|
| 41:39 | is the in homogeneous wave equation for scalar case. And I'm gonna write |
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| 41:47 | this way and i In the that's just another way of writing out |
|
| 41:57 | zero and it's got in here the of factors and it's got to be |
|
| 42:08 | arranged so that um it's dimensionally For example, this is gonna be |
|
| 42:21 | same VP as we have over here the left side. R zero is |
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| 42:26 | to be some sort of reference distance we haven't defined yet. Put that |
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| 42:31 | there to make sure that the whole is um dimensionally correct. And so |
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| 42:38 | the way we didn't hear this make sure it's because it's got pressure |
|
| 42:48 | . And so um um you tie me let me turn to your |
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| 43:00 | Is this thing dimensionally correct here? what it means is the right side |
|
| 43:06 | to have the same physical dimensions as left side. And just to remind |
|
| 43:11 | , we decided that these two terms the correct dimensions because uh this del |
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| 43:17 | has the dimensions of one over distance and one over distance squared times distance |
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| 43:25 | . Right by time squared leaves one times square. Which is the same |
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| 43:30 | the physical dimensions here. Now can tell me uh existing have the correct |
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| 43:38 | dimensions? Good job it's got to physical dimensions of pressure divided by times |
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| 43:52 | . And here's the pressure. Why it not correct? Well maybe it's |
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| 44:20 | . So that's why I'm asking you question. So if we write the |
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| 44:26 | signature like instead of a zero like and have the time variation in |
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| 44:33 | But meanwhile we put in these factors P squared at R zero. And |
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| 44:40 | asking it uh considering all of this , is that got the correct physical |
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| 44:46 | as the left hand side, beg ? Okay. So what we need |
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| 45:04 | to have pressure divided by times square we got pressure here. So uh |
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| 45:11 | and this and this better make up um um times square right one over |
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| 45:20 | is great. So um so here have distance squared divided by times |
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| 45:30 | And then distance here something distance cubed . Time squared. Okay, so |
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| 45:38 | that's the physical dimension of times Um that's the physical dimensions of this |
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| 45:44 | . So what are the physical dimensions delta? Hmm. What makes you |
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| 45:51 | that? Uh the reason you said , I'll answer the question for |
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| 46:03 | the reason you said that is because didn't go over it properly. So |
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| 46:08 | we're gonna do is get out of file and we're gonna go to the |
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| 46:18 | Cherie So am I still sharing properly zoom? Okay. So what we're |
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| 46:28 | do now is look up delta. yeah look up the direct delta |
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| 46:34 | Not not the del operator. Not one gonna go to direct delta |
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| 46:42 | Well, I got here to do . Here's the direct delta function. |
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| 46:48 | , this is in one day. . Now, so here's the definition |
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| 46:54 | the direct delta function. This is we talked about before. It's infinite |
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| 46:58 | Mexico zero and zero everywhere else. here's what we didn't talk about the |
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| 47:06 | under this spike From - Infinity to . That integral is one. So |
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| 47:16 | the spike and it goes up here infinity. But you fight take the |
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| 47:20 | from here all the way over And that integral has to be |
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| 47:24 | That's the definition of the dirac delta and one dimension. And so from |
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| 47:32 | definition it follows that if you take integral of the direct delta function times |
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| 47:37 | function gonna go over X. That gives the um the the value of |
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| 47:46 | function at X equals zero. Because thing is zero everywhere. And at |
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| 47:52 | at the location of zero, it's infinite. And what's the area underneath |
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| 47:59 | infinite spike? It's it's got infinity the width of it is zero. |
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| 48:05 | infinity times zero in this case is to be one. That's what it |
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| 48:12 | here. So that's why when you delta times F you get f M |
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| 48:18 | . Now that's all for one dimension three D. Very similar. This |
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| 48:31 | what we were talking about in connection the seismic source. We've got delta |
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| 48:37 | it's uh it's uh infinite at the zero elsewhere. And in addition we |
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| 48:43 | this uh condition when you make a integral over all of space, That's |
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| 48:51 | to be one. And when you a triple integral over all of |
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| 48:57 | The delta times some function. What get is the value of the function |
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| 49:01 | zero. Now look at this expression , that's part of the definition of |
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| 49:09 | . Now you talk tell me what is the physical dimensions of delta? |
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| 49:26 | correct. Not what you said earlier , you said its dimensions. Now |
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| 49:31 | can see from this definition it's got dimension, it's got the physical dimensions |
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| 49:36 | one over length to the Q Let's go back to the one dimensional |
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| 49:47 | . So here uh the physical dimensions delta of act is 11 over |
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| 49:53 | Right. And so uh so the dimensions of delta are are inverse um |
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| 50:03 | square. Okay so now let us back to um here and now. |
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| 50:10 | what we have here on the right side is pressure times uh X squared |
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| 50:16 | time squared. And uh here's another of X. We have pressure times |
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| 50:25 | cubed over time squared. And now dimensions of this are one over X |
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| 50:34 | . So the dimensions of the whole are pressure over time squared, which |
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| 50:38 | what it should be. And that's we uh renamed the source strength in |
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| 50:45 | way to show the pressure explicitly and show um um we didn't have to |
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| 50:52 | it the velocity this way but I that just for fun but I needed |
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| 50:58 | additional um uh parameter whose value uh dimensions are the link. So that |
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| 51:06 | uh huh So the physical dimensions work correctly. Okay now so uh we're |
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| 51:20 | uh now uh so now we have different equation before we had a zero |
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| 51:31 | , and we uh we're able to talk about solutions being plane waves, |
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| 51:35 | now we've got a source, such different equation. So we're likely to |
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| 51:40 | different solutions. And so what I'm to guess, here's what I'm going |
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| 51:44 | guess is the solution is a sum plane waves, but also with some |
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| 51:51 | normalization. Uh I'm gonna have one R here, and that's gonna give |
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| 51:56 | geometrical spreading. And then to make to make the units right. I'm |
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| 52:02 | multiply by that same quantity R. . So, uh this is |
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| 52:06 | And then just for fun, I'm put in here this end where n |
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| 52:10 | the number of terms in the um in the sun. So, and |
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| 52:17 | be three, or it might be , or it might be 560 who |
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| 52:21 | what it is, but it's gonna out. So, as I |
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| 52:24 | down here, it's just a four decomposition with a guess about geometrical spreading |
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| 52:30 | here. So, uh so then put that guess uh into the |
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| 52:42 | So, in the in the uh the equation on the left hand |
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| 52:47 | there appears to quantity um uh, operator times P. So, uh |
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| 52:55 | on that guest that we just made laplace operator uh with fearful symmetry. |
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| 53:03 | not talking about any variation with asthma anything like that. Uh uh if |
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| 53:10 | work through the uh the application of spherical combustion operator which I gave you |
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| 53:18 | , operating on the uh the uh uh uh term here, you're gonna |
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| 53:26 | up with a term like this which uh familiar to you perhaps. But |
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| 53:32 | here we have the uh operating on over R. And so without |
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| 53:46 | I'm just going to tell you that the closing of the one over our |
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| 53:50 | is the direct delta function where the sign. And you can see a |
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| 53:56 | of that. Let's see here. , the proof of that is in |
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| 54:00 | Gloucester. Okay, so putting this the wave equation, this is what |
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| 54:08 | just did right there. Here's the of the wave equation. So we |
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| 54:13 | on the left hand side, the with our guests in there, we |
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| 54:17 | uh the same some uh and we minus omega N squared because this is |
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| 54:25 | end uh something over in frequencies. for each term we gotta minus omega |
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| 54:34 | . And then uh from the uh this term you get from the previous |
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| 54:39 | , this quantity here, this is from the previous page. So we |
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| 54:44 | terms. And so this left side it here, left side begins to |
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| 54:51 | like this. And uh because this here, this vanishes. So the |
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| 55:02 | side is just this thing here, came from the source term here, |
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| 55:20 | only working on with the left side the wave equation. So we put |
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| 55:26 | the left and the right sides. here is the right side uh showing |
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| 55:32 | source term. And uh just cancel the common factors you're left with this |
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| 55:39 | here, cancel common factors out of and left it here. And if |
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| 55:44 | look uh this with a sharp eye by mr fourier, you see that |
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| 55:52 | left side is just the fourier decomposition the pressure pulse. So this uh |
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| 56:02 | actually is true, should have a mark here, which then I take |
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| 56:08 | . But so what it means is guest for the solution is valid. |
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| 56:17 | these are are radiating circular ways which geometrically according to one over R. |
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| 56:24 | that's the guest that we put in that's the uh and we just |
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| 56:34 | So what are these waves like? , they oscillate because of this thing |
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| 56:38 | . And as they go out as increases, these waves decrease in amplitude |
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| 56:45 | . For all frequencies Hugh medical spreading independent of frequency according to this |
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| 57:03 | Let me think about this. I we could have our zero is a |
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| 57:10 | of frequency that is not specified. Stephanie, is this true or phones |
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| 57:19 | is our description of radiating waves radiating from the source at the origin. |
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| 57:30 | you see it's got a sum of in terms have constants in here, |
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| 57:35 | respectful constants. And uh there was plane wave But each uh each one |
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| 57:43 | those is some of those decreases in according to one over r. Now |
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| 57:51 | question is because of this one over active decay geometrical spreading is just like |
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| 57:58 | gen u. Ation prove are That's a bit of an unfair question |
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| 58:07 | we haven't talked about continuation. Uh It's it's a bit of |
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| 58:16 | But you know the answer to that you look at seismic data. And |
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| 58:22 | when when you look at seismic real data on a workstation and you |
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| 58:27 | at long reflection time. So look the bottom of your screen. Is |
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| 58:31 | uh the same frequency content or lower content? Or higher frequency content than |
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| 58:37 | early arrivals? Yeah it's lower. why is it lower? It's because |
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| 58:48 | high frequencies have been attenuated away. our first approximation the attenuation is the |
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| 58:56 | for all frequencies evaluation per cycle. the high frequencies make more cycles per |
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| 59:04 | . And the low frequencies. So attenuate more per second. But per |
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| 59:10 | , they attenuate uh about the same per second they attenuate more. And |
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| 59:17 | that's why the surviving frequencies at long times have lower frequencies. Well that's |
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| 59:24 | true for this thing here here. geometrical spreading is the same for all |
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| 59:31 | . So it's a bit of an question. But uh it's a you're |
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| 59:36 | on your common sense. And on student level familiar familiarity same data and |
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| 59:47 | got it right? F crush? , that one's true. So yeah |
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| 60:06 | repeating the wave equation with the sorcerer I'm repeating our guests. And right |
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| 60:13 | is a little potion operator and it that when the boston operator goes about |
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| 60:19 | change this part of our gas that uh drag delta function. That's what |
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| 60:26 | needed to match the source turns. that one is correct. Now. |
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| 60:34 | . So here it appeals to your sense. If the source were vibrator |
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| 60:43 | of an explosion, the equation would the same in the solution. Still |
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| 60:47 | valid. True or false. Yeah course it's false because we assume um |
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| 60:59 | assumed right here that it's uh radial in all directions. So uh the |
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| 61:06 | is different. And the solution of would not be valid. Uh We |
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| 61:12 | have the same hue. Medical spreading all directions for for a radio source |
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| 61:19 | for a vibrator starts actually. I'm sure that's true. Hold on. |
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| 61:31 | think that uh sow vibrator source is to be focusing the energy downloads. |
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| 61:40 | I think that it uh decays geometrically one over r uh that same factor |
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| 61:48 | vertically downwards and oblique. Thank Yeah the equation is uh not the |
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| 61:56 | . So this statement is false like said. Okay. Yeah. Everything |
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| 62:04 | been talking about is waves. So now let's talk about race first |
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| 62:11 | all. Have you ever seen Have you ever heard or felt |
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| 62:19 | pray and probably not. Have you seen a ray of light? |
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| 62:26 | you haven't seen any ray of Look at a laser in a darkened |
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| 62:35 | . If there's no dust floating in air, you don't see anything until |
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| 62:39 | hits the opposite wall. The only you see the uh something that looks |
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| 62:44 | a ray is, if there's dust air and the laser light is scattering |
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| 62:50 | the dust ray, uh, without dust, there is uh nothing to |
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| 62:59 | seen. You having your laser here , You see a spot on the |
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| 63:03 | . It's all you see. You never seen a ray. You see |
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| 63:08 | of wave fronts. Uh, I think you've seen seismic wave fronts, |
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| 63:15 | when you throw a rock into a , you see the ripples. Uh |
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| 63:19 | so that's away from right? And the wavefront is physical and the ray |
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| 63:26 | a mathematical concept. Okay, so see how this uh, It |
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| 63:35 | So here's one churn of the solution the scalar wave equation with a source |
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| 63:42 | it |
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