00:00 | The side. So we're almost on this morning. And uh uh and |
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00:11 | has uh kindly brought me a cup coffee. So I am fixing the |
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00:18 | just now and then we're gonna get the questions. I see that each |
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00:23 | you as a sent in a I haven't looked at them yet but |
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00:30 | I was pleased to see them all . OK. So um let me |
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00:44 | here. OK. So uh here's first question from uh uh le le |
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01:13 | OK. So I don't know if can see this. Uh uh have |
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01:21 | uh Yeah. Right. Yeah, , I need to uh oh it |
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01:46 | I'm screen sharing but it's not showing . Why is that? Mm mm |
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02:05 | , I have coffee. Yes. you. Right. OK. Uh |
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02:11 | uh uh maybe just dash share your . So it was Yeah, so |
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02:21 | changed 22 to 21 to your uh , I don't think I'm properly. |
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03:10 | . Can they, how come I say it here? So doing |
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03:23 | it does not allow me. So were having trouble again. Audiovisual we |
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03:29 | here in the classroom. We have a large screen and my screen sharing |
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03:34 | not showing on that screen. Mhm one. Mhm I want to make |
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03:49 | that you played your screen by or it is, maybe it's not showing |
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03:57 | screen. It's extend this curse and back and forth between two screens. |
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04:02 | extends the bladder or uh if you I think I know how to uh |
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04:09 | it. Uh um Yeah. So display settings. So um so we |
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04:20 | it to do play. Keep is UK. Yeah. Yes, you're |
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04:29 | . And now shows good. I share with students. Yeah. |
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04:39 | Um So, uh you know, funny that uh we've been uh doing |
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04:46 | uh the Zoom meeting now for two , but we're still learning how to |
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04:50 | it. OK. So um uh are good questions here. Look and |
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05:00 | look here how she has uh uh me four questions, not just |
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05:05 | So she's getting four for the price one. So, uh that's |
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05:11 | So let us bring up uh uh slides and I think that um I |
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05:21 | you uh says slide 36. I that that's referring to the uh uh |
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05:31 | math 101. Am I correct? . So, uh that is very |
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05:38 | . You see here, she was to tell me which slide number because |
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05:43 | put the slide numbers. Oops. I, I've got to um uh |
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05:49 | get my can you see my, um Yeah, you, you can |
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05:54 | my pointer. So I, I the slide numbers here. So that's |
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05:58 | important thing. It's, it's almost . But whenever you make a presentation |
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06:03 | presenting some of your work, always slide numbers on it so that people |
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06:08 | say go to slide 36. So she took advantage of that because |
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06:14 | uh have done this before and I how useful it is. So let's |
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06:18 | down here to slide uh 36. uh too far. Oh, |
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06:35 | 34. Um Oh, ok. that's it. I'm in the wrong |
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06:54 | . Ok. I'm in the wrong . So uh uh let me uh |
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06:59 | out of this. Uh oops. . Yeah. Elasticity. Yeah. |
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07:11 | . So, ok. Ok. , uh we didn't get a chance |
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07:24 | do that uh uh to do this , but I do, I do |
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07:28 | to, um, say a few about this uh plot so I'm gonna |
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07:36 | it out here. Um Yeah. . So whenever you're working for a |
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07:44 | , normally the company is going to every meeting with uh an emergency notification |
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07:52 | because it's good practice. And so should do that here. And |
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07:57 | the emergency notification usually says there's no today, but it also gives instructions |
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08:04 | what to do in case there's an . So, uh uh what you |
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08:09 | to do is update your emergency contact uh with the university so that you |
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08:18 | uh uh so that the university always how to find you in an |
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08:23 | So there might be a weather emergency there might be a gun violence uh |
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08:28 | emergency. Uh And the, the has techniques to contact you to warn |
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08:35 | about all these things in advance. um uh so you need to make |
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08:40 | that the uh university knows how to in touch with you. So, |
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08:45 | that's true for everybody and including I, I think uh Carlos won't |
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08:50 | to worry about any uh uh emergen any weather emergencies in Houston. But |
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08:56 | , it's a good idea for the to have your phone number and your |
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09:00 | address and so on just to make that they can contact you. |
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09:07 | So do you, do you? , I do that. That's very |
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09:13 | . It's ok. It's not the look at your screen. So they |
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09:17 | looking, looking at this, aren't ? You know, I, I |
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09:20 | it because I want to show the before, but I need to. |
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09:27 | , it's very, very common. . Ok. Ok. Ok. |
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09:35 | then that's uh uh slide one and my uh uh um lecture two, |
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09:40 | one. So here is uh uh into the, the rest of the |
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09:46 | , which we already did. So now I want to um uh |
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09:51 | um move to slide uh 36 and see, I think the best way |
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10:00 | do this is to catch it. , OK, Honduras when his |
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10:22 | well, I'm not seeing the slide here. It is this side that |
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10:42 | . So I'm not sure why I see any slide numbers on this. |
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10:49 | , one more. Ok. Uh So I don't, I'm not sure |
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10:56 | I'm not seeing slide numbers here, um I'll look into that later. |
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11:03 | should be slide numbers showing on the . I know that I have them |
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11:08 | the f it might be that they're down here in this, in |
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11:15 | they're hidden behind this um uh uh frame here. Hey, but uh |
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11:23 | , you found it uh in your , right? Yeah, they're good |
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11:28 | you. Now, the question is do you do uh um uh a |
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11:37 | subtract? And so the way you that is you just uh uh multiply |
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11:42 | one vector you wanna subtract uh by one. So you reverse it and |
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11:47 | you do an addition, right? . So uh uh that's, that's |
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11:53 | easy. Uh OK. So now let us see here. So um |
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12:06 | , I'm having a problem here. Look, I can't move my mouse |
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12:12 | of this screen, see I can't my mouse outside that screen. |
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12:19 | Yeah. What, what did you ? I just pressed the oh |
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12:25 | Thank you. So, now what wanna do is uh get out of |
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12:29 | um presentation. OK. And I want to share the screen again |
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12:43 | um questions. OK. Slide Then the new distance vector has |
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12:53 | Who? And that's magnitude square, is the two means the definition of |
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13:02 | . Uh um uh in that uh let's, we don't have to |
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13:06 | it up, up, up Uh um In that slide, |
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13:12 | the uh um the magnitude was given the symbol L and uh uh so |
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13:26 | the two means to uh uh I showing you the square of, of |
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13:31 | , not the square of L square distance. Yes. Yeah. So |
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13:40 | why it was a sub. Uh it's uh when you have a superscript |
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13:44 | that, sometimes it's just notation and it means raising it to a |
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13:50 | In this case, it was raising to a power. What I was |
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13:53 | you on that slide was uh the of the distance. OK. |
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14:02 | um OK. Next question is on 40. The strain 10 is defined |
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14:10 | divided by two. There's a one in that definition. And so um |
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14:16 | see. Uh OK. OK. I think it's important to bring that |
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14:26 | . Um stop sharing and then I'm start sharing. OK. So uh |
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14:44 | everybody see that slide? That's where were before. So now she |
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14:49 | uh uh see uh see thi this is L prime square. OK. |
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15:13 | , what she's asking about is this half here? So, uh um |
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15:19 | because we put in a 1.5 that means we have to put in |
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15:23 | two here to cancel out the one . And so, uh uh very |
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15:28 | , uh uh you will see, you will see the reason why we |
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15:31 | that. Um Let's see here. I think, II I think uh |
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15:42 | , uh this is not a good for me to explain why uh I |
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15:47 | in a two but very shortly you find uh that was a clever thing |
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15:52 | do. OK. OK. uh the next uh slide 45 and |
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16:09 | how do you find the uh the of a vector? Is it this |
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16:24 | ? OK. So uh this is good flight. Can I uh uh |
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16:31 | everybody sees this? Uh Let me sure that everybody is seeing this. |
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16:40 | Yeah. So uh Carlos uh you're that I had that uh right. |
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16:45 | , um yeah, but not Not anymore. OK. Um |
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16:55 | Is that good? Yes. So, um um here we see |
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17:05 | a square which was uh uh the dash line is the original um |
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17:11 | orientation of the square. And after deformation, this uh the square has |
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17:16 | um changed into this shape which is square. Uh We call that parallel |
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17:22 | p it, I'm not sure if know that word in uh from your |
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17:27 | , but that's the, the uh uh um you remember back when you |
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17:33 | plane geometry as a sophomore, you that, that the, the name |
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17:38 | a shape like this is a parallel pip it. And of course, |
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17:42 | mean this line is parallel to this , this line is parallel to this |
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17:46 | . OK. So the square in , the dash square has been deformed |
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17:52 | this solid parallel piping. And so deformation is in the one direction. |
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18:00 | uh Can you see that uh uh in the one direction here and also |
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18:06 | here, it's in the one So you see as a function of |
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18:10 | X three, it changes from nothing something. So uh uh you |
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18:18 | I, I think this is a picture which shows that the uh the |
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18:24 | of deformation is in the one direction the one direction and it varies with |
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18:33 | three directions. So did, did answer your question lately? Uh |
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18:45 | it's not uh it, it's not uh excuse me. Uh uh it's |
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18:50 | the, the displacement you is only displacement you want, it's only a |
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18:57 | one component of the displacement. this number is zero. In this |
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19:05 | , uh uh The displacement in the direction is zero because it sees nothing |
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19:10 | this um um shape has uh moved the three direction, it's only moved |
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19:17 | the one direction. So this one uh that's what it's showing here. |
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19:22 | this part is zero, but the three direction is this one. And |
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19:28 | can see that it changes uh uh your new one, um uh changes |
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19:35 | uh from zero. And as you to larger X three, it's getting |
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19:41 | and bigger. And so this term is non zero. So then in |
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19:48 | case, the epsilon 13 is equal one half times this term here |
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19:55 | OK. OK. So uh I was thinking when I um um |
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20:06 | , I was thinking when I was this yesterday that this slide needs more |
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20:14 | and you just proved it, Lili uh your question. OK. So |
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20:25 | gonna stop sharing this screen and I'm go back and share the uh the |
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20:36 | . Yeah. OK. So thanks those questions. Let's uh look at |
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20:43 | next my question from yesterday's lecture. you see this? You know? |
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20:50 | , is about the stress tensor. is not clear to me why applying |
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20:55 | sheer stress would cause infinite spinning? . Um So that's not what I |
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21:06 | . I uh I, I didn't that when you apply sheer stress that |
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21:10 | cause infinite spinning. I can, can I rephrase that? Maybe I |
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21:16 | to II I miss to put that they are not the same. Hm |
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21:23 | . So uh uh uh if they not the, so, so uh |
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21:27 | we see that uh if the two of, for example Tau 12 and |
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21:32 | 11, et cetera, wrong tau and tau 21. If those two |
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21:38 | not the same then, um, , it would cause, uh, |
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21:43 | spinning. Well, so, um, I think that, |
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22:00 | uh, for me to explain um, clearly it's gonna take too |
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22:06 | time. So, uh, um, well, what I, |
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22:13 | , so what I'll do is I'm gonna defer that question and, |
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22:19 | , let's see how I can help anyway. Um uh Any textbook will |
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22:29 | a statement like that in, in . If you uh uh um uh |
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22:36 | you have a textbook on uh uh um geophysics, there will be a |
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22:42 | on stress and strength and uh in chapter, there will be a statement |
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22:49 | to this. I think it should . If not, you can uh |
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22:55 | um um look online and uh mm OK. Um uh So, so |
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23:15 | the suggestion Rosada, look online in . Do, do you know uh |
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23:21 | about Wikipedia? So just search online in Wikipedia using the search stress and |
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23:29 | will be an article about stress And I think that in that article |
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23:34 | be a discussion of this point. is pretty good. Wikipedia. I |
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23:40 | very skeptical when Wikipedia began to uh be uh built because I thought there |
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23:48 | so many opportunities here for misinformation that it's gonna be unreliable. But, |
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23:55 | know, there is a whole army people who volunteer their time to um |
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24:04 | uh uh to ensure the validity of , of, of the information on |
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24:10 | . And uh um the way it is that you can join the army |
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24:19 | a volunteer at any time, you sign up with Wikipedia. And then |
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24:23 | you can have uh the uh the to go into any page a Wikipedia |
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24:32 | make a change. And as soon you make that change, it's visible |
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24:36 | everybody worldwide. OK. So, um that is why I thought that |
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24:50 | system was so prone to misinformation because can edit it, they can edit |
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24:59 | uh uh uh out of and, they, they can uh uh put |
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25:04 | their wrong information. Uh And uh instantly visible worldwide, however, in |
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25:13 | background, uh there is a record um uh what you did and you |
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25:21 | find that record by uh when you into uh uh Wikipedia, click |
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25:26 | on, on any page, there's tab for called talk, talk |
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25:32 | And then you can see the history all the changes that have been made |
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25:37 | that um uh page. And um uh normally when you make a change |
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25:45 | that, it is noticed AAA flag hoisted somewhere and there'll be a member |
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25:52 | this army who's an expert in things stress and you'll see that uh uh |
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25:58 | that somebody named Leon Thompson made a and what the change is, and |
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26:03 | he's an expert and he looks at and if he decides that's a, |
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26:06 | good change, he leaves it and he, uh, uh, decides |
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26:10 | a bad change, he'll take it , take it away again, just |
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26:15 | that change. And if Leon Thompson a similar thing, uh, |
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26:19 | twice more then he gets banned from army. Uh, so, |
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26:25 | uh, in that way they ensure , um, the integrity of the |
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26:32 | on, uh, uh, on wiki, on Wikipedia. |
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26:37 | normally for a mathematical subject or a subject like this, there would be |
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26:41 | , um, only a small number people watching but, um, there |
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26:45 | many pages, uh, in Wikipedia are about, um, controversial |
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26:52 | for example, uh, I guarantee that if you go on to the |
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26:56 | today and search for Donald Trump, will find, uh, uh, |
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27:02 | notification that yesterday he was fined $355 . Uh, they are very quick |
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27:10 | put news like that into Wikipedia. so you can go in there |
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27:16 | uh, you can edit the page , uh, Donald Trump and you |
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27:21 | put in there. Um, Donald Trump is an idiot and a |
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27:26 | . So within seconds that one will , uh, um, um, |
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27:32 | because it's, uh, political and and if you do it again, |
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27:38 | , uh, uh, you will kicked out of the system and you |
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27:40 | be able to uh edit at So that's the way they maintain the |
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27:46 | of Wikipedia. So I encourage you actually um become a member of the |
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27:51 | to sign yourself up. And uh when you see uh a change that |
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27:58 | to be done, go ahead and it. And uh um uh you |
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28:05 | get hooked. You know, there thousands of people who uh spend hours |
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28:10 | day doing this. Now, I tell you that the seg has its |
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28:15 | SCG version of Wikipedia, which is in um uh only. And so |
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28:22 | can find that on seg.org. And uh if you look around, uh |
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28:28 | be easy to find, it's called seg wiki and it has stuff about |
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28:34 | uh geophysics and you might find that in this course. Uh uh If |
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28:41 | uh um if you go into the wiki and search for stress, you |
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28:48 | find um uh a description and I it will not be as um um |
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28:55 | a definition as you will find on worldwide Wikipedia because uh it will be |
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29:02 | by a geophysicist who will define things very simply and won't concern himself with |
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29:08 | like uh the symmetry of the stress . That's what I think. |
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29:17 | So let me go on with the uh for Bria, she says, |
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29:21 | is this then related to the fact some of the elements of compliance and |
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29:25 | are the same. Yes, it . Uh that symmetry is very important |
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29:30 | uh that is, that really helps uh uh reduce the number of uh |
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29:35 | elements from 81 which is, you , three by three by three by |
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29:41 | uh down to 21. And that in the number of independent elements comes |
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29:51 | of uh symmetries. And it's true uh any um anybody uh you |
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30:01 | a, a piece of glass, , a piece of rock, |
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30:06 | Uh And then further uh for the uh if the, if the body |
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30:12 | to be isotropic, then the uh list of, of uh independent elements |
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30:20 | down to two out of the 81 two are independent. And um um |
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30:30 | some of those are repeated. So , there's uh doesn't mean there's only |
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30:34 | nonzero terms. That means that there's two independent terms. So we, |
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30:40 | are gonna return to this topic of the number of independent uh elements for |
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30:46 | isotropic body uh later this morning. . So, thanks for that |
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30:53 | Next question is, and you explain physical concept of elastic stiffness and elastic |
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31:01 | . Well, now that's a very question. Um uh um let's just |
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31:07 | about stiffness first. Uh uh uh remember that uh stiffness is uh um |
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31:16 | a tensor with four indices and 81 , three times three times three times |
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31:27 | elements. What is the physical Well, that set of numbers is |
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31:33 | set of proportionality coefficients between stress and . So H Hook's law says that |
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31:43 | is proportional to strain and those uh 81 elements uh give the proportionality |
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31:52 | So when you have uh a large , uh uh a, a large |
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31:56 | of stiffness, that means that a um um oh uh uh given |
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32:05 | a small strain can uh be uh with a large stress because you have |
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32:13 | small strain epsilon times a, a stiffness coefficient on the right side of |
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32:21 | equation. And then on the left , you have the stress. So |
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32:25 | says the stress uh corresponding to that is a large number. So um |
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32:33 | compliance is just the uh the inverse that. If you have a large |
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32:38 | , it means that the uh the is, is uh weak, it's |
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32:44 | and you can uh deform it a with a small stress. So in |
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32:52 | , for example, you have a hard rocks, for example, think |
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32:56 | a granite. So a granite has , has a hard stiff rock and |
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33:02 | has large values of the stiffness of . It has small values of the |
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33:11 | , think of a sponge piece of and just think of a, of |
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33:16 | sponge which you can deform in your and that is highly compliant and it |
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33:22 | a lot when you give it just little bit of stress. So that |
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33:27 | high values of compliance. So does answer the the, is that a |
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33:33 | physical uh explanation for you, Yes. Thank you very much. |
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33:42 | . Well, thanks for these I, I like these very much |
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33:48 | great institution that more. Uh So don't know here uh some three late |
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34:04 | about the dot product when the vector itself is the result still a |
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34:09 | That's a very uh uh important So let us go back, let |
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34:26 | go back to the is a I'm gonna stop sharing this. |
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35:35 | OK. So I think I'm sharing screen with slide number M-16. Does |
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35:41 | see that? Uh uh Rosada? you see this? Yes, I |
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35:49 | ? OK. So this is from math 101 file. Let me see |
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35:54 | . So uh uh uh it and , in answer to Li's question, |
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36:00 | uh you can multiply two vectors together two different ways. So this is |
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36:06 | first one, the dot pilot is scaler not a vector. And let |
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36:11 | just go forward here. Next slide uh uh the second way and the |
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36:21 | product of two vectors is a So you see the difference, there's |
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36:26 | different ways to multiply vectors together. I'm gonna go back to the first |
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36:33 | . So that's the dot product. so uh the, the dot product |
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36:38 | defined this way you take the uh the, the first component of the |
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36:44 | times the first component of the other X two, Y two plus X |
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36:49 | Y three. That's the definitions This right here means def uh um |
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36:58 | . And uh remember when we, can write, uh we can write |
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37:03 | AAA complicated expression like this very simply by saying X of I times Y |
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37:10 | I remembering that when we have a index, we got a sum from |
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37:16 | to 3. So separately uh uh think of this as the definition and |
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37:25 | you can also show and this is to find in any book on plane |
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37:30 | . Uh that uh this uh some three terms is equal to the me |
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37:37 | length of one times the length of other times the cosine of the angle |
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37:42 | them. So I think that's fairly . Uh uh This first way of |
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37:49 | the dot product is fairly simple. second way is more complicated. This |
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37:55 | the cross product of two vectors and a vector. And uh uh here |
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38:00 | here is the result of that for three component vector. You can see |
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38:05 | has three components and each component is difference between two terms. And it's |
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38:10 | of complicated. And then furthermore, uh uh plane geometry book will show |
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38:17 | that this thing is, this thing out to be the uh uh the |
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38:22 | of X times the length of Y the sign of the angle between them |
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38:28 | a unit vector, which is perpendicular both of these. So if you |
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38:34 | X and Y in the plane of screen, then X, then the |
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38:39 | uh vector is gonna be pointing out the screen. And uh well, |
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38:45 | cross product X cross Y is gonna pointing a vector pointing out on the |
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38:52 | . So um uh um I know did a lot of mathematics yesterday and |
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39:02 | lot of notation and I think you're um uh unhappy with that, but |
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39:09 | will be a payoff because the notation we develop is gonna make things easy |
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39:15 | us uh in the future. So um let me stop sharing |
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39:30 | Oh Start sharing. Oops. Um see what I want to have |
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41:06 | OK. I think you can all this, these thumbnails of the |
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41:12 | Uh We talked about yesterday and I'm go back to this one and I'm |
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41:19 | to uh OK. OK. I I'll just remind you that uh uh |
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41:26 | what we found yesterday by considering uh different stresses on um a cylinder of |
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41:35 | rock in the laboratory. We, figured an isotropic. Uh We figured |
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41:42 | that the uh the matrix, the by six matrix which contains all the |
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41:49 | of the uh um compliance tensor that these expressions only. So, uh |
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41:58 | it has a lot uh remember the triangle is the same as the upper |
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42:03 | has a lot of zeros. And the, the main axis here, |
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42:07 | has one over the young motos that's the one direction that's for squeezing it |
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42:11 | the ends. And because it's it's the same in the two direction |
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42:17 | the three direction. And uh when squeeze it like that on the uh |
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42:23 | the ends, it also uh gets , it expands in the perpendicular |
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42:30 | And that's described by these terms here this is protons ratio divided by uh |
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42:38 | young models and the minus signs are here in order. So that pros |
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42:43 | will be a positive number and epsilon be positive number. And of |
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42:48 | uh what uh the minus sign comes when you squeeze a cylinder, it |
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42:53 | shorter and it, but it gets sadder also. So uh obviously, |
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42:59 | strain is the negative of the other , is uh the, when you |
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43:04 | it, it gets shorter. So a negative strain. Um uh And |
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43:09 | we uh it gets fatter, so positive strain. So that's why I |
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43:16 | a, a minus sign here. then uh uh if you uh uh |
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43:21 | it, it uh it responds according the sheer modules like that. And |
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43:26 | uh that's easy to, to show with uh one cylinder, yes, |
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43:33 | putting on there uh a 12 And then because it's iso it's the |
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43:39 | in the other directions. So, in that way, we constructed in |
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43:44 | very intuitive way, this uh compliance which shows all the information necessary in |
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43:52 | compliance tension. Uh But uh uh bad. This is not what we |
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44:02 | for wave propagation uh uh as Uh Well, uh you know, |
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44:08 | have here at the University of we have a world class laboratory, |
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44:12 | uh uh rock physics where we do , all kinds of squeezing of |
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44:18 | And uh uh we, we uh yeah, it's being run by Professor |
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44:25 | and he's got uh a number of in there who are all becoming experts |
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44:30 | experimental rock physics. And so these the sorts of things they do, |
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44:36 | also they do wave propagation experiments. they will um uh send a wave |
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44:45 | these required samples. And we are shortly learn that this compliance matrix is |
|
44:53 | what we need for wave propagation before pass on here. I just want |
|
45:00 | remind you that we have here uh current three independent numbers, Young's models |
|
45:08 | ratio and she models. However, it's isotropic, you can show that |
|
45:14 | this relationship between the two. So um uh in fact, there's uh |
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45:21 | uh excuse me, there's this, relationship between uh uh sigma and the |
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45:27 | two. So in fact that there's two independent numbers which characterize the isotropic |
|
45:34 | and eventually that's gonna lead to our a conclusion that two types of waves |
|
45:41 | in those crops, uh uh B and sheer waves. So that's |
|
45:48 | If, if, if we didn't this uh relationship here, there would |
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45:52 | three kinds of a way which And uh remember that I said we |
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46:02 | find this relationship because the material is . So most of our rock materials |
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46:09 | not isotropic, most of them are , which means that this is not |
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46:16 | for most rocks. Most rocks have kind of wave which is propagating in |
|
46:23 | rocks. So that's a AAA big . And so we're gonna ignore that |
|
46:33 | for a while and we're going to uh soon that the rocks are |
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46:41 | And then later on, once we to be smarter, then we'll uh |
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46:47 | say, OK, uh Back at beginning, we have made an unrealistic |
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46:53 | , we assume that the rocks are . Now, let's think about uh |
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46:58 | rocks. OK. So, uh that's gonna happen at the end of |
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47:04 | , towards the end of the Now, we went on to discuss |
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47:11 | happens, not uh uh uh what when you squeeze the wrong equally from |
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47:16 | sides. And you can imagine doing in the laboratory uh and squeezing it |
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47:20 | all uh um uh equally from all . And from that, we decided |
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47:27 | uh uh when you do that, uh uh uh uh design the, |
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47:34 | book mogul is K and it turns to be uh this number here, |
|
47:39 | comes from summing all these others. I encourage you to um um to |
|
47:46 | up on this uh in the uh refresh your memory so that you are |
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47:53 | you understand where that come comes Uh Then we, we had a |
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48:04 | quiz here and I think we did quiz and uh uh you all uh |
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48:09 | did well on the quiz. And , and now we came to elastic |
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48:15 | . Uh This is uh uh what gonna need for wave propagation. Remember |
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48:21 | we had Hooks law in two forms where the strain is proportional to the |
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48:27 | and one where the stress is proportional the strain. And so um uh |
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48:32 | this is the compliance matrix and this the stiffness matrix and they must be |
|
48:39 | to each other, right? Because uh this uh there's just two ways |
|
48:43 | saying the same thing. And so learned that the uh um uh we |
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48:55 | that the stiffness ma the stiffness matrix we're gonna need for wave propagation is |
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49:02 | inverse of the compliance matrix, which just figured out by thought experiments involving |
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49:08 | rocks. So I think what at point, what I want to do |
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49:14 | change into the uh slideshow mode. . So is everybody in the slideshow |
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49:23 | ? And here's the other form And so now what we wanted to |
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49:27 | since we know this uh these things uh thought experiments. We're gonna use |
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49:33 | , uh use this expression for um , I take it to do |
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49:41 | the set of ST stiffness coefficients. . So this is the inverse |
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49:48 | So, so, so, oh . OK. So we're gonna stop |
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50:00 | and start sharing again. OK. that good? We're gonna put |
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50:16 | So it's not Sharon and um then sharing screen two. Sure that |
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50:36 | Are we good? OK. So we decided uh uh uh yesterday that |
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50:45 | complicated product is equal to the um I know the identity tensor, |
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50:55 | the identity tensor with four indices. let's look inside here. We got |
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51:00 | a repeated M oop second, I a pointer comes Mike Porter repeated M |
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51:10 | repeated N. So we're summing over and NS and that comes out to |
|
51:15 | the uh uh the uh the identity uh And in fourth rank uh um |
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51:23 | four indices, what does that In terms of things that, you |
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51:27 | , intuitively, it's this combination of delta functions and remember this delta function |
|
51:34 | equal to zero if J is um is not equal to P and it's |
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51:40 | to one if J equals P. it looks like um a three by |
|
51:47 | tensor with ones on the diagonal and off diagonal. And the same sort |
|
51:52 | thing cohere with different indices and so here. And So that's the |
|
51:57 | So let's show how to implement that . Let's first look at the 66 |
|
52:04 | . Uh uh we, we're gonna the 66 component of the stiffness out |
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52:10 | remember that 66 comes from the four notation 1212. So that means we're |
|
52:19 | put in here um uh One here two here and one here and two |
|
52:26 | . That's this. And then we're sum over all the MS and N |
|
52:31 | so when you do that, um you find uh that, that is |
|
52:37 | to the fourth, the rank four matrix uh uh putting in the, |
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52:43 | ones and the twos here on, the deltas just like it in here |
|
52:47 | that uh uh we have J equals . So we put in here, |
|
52:54 | equals one right here and we put equals one right here. So putting |
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52:59 | that together, you see that we this term and so these are both |
|
53:04 | . So that's a zero and this one times one. So uh uh |
|
53:08 | all of that, that comes together make one half. That is this |
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53:15 | um oh uh this product here. uh we know what all these |
|
53:25 | And so uh uh when we, when we stick in here, all |
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53:34 | compliance elements that we just learned about a, a lot of them are |
|
53:42 | . So uh the only terms surviving the sum are these two terms. |
|
53:47 | we have ac 12, that's the here and we have M equals one |
|
53:52 | N equal A and N equals And uh uh again here N equals |
|
53:59 | and NN equals two. See, this one comes from those values and |
|
54:06 | , uh and there's another term like , these are the only ones that |
|
54:11 | all these sums because a lot of uh compliance elements or zero, look |
|
54:20 | over for yourself. You will see the only terms out of this |
|
54:24 | which are non zero are these OK. The C 1212 and the |
|
54:30 | 66. And so you might think S 1212 is an uh an S |
|
54:37 | . But no, that's not We also, when we convert the |
|
54:43 | to from a four index notation to index notation, we need to have |
|
54:49 | , a quantity of 1/4 in The reason for that I skipped over |
|
54:54 | and I don't wanna go back to today. But if you're concerned about |
|
54:58 | , you should go back and look the uh slides that I skipped over |
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55:02 | , which explains why we need AAA here. And so for this term |
|
55:09 | again, and we have C Oops, sorry about that. |
|
55:28 | Ok. Ok. The C 1221 also AC 66 because of the symmetries |
|
55:46 | we talked about earlier. And so , uh when we put in uh |
|
55:52 | when we convert from the four index to the two index notation for compliance |
|
55:57 | we need, we need 1/4. combining these two terms were left with |
|
56:02 | . And so uh that's the, left side of this equation, the |
|
56:07 | side is one half. And so the left side with the right |
|
56:11 | we deduce that C 66 is the of S 66, which is um |
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56:22 | the sheer marvelous mute. OK. what that means is that we have |
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56:33 | uh uh the uh uh from the a stiffness tensor distance matrix will deduce |
|
56:41 | M equal zero. And that these terms are all mute. And just |
|
56:51 | with the compliance makers, we have lot of zeros off of uh off |
|
56:56 | the side, you can, you convince yourself of that just by following |
|
57:00 | previous logic. And now let's consider 11 component. So we put in |
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57:06 | 11, put in here, make all this sum. And on |
|
57:09 | right hand side, we're gonna get one. And when we carry out |
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57:15 | sum with all the uh the zeros in here from the compliance make we |
|
57:21 | with these three terms and converting from indices to two indices were left with |
|
57:27 | two terms. And in, in of, of uh young smarts and |
|
57:34 | ratio. We uh uh we get of the compliance matrix using these uh |
|
57:44 | English words and using the expression that had before we uh uh excuse |
|
57:51 | um uh uh th this is simply uh uh multiplying both sides of this |
|
57:58 | by young's models. And so we have f finally, young's models is |
|
58:05 | by this difference here and we don't want the uh Young's model. We |
|
58:11 | modules appears in the, in the matrix. What we want is C |
|
58:17 | and C 12. And so, let us say, OK, this |
|
58:25 | a useful result to begin with. , what are we gonna do |
|
58:28 | Well, we're gonna do the we're gonna do it with, with |
|
58:32 | ones and two twos and um go a similar logic and we find this |
|
58:39 | relationship between C 11 and C And so you solve those two expressions |
|
58:48 | , separately for C 11 and C . And so we get here for |
|
58:52 | 11, we get an L uh gonna call the result Emma and um |
|
59:00 | it's the same uh for 22 direction for 33 direction and the C |
|
59:07 | we uh get by solving those two uh uh is equal to M minus |
|
59:13 | mu where the view is here. what is M, you don't know |
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59:18 | M is until I show you that is given by this. And suddenly |
|
59:25 | think, oh, now I see familiar because I know that K plus |
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59:31 | thirds mu governs the P velocity. furthermore, I know that the sheer |
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59:38 | M governs the sheer modules. And uh uh where did the density come |
|
59:44 | ? Well, I, and you know yet, but uh we will |
|
59:48 | that in the next lecture to show uh uh uh uh these things come |
|
59:55 | the wave equation. So I'm gonna up. Now, we know what |
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59:58 | that these three govern heat wave these three govern uh shwa population and |
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60:08 | three it's not quite clear, is what those three are, are good |
|
60:14 | ? But at least we know it in terms that we are familiar with |
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60:18 | other courses where we have the U the K. Now this, this |
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60:25 | a good uh uh uh and go it. This is a good want |
|
60:36 | remember that when we say a P as a P wave goes through a |
|
60:42 | , it does not make your uh on all sides of the rock. |
|
60:47 | makes longitudinal stress. So the stress from uh isotropic compression because there's some |
|
60:54 | in there. And that's why you the sheer module syndrome. And now |
|
61:01 | way of propagation, the young smokes personalization do not appear this off diagonal |
|
61:09 | M minus two new is called the parameter. And you can look up |
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61:13 | lema parameter um uh in the Uh you can do that on your |
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61:19 | . LeMay was actually a priest. you imagine that a priest during elasticity |
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61:25 | in the 19th century? Uh you , all his buddies were uh uh |
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61:32 | chanting uh uh mhm Gregorian chants in choir and some of them were out |
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61:40 | the fields um uh growing grapes. And then uh LeMay was sitting up |
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61:48 | his little cubicle in the monastery during . And to think about that, |
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61:57 | you will find out that uh when look at the wave propagation equation, |
|
62:02 | does not appear anywhere even. So can uh uh if we put that |
|
62:08 | hit in here, it makes the element look uh simpler and remember that |
|
62:15 | are all zeros and these are all same as the upper tri and as |
|
62:23 | spreadsheet. Well, we don't have board anymore. We have canvas. |
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62:29 | um uh uh I gave you a uh uh in the com canvas module |
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62:37 | that works as an Excel spreadsheet. can download it and uh look at |
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62:41 | . And the first worksheet in that you to calculate any of these parameters |
|
62:48 | from the seismic parameter. So you uh you probably have a good intuitive |
|
62:55 | for AAA different type of rock, VP should be and vs should be |
|
63:02 | you can just put it in there it will uh calculate for you. |
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63:07 | I actually forgot whether uh the input in terms of feet per second or |
|
63:12 | per second. Um There might be way to switch between the two. |
|
63:20 | . So here's a little um uh . So how many independent components does |
|
63:30 | stiff stiffness tensor have? The charges 81 36 21 3 or two, |
|
63:39 | ? How many independent components? So , le uh it says for isotropic |
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63:50 | . So your answer is correct for isotropic rock. But that's not what |
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63:56 | question is. So how many independent components are? Um le let's go |
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64:05 | and look at this. OK. it is. Yeah. So two |
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64:14 | and in this matrix, uh it's um nine of them to be |
|
64:21 | Nine of them are nonzero. But uh we've got these repetitions because it's |
|
64:27 | . When we get to an these will all be different. But |
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64:30 | , uh you were correct. Uh um nine of them are nonzero. |
|
64:36 | uh three different uh symbols appear but we have this uh relationship between |
|
64:43 | two. So only two of them independent. So uh um when I |
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64:50 | a question like this, uh uh need to read it carefully because it |
|
64:54 | be a trick question. By the , this is a good time for |
|
64:57 | to stop and tell you what the uh uh the, the grading, |
|
65:05 | , process is gonna be like in course, I'm gonna have to show |
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65:10 | a lot of these quizzes, uh, uh, every day I'll |
|
65:13 | showing you quizzes. But, uh, they don't contribute to your |
|
65:18 | . The only thing that's gonna contribute your grade is the final test. |
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65:24 | . And the final test will be test of that, uh, of |
|
65:28 | sort that you might not have ever before. It's gonna be uh um |
|
65:34 | book. So you'll be here when sit down and take the test, |
|
65:38 | can have all of this stuff in of you, everything that I gave |
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65:43 | and whatever uh you check out from library or whatever you buy from |
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65:47 | you can have anything is open to . So, uh uh uh the |
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65:54 | is gonna be a test of your , not of your memory, |
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66:00 | Because if you're uncertain about some you can always look it up because |
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66:06 | your books are gonna be open it's gonna be unlimited time. So |
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66:13 | can take three days to take the . So, uh uh the only |
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66:23 | uh uh there's only two conditions, , uh uh there's only two |
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66:29 | one, you have to do it yourself. You, you can't do |
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66:32 | together. And by the way, encourage you all to get together uh |
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66:37 | outside of class, maybe by zoom talk these things over, you have |
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66:43 | uh uh you have uh you know to get in touch with each other |
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66:48 | I think you are able to um together and if you can't zoom |
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66:53 | Utah is gonna help you zoom together or you can meet uh somewhere and |
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66:59 | a, a AAA drink while you're talking. Um Probably that would be |
|
67:05 | for Carlos but uh uh I, recommend that you zoom together the three |
|
67:11 | you and get you t to, , if you have a question, |
|
67:15 | call him up. Uh, uh he's gonna be somewhere else, but |
|
67:19 | be happy to uh advise you. , uh I talk about these things |
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67:24 | also, uh at the end of lecture at, at uh, we're |
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67:29 | go today till one o'clock. So the end of this lecture, uh |
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67:33 | you're gonna have, uh, submit a question and, um, uh |
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67:41 | at one o'clock we're gonna be breaking lunch uh until two o'clock and then |
|
67:46 | gonna resume at two and we'll go six. So you might not have |
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67:51 | to submit a question between one and . But, uh this evening, |
|
67:56 | after uh class, you're gonna email with a question from the morning lecture |
|
68:02 | a question from the afternoon lecture. . So back to the final |
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68:08 | it's gonna be open book unlimited do it by yourself only when you |
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68:16 | a test only. And uh, , uh, here's one more |
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68:20 | You have to do it in one . So you can't do half of |
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68:24 | now and half of it the next do it all in one sitting. |
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68:29 | so you, I'm gonna hand it on the last day of class and |
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68:33 | give you about five days before you to turn it in. So, |
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68:37 | , uh, you choose a time included in there will be a |
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68:42 | And so you choose a time when have several hours of, uh, |
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68:49 | , and of no interruptions. So you have a AAA family, uh |
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68:56 | find a place where you can be and concentrate with your books open and |
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69:02 | computer open, everything is open and do the, uh, the exam |
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69:07 | M Juan City uh on your on our system. Hello, I'm |
|
69:17 | to give you a number of It'll be, you know, about |
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69:22 | 20 questions with, uh, some of them will have several parts |
|
69:27 | that question. And so I'm not good at, um, making |
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69:38 | So, what I try to do I try to make exams, |
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69:42 | which if you're, uh, if you understand everything, you can |
|
69:48 | the exam easily in two hours. if you have to consult with your |
|
69:54 | and so on, it might take three hours but you might want |
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70:00 | So choose a time when, you have a, a all the |
|
70:06 | you can devote to it and, , uh, uh, uh, |
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70:10 | normally I find that people who spend time do better. So, but |
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70:17 | people spend hours and hours and they do well at all. And so |
|
70:21 | people have not understood the material. , uh, I'm, uh, |
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70:27 | gonna be writing questions, which some them will be easy and some will |
|
70:33 | hard and I really don't expect that is gonna get everything right. |
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70:41 | I'm gonna expect that you're gonna be , uh, uh, uh, |
|
70:46 | , uh, out of a maximum of 100 I think that the scores |
|
70:50 | probably be, be between 8050. , uh, uh, so |
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70:59 | uh, I will decide how uh, uh, assign letter grades |
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71:05 | that. So, uh, if, uh, recognizing that I'm |
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71:12 | very good at writing an exam, if we have two of you |
|
71:17 | you know, uh, uh, and 80 the third one gets 40 |
|
71:22 | that's clearly a difference in learning. if, uh, if two gets |
|
71:28 | and 80 the other one gets, , uh, 70 that's pretty |
|
71:32 | And so I will make a judgment , uh, uh, how to |
|
71:38 | a letter break that if we had large class, if we had 50 |
|
71:44 | in the class, I would be on a curve and I would say |
|
71:49 | average grade is gonna be a B so that it's gonna mean a, |
|
71:53 | certain number of A's and a certain of CS and a lot of BS |
|
71:58 | a small class like this doesn't, does, doesn't make sense. So |
|
72:03 | will uh uh be making um a about how to assign a letter grade |
|
72:12 | you turn in the exact. Mhm. My experience is that when |
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72:22 | have an exam like this, which open book, unlimited time. You |
|
72:28 | a lot during the exam. Oh was my experience. The first time |
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72:34 | took an exam like this, I lost for the entire course. But |
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72:38 | the exam, uh uh I finally because it was open book unlimited |
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72:46 | And finally, I figured it So I hope the same is true |
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72:49 | you know. So that's the way grading is gonna go in this |
|
72:55 | So let's look at the next uh quiz questionnaire. So here's a statement |
|
73:02 | isotropic rocks, the normal stiffness component controls the uh the heat wave uh |
|
73:09 | velocity. Is that equal to Is that true or false? So |
|
73:15 | me turn to uh uh Mesa Can you hear me? Can you |
|
73:26 | ? I'm thinking, I think it's . Uh So that uh uh that |
|
73:36 | controls the p velocity. And so uh we know that K measures the |
|
73:42 | to pressure. So I think that's you're thinking of here. Uh But |
|
73:47 | go back we uh uh uh if had a question like this on the |
|
73:51 | test and say, oh, let's , let's go back to the uh |
|
73:55 | the book is open. And so it is. And here is the |
|
73:59 | position and that's not A K, an M, I remember the M |
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74:05 | related to uh image related to K this way, K plus four |
|
74:13 | You, so you uh uh uh answer was wrong. You, you |
|
74:17 | sort of guessing, but I, you were doing this on the |
|
74:21 | you wouldn't be guessing, you would looking it up. Uh But as |
|
74:26 | look it up, uh and that's some time. And so if you |
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74:30 | that on every question, you after two or three or four or |
|
74:34 | hours, you would sort of run of time and run out of |
|
74:39 | And so it, it's really important you to understand these things. So |
|
74:44 | don't have to look it up. PC three is equal to M not |
|
74:55 | to K. That would be OK. So we're almost finished with |
|
75:02 | first lecture and here we are uh hour and a half into the second |
|
75:09 | . That's OK. We're uh we're gonna be fine. So let's |
|
75:12 | about anisotropy. So, uh if have um uh a crystal, if |
|
75:19 | look at any crystal, it has , a, an uh uh uh |
|
75:22 | exterior shape with uh uh all his on the, on the side. |
|
75:28 | so, obviously, that is gonna to uh uh that's obviously due to |
|
75:35 | internal arrangement of the atoms and those uh arrangements are going to make for |
|
75:42 | isotropic wave propagation. And in that , we need all of these 21 |
|
75:47 | . So that's mm clearly, it's nothing we can deal with in the |
|
75:55 | . We uh uh when we do experiments, normally, we measure only |
|
76:00 | waves, we might also uh uh shear waves. And Utah is measuring |
|
76:07 | surface waves which uh uh a combination those uh uh uh you just can't |
|
76:17 | in the field ever measuring 21 Uh So uh uh lucky for |
|
76:25 | uh they don't have to, even the uh uh uh uh rocks are |
|
76:31 | out of these kinds of crystals which the crystals require this kind of analysis |
|
76:40 | . So, for uh uh for rocks, uh the uh the grains |
|
76:46 | not all aligned. So the grains a lot are uh oriented in a |
|
76:54 | way. And if it's perfectly it comes out to be isotropic, |
|
76:59 | um uh it's not always perfectly So for the simplest geophysical case, |
|
77:05 | gonna find out that the stiffness sensor not 21 elements but five different |
|
77:13 | And you can see here the bunch zeros out here. You see |
|
77:17 | there's a 11 C 11 is different C 33. So what that means |
|
77:23 | that the P wave velocity in the direction and the horizontal direction is different |
|
77:29 | P wave velocity in the uh vertical . And similarly, there are two |
|
77:37 | cheer mod line here. So we'll more about that later. And then |
|
77:42 | 1/5 parameter. C 13, which off to the side here. And |
|
77:46 | there's a calculated parameter right here. this is the simplest geophysical case. |
|
77:53 | this is corresponding to shales un fractured . And it's also um applies to |
|
78:04 | bedded sequences. Think about this. you have a bunch of layers where |
|
78:08 | long wavelength seismic waves traveling through those many thin layers with a long wavelengths |
|
78:15 | many bed thicknesses, it's gonna travel a different velocity uh depending on the |
|
78:24 | it's gonna travel with a different velocity and horizontally in between. So we |
|
78:31 | that uh in such a situation, uh long wavelength cy waves going through |
|
78:39 | thin sedimentary beds, that wave is as though it were a uniform anisotropic |
|
78:49 | . So you know that's a pretty um description of most sedimentary rocks uh |
|
78:54 | thin layers compared to the seismic So that's why we say that in |
|
79:00 | , in almost all seismic situations, the waves are propagating and isotropic. |
|
79:08 | that means that everything we learned uh and today. And for the next |
|
79:12 | lectures about isotropic rocks. That's only beginning and real locks are mostly an |
|
79:23 | . Yeah. Um I know a about that. My uh uh uh |
|
79:29 | an expert in such matters. And uh I will share some of my |
|
79:35 | with you on the in the 10th . But for now I'm just gonna |
|
79:39 | you this glimpse, to show you it's gonna be complicated and to warn |
|
79:45 | that riddle walks are more complicated than um uh are learning about in these |
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79:53 | lectures of S Boys and Race. as a summary of that, uh |
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79:59 | we learned at the very beginning is is the study of the deformation of |
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80:04 | materials under stress. And that simple means materials like and glass are like |
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80:12 | doesn't mean rocks. So everything we out here is going to be uh |
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80:18 | directly applicable to rocks, but I that you have been applying it uh |
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80:24 | rocks uh in your entire career. that should uh bother you. Uh |
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80:30 | if Hook was doing stuff only for materials, how can we get away |
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80:36 | applying it to complicated materials like rocks if they're isotropic, consider a sandstone |
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80:44 | . Uh it's got grains and pores uh uh so that's not a simple |
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80:51 | like uh Hook was thinking of. , the grains are little crystals, |
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80:58 | don't have polished faces, but they have internal uh arrangement of the |
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81:03 | which means that every single grain on smallest scale is an isotropic and a |
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81:09 | , those are all gonna be oriented . So the sandstone itself is gonna |
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81:14 | out to be isotropic. But there's situation that uh hope never dreamed |
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81:21 | So we're gonna deal with those issues in the course. So what is |
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81:29 | and what is strength stress is the bringing an area applied to material since |
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81:36 | forces of the vector and since the is specified by a vector, which |
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81:42 | normal to itself, that specifies the of the area. Because of |
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81:47 | the stress is a symmetric three by tension. As we learn that strain |
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81:55 | a measure of deformation, a non measure of deformation. It's defined independently |
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82:02 | the coordinate system. So we talked links, we didn't talk about uh |
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82:06 | uh deformation in the one direction and so on. We combine those |
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82:11 | different into uh showing how the length . It's also a symmetric three by |
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82:17 | tensor. And Hook's law is crucial us. Uh back in the uh |
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82:24 | the 17th century. Imagine that hundreds years ago, h had some |
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82:30 | did some thinking, sitting in his , I think in London and his |
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82:37 | have come down to us hundreds of later, enable us enabling us to |
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82:42 | oil and gas. That's kind of to think about that. So what |
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82:45 | he say? He said he, he made the assumption that stress and |
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82:49 | are proportional to each other in a way. And he did not specify |
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82:56 | one is either cause or effect. that should bother you. Uh uh |
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83:01 | had some disagreement in the, in small class, we had some disagreement |
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83:06 | whether stress causes strain or strain causes . So we did not resolve those |
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83:13 | . You should continue to think about on your own. And we'll resolve |
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83:18 | question by the end of the So then we de define elastic compliance |
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83:27 | the ratio of strain distress. It out to be a complicated tensor with |
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83:32 | different indices. And we can't think that in our minds because we can't |
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83:38 | it down in a simple way and visualize it. But lucky for |
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83:42 | , we can write all that information in the sensor as a six by |
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83:47 | matrix. Now, for isotropic these various components are have contained only |
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83:54 | independent parameters, Young's models and sure and also some of the elements are |
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84:01 | of these two. But that's not we need for wave fag. We |
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84:06 | don't know this yet, but we're need for wave fabrication, the inverse |
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84:10 | compliance, which is the stiffness also complicated tensor. But uh for isotopic |
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84:18 | , we uh I need to only two independent uh parameters, the longitudinal |
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84:25 | m and the sheer modulus m And then uh uh there are functions |
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84:32 | these which uh uh appear in the in the uh the six by six |
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84:39 | , but we don't need them And then we saw briefly that elastic |
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84:46 | is more complicated. So that's the of the first lecture. So, |
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84:53 | I wanna do is um uh I what I wanna do is stop sharing |
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84:59 | and then I want to um stop presentation, art another presentation, but |
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85:28 | gotta find it. So, um . So it's not here. And |
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86:00 | well, I'm gonna have to browse thoroughly. Uh don't uh despair |
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86:07 | I'm gonna find me. Oh, . OK. So, um now |
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86:35 | want to um share this. Um Can people see that now? |
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86:59 | three, the wave equation? You OK. Um Carlos, do |
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87:08 | see it? Yes professor. So I'm gonna go into uh uh |
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87:15 | mode here. OK. So now know about stress and strain and how |
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87:26 | are related. So now we're gonna that into, put them together and |
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87:33 | uh um how they make waves. , I have to uh reshare |
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87:48 | The screen two. OK. Are we good? So uh |
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88:04 | so um it's not responding here. we go. OK. So here |
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88:16 | uh uh um list of course OK. So, um the previous |
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88:24 | are, are gonna lead to what call the scalar wave equation for wave |
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88:29 | in fluids like the ocean. And then making more realistic assumptions. |
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88:38 | going to then find out how uh uh um uh a generalization of the |
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88:45 | WAV ation. We're gonna find this WAV ation with second. And then |
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88:49 | get my, we're gonna generalize this wave equation to the vector wave equation |
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88:57 | is gonna apply to solids. And gonna be uh uh isotropic solids and |
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89:05 | isotropic solids. So, you that, that's still not what we |
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89:09 | is it for applying for our field ? Uh But it's a big step |
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89:17 | from here. We're gonna only deal uh uh uh waves and fluids. |
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89:23 | here, at least we got solids , you know, we're, we're |
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89:28 | more realistic uh as we go. so then uh these equations are, |
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89:38 | not gonna have any source in They're gonna explain how waves it um |
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89:47 | know, propagate without even thinking about source. But we do have to |
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89:52 | about the source. So that's what gonna show next. And then uh |
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89:58 | gonna uh uh think about in homogeneity good because uh uh rocks are in |
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90:05 | , rock formations are in homogeneous. those equations are gonna have lots of |
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90:11 | solutions. And when we do AAA survey, we're gonna find rays which |
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90:19 | to our um uh receivers uh uh lots of different pathways, lots of |
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90:28 | pathways. Uh uh So uh it can get really complicated. And |
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90:38 | our geophysicists have learned over the last years how to deal with all. |
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90:44 | so we are going to see some the, uh, some of the |
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90:48 | of dealing with it. Now as , um, as a further |
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91:01 | we're gonna talk about the concept of reciprocity. And so I suppose you |
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91:09 | have heard of that and you might you understand it, but I |
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91:15 | uh I'm gonna show you something which gonna shock you, which will think |
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91:20 | make you uh think. Uh maybe don't understand it after all. But |
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91:25 | that's later, uh that will become in the afternoon. So let's begin |
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91:31 | , let's begin with the uh the year we're gonna find the scalar wave |
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91:36 | now. So I Newton was the first one who wrote down this expression |
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91:43 | uh uh how particles react to So, if you put a force |
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91:48 | a particle, it's gonna accelerate uh on the mass. Now, that |
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91:58 | was um uh uh a revolution in name because uh before Newton wrote that |
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92:11 | thought that when you have a force makes for a velocity. So you |
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92:16 | you uh you're pushing a, pushing , a um a block of wood |
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92:21 | the table, you're applying a force you make a um uh velocity knock |
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92:28 | acceleration. But uh Newton realized that situation of pushing, pushing a block |
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92:38 | the table means that the uh the forces on the block from the table |
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92:45 | canceling out the, the force that pushing on the side. So there |
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92:50 | uh constant velocity comes from zero zero, total force including the resistance |
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93:03 | on the uh uh from the uh on the bottom of the block. |
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93:11 | you don't have that resistance from, you get an acceleration instead of uh |
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93:17 | velocity. So that was really quite revelation back in those days. But |
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93:22 | not what we need for our We need to uh recast this for |
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93:26 | bodies. And I want you to about uh uh in the first |
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93:32 | we're gonna think about uh water. . So first we're gonna do |
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93:39 | So imagine inside of a fluid, have AAA volume element called a |
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93:46 | You might have uh you might not familiar with this term, but here's |
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93:51 | picture of a Voxel and it's a cube. And of course, |
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93:57 | since we're inside a fluid, it doesn't matter how the cube is |
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94:04 | Uh This word Voxel is uh a of a word, you might be |
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94:10 | familiar with pixel. So when you a two D image uh into its |
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94:16 | pieces, those smallest pieces are called . And so the 3D generalization is |
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94:22 | a Voxel, OK. Now, Voxel has uh uh uh it's a |
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94:28 | uh uh with the edge length that gonna call D and its mass is |
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94:34 | by uh uh uh D cube, is the volume times the density. |
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94:43 | , operating on this Vauxhall, there both body forces and surface forces. |
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94:48 | , uh the, the body forces , are like gravity. And so |
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94:55 | uh uh I was sitting here uh pulled down by gravity but uh canceled |
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95:01 | uh uh uh pressure from below surface from below. So we're gonna ignore |
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95:07 | body force and we're gonna consider only uh transmitted across the surface. |
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95:17 | yeah, where is this Voxel? , let's imagine an or a co |
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95:22 | quarter system here. We're gonna agree a, a right handed uh Cartesian |
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95:28 | system that has an origin over And um uh where is this Voxel |
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95:37 | with respect to the origin of Well, according to some of vector |
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95:42 | this vector stretches from the origin to center of the vox and the pressure |
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95:50 | the center of the Voxel here, gonna call E as a function of |
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95:54 | vector and we're gonna presume that the uh is uh may be variable in |
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96:00 | places. Yeah, it's gonna be at other places. For example, |
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96:06 | uh uh Here is the place uh is um as a um it differs |
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96:13 | the or from the center of the by this vector here, which is |
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96:20 | by uh three components, 00 and over two. You see the, |
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96:25 | full length of the edge here is . So this difference must be D |
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96:30 | two and it's in the three direction . Uh this uh uh uh delta |
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96:38 | is a three dimensions, the three in the three direction only. And |
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96:45 | doesn't have any com any component in one direction or in the two |
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96:50 | And so what is the pressure up ? The pressure is denoted by the |
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96:57 | at X plus this different fact. similarly, the pressure at the bottom |
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97:06 | given by this, where we have minus D over two here. And |
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97:10 | pressure on the side is given by uh I distance vector. In the |
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97:20 | direction. See this is in the direction, this is the one direction |
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97:24 | in the backside, it's a minus and then on the front uh uh |
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97:29 | the back uh uh we have uh different sections are in the two |
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97:36 | This is minus D and this one is D. So this one |
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97:41 | that's in the back and this one in the front. And remember that's |
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97:46 | the center of the Vauxhall center. Vauxhall is uh not showing in this |
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97:53 | . And what are the corresponding Well, they are uh uh uh |
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98:00 | time is the square of the um of the length. So this square |
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98:07 | the surface area and that this is pressure, this is the force pit |
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98:12 | . And here is the amount of units of the unit area. And |
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98:17 | see that we, we put in ad squared in front of all of |
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98:21 | things. So these are the So uh uh this is the force |
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98:27 | the one direction at the uh at , at the left side. And |
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98:32 | here is the force in the one on the right side. Seeing how |
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98:37 | notation works here is the force direction at the top. And here's the |
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98:44 | at the bottom of this one over . So that this has a minus |
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98:49 | here, this has a plus. adding up all these forces, we |
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98:56 | this expression. This is not a . This is just a comp this |
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98:59 | a sum with uh three components and is just adding up all those contributions |
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99:06 | we did before and look here, have some minus signs here. Let's |
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99:10 | sure where that comes from. See , we have a minus sign here |
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99:15 | it's pushing to the left, no sign here. It's pushing to the |
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99:20 | in the same way, we've got minus sign here, but we do |
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99:23 | a minus sign here. So uh when I put together this slide, |
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99:29 | was careful to get all of these signs in the right place. So |
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99:33 | have some minus signs here and then have places like this with a minus |
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99:38 | here. So all of that is correctly. Uh step by step, |
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99:42 | can check me out by going over uh uh afterwards. So here's all |
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99:47 | list left and right is here, and back is here, top and |
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99:50 | is here. Now, that's uh um it's for any pressure distribution. |
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99:59 | now let's figure out the sound way traveling vertically. So traveling vertically uh |
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100:06 | uh a vertical, either top to or bottom to top, we haven't |
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100:11 | yet. But in those cases, pressure did not, does not vary |
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100:16 | the horizontal direction. So that this cancels this one. You see there's |
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100:21 | minus sign here cancels this one. we have only these two terms and |
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100:30 | the pressure at the top pressure at bottom that comes from a sound wave |
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100:35 | vertical. Now, since this V is gonna be small, we're assuming |
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100:39 | gonna be small compared to the seismic we're going uh to approximate this |
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100:47 | which is what we had right here terms of use the tailor expansion, |
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100:53 | we talked about before. Remember the expansion says that when you have functions |
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100:59 | P uh and you know the value a certain position, you can find |
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101:06 | value at a nearby position by taking amount of the displacement, that's the |
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101:11 | two that's coming from here. And by the uh derivative of the function |
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101:21 | with respect to X three because this uh in the three direction. And |
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101:26 | gonna evaluate this derivative here back here New York or using that Taylor expansion |
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101:36 | uh the previous expression we have the forces given by this. Uh |
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101:42 | um right. Mm in the seg of this coalition, this is an |
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101:51 | slide and you can click here and yourself back to the previous slide. |
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101:56 | can't do that here. And we're put in here the Taylor expansion that |
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102:02 | just arrived and then notice that this cancels this one. And so um |
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102:08 | we're left with this and these two are the same. So we're gonna |
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102:12 | able to simplify further. And we that the force in the three direction |
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102:17 | given by minus the cube of the edge times this derivative, where does |
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102:25 | tube comes from? I see we these D squares here and then we're |
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102:29 | uh another factor D here and, uh we have twos uh uh |
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102:36 | So we get uh oh de cubed the derivative with a minus sign. |
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102:44 | you can see where that minus sign from. You can cancel all that |
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102:48 | follow and see where that minus sign from. And um so putting all |
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103:01 | in the way into the equation of , we have that force and the |
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103:07 | that we just arrived equals mass times acceleration, imagining this uh Voxel as |
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103:13 | particle and it's gonna be accelerating in three directions. So we rearrange |
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103:21 | So we uh uh find that the is equal to minus one over the |
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103:26 | times this derivative. Now, because this notation, we uh uh because |
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103:36 | the um time we spent yesterday to about uh vector notation, we can |
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103:44 | generalize this for motion in any direction by changing these threes into eyes. |
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103:52 | we have uh uh uh I uh and we can use any I |
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103:59 | we, we were restricted to IOS . But now because we're so |
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104:03 | we spent the time yesterday to, develop this notation. Uh We can |
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104:11 | it is for a P wave traveling any of the three directions. And |
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104:18 | have a name for this here. called the gradient operator. OK. |
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104:25 | this is a good time to um to follow this link and we're gonna |
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104:37 | this link back to uh uh the 101. So I'm gonna stop sharing |
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104:45 | . Yes, ma'am. And I'm share my screen, let's see |
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105:00 | Um Not sharing the screen yet because want to bring up, I know |
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105:15 | math zero and um oh you Sure. Um OK. Yeah. |
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105:38 | when we, when we did math yesterday, we didn't do the whole |
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105:51 | um because we didn't quite need it . I thought I had myself set |
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106:11 | there but I did not mm You stop present the old one. Shut |
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106:24 | . So that, and what was one? Those three things? |
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106:41 | That sounds good. Yeah. yes. I, I was gonna |
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106:53 | you if we can take a five break. Yeah. So, um |
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107:00 | uh that's a good idea. Let's a five minute break right now. |
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107:03 | by the time you come back, will figure this out. She was |
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107:11 | . So resuming now, uh really you see this on your screen? |
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107:22 | . So, uh what we have now, uh during the lecture, |
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107:27 | encountered a new notation which we uh have not seen before. I actually |
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107:34 | you have seen it before, but maybe some time ago. And so |
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107:40 | we need to do is discuss what notation means. So let's go back |
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107:45 | the um uh math 101. And uh uh I'm on slide 65. |
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107:54 | Well, just a second. I a pointer. So uh on slide |
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108:01 | of the math 101 file and there begins, we did all this other |
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108:06 | earlier. And so now we want talk about vector calculus. So we |
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108:12 | gonna define a symbol. Uh It's Dell. It's like an upside down |
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108:19 | . And uh um it's, it's as a vector and w we didn't |
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108:26 | a uh um uh an error on because uh nobody does the, the |
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108:32 | notation for Dell is without the So we're gonna stick with the standard |
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108:38 | , but it is a vector and , the different components of that vector |
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108:42 | partial derivatives with respect to the different um directions, either two directions or |
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108:51 | directions. And this operator operates on and tensors only from the left |
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108:57 | So we're gonna go here next. uh um if it's applied to a |
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109:05 | , I hear is Dell operating on scaler phi then um uh um that |
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109:15 | uh uh a vector which has these three components, let's go |
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109:21 | And you can see that if you this quantity uh uh uh like a |
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109:28 | of a vector and you're operating on scr, then that vector looks like |
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109:36 | partial derivatives in each of the three directions that can also be applied to |
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109:44 | vector. And uh uh uh uh that happens, it makes a scaler |
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109:52 | uh because uh uh according to the product which we defined yesterday, uh |
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110:00 | we're summing over uh uh equals And so that's uh uh there are |
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110:06 | unpaired uh indices on the right side that. So this is a scaler |
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110:12 | dot vector. And also we can del cross a vector. In which |
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110:20 | , it, it has this more expression as a as a vector can |
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110:28 | applied. Uh uh uh It can be applied to a vector uh to |
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110:34 | uh this gradient tensor. So you here's del with uh uh uh Here's |
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110:46 | dot U, here's Del Cross here here is Dell times you in which |
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110:56 | oh we have a um a This is a two by 210. |
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111:05 | I probably should have made this a by 310. And it can be |
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111:14 | to a tensor to make a, gradient operation on the tensor. And |
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111:18 | leads up to a vector. You how that's a vector we're summing from |
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111:23 | over eyes. And we have here loose J on the side. So |
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111:28 | is has a single, un And furthermore, it can be applied |
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111:39 | to a scaler to make another So if you work through our previous |
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111:45 | uh definitions of del dot and del the scalar, you, you'll see |
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111:54 | uh it's equal to the uh the equals 123 of the second derivative |
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112:00 | of this uh of this, of scaler. And that's also a |
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112:07 | Uh And so this is named after , that's a picture of him and |
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112:16 | can be applied twice to a So you see how um um this |
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112:23 | as we defined it um uh six slides ago can actually be uh uh |
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112:32 | in a variety of different ways. uh just before we go back to |
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112:37 | lecture, I will show you that are various calculus identities uh which uh |
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112:45 | will not prove, but we uh uh universally true. So I'll just |
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112:51 | show here that if you have del a gradient that's a Z that always |
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112:59 | out to be a zero. And del dot uh uh uh this operation |
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113:05 | called the curl operation del cross that's also a zero. And that's |
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113:14 | scaler. And then here's a, vector quantity here. And so, |
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113:20 | words, we say that this one the curl of a gradient. So |
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113:26 | del cross is called a curl This is called a gradient operation. |
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113:32 | what the equation says, the curl a gradient is zero. And this |
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113:37 | uh here's another curl because we're at dell cross uh a vector. And |
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113:45 | we're taking the dot product of And so that's called a di the |
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113:51 | , a divergence of a curl and . And why would we introduce this |
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113:58 | notation for uh uh for vector calculus dealt with all its different manifest |
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114:10 | Well, here's a good reason, fair. Uh uh We don't always |
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114:20 | , um we don't always use a coordinate. Sometimes we use spherical |
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114:26 | And in that case to a an operator looks like this in terms |
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114:31 | various derivatives with respect to R in to angles and respect to the other |
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114:38 | . And uh so why would we to do this? Why would we |
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114:44 | to use F coordinates? We were fine with um uh Cartesian coordinates. |
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114:51 | was easy let me just back up . I'm gonna back up. |
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114:56 | So uh the laplacian operator L squared the scalar has a simple expression like |
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115:04 | . Like so um in um uh co system, just the sum equals |
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115:14 | of the second derivative of that same three terms expressed in one expression. |
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115:20 | like, so that's for Cartesian um um Lalas operator and trust me, |
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115:28 | will uh this Laplace operator will show in our equations very soon. And |
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115:34 | has a simple expression and artesian But in TRL coordinates look how complicated |
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115:45 | , it still has two derivatives but complicated. So why would we want |
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115:51 | do that? Previously, we talked using spherical coordinates for uh uh uh |
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115:59 | academic geophysicists, friends who study earthquakes throughout the sphere of the earth. |
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116:06 | there's an, an example, there's AAA better example more applicable to us |
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116:12 | aspiration. Ge thinks right here when have a, a source say uh |
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116:20 | uh a a dynamite source think of dynamite source in AAA conventional survey. |
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116:28 | that uh is um a source where waves are spreading out in all directions |
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116:35 | like spherical waves. Uh maybe uh uh is a little bit better. |
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116:46 | think instead of dynamite on land, think of dynamite in the ocean. |
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116:52 | he put a, put a charge dynamite in the ocean, making a |
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116:56 | source and it sends out waves in directions, uh And some of them |
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117:01 | down and some of them go up , and uh hit the, um |
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117:06 | surface of the water and get reflected down. So that's a complication that |
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117:09 | need to deal with later. But uh uh aside from that, you |
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117:14 | see the waves are expanding from uh point source in all directions. And |
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117:22 | it's gonna be convenient for us to that in terms of spherical coordinate |
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117:28 | And we can do it trivially, we uh uh uh once we find |
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117:34 | laplacian operator, you're gonna find that up in the wave equation. And |
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117:39 | can trivially apply that either to partition or to circle coordinates if we use |
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117:46 | symbol here. OK. Now, of course, uh you, you |
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117:52 | that we don't typically use dynamite for in the ocean. We use |
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117:57 | but it's the same stuff. Um , um in the borehole, we're |
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118:05 | be uh using uh we're gonna imagine traveling in a borehole. And |
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118:09 | in that case, the, the and operator has this form in terms |
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118:13 | derivatives with respect to axial direction and to radial direction and with respect to |
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118:20 | direction. And so, uh I pose a question and of course, |
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118:26 | uh could be useful for um uh the bar. Now, I'm gonna |
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118:35 | show you here on the uh in math 101 file there is a discussion |
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118:41 | compliance and you might have seen this and you thought, oh, that's |
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118:45 | . This is strictly elasticity whereas all other stuff has nothing to do. |
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118:51 | , uh, much more broadly applicable is elasticity. And right here is |
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118:56 | explanation of why we had those, , strange factors. 1/4 coming up |
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119:03 | the earlier discussion, this lecture and gonna leave that here for you to |
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119:10 | on your own. Not gonna do . And uh in front of you |
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119:16 | , I just point out that here's it is. OK. So what |
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119:20 | gonna do now is stop sharing And I'm gonna uh start sharing singer |
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119:50 | I want insulin. Wait. So folks can you see this? |
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120:06 | , that's presenting our progress. Mhm mhm This one. No uh |
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120:26 | one and OK. And I can . Yeah. OK. I've got |
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120:39 | learn how to do this better. uh this is where uh uh |
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120:44 | we switched back to uh uh the file. We uh switch when we |
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120:57 | this. And can you see Uh This uh Dell symbol is now |
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121:04 | uh the same as we showed before we showed it, Dell operating |
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121:08 | a scale of five here, Dell operating on a scale of P for |
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121:13 | . And so this expression here, I need to order this expression here |
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121:23 | uh expressed in different ways using the uh gradient operator. Dell and that's |
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121:33 | the definition of, of DELL times scr it's a vector and the components |
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121:40 | the vector are given by this. , all of this discussion is valid |
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121:47 | for fluid media. Why is Because we assume that the pressure is |
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121:51 | same on all sides of the how I have a um quiz |
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122:00 | So um uh Carlos uh is this statement true or false? So the |
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122:14 | of motion from Isaac Newton, is the starting point for this derivation of |
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122:21 | scale of ever? Let me, me just back up one slide. |
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122:25 | here, here we have um uh no. Um the scalar wave equation |
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122:33 | not in its final form, but almost final form. And so now |
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122:38 | question is, did we uh get from Isaac Newton? Is that true |
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122:44 | false? I would say it's Yeah, that's true. Of |
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122:48 | Yeah, that question. No sound . Next question is this true or |
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122:55 | ? Sound waves are driven by So they would be slower on the |
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122:59 | where gravity is less? Is that or false? Uh uh Lily, |
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123:05 | one's false. Of course. Next is uh on the equation the |
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123:14 | the pressure at the left center of voxel is given by which for these |
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123:19 | um here's our coordinate system and imagine voxel in here. And uh you |
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123:26 | answer this question by going back to the original pictures, uh, that |
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123:33 | , uh, showed or maybe you just examine it. Um, |
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123:41 | um, verser, let me call you for the answer. Is |
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123:45 | is it a ABC or D talking the left center of the fox? |
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123:54 | the center and a, and here , uh, the, uh, |
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124:00 | , place. Uh, it's, a so, no, so left |
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124:07 | no, it's C, yeah, because of the minus here. |
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124:11 | uh uh A is the right center the Vauxhall and the uh B is |
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124:17 | top of the Vauxhall and D is , the back side of the box |
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124:23 | the rock. So, because this in the true position, very |
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124:28 | Now, let me see here. question is um now, I'm gonna |
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124:43 | on Carlos for this and there's this , true or false. The equation |
|
124:47 | for a wave traveling right to left that given by this expression here. |
|
124:54 | , uh this is a, a challenge for Carlos because we didn't, |
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124:59 | didn't present this. He can't find answer to this by looking back in |
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125:05 | past uh derivation because in the past , we only considered waves traveling |
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125:12 | Now, this is a question about traveling right to left. So um |
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125:18 | uh uh to get this right, is gonna have to understand what we |
|
125:23 | about before and apply that to the uh question of traveling right to |
|
125:30 | So, uh uh Carlos what do think? I, I would say |
|
125:34 | true. Yeah. Yeah. The difference between this expression and the previous |
|
125:40 | is we got ones here instead of . So you see how easy it |
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125:44 | we uh uh we tr we jumped vertical uh propagation to um uh hm |
|
125:58 | horizontal propagation just by changing index. , no sweat throat. No. |
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126:09 | here's our expression for propagation in any . Now, it's a vector |
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126:16 | See it's got vectors on the left , and vectors on the right. |
|
126:19 | uh uh uh let's try this, us operate from uh uh let's take |
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126:26 | derivative of each of these things by from the left on the, on |
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126:33 | left side and on the right side the operator partially with respect to X |
|
126:38 | I. So on the right, expression is spread in terms of |
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126:46 | of the Dell operator. Um in this way, go back and |
|
126:51 | what we uh said about Dell dot vector is exactly what this is. |
|
126:59 | so the same thing here del dot and that's in vector notation. So |
|
127:05 | left side of this equation is uh we just did is a derivative with |
|
127:11 | spec to X I of this derivative respect to time of the displacement, |
|
127:17 | can interchange the order of these. so now we have the derivative of |
|
127:22 | , of the uh with the spec X I is operating directly on here |
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127:28 | , on the displacement, the I of displacement. And all of |
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127:33 | we're taking the second rib. this quantity is the strain. If |
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127:46 | look back at how we define uh uh uh the uh uh this |
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127:52 | uh the, the sum of the the sum of the derivatives uh uh |
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128:01 | uh a 12 and three that expressed terms of strain like this. And |
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128:05 | had a separate um um word for , this some of the diagonal string |
|
128:12 | is called the dilatation. And we found the in compressibility was related |
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128:22 | the dilation in this way. And the left side of the wave equation |
|
128:28 | looks like this and that's really uh because we don't have the displacement |
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128:37 | We have the, the pressure and don't have any uh uh uh uh |
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128:44 | any special derivatives at all. On left hand side, all we have |
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128:48 | is uh time derivatives. And we the material properties right here. Now |
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128:56 | that back together where the right hand let's go back here. Here is |
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129:04 | , the right hand side here that question index notation, it looked |
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129:08 | this. And so by, by that, uh uh by doing this |
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129:14 | , we convert it, we we transform this displacement here into a |
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129:21 | term. And now we have only . Now we have the uh the |
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129:26 | the unknown here is clearly pressure. , one more thing, let's assume |
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129:35 | the uh the medium is uniform. in the ocean, maybe that's not |
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129:40 | bad uh uh uh uh approximation. that, that means that uh this |
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129:46 | is the same um uh at the of the Vauxhall, the bottom of |
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129:52 | voxel left and right. And this does not change on the small |
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129:59 | And let's assume the same thing about material property K. So we can |
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130:03 | these material properties outside of the Like we show here. And you |
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130:10 | how this is getting to be pretty uh uh uh uh much, much |
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130:15 | simple. We find after this we find that the, the uh |
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130:21 | derivative of the pressure, this time proportional to the second derivative of uh |
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130:28 | the pressure with respect to these three of strain, three components of |
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130:34 | three components of position. That's what trying to say. And what is |
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130:40 | uh what is the proportionality constant is which is K called K over |
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130:49 | So the unknown here is the pressure width, position and temperature. We |
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130:56 | two derivatives. We spend the time derivative of two derivatives with spec to |
|
131:02 | . And here is the proportionality And so this is the actual equation |
|
131:08 | governs the waves which arrive at a in marine acquisition. So imagine marine |
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131:18 | , you have a, a boat through the ocean. It's got it's |
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131:22 | a source behind it. An air source, that air gun source is |
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131:26 | out waves and the uh uh propagating through the ocean also up through the |
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131:33 | of the surface reflecting back down, it goes into the rocks where reflects |
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131:38 | the rocks comes back through the water the receivers which are towed behind the |
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131:46 | source boat in a long string of may be as long as 10 kilometers |
|
131:54 | . And since the way, since uh uh receivers, we call them |
|
131:59 | , they're sitting in the water being through the water. And um, |
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132:09 | the waves are arriving through the water in the propagation history, they've traveled |
|
132:15 | , through the rocks, but as hit the waves as they hit the |
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132:20 | , they're traveling through the water. this is all we need to |
|
132:25 | This equation governs waves arriving at a in ma in standard marine acquisition. |
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132:33 | , uh there's a complication here which the, uh the boat is |
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132:38 | It's not just sitting there, it's uh uh through the ocean, uh |
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132:44 | out uh uh uh energizing the source few seconds, maybe every 30 |
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132:51 | Uh And so the receivers are also . And so we've ignored the fact |
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133:00 | the receivers are moving here. But the, the velocity of the, |
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133:05 | the receivers through the water is slow to the um uh velocity of the |
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133:13 | coming through the water. So we ignore that fact. Now, so |
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133:22 | the wave equation or in more compact notation, we can write it here |
|
133:28 | terms of the Laplace operator. Remember , I warned you that the Laplace |
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133:33 | was gonna show up and right it's all the complications hidden inside the |
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133:45 | Pasian operator are hidden inside the notation we call it DEL square and everybody |
|
133:52 | that's the Lalan operator and in our coordinates, it's this sum here. |
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134:00 | as you saw previously for in other systems, it can be a lot |
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134:05 | complicated. So uh we are clever to hide all those complications uh inside |
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134:13 | little plus an operator. And by way, the waves don't know anything |
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134:19 | our Cartesian Corridor system. That's all our imagination. So the waves don't |
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134:26 | or care about these indices here. the laplacian operator is designed in such |
|
134:32 | way that it doesn't care either. gonna be the same. Uh whether |
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134:38 | have uh a Cartesian coordinate system with vertical axis, vertical or horizontal or |
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134:45 | in between all those complications arising from choice of coordinate system. That's all |
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134:54 | inside this laplacian operator. So that's very clever thing we did to uh |
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135:01 | make sure that the wave equation is dependent on our choice of court |
|
135:11 | analyzing the waves. Uh It's gonna uh easier um in one coordinate system |
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135:18 | the other. But remember the waves know or care about what we think |
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135:23 | the origin of this cord system. on the left side, we have |
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135:27 | second r respective time. So you know, corresponding to the |
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135:34 | But here we don't have acceleration of , we have the change of pressure |
|
135:39 | divided by time. I look at uh um this proportionality constant here, |
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135:51 | got to have the di the dimensions length over times squared. Why? |
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135:57 | because we have uh uh this has have the same physical dimensions on both |
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136:02 | . So we have the pressure on sides. But here we have times |
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136:06 | and here we have distance squared. this thing must have the dimensions physical |
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136:11 | of length over times square. So the same as velocity square. And |
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136:16 | we can just give that in your V squared. And here's the definition |
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136:22 | V square, it's trail a And who knows what that velocity |
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136:27 | All we know is that it's a with the dimensions of uh velocity |
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136:36 | What does that mean? Does that the velocity of the wave, the |
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136:39 | of the particles within the wave, velocity of the boat as it goes |
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136:44 | the water? What does that We do not know at this |
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136:49 | All we know is that we've derived expression and we've given the proportionality constant |
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136:55 | you name the square. So we up with this scalar wave equation. |
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137:05 | , we can do this in another . Uh where the unknown is pressure |
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137:09 | . That kind of makes uh the for us, doesn't it? But |
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137:14 | could have done it another way so we had derived a wave equation with |
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137:19 | same form except the unknown is, the dilatation. And uh it has |
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137:27 | , it has exactly the same exactly the same coefficient here. But |
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137:32 | the unknown is dilatation instead of That's interesting, isn't it? You |
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137:37 | do it one way or the other out the same. But we're gonna |
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137:41 | to analyze the pressure version. Now, let's consider waves only traveling |
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137:53 | . So if it's only traveling this uh vector operator simplifies and we |
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138:00 | uh uh show it this way everything in this equation is the same. |
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138:05 | this is your only way of traveling . Now, the, the solution |
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138:13 | gonna be a function which varies only uh uh uh uh with Z and |
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138:18 | T for this situation where we're failing , we're not gonna have any variation |
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138:26 | the spect action. Why? And this equation says is that whatever the |
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138:32 | is, it's gonna depend only on difference between a time term and a |
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138:39 | term. And uh uh the uh can have either a plus or a |
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138:46 | here and the time is gonna be by um parameter which we call omega |
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138:52 | the depth uh is gonna be uh by a term that we call K |
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138:58 | . OK. So what, what this mean? That means I'll say |
|
139:03 | again. The solution is gonna only variations in this quantity phase. No |
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139:13 | combination of T and Z are gonna only this combination. And that's |
|
139:23 | Yeah. Uh So that's true for way of traveling vertically. So we |
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139:30 | names for this, this omega is the angular frequency. And uh the |
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139:36 | um we don't say angular and we frequency, that means frequency, the |
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139:42 | per second and the angular frequency is the cycles per second by a two |
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139:49 | which means as uh you can think it as radiance per second. Oh |
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139:57 | , this uh arrows uh should be over here to K three, not |
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140:03 | time. Uh This is point in center. So this box here, |
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140:08 | call out box is referring to K . That's the vertical wave number. |
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140:13 | if you want to know more about wave number, you can go to |
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140:16 | glossary, look up wave number and do that on your own time. |
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140:23 | , let's just verify that this um function actually solves the wave equation. |
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140:30 | uh um uh it's uh I'm gonna here this solution uh with respect to |
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140:40 | and space, we can only have in terms of five. That's what |
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140:45 | said back there. So let's just that. So uh on the left |
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140:49 | of that equation, we've got um uh uh two derivatives respect to |
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140:56 | Let's separate them out, dr them at a time. And so, |
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141:00 | this uh question, um um really pressure with the strength of time can |
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141:07 | expressed in this way, uh using rule calculus. So all we do |
|
141:14 | put in a, a partial inspector here, partial inspector Phi here. |
|
141:21 | um, now let's do the same of thing. Um um Let's uh |
|
141:28 | me. What is the partial R the Phi where Phi with respect to |
|
141:34 | , I'll back up. And here's definition of Phi, so that a |
|
141:39 | of Phi with respect to time is omega. So that's why we put |
|
141:44 | the omega here for this. And we do that same sort of thing |
|
141:50 | , uh uh uh uh we're gonna a, another d of the spec |
|
141:55 | time, we do it again and end up with the left side of |
|
141:58 | wave equation is omega square times the repeat with respect to thought. Uh |
|
142:09 | converted a con uh we converted a of the spec of time into a |
|
142:14 | with respect to five. Likewise, right side of the equation uh uh |
|
142:20 | to a similar thing and we get uh on the right side, it's |
|
142:25 | same V squared as we start out plus we have an additional K three |
|
142:30 | because uh and it's gonna be good either a positive K three or a |
|
142:35 | K three. And the same derived respect to uh uh phi of |
|
142:41 | And so these two sides, the side and the left side are gonna |
|
142:48 | equal for any function pressure as any of five if and only if we |
|
142:56 | this expression here relating K three to over V. So we gotta have |
|
143:03 | squared equals V squared times K three . And that condition relating the angular |
|
143:12 | with the vertical wave number through the . And uh uh uh and you |
|
143:21 | a plus or a minus here. what is this solution? No, |
|
143:28 | think that you are familiar with this wave solution here. So here uh |
|
143:36 | uh uh this, what I've written here is called a the plane wave |
|
143:43 | for the pressure. And it's expressed a constant time is an exponential with |
|
143:50 | in the exponent we have I times recognize this right here is the five |
|
143:58 | this um uh uh this wave, uh plane wave solution is good for |
|
144:06 | value of fry here. Now, just pause here and I talk about |
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144:14 | number E. That's, this is first time we've encountered E in this |
|
144:20 | . Are you familiar Lily with Oiler ? E? Have you seen it |
|
144:26 | ? It has very special properties invented this guy Oiler? Yeah. |
|
144:36 | So uh uh I think your thanks for uh being uh direct and |
|
144:42 | here. You've seen it before, you're not quite sure. So Oiler |
|
144:47 | is a special number like pi, think all of, you know, |
|
144:51 | pi is 3.14159 blah, blah, . It's a very special number. |
|
144:58 | Oiler number is another very special It has a value of 2.715 blah |
|
145:05 | blah. And those uh it's a i it's an irrational number and those |
|
145:11 | go on forever. And it it's a special number which was discovered |
|
145:16 | Oiler uh which was a, a genius. He made the smartest guy |
|
145:22 | ever lived in the 19th century in . And he discovered this uh uh |
|
145:29 | number which happens to be close to , you know, uh that's kind |
|
145:34 | um uh uh interesting. Why should be close to P it, it |
|
145:38 | very special properties and um um you , why, why this is very |
|
145:46 | number? Why is it not uh know, uh 37 0.251 you |
|
145:55 | it happens to be close to pi I would say nobody knows why it |
|
145:59 | to be close to pi but it's very special number. And so it |
|
146:04 | very special properties which you can look in the glossary. So I encourage |
|
146:09 | to look up that uh uh not , but uh uh uh look it |
|
146:14 | later and has lots of magical properties we will be uh showing some of |
|
146:20 | properties uh later in this course. uh it's such a special number that |
|
146:30 | we remember uh Oiler most, mostly remember Oiler by because he discovered this |
|
146:37 | and we call it e in his . Now, e is raised to |
|
146:46 | power I to the five. And you should know about I, which |
|
146:51 | the square root of minus one. let me ask, uh um uh |
|
146:56 | . Carlos, are you familiar with concept of square root of minus |
|
147:04 | Um I think I have seen it but uh no, OK. |
|
147:10 | yeah, so uh uh um uh . Have you, are you familiar |
|
147:14 | Enn? I? Yes, I . OK. Well, that's |
|
147:19 | Uh uh because uh you are working slumber, you will have uh encountered |
|
147:25 | many times. Uh Carlos is a . So he uh is not so |
|
147:30 | with this, but I is called imaginary number. And so if you |
|
147:36 | a square of I, that gives to be minus one. And so |
|
147:41 | can it possibly be true that the of any number is negative? |
|
147:47 | there are, uh there are a bunch of people who have wondered about |
|
147:54 | for centuries. So let's sort of through the um uh the elements of |
|
148:04 | theory. So let's start off with idea, one plus one equals |
|
148:11 | So uh uh let me say one two equals three. So what are |
|
148:16 | numbers? 12 and three? Those abstract ideas, abstract ideas. |
|
148:25 | we apply them in physics by attaching to that. So we can say |
|
148:34 | apple plus two apples makes three OK. So that suddenly removed from |
|
148:42 | mathematics into physics says if we have apple and you add together two apples |
|
148:50 | three apples. But what if one is uh uh uh uh what if |
|
148:55 | one, the, the, the apple is a big apple, a |
|
148:59 | delicious apple, a type of Uh and uh uh the, the |
|
149:04 | other apples, suppose those are um honey crisp apples. So what's the |
|
149:11 | of one delicious apple and two honey apples? Is that still three apples |
|
149:18 | not? What if the honey crisp are small and the diligence apple is |
|
149:23 | ? Do we still get three apples the right side of that equation? |
|
149:27 | see that gets you into all sorts complicated discussions and mathematics. You don't |
|
149:33 | any of that, you say one two equals three because we didn't associate |
|
149:39 | was counting. OK. Now what that, so, but it was |
|
149:48 | uh Greeks back in Pha Pythagoras thought about these issues and then they |
|
149:55 | about what happens if you have. there any numbers between one and |
|
150:02 | Well, sure, there's 1.5, one and two thirds. And so |
|
150:07 | , they invented fractions and those are rational numbers. And there's an infinite |
|
150:15 | of rational numbers like that between one two, an infinite number. |
|
150:22 | And how about, if you think , uh uh uh are there any |
|
150:28 | between these rational numbers? Suppose you a rational number? One divided by |
|
150:36 | and right next to it, you one divided by 213. Are there |
|
150:40 | numbers in between there? Well, , there's an infinite, it turns |
|
150:47 | there's an infinite number of rational numbers those two numbers. And also there |
|
150:53 | uh a, an infinite number of which we call irrational numbers between the |
|
151:01 | numbers. And those are numbers like and E where you uh when you |
|
151:06 | them out in a, in a form, the digits go on forever |
|
151:11 | repeating. Uh You can uh define value of P uh to a million |
|
151:25 | significant figures. And you can uh can look that up. Uh um |
|
151:31 | If you go to Google scholar, can find a paper where they calculate |
|
151:36 | , the uh the digital representation of up to a million significant figures and |
|
151:44 | doesn't repeat anywhere. So those are generalization of the counting number. The |
|
151:52 | we can call those in integers or natural numbers. And uh uh it's |
|
151:59 | of mind boggling to think of how of them are there between one and |
|
152:04 | ? And, you know, uh more between two and three, there's |
|
152:07 | infinite number of them wherever you So, are we done, are |
|
152:14 | , uh this is all part of theory? And are there any other |
|
152:19 | of numbers? You take all of , if you, um, take |
|
152:24 | of the numbers that we just described take a square of that, you're |
|
152:27 | get a positive number. Are there numbers at all? Can you even |
|
152:34 | any numbers at all? Or you the square of it? And you |
|
152:38 | minus numbers? Wow, you can't that with apples and oranges, but |
|
152:45 | can do it mathematically. And all say is let's just assume that there |
|
152:51 | such a number and we'll call it , where you uh uh it's, |
|
152:57 | it's defined as the square root of one. Oh, we didn't talk |
|
153:02 | minus numbers. Everything I said about numbers. You can think about ne |
|
153:07 | minus numbers. And I still remember my mind's eye, my, when |
|
153:14 | was playing with my nephew who's now uh uh a young man uh uh |
|
153:21 | in a high tech uh um uh in California when he was a boy |
|
153:27 | four, I was playing with him the rug and I was playing with |
|
153:33 | uh uh uh teaching him mathematics, know, I was teaching him how |
|
153:37 | count. Uh uh uh what happens you have uh one apple uh plus |
|
153:42 | apples and so on. And that fun. Uh We plan for and |
|
153:46 | I said, what happens when you uh uh three apples, take away |
|
153:52 | apples and he figured out you get apple, pretty smart kid. And |
|
153:58 | I said, what happens when you three apples, take away four apples |
|
154:03 | the spot. This kid four years invented negative numbers. I was amazed |
|
154:10 | a smart kid he has and he out to be smart as well. |
|
154:13 | So that's negative numbers. Everything we about positive number, it has a |
|
154:19 | thing about negative number. And you see that it gets amazingly complicated. |
|
154:25 | it gets even more amazing when you that there's uh you can generalize these |
|
154:30 | in another way by imagining that there a number called I and when you |
|
154:38 | that number, you get a negative . Wow, a whole new class |
|
154:43 | numbers, imaginary numbers and any imaginary uh can be represented as a, |
|
154:51 | AAA real number times this odd. you can have imaginary numbers of any |
|
154:58 | . Wow, a whole new class um how the numbers. And so |
|
155:06 | next thing you say, oh What happens if you uh if, |
|
155:10 | you consider something where the, the the square root is um uh the |
|
155:19 | is consider number, call it. OK, we say that call it |
|
155:31 | so imagine a number. And the root of that number is I |
|
155:36 | the square root is not a minus . But this, if you square |
|
155:40 | number you get I, is that new class of numbers? Well, |
|
155:43 | turns out no, it turns out I is as far as you can |
|
155:48 | in this. Um And in this process, you can only go to |
|
155:55 | numbers. And there's nothing you can't anything more complicated than imaginary numbers until |
|
156:04 | get to vectors and matrices and things that. But I thinking about |
|
156:11 | there's nothing more complicated than imaginary to . That's quite uh amazing. Uh |
|
156:18 | would think that when you take the root of an imaginary number, it |
|
156:23 | be another faster number. No, , it's oh it's another imaginary |
|
156:30 | So that, that's worth thinking about it is that this process of um |
|
156:35 | a generalization ends at imaginary numbers. . So uh more discussion about I |
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156:44 | in the um um in the glossary that I is appearing here in our |
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156:51 | equation. And so I'm going to uh uh uh so because I is |
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157:01 | in the exponent up here, that that the pressure is not real. |
|
157:08 | pressure is I imagine has an imaginary to it. And so at this |
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157:13 | , you should be saying, how can I be measuring imaginary |
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157:21 | And so the answer to this is gonna ignore that question when we're dealing |
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157:34 | plane waves. And we're only going insist that we get real answers when |
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157:42 | propose an experiment. And uh we're do all of this manipulation using imaginary |
|
157:50 | and they're gonna be imaginary uh uh in the uh exponent. But that |
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157:56 | this, this whole thing ha it uh is uh we call it a |
|
158:00 | number. It has a real part , and an imaginary part. But |
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158:05 | instruments are real instruments. We can't imaginary quantities with our real instruments. |
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158:12 | I guarantee you whenever we work with equations and uh uh we have complex |
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158:22 | all through with the derivation. When end up at the end with uh |
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158:28 | uh an observable quantity, that one gonna be real. No, this |
|
158:36 | is not real because you can see uh the imaginary quantity right there. |
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158:42 | whenever we're actually gonna measure something that's uh be real, I guarantee |
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158:48 | So you should keep that doubt in mind. And uh uh it might |
|
158:55 | , you never thought about that, you should think about that. How |
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158:58 | we have this imaginary quantity showing up our equations when we have real |
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159:05 | And so, uh the resolution of is what I said, whenever we |
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159:10 | these complex formulations to um uh account observable quantities, the, the observation |
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159:20 | always gonna be real. Yeah, this plane wave solution, the wave |
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159:30 | is related to the wavelength by this here, the wavelength uh uh uppercase |
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159:36 | with a subscript C because we're traveling vertically. So that's equal to the |
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159:42 | um uh divided by the frequency of it's uh that's um I think, |
|
159:51 | know that the, the uh well we're actually looking at a a |
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159:57 | Uh but what you will be aware the velocity is given by the wavelength |
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160:02 | the frequency. So the wavelength is like this. And so it's limited |
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160:07 | the, the wave number by So uh uh uh here you see |
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160:13 | K three as the dimensions of one Z, you can see that right |
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160:19 | K three has the dimensions of one length just like Omega has the dimensions |
|
160:23 | one over top. So this plane solution uh uh uh has all these |
|
160:31 | features that I just uh just spent minutes discussing. It's got this strange |
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160:38 | . E it's got this strange I it's got new parameters, Omega |
|
160:43 | K three, but it's gonna be useful to us like, so trust |
|
160:48 | , you, you will see this and over again, you need to |
|
160:52 | very familiar, very, um in , highly acquainted with, very friendly |
|
161:01 | this wave, this plane wave solution the wave equation. OK. Oh |
|
161:17 | . So let's make a guess about before we're we're gonna rewrite the phrase |
|
161:22 | , uh the phase in this So we factor out the omega and |
|
161:25 | left with uh uh uh uh uh over V is equal to plus or |
|
161:33 | K three. That's what we decided . Now, since this is a |
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161:38 | for any omega assume that both uh uh uh uh the, the solution |
|
161:46 | P with one frequency. And here have the solution for another frequency. |
|
161:52 | uh I assume that uh both of separately are solutions. So now we're |
|
161:59 | guess the sun is also a So you can verify for yourself that |
|
162:07 | is true. If, if this a solution and this is a |
|
162:12 | then the sum is also a This happens only because the wave equation |
|
162:21 | linear. What do we mean by ? Let's back up here. Here's |
|
162:30 | way we, we say it's linear the unknown appears only to the first |
|
162:36 | . We've got squares everywhere through but the unknown appears to the first |
|
162:40 | . And so there's no term out and you know, involving P |
|
162:44 | Now, because of that we have solution here. If we have one |
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162:53 | at one frequency and another solution at frequency, the sum is also a |
|
162:58 | , it is also a solution that like it's uh an amusing mathematical |
|
163:04 | But that's essential to us in geophysics all of our rays, all of |
|
163:10 | waves are composed of sums like this different comp uh different frequency components in |
|
163:18 | wave. Any measure that we ever has some uh uh high frequency components |
|
163:25 | some low frequency components. And the itself is a salt of all these |
|
163:30 | components. By the same logic, you have a weighted sum, uh |
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163:37 | we have a, a one is big number and a two is a |
|
163:40 | number, a smaller number. And also a solution. And of |
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163:44 | you have, if we have a with many terms in here, maybe |
|
163:47 | uh is gonna sum uh seem to a different coefficient for each one of |
|
163:52 | uh uh different frequency components and sum all up. Um I can have |
|
163:58 | a 17 of them, we can 1000 of them. Uh That's also |
|
164:01 | solution. And what are these They're gonna be determined by initial conditions |
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164:09 | boundary conditions conditions at the source and uh on the boundary of the |
|
164:16 | for example, at the surface of earth. So those kinds of boundary |
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164:22 | are gonna determine these. And so these coefficients are called uh uh the |
|
164:33 | of the solution. And I know know this term from before of excellent |
|
164:40 | when we uh when we give AAA showing how much uh energy is in |
|
164:47 | one frequency and another frequency and so that is this set of coefficients a |
|
164:55 | now, all of that was for . So um it's easy to generalize |
|
165:02 | uh for uh um uh propagation in 3d directions, all we have to |
|
165:08 | is change the Z squared to a, an X I square. |
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165:14 | this arrow here uh cars you back the previous generation. If we're gonna |
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165:21 | that, we have to generalize the of phase. So instead of having |
|
165:25 | K three times Z, we have K vector times X vector, they |
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165:33 | the same omega T will have the plus or minus here. So then |
|
165:40 | you put this definition of a phase here, we find uh uh uh |
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165:47 | any solution uh where the pressure is only on phase. We have um |
|
165:55 | uh uh this uh you know, grind through the uh through the um |
|
166:04 | rule calculus. And you'll find this not a bad grinding, it's easy |
|
166:09 | do. And so we find then Omega squared is equal to V squared |
|
166:14 | the sum of the squares of the of K. And so uh this |
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166:24 | assumption that he depends only on phi for any phi if and only if |
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166:32 | length of K is given by omega the. See, this doesn't say |
|
166:37 | about the components. It, the of the, of the, of |
|
166:42 | wave vector is uh related to the in this way. So, |
|
166:57 | it's still a question. It says scalar wave equation has and all these |
|
167:06 | . So let's um let's uh work way down through this. Le le |
|
167:12 | does it have two derivatives with respect time? It says yes. Um |
|
167:19 | per it do, does it have with respect to position? Yes, |
|
167:26 | says yes. Uh It ha does have the unknown function to the first |
|
167:31 | ? Only Carlos? Not sure, sure. I would say not. |
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167:43 | . Uh uh uh So that is crucial thing. Uh You've got to |
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167:48 | uh have that clearly in your So we're gonna go back here. |
|
167:52 | the wave equation right here. No, exactly. Here's the way |
|
167:57 | the unknown is peak that only appeared the first power. Everything we know |
|
168:03 | wave propagation would be destroy if there a term in P squared. So |
|
168:10 | very crucial that we understand that this has the unknown only to the first |
|
168:16 | . So that makes it a linear that's here. Now, back to |
|
168:22 | uh uh does it have a single which describes the medium, a single |
|
168:32 | ? She says no. Um let's go back here. So here |
|
168:36 | have a single parameter. We're back . See there's the parameter. It's |
|
168:42 | V squared. I know you were that inside the squared is KO |
|
168:50 | But though it only appears in that combination K over row, it never |
|
168:54 | , you know, uh K plus thirds mirror row. It uh |
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168:58 | it doesn't appear in any other combination K over row. So we gave |
|
169:04 | a name. It's one parameter V . So the answer to this question |
|
169:10 | a single parameter, yes, which the medium. And, and we |
|
169:14 | describe that we can either call it or rope, but that's not really |
|
169:19 | parameters. That's only one parameter. we give it a name. |
|
169:23 | we don't yet know what V uh means. But uh uh it, |
|
169:29 | know it has the dimensions of So we call it A V and |
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169:35 | it has all the attributes above. uh So the answer to this would |
|
169:40 | , wouldn't it? Uh uh And is false. Mm OK. So |
|
169:47 | let's go to Carlos uh the scalar equation. Is this true or |
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169:52 | Can it be written either in terms pressure with the unknown function or with |
|
169:57 | location of the unknown function? Is true or false? I think |
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170:04 | I didn't hear you. Yeah, think it's true. Yes, that's |
|
170:09 | . Uh uh We showed that very uh back about 10 slides. That's |
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170:14 | . OK. So brice that uh or false, uh it's only valid |
|
170:19 | propagation in the vertical direction since fluids vary in the vertical direction. Is |
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170:25 | true or false? True? now it's true that um that fluids |
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170:36 | in the vertical direction. Let's think the ocean the ocean at the top |
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170:41 | the ocean, the water is warm the Gulf of Mexico, uh the |
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170:46 | is warm and it gets colder as go down. And furthermore, the |
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170:51 | salinity, the salt content of the changes as you ruled out. So |
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170:56 | is definitely true that in the the water, uh the uh the |
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171:03 | properties of the water very in And by the way, they also |
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171:08 | laterally also ve very laterally. Uh uh in a, a complicated |
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171:16 | whereas it varies vertically in a simple , sort of like layers of |
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171:21 | And we frequently ignore that variation but there. And uh uh uh some |
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171:29 | uh uh can actually do our kind reflection seismology. Um um looking at |
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171:38 | in the water color, think about , think about a reflection of a |
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171:44 | wave from the vertical variation of properties the water. So normally we ignore |
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171:53 | possibility, but I assure you that if you look for it in the |
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171:58 | way, you'll find it. Uh uh in all kinds of experiments, |
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172:02 | that's uh um not a common mode with fabrication. So it is true |
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172:11 | it slid vary in the vertical But the question says, is it |
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172:19 | only for propagation in the vertical Well, let's go back here |
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172:28 | we have uh uh uh a propagation 3D in any direction in 3D and |
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172:36 | we had to do to uh make ation was to um what let a |
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172:46 | uh change from X squared in this to uh X I square. And |
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172:53 | we had to change the direction the the definition of, of uh phase |
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173:00 | to a vector X. And uh had to have a vector X here |
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173:06 | a dot product with, with what call the wave vector. It's gonna |
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173:10 | out folks that this wave vector give direction of the wave and the X |
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173:16 | gives the position inside the um uh uh the ocean. So the answer |
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173:22 | this one is false since it's valid any propagation, not just in the |
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173:33 | direction. And if you go through uh our derivation, you will confirm |
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173:39 | in that last form here. Uh generalize from ver from the vertical to |
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173:45 | direction and it all worked because we clever notation. Next question is um |
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173:53 | see, I think um it's up you uh back to you. Uh |
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173:56 | No. Is it to you le uh uh if you have a 17 |
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174:01 | solutions uh uh make a sum of those is that also a solution? |
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174:11 | , we showed that if you had different solutions, that's also a |
|
174:15 | right? So when you think about uh uh go uh think about |
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174:20 | And uh um um uh you can that argument to make 17 or, |
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174:28 | 39 or whatever you want. And also a solution. If you have |
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174:33 | solutions for uh uh a as many solutions as you want to make a |
|
174:40 | , that's also a solution. And really important for us because all these |
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174:46 | solutions are gonna have different frequencies and different wave uh vectors and they all |
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174:54 | together it's still a solution and that's to be a solution which hits our |
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175:02 | . And we got to uh to figure it all out what it means |
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175:06 | you know, basically what we're gonna is a complicated wiggle. And we |
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175:11 | a different wiggle at a nearby hydrophone we got uh uh figure all this |
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175:16 | uh uh as a function of position as a function of uh frequency. |
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175:25 | all of that is uh and by way, these sorts of questions are |
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175:29 | unlike the sort of questions that you find on the final exam. So |
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175:36 | encourage you to go back uh through uh questions and verify that you're confident |
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175:45 | your mind, what the answer And uh if you're not confident, |
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175:52 | send me a question uh tomorrow morning o on this point. Now, |
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175:58 | um here's one thing that we didn't about. Uh it, it's useful |
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176:05 | you all to have a printout of lectures. And as I gave them |
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176:12 | you in canvas, you can print out. Uh and then have them |
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176:17 | in front of you while you're listening me talk. And, uh, |
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176:25 | , uh, as you go through quizzes, you can mark right on |
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176:29 | , uh as we're going through the , if you have a question, |
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176:32 | write your question there uh on the in front of you and it, |
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176:37 | the questions you can, you circle the correct answer. And if |
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176:42 | go back to the, the um and you don't understand why that |
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176:48 | given as the correct answer, then can bring it up with me uh |
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176:54 | your email to me overnight. So why it's useful to have a print |
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176:59 | in front of. OK. So we understand um uh now, now |
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177:09 | understand uh propagation, the wave equations propagation in the ocean. We still |
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177:22 | know anything about sources, right? , we just found out that the |
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177:26 | waves propagate, who knows where they from and once they're there, they |
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177:32 | from here to there following the equations uh that we just said that's called |
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177:37 | scalar wave equation because uh the wave as a scalar wave and the unknown |
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177:45 | the pressure, that's a scalar. . Now, let's think about uh |
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177:53 | that's not gonna be good enough for because we have uh beneath the |
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178:00 | we have rocks, the waves go into the rocks like and come |
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178:05 | So we have to know how waves in solids So that's the next point |
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178:11 | , we have a voxel inside of solid where we're gonna consider the stress |
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178:19 | this voxel, not just the because in a solid we have, |
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178:24 | not common to have equal pressure from sides is very common to have unequal |
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178:31 | on the very side. So the is calculated like before the uh the |
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178:39 | is gonna be different up here than is in the center. And |
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178:44 | in particular, this is gonna be uh uh the Tau, we're gonna |
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178:49 | this the Tau three J stress. what this three means that on the |
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178:55 | face, uh uh uh the fact it's uh uh has the, the |
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179:02 | to this top face is pointed in three direction. So the orientation of |
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179:07 | top face is given by unit factor the three directions. So that's why |
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179:11 | call that A three here. And any um uh uh well, uh |
|
179:19 | J component of stress on this top , it's given by um uh uh |
|
179:27 | know, it's gonna be a function the central position plus um this distance |
|
179:35 | from the sand to the top. here's the same thing at the center |
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179:39 | the bottom with the minus side. the center to the right side where |
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179:44 | distance vector is on the, in one position. Here's on the left |
|
179:48 | with A minus D over here, the uh front and the back, |
|
179:53 | minus D in front and A A D in the back. OK. |
|
180:00 | the corresponding forces are um multiply these by D squared in every case. |
|
180:10 | in some cases, we have a . And in other cases, we |
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180:13 | a plus here uh depending on uh uh which face we're talking about and |
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180:21 | all these together like we did Remember, this is not AAA um |
|
180:28 | matrix. This is uh uh just single, a single equation with six |
|
180:35 | in it. So the index one gives the in direction of the area |
|
180:40 | the area. So it's the same here as it is here down |
|
180:46 | It's a two instead of a And it's the same two over here |
|
180:50 | indicates front and back down here. a three instead of a one and |
|
180:55 | same three over here that indicates top bottom. The second index indicates the |
|
181:01 | of the fourth. So here we a force in the three direction uh |
|
181:11 | at all, let's see. Oh is that force in the three |
|
181:17 | Because it says, so here, says that we're only interested in the |
|
181:21 | three and the force in the three . So that's the direction of the |
|
181:25 | as the three everywhere because we're concentrating the vertical form. And so you |
|
181:33 | compare that with a scalar case by back through the, through the |
|
181:39 | So managing this is it, this uh this is the scalar case we |
|
181:44 | have it's the same force in all in that case. Uh But in |
|
181:48 | case, it's uh it's a force the three direction only. OK. |
|
181:55 | now we specialize to a sound wave , verify the force does not vary |
|
182:00 | the horizontal direction. So we're left these two terms only. So you |
|
182:05 | compare with the scalar case here. , I did that for you next |
|
182:09 | here is the scalar case is only pressure at the top of the |
|
182:16 | As before we're gonna use the tailor to simplify the previous expression. And |
|
182:22 | comes down to the force in the directions is given by the cube of |
|
182:29 | um uh edge times the X three of the cal 33 source cal 33 |
|
182:40 | . And that derivative is evaluated at center. So that's for the vertical |
|
182:48 | only. So you can do the thing with uh force in the two |
|
182:53 | in the one direction and they end with this. But this is what |
|
182:58 | just derived here is the two direction the three direction you see the uh |
|
183:03 | all cases, since we're talking about propagation, we've got vertical variation down |
|
183:09 | and we have index 32 and one . It so we have the variation |
|
183:18 | in the X three direction because we've the vertical propagation. And notice here |
|
183:24 | the stress is only on the X planes. We can have uh force |
|
183:30 | the one in the three direction force the two direction or force in the |
|
183:36 | chimi. Uh we can have the are in all three directions. |
|
183:47 | it's easy for us to uh um generalize that uh for propagation in a |
|
183:56 | because of the clever way that we've up this notation. Uh All we |
|
184:00 | to do is replace the three with and then recognize and what we have |
|
184:06 | the derivative with respect to J of IJ where the J is uh repeated |
|
184:12 | . So that's why we have to uh over Js. And so in |
|
184:23 | vector notation, we can say that force divided by the volume is given |
|
184:30 | the gradient of the strength for Yeah, grading of the, of |
|
184:35 | stress tensor and go back to the 101 section. And you'll see what |
|
184:40 | , that would when we have the or tensor, it is just exactly |
|
184:45 | we're showing here sir. For being environment is the gradient of the |
|
184:54 | So put that in the equation of uh uh F equals ma uh in |
|
184:59 | words, a vector equals F vector by M. And we find that |
|
185:04 | A vector is given by the gradient the stress divided by the density. |
|
185:12 | we're gonna express this good way. uh we're gonna express the acceleration in |
|
185:26 | of displacement in this way. And that's the left side of this and |
|
185:30 | right side here and the same side the right side here. OK. |
|
185:35 | this is the vector equation of motion from uh this is gonna be valid |
|
185:40 | the subsurface rocks and with the stress . So what's the, what's the |
|
185:48 | OK. The unknown here. Is displacement or is it uh stress? |
|
185:54 | , um uh we're gonna have to one or the other and we're gonna |
|
185:59 | Hook's Law to eliminate the stress. here's the stress and here's Hook's law |
|
186:04 | the stress that this equation is what had on the previous slide. And |
|
186:09 | uh here's Hook's law for the Now, what we're gonna do is |
|
186:15 | yeah, that's right. Express the as we did before in terms of |
|
186:21 | of displacement. And here's the one that we saw before. Uh And |
|
186:28 | I dodged the question of why, are we, why did we put |
|
186:33 | one half in there or that, one half came in in, in |
|
186:37 | , in the definition of epsilon? there it is now because of the |
|
186:45 | that we talked about before, in distance element, we can uh we |
|
186:50 | convert this expression here which from the slide to this expression and look the |
|
186:58 | had disappeared. Why is that because this derivative is obviously different from this |
|
187:05 | . You see the MS and the are in opposite places here. But |
|
187:09 | of these symmetries, we can uh uh uh um uh combine those together |
|
187:18 | doing so the two disappears. let's assume that the, the medium |
|
187:25 | uniform. So we take this outside the derivative and we're left with |
|
187:30 | And you can see on the left side of the second root of the |
|
187:34 | time on the right hand side, ro respective position. But it's not |
|
187:40 | the wave equation because it has the form. It has too many different |
|
187:46 | here to what we want is the posse to appear. OK. |
|
188:02 | now we turn to uh another 19th German mathematician, his name is |
|
188:13 | And uh we have an actual photograph Helmholtz. And so Helmholtz's theorem says |
|
188:21 | uh for any um uh for any field such as the displacement, we |
|
188:28 | express that a sum as a sum a part which has a zero curl |
|
188:34 | a sum that has zero divergence. we've skipped uh lightly over this. |
|
188:41 | uh But earlier in the day, , I talked about uh uh a |
|
188:47 | calculus and uh uh we uh we that if you have a vector, |
|
189:00 | has zero curl, that means that can express that as the gradient of |
|
189:07 | scalar. Remember at the end of discussion about math, 101 vector um |
|
189:16 | uh vector calculus I showed three um identities involving Dalton involving Dell. And |
|
189:34 | of those said that if you have , a part here which has zero |
|
189:41 | , then that means that it can expressed as the divergence of as the |
|
189:46 | of a scr because the greed of scr uh operating with the curl operator |
|
189:52 | zero. So this one has curl curl free and it can be expressed |
|
189:58 | the gradient of a scaler. And next of those three vector identities was |
|
190:06 | if you have AAA if we have um a vector field like displacement, |
|
190:16 | if you ha if it has zero , then that can be expressed as |
|
190:21 | curl of a of another factor. so this is called, this one |
|
190:29 | called the scalar potential and this is the vector potential. So those are |
|
190:35 | we can uh separate the uh the into a part described by scalar potential |
|
190:44 | a part described by a vector OK. So the displacement I think |
|
190:51 | have a good um um a good for it. You can imagine in |
|
190:57 | mind, the displacement of um the inside of a solid. And you |
|
191:03 | imagine how that displacement is gonna vary position. I doubt if you can |
|
191:08 | have a good mental uh picture of scale of displacement, it's gonna be |
|
191:13 | scale of potential even. So even , I think you don't have a |
|
191:19 | um uh idea in your mind of vector potential. But Mr uh Helm |
|
191:27 | has told us that we can always any displacement field. And the sum |
|
191:35 | these two parts, one has zero and the other has zero divergence |
|
191:42 | always. And so, since this has zero curl, we can describe |
|
191:47 | uh in terms of the sca scalar and here we can describe this |
|
191:51 | we can describe as a vector Now, why is this important, |
|
191:58 | of these is gonna lead to key and the other one is gonna lead |
|
192:01 | sheer waves. So what we're going do is, oh so uh if |
|
192:15 | follow uh uh if, if you back to uh uh the, the |
|
192:20 | 101 that we did earlier this morning look up the definition of curl and |
|
192:25 | definition of divergence you will see spelled in better detail what I just |
|
192:34 | And I'm gonna bring you back right to uh uh the glossary and uh |
|
192:42 | uh um where we defined Dell. so we said there in the |
|
192:48 | we said Dell can be applied to vector to make a scalar. So |
|
192:53 | a vector, it has three components I and we uh uh operate in |
|
192:59 | , in a dot product fashion with and we have the partial R |
|
193:03 | with X I and the sum of . So my equal 1 to 3 |
|
193:07 | is, that makes a scalar. we also said that when you operate |
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193:15 | uh uh with dell on a you get this vector and this is |
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193:19 | the curl operation. So applying these to the wave equation, say the |
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193:30 | free part of the wave equation is by the gradient of this scalar |
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193:37 | Why we say it's the curl push The superscript P means that it's a |
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193:43 | wave and it has zero curl to . So imagine now what does that |
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193:53 | curl free as the P wave is watch my hands as the P wave |
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193:59 | going through the rock, it's uh the compressing and um and decompressing the |
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194:08 | as it goes through. It's not any of this. It's not making |
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194:12 | sheer of this sort. It's making a, a long compression. So |
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194:19 | can see that uh when we uh can see that the a good English |
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194:25 | for describing this part without this kind motion to it that has no curliness |
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194:32 | it, it only has longitudinal uh to it. So that's why we |
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194:37 | the curl freak part is a P . And so we'll come to the |
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194:42 | part of the sheer wave in a . So let's designate to designate this |
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194:51 | free part. And we're gonna call A PW. And it's gonna be |
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194:58 | uh you know, uh expressed as , the gradient of a of some |
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195:07 | potential function, which we can see mathematically. But I doubt if any |
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195:13 | us have a good mental picture of that scale of potential is. In |
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195:19 | same way, the divergence free part an S way indicating it here. |
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195:24 | I'm gonna go back here and this here has zero divergence that would be |
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195:47 | . The part which has zero divergence an S wave. So let's think |
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195:51 | an S wave going through a solid this. It's not squeezing wa watch |
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195:58 | hand. It's, it's going this horizontally and it's not as it's going |
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196:02 | a saddle like this and it's not the rock at all. So |
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196:07 | it's not converging or diverging the rock it goes through. It's only sharing |
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196:13 | rock. So, uh uh so why a weak site. So that's |
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196:18 | free, only a shear wave. so we give, uh we just |
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196:22 | the notation here. So instead of zero divergence, we say superscript |
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196:28 | for she and that's given by the of some scalar potential vector field or |
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196:37 | potential, which again, I think don't have a good physical intuition for |
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196:43 | but you do have good f physical intuition for these displacement fields on the |
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196:50 | side. So here's the displacement inside P way and here's the displacement inside |
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196:54 | share way. Now, um all of this is uh valid for |
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197:06 | isotropic rocks. For an isotropic rocks a complication. So we're gonna be |
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197:11 | something different or analyze the trapping Not now, but later. And |
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197:19 | look for heat wave solutions. And uh uh uh uh we're looking for |
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197:25 | , here's our wave equation, these terms. And you see we have |
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197:32 | second group of respective time. And just put in here the uh the |
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197:37 | of uh of the scalar potential phi that's gonna be the gradient of the |
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197:51 | uh OK. Yeah. Oh So is the second route of the spec |
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198:04 | time. On the right side, have uh uh uh the green of |
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198:10 | curl free part of um of, the stress. Let's, let's uh |
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198:19 | back up here. OK. If gonna go back earlier right here. |
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198:31 | , uh on that. So on left hand side, we have the |
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198:34 | ro in respect to time. On right hand, we have gradient of |
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198:39 | gradient of stress. So now let's forward again. So right here we |
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198:50 | the gr uh we're, we're specifically that uh uh uh we're taking the |
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198:57 | of the curl free part of the . So that means that uh uh |
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199:01 | the part of the stress which depends the scale of potential F. So |
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199:08 | index notation, um uh this looks this where you see here, |
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199:15 | the index eye here, this, shows only the I component. This |
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199:20 | a vector equation which shows all But for the I component we have |
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199:24 | the left side, this and in right hand side, we have this |
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199:29 | of this stress tensor uh uh uh in this way. Now we're gonna |
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199:37 | the stress using hook flop. So not take the stress and put in |
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199:44 | flaw right here. So again, have the factor of one half. |
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199:49 | have the strain here. We have stiffness tensor here. And uh |
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199:57 | on the right hand side, we're to um uh put in here for |
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200:03 | uh the C component of the B . We're gonna put in uh uh |
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200:11 | a gradient with a spec to K K as we have here uh of |
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200:18 | scale of displacement. And this, uh here is the uh uh |
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200:25 | I think the, the end component the G displacement is given by the |
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200:32 | of the uh with respect to XM the same uh scale of potential. |
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200:38 | , look here um uh we have really with respect to mxmnxk here, |
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200:48 | have XK and XM is the same . And so we can combine those |
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200:54 | and when we combine those two, can get rid of the two. |
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200:57 | now you see how we get rid the one half. And so now |
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201:01 | see why it was clever for us put that one half in there. |
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201:05 | that it comes out this point. , what we're gonna do is is |
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201:11 | that the uh uh the uniform is , it, the medium is |
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201:17 | So this thing does not depend upon , bring that outside the derivative, |
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201:23 | it over here. And uh uh we have um a, a simpler |
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201:29 | involving multiple um uh components with all xjs and X MS and XK. |
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201:39 | looks like it's pretty complicated. I no fear it's gonna simplify shortly because |
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201:46 | this scalar potential fly. But it complicated three equations. One for each |
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201:54 | of I and with 27 terms in , all these sums, you |
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201:59 | we got a sum over J some M, some over K 27 terms |
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202:04 | , in each. So, oh see because of the clever notation, |
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202:14 | of this is gonna simplify very Let's look first at one of these |
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202:23 | for third component only. And so we have uh uh uh a three |
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202:31 | , a three here. And so have three here and, and we're |
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202:36 | have a sum from JK and M um 27 terms like this. |
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202:45 | So um if we show this term uh over J explicitly, so we'll |
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202:53 | the derivatives of spec to XM and put them out here, we're showing |
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202:57 | only with respect to uh only variations respect to um uh from the sum |
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203:05 | sum over XJ explicitly here's X one two and X three. And uh |
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203:12 | see no more Jays here so that j um one equals 12 and |
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203:24 | So now we're going to uh uh uh isotropic elasticity. So remember we |
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203:32 | uh we found this before that uh for isotropic media, the uh the |
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203:40 | uh are quite simple because of all zeros out here. And so, |
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203:45 | of all these zeros, most of terms in this previous equation are |
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203:50 | So we had here, all these in here, most of them are |
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203:55 | because of the properties of the uh stiffness sensor for isotropic media. And |
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204:03 | work only these only these terms So 1234567 terms, I have no |
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204:16 | . It's gonna get simpler still. . Next, we're gonna uh uh |
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204:21 | transfer to two index notation. So example, this 3131 becomes a |
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204:30 | its 3232 becomes a firefox. And similarly down here, so we're left |
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204:38 | with these only these five terms. uh and, and if you look |
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204:46 | here uh uh uh up there, we can collect all these terms. |
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204:51 | so um we uh after we collect the terms, we um uh simplified |
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204:59 | expression to this and using the common using the common names for the elements |
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205:06 | uh uh C 31 and so Uh C 31 is equal to Lambda |
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205:11 | two mu, again, Lambda plus mu and this one is AMP so |
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205:17 | are simply the common name for these components. And because lambda equals M |
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205:26 | two new, all of these terms proportional to M. And so look |
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205:31 | we have everything uh uh uh comes uh uh M and we're left with |
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205:39 | M over on the right side times lobos and of the vertical component of |
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205:45 | displacement. Remember that we're talking about traveling P waves. And on the |
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205:50 | side, we have the secondary respective of the vertical component of uh of |
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205:58 | the displacement. And if we repeat for the other two components, we |
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206:03 | the vector wave equations from P waves nicely like this very simply. And |
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206:10 | remember that M is given by K four thirds mu. And so |
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206:18 | III I said that wrong, let define it is defined uh uh um |
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206:28 | Emma Moreau, let's define that as square of something we call VP. |
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206:35 | finally, we get this expression. after all that um struggle, it |
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206:40 | out to be very easy. Um you go back over this next |
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206:52 | you will find that the struggle wasn't that struggling. It, it's very |
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206:56 | . It's very straightforward because of the we used now. So this is |
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207:05 | be propagation for P waves in any and it's gonna be easy to convince |
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207:13 | that all of these uh uh uh solutions depend upon this phase factor with |
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207:19 | this uh frequency. And this, dot X so long as the length |
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207:26 | K, never mind the components of , the length of K is related |
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207:31 | to uh VP, which we just uh defined. Still, we don't |
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207:36 | what VP is, right? We know whether it's the VP of |
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207:42 | of, of the wave or the or what we don't know yet. |
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207:46 | gonna find out soon. We don't right now. It's just a notation |
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208:04 | . Now, we could do the sort of thing instead of uh uh |
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208:08 | of having the displacement here, we , we could get a wave equation |
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208:12 | the scalar potential. Let me just up here. So here's the equation |
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208:20 | just came up with, we could this in the uh well, the |
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208:25 | of potential back in here right here for the, for the uh uh |
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208:31 | after we got rid of it some ago, but let's put it back |
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208:35 | and now let's take it outside this . So the, the gradient operation |
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208:42 | be uh uh interchanged with the, time derivative like so, and uh |
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208:48 | a similar way, this green operation be uh interchanged with the low cross |
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208:54 | . So we have the uh the operator outside of everything. And so |
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209:00 | means that uh well, the equation good even without the grading operation. |
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209:07 | uh uh we have the same wave for the scale of potential as we |
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209:12 | here for the displacement. And you see there's a big advantage here in |
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209:18 | computation because this is three equations in . And this is one equation. |
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209:24 | is three equations for the three components the displacement. I know you have |
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209:28 | good intuition about the displacement. You have a good intuition about the uh |
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209:35 | the there are potential. But the uh uh the same solution is gonna |
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209:42 | given much simpler by this scalar equation opposed to this vector equation. And |
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209:54 | we uh uh uh uh I remember uh he instruments, our receivers are |
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210:00 | gonna be receiving the potential, they're be receiving the displacement. So uh |
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210:08 | you are operating in the computer uh the uh Gaylor Wave equation, in |
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210:14 | of the potential at the end, got to find out uh you've got |
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210:19 | take the gradient of that potential solution find the displacement to find what the |
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210:26 | is gonna look like. That's what says here. The gradient of the |
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210:30 | here yields the observable displacement. So that's for P waves. And so |
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210:40 | most of what we do in exploration physics is with P waves, but |
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210:45 | shear waves are important. So let's at the same sort of thing with |
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210:50 | shear waves, it's easy to show for the vector potential you get a |
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210:55 | or you get an equation where instead VP here you get something we call |
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211:01 | and that is shorthand for New over , you can um uh convince yourself |
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211:12 | . Now have we have a vector . Uh you know that in um |
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211:18 | , the displacement is perpendicular to the uh direction of propagation. Well, |
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211:25 | because you know that uh it follows this scale of potentials this vector potential |
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211:32 | in in the direction of the And this thing has uh uh solutions |
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211:41 | depend upon the phase defined in this . So long as the wave vector |
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211:46 | related to the frequency by um the velocity instead of the P velocity. |
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211:56 | uh take the curl of this expression , the curl of this expression here |
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212:02 | get the vector wave equation for sheer where now we have the shear wave |
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212:10 | . And uh so here it says uh uh uh uh and it says |
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212:18 | the deice is perpend color to that . And uh um it's a transverse |
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212:28 | no, we defined and we have vector wave equation and previously had a |
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212:51 | wave equation for the P waves. , it says that the solutions to |
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212:55 | vector wave equations have the same properties those of the scalar wave equation, |
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213:00 | are the solutions do have three parameters be fixed by initial or boundary |
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213:06 | So we're gonna be able to um represent the solution as a sum of |
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213:13 | waves. And uh uh I in sum there's gonna be three parameters which |
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213:19 | specify using the initial conditions and the conditions, the functions of the |
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213:27 | But we have either VP or vs here depending on what we're talking |
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213:33 | PR S and a sum of solutions a solution and a weighted sum of |
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213:38 | is a solution. So these are properties of the vector wave equation, |
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213:44 | is the way the waves are propagating the earth. And if we're doing |
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213:48 | land survey, not a marine but a land survey, you |
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213:53 | we're gonna be putting out um uh on the land on the surface |
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213:59 | of the land and uh um coupled the ground. As a matter of |
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214:04 | , my first job in Geophysics was that job, uh uh deploying um |
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214:11 | receivers Trump a quick quiz here. Let me see here. Uh Let |
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214:23 | do the following. I missed a from my wife just now. If |
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214:35 | uh um if you will permit I'm going to send her a |
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215:03 | I'm here. Uh 104, send that note. So yesterday after |
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215:11 | I was uh uh maybe two days , I was uh sitting on the |
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215:16 | just outside the building here waiting for to come pick me up and a |
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215:21 | came along and he said, you're waiting for somebody. And I |
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215:24 | , yeah, I'm waiting for my . He said, you, you're |
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215:27 | that you have a wife to come you up. And I agree that |
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215:33 | am a very lucky guy. uh, um, not everybody has |
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215:38 | wife to come pick them up or husband to pick them up. |
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215:42 | but, uh, I'm very I've had the same wife for 58 |
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215:47 | . And uh so I think she's trying to pick me up. Um |
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215:53 | I think about that 58 year longer you all have lived, maybe longer |
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215:58 | your parents have lived. So I've a wife who comes to pick me |
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216:03 | all that time. I'm a very guy. So uh I just sent |
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216:09 | that note. So, so she'll here at one. So we need |
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216:13 | finish up here at one. um let's uh uh look at this |
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216:20 | . Uh question says, is this false? The equation of motion is |
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216:24 | starting point for this derivation of the wave equation. So this is a |
|
216:31 | equation, a similar question to what had for the scalar wave equation if |
|
216:36 | remember. But this is about the wave equation and that's it, is |
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216:41 | true that the starting point for the wave equation is, is new or |
|
216:47 | ? So le le le le, that true? That's true. She |
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216:51 | that's true. And she's correct. . So uh turning to Carlos, |
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216:57 | uh here it exists to or A principal difference between the scalar wave |
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217:03 | and the vector wave equation is that former ie the scalar wave patient analyzes |
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217:09 | on six faces of the voxel. the latter vector wave patient analyzes the |
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217:15 | stresses on the six faces of the . Is that true Carlos or false |
|
217:28 | ? We didn't hear you. Yeah. Yeah. Yeah, I |
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217:30 | thinking and then yeah, so what he's doing right now, he's |
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217:36 | over the uh uh uh statement very . It might be a trick |
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217:43 | And so he's reading over it very . Yeah, I think it's |
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217:47 | Yeah. And, and he came the right conclusion. Uh good uh |
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217:51 | good answer. Carlos. OK. versa. This one is for |
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217:55 | the helm theorem separates out the divergence part of this placement leading in the |
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218:02 | case to pure P waves. Is true or false? I think it's |
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218:09 | . Uh Yes. Very good. separates out the con the curl free |
|
218:15 | . Yeah. The P waves come the curl free part. So uh |
|
218:19 | was a trick question. You it looks like it, the an |
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218:23 | uh it's uh uh you could have fooled. But uh uh you read |
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218:27 | carefully and you say that uh the waves do have convergence and divergence. |
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218:33 | it's the uh it has zero curl not zero divergence. So that one |
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218:38 | false very good. OK. Back you Lili, the scalar wave equations |
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218:44 | the pressure P in a fluid is from the scalar R equation for the |
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218:51 | um fry in a solid for uh uh uh uh these answers here. |
|
218:59 | let, let's uh uh think about . I'm gonna uh uh uh Lily |
|
219:05 | uh tell me about a uh uh know, is this true that the |
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219:09 | called pressure in a P wave in solid is really launch stress, not |
|
219:15 | pure pressure. Is, is this true understanding by itself. Yeah, |
|
219:29 | statement is true. The so-called pressure a P wave in a solid is |
|
219:35 | a long stress. It's not pure . That's true, but it doesn't |
|
219:39 | the question. So we uh uh so we don't wanna pick a, |
|
219:44 | uh Carlos, the next one is uh mhm No. B we program |
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219:51 | computers with the equation for five for uh uh for increased efficiency. Number |
|
220:02 | , tell me just thinking about this uh uh statement by itself be, |
|
220:07 | that true or false? Yes, true because it's easier to program computers |
|
220:14 | uh uh scalar equations than for vector . So we do do this. |
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220:20 | , the question for colors is, this answer the question? OK. |
|
220:42 | . And while Carlos was thinking about and uh you know, you're coming |
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220:47 | with number C. So you'd be about number C while he's thinking about |
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220:53 | B, I think. Yeah, think about number B, um, |
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221:02 | , let's concentrate on B is uh, uh, uh, answer |
|
221:07 | question? Yeah, I think Number B, um, well, |
|
221:17 | , I would, uh, I'm, uh, I'm gonna say |
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221:21 | because, uh, um, this business of computers, the, |
|
221:27 | equations don't know anything about computers, ? All right. So, |
|
221:34 | so the equations are on their own of where they were solving it by |
|
221:39 | or paper and pencil or whatever. I'm gonna say that, uh, |
|
221:43 | is a true statement. B but doesn't answer the question. So now |
|
221:48 | turn to c, it says the P is the observable whereas the potential |
|
221:53 | is not an observable. Now, . And, uh, uh, |
|
221:59 | , uh, thinking about this see by itself, is that true |
|
222:04 | false? Uh, uh, it's . Yeah, that one is |
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222:09 | Now doesn't answer the question. And now we, we, we've already |
|
222:14 | A and B and so if this , uh, uh, if we |
|
222:19 | C, we're left with D. so I think about that. Do |
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222:23 | really want this? But you gave the answer that yes, this |
|
222:27 | um, uh, answer the uh uh, that, uh, |
|
222:31 | , uh, this is, a, a fact, a true |
|
222:35 | which explains the difference between, the scalar wave equation and uh uh |
|
222:41 | heat and the scalar wave equation for very good. So now the next |
|
222:47 | in here um uh back to you le le uh vector equations for the |
|
222:56 | wave differs from the vector wave equation the P wave because A B is |
|
223:03 | all of the above. So um A is true. Yeah. Uh |
|
223:09 | , um uh uh the velocity depends the sheer marginist view, not the |
|
223:15 | margins. And I, is that ? A statement by itself? And |
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223:21 | think, does uh does that answer question? Mhm I didn't hear |
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223:35 | So uh uh uh let's uh uh this one at a time. I'm |
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223:39 | you about A MM Number one is statement true. And does it answer |
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223:46 | question um is, is the statement ? OK. She says the statement |
|
223:57 | true. And now uh uh the is it, you say it does |
|
224:02 | answer the question, tell me why doesn't answer the question. I, |
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224:13 | didn't hear you DC. Well, uh uh uh before we get to |
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224:19 | let's think about A. OK. uh so what I'm gonna say is |
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224:24 | this is true statement and it's uh uh it answers the question, but |
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224:30 | some of the others do too. now let's uh uh uh uh look |
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224:36 | um um uh question B and answer and uh Carlos it says the displacement |
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224:46 | in the sheer wave is perpendicular to wave vector, not parallel to it |
|
224:51 | in P waves. Uh um is this uh the statement true? |
|
224:56 | one and number two, that it answer the question? Yeah, the |
|
225:00 | B is also true also true. . So we got two are |
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225:05 | So we're suspecting that we got all the above. But uh uh uh |
|
225:09 | we do that, uh um let's to Brisa and ask her about uh |
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225:15 | point C uh is uh the displacement , does the displacement feel of the |
|
225:23 | wave? Does it have divergence or ? It's, it's true. It's |
|
225:29 | . It has, it has no . And so that's the difference here |
|
225:33 | sways and p. So we got of the above. Very good. |
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225:36 | would say folks, this is not easy question, but that's the sort |
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225:40 | question which you might find on the exam. So you, you wanna |
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225:45 | sure you understand all that. here's a good place to stop for |
|
225:50 | morning. Um uh So we're gonna up, we're gonna pick up at |
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225:54 | o'clock and go until six. And , um let us stop at this |
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226:02 | . You can stop the |
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