| 00:00 | Afternoon folks. Can you uh your mute is on. So if you'll |
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| 00:06 | your mute off for a moment and us uh um talk I that |
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| 00:28 | So uh good afternoon Carlos uh Carlos are muted and uh one second minute |
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| 00:42 | change it to set student. I know why they cannot hear you, |
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| 01:00 | ? Minimize this here. As you see the zooms going through. I |
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| 01:04 | drop these two and Hello. Hello. Hello? No. Can |
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| 01:33 | hear me? Hello? No. . The medical, what else |
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| 02:07 | Can you hear anything right now? . OK. We have, we |
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| 02:15 | microphones somehow they find microphone. So sort of every dish. |
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| 02:52 | speaker, microphone, nodded. Can you hear anything right now? |
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| 03:04 | . Why? Hello? So let know. OK. It's good. |
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| 03:16 | you can try to say something. ? Can you hear me? Wave |
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| 03:20 | hand? Can you hear me? can hear? OK, then try |
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| 03:26 | say something and we can test the . Can you hear me? |
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| 03:34 | Can you hear me? Yeah, can hear. But the speaker kind |
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| 03:42 | that one. Yeah. Ca can hear me? Yes, sweet. |
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| 03:49 | . We try, can anyone see , we can test the speaker? |
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| 03:54 | ? Hello? Hello? Hello? you hear me? I can hear |
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| 04:03 | . Uh, we speak. Yeah. Yeah. Ok. |
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| 04:17 | give us a few minutes. Don ? Hello? Ok. Let |
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| 04:30 | so the speaker. Ok. Uh speaker. Ok. Yeah, |
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| 04:37 | we're good to go right now and . Ok, let's verify that |
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| 04:43 | Can you hear me? Yes, . Uh Brisa, can you hear |
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| 04:49 | ? Yes, I can. And we have uh here in the |
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| 04:53 | , we have Lili and she can . Ok. Uh So um oh |
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| 05:04 | , welcome to the class folks. Did, did you uh receive the |
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| 05:10 | from um John Van Yen who this um giving you access to the files |
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| 05:19 | I had uploaded earlier? Ok. um mhm. Those we will have |
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| 05:27 | a total of uh 10 lectures um 14 hours. And so um they're |
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| 05:38 | one through 10 of which you have first four now available to you. |
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| 05:45 | The first one is uh an very uh uh light stuff. So |
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| 05:51 | not gonna go over that in So I'm gonna ask for you to |
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| 05:54 | that one for yourself. And so gonna start today with lesson two |
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| 06:01 | So before we do uh uh let's do some uh uh introductions. Um |
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| 06:09 | we have here in class Yy So uh y li will, will |
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| 06:13 | tell us, uh, uh, about yourself. Tell us, |
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| 06:17 | uh, um, ok. Where you stand with the U of |
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| 06:24 | , are you a new student? you been here for a while or |
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| 06:27 | ? Uh, uh, are uh, do you have a |
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| 06:30 | uh, uh, here in Te, tell us about yourself. |
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| 06:36 | from Calgary. Yes. Uh, , uh, but, uh, |
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| 06:43 | , most recently from Calgary. you know, they have a very |
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| 06:47 | uh university in Calgary. So we're very happy that you passed by |
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| 06:52 | and came all the way to Houston uh uh study with us. |
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| 06:57 | uh when, when uh how long you been at University of Houston last |
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| 07:03 | ? So, this is your second ? Yeah. And so uh uh |
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| 07:08 | you a, a master's student or p master's student? And so, |
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| 07:13 | your plan is to get a master's in Geophysics and then to look for |
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| 07:19 | job. Yeah, here in Yeah. So uh it may be |
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| 07:24 | uh um there are good in Houston , good jobs in Calgary. Uh |
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| 07:31 | we were you born in Canada? , in, in, in |
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| 07:38 | Ok. Uh Well, your English very good. Uh Right. |
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| 07:53 | So uh next up is uh Carlos tell us about yourself. Yeah, |
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| 08:00 | afternoon professor. My name is Carlos . I'm from Colombia. Uh I |
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| 08:06 | a geologist and I have been working the oil industry for like 17 years |
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| 08:12 | . And yeah, I am currently master program, the professional master |
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| 08:18 | Um Yeah, for geophysics. Oil controlling Geophysics is the name of |
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| 08:24 | program and currently I am working for for an oil company for an operator |
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| 08:32 | . Colombia. Mm I think that be. Uh so you, you're |
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| 08:37 | for a Colombian company? Uh uh and where are you now? |
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| 08:43 | where are in Colombia? Yeah. . In Bogota. In the capital |
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| 08:48 | of Colombia. Yeah. Yeah. I like that. Uh So uh |
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| 08:52 | modern uh technology you can uh uh get a further education uh in Houston |
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| 09:00 | you're still at home in Bogota. what's the name of your, what's |
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| 09:05 | name of your company? Yeah, the company. Uh the name of |
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| 09:08 | company now. It's Sierra Cole but actually it was occidental oxy but um |
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| 09:16 | was OXY, Colombia but OXY sold all the assets here and it was |
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| 09:21 | a new company with this name, Cole. Sierra Co and so what |
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| 09:26 | you do for them? Yeah, am a reservoir geologist now but I |
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| 09:31 | like involved in the interpretation part, interpretation sometimes. Yeah. And sometimes |
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| 09:37 | have been out of the field in doing seismic acquisition too. Yeah, |
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| 09:42 | have have some exposure to the to the geophysical part, let's say |
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| 09:48 | , good. So uh um this an, an introductory course and wave |
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| 09:54 | and uh you will find this uh . Um And so, uh because |
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| 09:59 | your background, very different from uh , I think because uh you are |
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| 10:05 | , um uh uh but your experience in geology. So, uh, |
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| 10:10 | , you will find this very uh and uh surely, you |
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| 10:14 | things that I don't know. Uh you have skills that I don't |
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| 10:18 | but also, uh you'll be able learn something uh in this course. |
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| 10:23 | , um now, um do you access to the stuff uh that I |
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| 10:31 | earlier? Yeah. Um uh Yes. Yes, sir. |
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| 10:35 | it's like like uh uh 30 minutes . I, I could download uh |
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| 10:43 | , the, the, the So, yeah, I, I |
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| 10:46 | it here now I have here. . So what you have there in |
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| 10:51 | module, you have uh four lectures eventually there will be a total of |
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| 10:56 | lectures. And also you have a uh called glossary which uh gives uh |
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| 11:03 | certain, uh which is like a of terms that we use in technical |
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| 11:11 | . And uh uh we'll use that file from time to time. Um |
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| 11:17 | you have a file called Math And so we'll uh talk about that |
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| 11:24 | today. And uh then also there a spreadsheet which we will be using |
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| 11:30 | later in the course. So I you have all of that available to |
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| 11:35 | and you can download it, do you want with that? And uh |
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| 11:39 | , so now let's turn to uh Bria. You are muted. |
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| 11:46 | Hi. Uh My name is Brisa . I am from Mexico but I |
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| 11:52 | here now in Houston and, and , I, I am geophysicist, |
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| 11:59 | background and I work for Slb Schlumberger I work for uh Western Deco but |
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| 12:09 | work as a um interpreter, seismic . So I work with the interpretation |
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| 12:17 | the imaging. OK. Uh But this, of course is uh uh |
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| 12:23 | first course in geophysics. So this be old for you. Well, |
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| 12:28 | a good mind there. Some things , yeah, so you're working for |
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| 12:35 | in Sugar Land. Uh No, Richmond in Richmond. Yeah. So |
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| 12:43 | I have many friends in Schlumberger and uh uh it's a very good |
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| 12:49 | Uh So, uh we're, we're that you're joining us here and uh |
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| 12:55 | um um uh maybe we'll find this in geophysics uh useful. So, |
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| 13:06 | so then let's, so I, should tell you something about myself. |
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| 13:11 | So, uh uh you know, name uh Leon Thompson. So uh |
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| 13:17 | I am an old guy. Uh You might have noticed that I don't |
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| 13:22 | so much hair. Uh But uh worry about that. Uh My brain |
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| 13:27 | still good. Uh uh When I a much younger man, I was |
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| 13:31 | professor uh in New York, the of New York. And then after |
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| 13:37 | years, I left. Um uh I was a professor of Geophysics and |
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| 13:44 | those days I was mostly interested in in what we call curiosity driven |
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| 13:53 | studying the deep, deep interior of earth and studying earthquakes and things like |
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| 13:59 | . And the origin of the all, all those things were uh |
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| 14:03 | very interesting uh uh to me and still are, of course. But |
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| 14:07 | then um I uh uh eventually I uh uh uh promoted. That was |
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| 14:16 | first real job as an as, an Assistant Professor of Geophysics in the |
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| 14:21 | University of New York. And then seven years, uh uh I was |
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| 14:26 | and I was given a sabbatical You know what that is? |
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| 14:31 | that is where the university says uh you can uh uh go away from |
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| 14:36 | , we'll still pay your salary and can go away from here for six |
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| 14:41 | and go anywhere you like. And some people go to the beach. |
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| 14:46 | uh it's better if you go take uh professional um appointment somewhere. So |
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| 14:52 | did uh uh uh an appointment in and then I came back uh and |
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| 14:58 | months later. And um during my in Australia, I was working with |
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| 15:06 | very famous guy who was famous for um uh cur curiosity driven geophysics. |
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| 15:16 | late in his career, he had idea that which he thought could be |
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| 15:21 | to society and that was an idea how to dispose of radioactive waste. |
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| 15:28 | , you know, when we have , a nuclear reactor, uh uh |
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| 15:32 | there's uh inside is all sorts of reactions going on. And um uh |
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| 15:40 | kind of leaves it, uh a residual behind and uh uh that |
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| 15:46 | has to be disposed of and it radioactive for thousands and thousands of |
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| 15:53 | So you can't just uh toss it in the backyard and you can't uh |
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| 15:58 | throw it in the, the You got to do something clever with |
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| 16:03 | . And so he thought he had clever idea. And so, um |
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| 16:08 | everybody agreed, that idea is still debated now, 50 years later. |
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| 16:12 | for me, it was a new that maybe you can do geophysics, |
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| 16:20 | not by curiosity but motivated by the to do something useful. And so |
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| 16:27 | uh uh when I came back to university, uh uh it was a |
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| 16:33 | when the oil business was booming and recruiters were coming around and hiring all |
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| 16:38 | our students. And I'm sitting there the audience thinking, yeah, why |
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| 16:42 | these students have all the fun? could do that. And so I |
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| 16:46 | up my hand and they hired me . And so um they uh uh |
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| 16:53 | company that hired me was called Amaco uh they hired me uh um number |
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| 17:01 | because uh I'm so good looking, maybe also, maybe they hired me |
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| 17:08 | they had, he previously hired my , my father worked for Aero for |
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| 17:13 | and years and years and he had very good reputation inside Amao. So |
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| 17:18 | might be that they hired me to , uh, uh, somebody like |
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| 17:24 | . So, uh, uh, , and he was still working |
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| 17:28 | Uh, no, no. Excuse . He was retired by that |
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| 17:31 | But, uh, some of uh, young proteges were still working |
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| 17:36 | the company and I think they probably to get somebody like him. So |
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| 17:41 | hired me. Well, I'm sure were disappointed but, uh, |
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| 17:45 | they did keep me on for a of years until Amoco was bought by |
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| 17:52 | . And that was about 1999. , uh, um, so I |
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| 17:59 | along uh with a new company and new company was called BP, |
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| 18:04 | And about three or four years they decided to, to change the |
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| 18:08 | again and they changed it back to . So that's the BP that, |
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| 18:12 | know, today. Uh It used be that BP was a short, |
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| 18:18 | a nickname for British Petroleum, but no longer true. Uh, |
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| 18:24 | uh uh BP is a nickname for Incorporate. And, uh, |
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| 18:36 | uh I worked for BP for maybe years and then I retired at age |
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| 18:43 | . So during that time, uh I, um, uh was |
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| 18:49 | in research, so you people who doing uh interpretation, you know, |
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| 18:54 | lot of things about exploration uh uh that I don't know because I was |
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| 19:01 | in research and I have to confess you that in all my years, |
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| 19:06 | think I spent 25 years with Amaco BP, never did find a single |
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| 19:13 | of oil. But what I did was I found ideas and those ideas |
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| 19:20 | been very useful for lots of And so, uh uh of |
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| 19:26 | I published those ideas and we will about those ideas later in this corse |
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| 19:34 | the beginning because this is an introductory and those are advanced ideas. And |
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| 19:41 | um uh we'll talk about those ideas in this course. Let's turn now |
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| 19:49 | um uh to an outline of the . So that's where we stand |
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| 20:02 | You can see there are 10 lessons we're um all we're now starting off |
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| 20:08 | the second lesson. And so the that I was talking about uh uh |
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| 20:13 | ideas involving an isotropy. And so uh uh if you look at any |
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| 20:21 | , you can see that uh crystal a AAA special shape to it and |
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| 20:27 | shape is determined by the internal arrangement the atoms. And uh so uh |
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| 20:36 | had the, the crystal has various and, and faces and so |
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| 20:40 | And that's all determined by the internal of the atoms. So surely it |
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| 20:46 | be true that, that same internal of the atoms means that the uh |
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| 20:53 | propagation of sound is gonna be in directions, whether it's across the faces |
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| 20:58 | along the edges or whatever. Uh uh uh it, it would be |
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| 21:05 | . And so that's the subject of . So it turns out that all |
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| 21:10 | rocks, not only minerals but rocks anisotropic. So, um we are |
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| 21:17 | to ignore that for most of this . Uh We're gonna start off uh |
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| 21:23 | way up here. Let's see. Let me get a, a pointer |
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| 21:26 | myself. Oops, that's not what want. I wanna um Yeah, |
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| 21:35 | gonna start off here. And so we're gonna be talking about um isotropic |
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| 21:41 | uh uh all down through here. then when we get to this point |
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| 21:47 | , we're essentially done with a standard in seismic ways and rights. But |
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| 21:54 | many uh assumptions that we made along way, which are really not |
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| 22:00 | And so then we're gonna take up three advanced topics down here. Uh |
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| 22:06 | oral elasticity as opposed to elasticity, of sound and anisotropy. So, |
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| 22:14 | are advanced topics in wave propagation uh uh go beyond uh the standard uh |
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| 22:23 | in uh seismic uh uh waves and , but they're important for modern |
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| 22:30 | No. Um mhm I did not uh when talking about myself that um |
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| 22:46 | uh was very active in the International of exploration geophysicists. And so let |
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| 22:53 | pause now and uh ask you, , are, are you guys um |
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| 23:01 | of the Society of Explorations just as it is, Carlos is not. |
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| 23:11 | you should join Carlos. Uh And uh not only uh is there. |
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| 23:16 | uh So, uh we have um uh we have a uh an affiliate |
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| 23:25 | in Colombia. I think the headquarters in Bogota and I think it's called |
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| 23:29 | Colombian Geophysical Society, something like And so you should join your local |
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| 23:35 | as well. And so Rosetti, should also join the Geophysical Society of |
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| 23:41 | . So it will cost you a dollars or maybe not. I think |
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| 23:46 | will pay for the Duke. I am, I am also a |
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| 23:51 | of that one. And how about , Eli both and I are you |
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| 23:58 | uh a member of Seg Canada? . OK. So these are all |
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| 24:03 | um uh associations for you to you should be active members because your |
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| 24:09 | job could come from somebody you meet the society, right? If you |
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| 24:15 | if you are an active member and attend meetings and you uh uh uh |
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| 24:20 | have um many sorts of meetings, technical meetings and social meetings and business |
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| 24:27 | and so on. You should be in your society and it might be |
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| 24:32 | you meet somebody there who thinks that want to hire you for your next |
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| 24:37 | . So uh um it's well worth money that you spent. Now, |
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| 24:45 | mentioned that because you can see you can see that uh we have |
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| 24:51 | seg mentioned here on this slide. , yeah, you can see it |
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| 24:57 | here. I see. So, yeah, um I developed this |
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| 25:05 | the first version of this course for seg and their plan at that time |
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| 25:12 | to offer it uh as uh uh uh a stand alone course without any |
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| 25:22 | present. And so we set up course so that the course it did |
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| 25:28 | require. And um uh uh an as anybody could uh uh buy the |
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| 25:37 | from the SCG and it didn't cost much money. I forgot what it |
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| 25:40 | cost uh probably less than you're paying for tuition for this course. And |
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| 25:47 | they could do the studying um um anywhere in the world. And so |
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| 25:54 | me show you here uh uh uh things that we have in this background |
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| 26:00 | , un unhappily what we have what I'm showing you here is just |
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| 26:04 | static background. So you won't see page number changing as we go through |
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| 26:10 | . Uh uh Because uh it's just static thing. Uh Because since I |
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| 26:17 | this course, I've uh uh uh years ago now, maybe 10 years |
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| 26:22 | , by now, the course has enhanced a lot by material which is |
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| 26:27 | included in the SCG version of the . And so I'm teaching you uh |
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| 26:36 | uh this in the next few I'm teaching you um the modern version |
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| 26:41 | this course. But I wanna show uh uh some features that we |
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| 26:47 | Uh uh we have here, for , you can see down here, |
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| 26:50 | You can see the uh the uh uh control buttons for, for moving |
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| 26:56 | the course. And can you see button right here? Uh If uh |
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| 27:00 | you, if we were doing if you had bought the course from |
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| 27:04 | SCG and we're listening to it, in your bedroom in China, why |
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| 27:12 | ? Uh uh uh you could uh this on and hear my voice |
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| 27:19 | Actually, it's not my voice. hired an actor. Well, that |
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| 27:23 | a mistake. The actor uh uh uh didn't do a good job. |
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| 27:27 | should have hired me uh but they an actor. And so uh that |
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| 27:32 | hear that they're gonna hear the actor's talking about this stuff. Now, |
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| 27:40 | the uh uh uh if you're as we get down here, suppose |
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| 27:43 | down here in surface waves and we back to something uh in an earlier |
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| 27:49 | . Uh uh you, you would able to click on this up here |
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| 27:53 | see all the chapters and go back the previous chapter and, and uh |
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| 27:58 | yourself what happened in that chapter. , you could, you could click |
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| 28:03 | here um for uh the glossary. so if we use the term uh |
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| 28:08 | this course, which uh you're not with, uh you could find an |
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| 28:15 | in that glossary so that you will all of this as a fossil. |
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| 28:21 | uh uh it's uh not active in version that, that we're gonna be |
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| 28:26 | about today. But actually, your is better, what I wanna do |
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| 28:30 | I want to uh persuade the seg uh update uh of that. Um |
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| 28:40 | that course that they have, it's uh uh 10 years old and it |
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| 28:45 | to be updated anyway. So, , so then, uh uh |
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| 28:52 | I, I'm going to now advance the next slide here and uh you |
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| 28:58 | see here that in, in uh the outline here, this is |
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| 29:02 | . And so, uh uh this changing. Uh uh It's, it's |
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| 29:07 | a dead outline right now. So are the, the uh uh |
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| 29:12 | And so this is the objectives of uh lecture only. So we're gonna |
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| 29:18 | four hours of lecture here. And the way, uh uh uh four |
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| 29:23 | is a long time to sit in place. Uh You, you're |
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| 29:26 | but it is gonna get tired. we are gonna break, um, |
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| 29:32 | halfway through, we'll find a convenient , we'll break halfway through and |
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| 29:38 | uh, uh, you know, for 10 minutes. Uh uh Maybe |
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| 29:42 | get, got a bite to eat something. Uh, I think everybody |
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| 29:48 | is more or less on the same zone. So that's not gonna be |
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| 29:52 | problem for us. Now, at the end of this, |
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| 30:02 | four hours today, we, we will finish at, at five |
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| 30:06 | , Houston time today and we will up, uh, uh, tomorrow |
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| 30:14 | a full day of instruction, 24 lectures tomorrow. Now, the, |
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| 30:22 | , uh, uh, the syllabus that we're gonna start at 8 a.m. |
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| 30:27 | time. Let me pose it to . Would you prefer to start at |
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| 30:33 | a.m. Houston time and go until So we'll still get the, |
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| 30:40 | the, uh, eight hours of tomorrow and subsequent Saturdays also. But |
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| 30:47 | start at nine o'clock instead of eight . Is that ok with you |
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| 30:51 | Yeah, he says, yes. about you, Brisa? Is that |
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| 30:55 | with you? Yes, I'm ok that. Yeah. And Carlos, |
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| 31:00 | , you have. Ok. So will, uh, is that ok |
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| 31:03 | you to? Ok. So we'll again at, uh, uh, |
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| 31:08 | , nine o'clock tomorrow and you will coffee for it. Ok. And |
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| 31:15 | , uh, I'm sorry, you uh in uh coming remotely, |
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| 31:19 | have to provide your own coffee, uh Utah is gonna uh bring coffee |
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| 31:26 | me and you leave. So now might happen at the end of, |
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| 31:32 | five o'clock today. We, we not finish with all the stuff we |
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| 31:37 | talk about today, so we'll just start up again. Uh um uh |
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| 31:43 | , on Saturday morning, we're not worry too much about how the uh |
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| 31:52 | how the modules match up with the slot. But uh by the time |
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| 31:58 | get to the end, we will in sync. Now, there's gonna |
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| 32:02 | one more thing which is very We will begin the lecture tomorrow morning |
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| 32:11 | nine o'clock. Not with, not we left off. We're gonna begin |
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| 32:19 | questions from you because uh my experience that um uh in a course like |
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| 32:28 | , the, the students are frequently to interrupt the professor with a question |
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| 32:37 | , but they, but everybody does questions in their mind. So, |
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| 32:41 | what I want for you to do um uh if you think if you |
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| 32:49 | a question, do not be go ahead and hold up your hand |
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| 32:52 | me. And uh uh I uh you can be certain that other people |
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| 32:58 | the class will be pleased that you the one who interrupted the uh the |
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| 33:05 | because they probably have the sa very question in their mind. But if |
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| 33:09 | are other questions which you don't want pose verbally during the lecture, you |
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| 33:18 | them down and send them to me email overnight and we will start |
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| 33:27 | uh the, the uh le shirt morning with your answering your questions |
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| 33:36 | And so that's a course requirement. ha uh uh it's not optional. |
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| 33:41 | there, we have three students here we're gonna, I'm gonna expect to |
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| 33:46 | uh um in my uh uh inbox um three emails, one from each |
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| 33:56 | you and you two time if, , if you have one. So |
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| 34:00 | of course, is an advanced But uh I would expect you ty |
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| 34:05 | you might learn something also in this which you had not uh uh learned |
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| 34:12 | , even though you had a course this. So I'm expecting that everybody |
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| 34:18 | going to send me a question and I will begin the lecture tomorrow morning |
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| 34:26 | those questions. So this is your to get uh uh uh uh 1 |
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| 34:31 | 1 answer uh to your questions just if you were uh uh uh coming |
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| 34:38 | my office here at the University of and sitting down and saying uh uh |
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| 34:43 | have a question about what was uh yesterday. Uh Could you please explain |
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| 34:48 | further? Something like that? I need for you to uh uh uh |
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| 34:56 | but since we got uh be doing uh by email, of course, |
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| 35:01 | will know your names uh uh associated each question. But uh I |
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| 35:07 | don't be shy. Um we'll uh arrange the uh the questions uh without |
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| 35:17 | things. OK. So let's now when the uh let's now begin with |
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| 35:30 | substance of this course. So that's first topic is what is elasticity. |
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| 35:38 | , elasticity is a physical property of material. So here you see a |
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| 35:46 | and the elasticity is a property of steel heat. It's not a property |
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| 35:52 | the spring because you could have the steel with different windings for the uh |
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| 36:01 | . So the elasticity is AAA property the material inside here. And it's |
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| 36:08 | , uh it's a property which makes material deform when a stress is |
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| 36:15 | And furthermore, when the stress is , that deformation disappears 100%. So |
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| 36:23 | what we mean by elasticity. this is not a, a perfect |
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| 36:36 | of the way rocks behave. If squeeze a rock, uh deforming in |
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| 36:45 | rock and then let go of that rock might not deform, might |
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| 36:51 | recover the original shape. Exactly. it might not recover the initial shape |
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| 36:59 | . It might be, there might a little delay even you suppose. |
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| 37:06 | suppose it does come exactly back to state. It was before you apply |
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| 37:14 | stress, you should still ask yourself , is that recovery instantaneous or does |
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| 37:21 | uh uh require a little bit of ? That turns out to be a |
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| 37:26 | profound question. You might not have about that when you release the stress |
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| 37:31 | a rock, what happens? Now, of course, it's this |
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| 37:40 | of elasticity which makes springs to be . Yeah, in the 19th |
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| 37:48 | this was a major um topic of by physicists. And back in those |
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| 37:56 | , in the 19th century, there not any geophysicists. The first |
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| 38:04 | whoever called himself a professor of he took up that job in |
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| 38:10 | So in, in the previous century the 18th century uh um in the |
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| 38:18 | which is uh 19th century, um were no professors of geophysics. |
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| 38:30 | the last important contributor to geophysics was guy named Love and you know his |
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| 38:36 | probably because you have heard of Love propagating uh in the earth. So |
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| 38:43 | the same guy love. And he went by his um uh initials A |
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| 38:52 | love to tell you the truth. don't know uh what it stands |
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| 38:57 | But so let us let us uh uh look here you, you see |
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| 39:03 | is underlying. So that means that name appears in the glossary. So |
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| 39:10 | we're gonna do is we're going to um right. Get out of |
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| 39:22 | Oh, and we stopped sharing. I'm gonna uh I don't know. |
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| 39:26 | I'm, I'm gonna need your We try, I think to re |
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| 39:32 | . Hold on. Not yet. yet, not yet. Um So |
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| 39:37 | I'm gonna do folks is I'm going bring up the glossary. OK. |
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| 39:57 | uh connect me properly here. Go . Mhm Right. You, you |
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| 40:26 | a great, yeah, I share too. So everything you share there |
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| 40:31 | can see it. Let me OK. So can everybody see the |
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| 40:38 | now? Yes. Ok. So uh um oh, she had 91 |
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| 40:49 | in the glossary. So, uh like this, things like this. |
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| 40:58 | let us um um, so in slide show and I'm going to uh |
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| 41:16 | you see my screen now? So I'm going to uh I, |
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| 41:21 | in uh what it calls uh uh view. OK. So there is |
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| 41:33 | glossary entry for Love and I thank . They presented that I share their |
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| 41:47 | . Yeah. OK. So there the, the glossary entry for Love |
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| 41:54 | you can see that it's in alphabetical . Here's the previous one and here's |
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| 42:00 | uh uh Love and here's the next . You, you see how this |
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| 42:04 | gonna work. So then um uh now we find out what his initials |
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| 42:09 | for. Um um So he published in 1928. And as a matter |
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| 42:17 | fact, you can buy that um on Amazon and uh there are several |
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| 42:24 | you, you might find the uh 1927 edition and so on or you |
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| 42:29 | buy the uh the uh 2008 reprint uh by Dover publication. And so |
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| 42:36 | are old books and uh I don't they're too expensive. So, uh |
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| 42:44 | you, you might want to buy . Now, um uh Utah come |
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| 42:48 | here and show me how to uh back to the lecture gracefully, you've |
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| 42:55 | stuck and go to the this one present it. OK. OK. |
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| 43:05 | , um so here we are. now, so that is elasticity. |
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| 43:20 | uh we are not really interested in for itself. There, there used |
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| 43:27 | be a lot of physicists who were in that, not so many |
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| 43:31 | And I would say that in mostly we are not interested in elasticity |
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| 43:38 | , but mostly we're interested in um a consequence of elasticity which is seismic |
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| 43:46 | . So, seismic waves are waves stress and strain inside the rock formations |
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| 43:52 | the earth. Uh That's why uh we say seismic, we really imply |
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| 43:56 | the earth. Oops I, I to get myself uh a pointer |
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| 44:02 | So when we say seismic, we are referring to the earth. And |
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| 44:07 | uh these seismic waves are approximately So you are familiar of course with |
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| 44:13 | waves. And here we have AAA of a P wave see this hammer |
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| 44:20 | uh uh hammering on the bottom of uh cartoon and the wave is going |
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| 44:25 | , see there's a zone of compression another and this goes up. And |
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| 44:31 | uh if the hammer hits on the , then it makes a a torsional |
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| 44:37 | uh wave and that one also goes up at a different speed. |
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| 44:41 | those are body waves and those are ones which we are most interested in |
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| 44:47 | they go down into the body of earth, they reflect and they come |
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| 44:52 | to our instrument. But tho even though those are the ones that |
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| 44:56 | most interested in, they are um not the, the ones which are |
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| 45:03 | prominent in our data. When we out our uh seismic receivers and receive |
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| 45:10 | waves. Suppose we're just doing a survey and we're going around nearby with |
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| 45:15 | uh um a vibrator uh or we're around nearby with an air gun at |
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| 45:23 | at sea generating seismic waves. Most what we uh um record the uh |
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| 45:33 | the most energetic arrivals are not but they are surface waves and these |
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| 45:40 | travel along the surface at speeds which not P wave speeds and they're not |
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| 45:49 | wave speeds either. And so there um um two kinds of uh surface |
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| 46:04 | . And you see, well, in the first place, uh uh |
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| 46:09 | one is pretty clear. Uh You see that it's, it's going up |
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| 46:12 | down in a, in a shearing as it moves from left to |
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| 46:17 | but look more closely at this You can see these little arrows here |
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| 46:21 | there. That means that the deformation into the screen and out of the |
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| 46:27 | as the wave moves to the So you can see that surface waves |
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| 46:33 | kind of complicated and we are, are really not interested in surface waves |
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| 46:39 | of us, most of the time regard these surface waves as uh uh |
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| 46:45 | that we don't even want to look . And it's a bummer that for |
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| 46:50 | data especially, they are the strongest on our data. And why is |
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| 46:56 | ? It's because we have our receivers the surface and these waves are confined |
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| 47:02 | the surface. Can you see on uh surface way down in particularly, |
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| 47:06 | you see the decoration here is big the deformation down here is less and |
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| 47:11 | defamation down here is even less as de definition down here is zero. |
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| 47:17 | these are waves which are confined to surface. And that's kind of uh |
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| 47:25 | to think about why it is that coupling of the wave to the surface |
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| 47:34 | it's gonna travel with a different Interesting. OK. So no, |
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| 47:45 | this study uh of uh sewa it's be uh fairly mathematical and that might |
|
| 47:59 | a problem for you all who are uh interpreters who are uh has been |
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| 48:05 | years since you took mathematics. So to make sure that we're all on |
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| 48:10 | same page, I wanted to ask some questions. So here is |
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| 48:15 | a little quiz. So here we a vector which is expressed on the |
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| 48:21 | side by this formula um by this on the right side by this |
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| 48:27 | And the uh the question is the of the vector is what you get |
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| 48:32 | questions, uh uh answers ABC or . So let me turn to you |
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| 48:38 | uh E Lee and uh, w answer is correct? B OK. |
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| 48:45 | , uh I think that you folks cannot hear her, right? |
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| 48:51 | uh what she says is, is uh b um and yes, |
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| 49:00 | does anybody disagree. So everybody agrees , and in fact, I agree |
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| 49:07 | very good. But before we pass here, notice here, there's a |
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| 49:11 | of notation, you see this vector denoted here with an uh an arrow |
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| 49:20 | the top. And uh um of , on the right hand side of |
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| 49:26 | equation, it indicates the square brackets two indices in between two components in |
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| 49:32 | , indicating that this is a, vector with only two components which are |
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| 49:38 | by the sub. OK. So notation is gonna be uh uh we're |
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| 49:45 | see it over and over again. In in this course, let me |
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| 49:49 | on to the next one. So how about this, the length of |
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| 49:54 | vector is uh uh as a human . So I if you look, |
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| 50:00 | you remember the previous one, these ABC and D are the same answers |
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| 50:05 | we had in the previous one. the specification of the vector is |
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| 50:10 | you see this one is specified like column, this one is specified like |
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| 50:15 | row. So let me ask you uh Carlos, uh what's the length |
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| 50:20 | this leg professor to tell you the ? I don't know. I think |
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| 50:27 | be the thing. I, I say that is the also but the |
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| 50:30 | , very good. So, I like that number one, you're |
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| 50:34 | a uh uh your ignorance. And , we understand that because you've been |
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| 50:39 | as a geologist for 17 years. bet it's been 17 years since you've |
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| 50:43 | uh uh uh since you had a in mathematics. So, you |
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| 50:48 | this always happens when I teach this that there's a wide range of experience |
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| 50:55 | the classroom. And so um it's calm that most, that many of |
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| 51:03 | students um have been some time since last course in mathematics. But mathematics |
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| 51:12 | a big part of this course. , you know, so uh I'm |
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| 51:15 | give you all a lot of help the mathematics and uh um uh at |
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| 51:23 | end of this course, uh you're be comfortable with things like that. |
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| 51:26 | , the next thing I like about answer, Carlos is you and you |
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| 51:29 | an intuition and you said uh uh , the length of it has got |
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| 51:34 | be independent of how we write it . And so you're, you're absolutely |
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| 51:39 | . This is the right answer again this is AAA just a different way |
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| 51:43 | write an effect. OK. Next , how about the length of this |
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| 51:50 | ? Now, you see this you can't tell on the left, |
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| 51:53 | on the right hand side, you tell it's got three components. So |
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| 51:57 | me turn to you Brisa and uh uh what is the length of, |
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| 52:02 | uh of this vector? I would the same B Yeah, it's B |
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| 52:08 | of course, uh B is now than it was. See B is |
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| 52:11 | square root of the sum of three instead of the sum of two |
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| 52:16 | So that's because this is a three vector and so on. But |
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| 52:21 | your intuition is correct. Uh uh would be really stupid if we uh |
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| 52:27 | uh if we uh change our definition the length uh in an important way |
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| 52:35 | because we have a third component Now, how about this one |
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| 52:46 | we um uh uh saying the length a vector is given by and by |
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| 52:52 | way, you see this has no on. So uh uh let me |
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| 52:56 | up here, see this one has arrow uh When we have an arrow |
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| 53:01 | means it's a vector without an it's a scr, you know, |
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| 53:05 | and, and uh doesn't have any to it. So now, going |
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| 53:11 | now, so here the, the length is given by this component |
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| 53:16 | Is this true or false come back you? Y le mm She thinks |
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| 53:25 | false. OK. So tell me thinking here. Mm uh Well, |
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| 53:34 | can't tell from this. You, don't know whether these are two dimensions |
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| 53:38 | three dimensions, right? Yeah, could be three dimensions. Uh uh |
|
| 53:45 | what we've done is introduce some notation . So this is called the dot |
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| 53:51 | of, of a vector with So you take a vector X vector |
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| 53:55 | and make it uh uh uh and it by itself with what we call |
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| 53:59 | dot product. So she's uh uh gonna make a guess of B So |
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| 54:06 | me turn to you Carlos. what do you think is the answer |
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| 54:12 | ? But I think it's also be . OK. And brace. What |
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| 54:16 | you think? I think it's OK. So she's so excellent. |
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| 54:29 | like your courage. You, you are, you're going against the |
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| 54:32 | , you got two of your And I guess you all have been |
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| 54:36 | now for uh for several courses, ? You know each other. |
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| 54:40 | And so here's Brusa having the courage vote against her friends. OK. |
|
| 54:46 | how do we know the answer? , what, what we can do |
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| 54:50 | we can see here. This is , isn't it? And so um |
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| 54:54 | we can go here to the um OK. Uh to the glossary. |
|
| 55:04 | uh uh uh remind me, what I supposed to do here? Uh |
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| 55:08 | Utah. I um do I mm uh Utah, what do I do |
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| 55:19 | get out of this? Uh And back in uh uh my name is |
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| 55:27 | . Ok. Ok. That's So, so uh so now uh |
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| 55:32 | stop sharing. Yeah, so you can't see anything. Uh But now |
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| 55:38 | gonna start sharing. Oh OK. . So this is back in the |
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| 55:49 | uh in, in the glossary OK. So um uh uh again |
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| 55:56 | gonna uh I'm gonna stop sharing. messed up. Mhm. Ok. |
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| 56:36 | So which size you want this OK. OK. Now um |
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| 56:42 | Yeah, that the, the right . OK. You hear screen? |
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| 56:54 | , no, that's right. That's . You can uh you share the |
|
| 56:59 | so you can when you present Mm Let me see and like |
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| 57:17 | so yes, still there. 32. Hm. No, that's |
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| 57:39 | , that's not right. And you present the one you want. Just |
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| 57:51 | , they can see your script. . Yeah. But II I did |
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| 57:56 | wrong again. I'm gonna uh stop , stop present. Mhm Like the |
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| 58:06 | walk, right? Mm No, don't, you don't, you don't |
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| 58:24 | to go to zoom anymore. It , you know, just present. |
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| 58:30 | know. H Yes. OK. So um I can, can you |
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| 58:47 | remotely see this now? OK. this is an uh uh um a |
|
| 58:56 | from the glossary. See down this is glossary slide 31. And |
|
| 59:01 | here it says it says the definition said when you see three equal sign |
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| 59:08 | this oops, wait a second, need to get myself a partner. |
|
| 59:13 | this means it's a, it's a . And you see three lines like |
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| 59:18 | , two lines is an equation, lines is a definition. OK? |
|
| 59:22 | so this is the definition of the product of two vectors, X vector |
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| 59:27 | Y vector. And you see it's sum of X one times Y one |
|
| 59:32 | X, two times Y two plus three times Y three. Now, |
|
| 59:36 | can see that if this is true any Y but suppose that we have |
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| 59:45 | uh Y is the same as then this is X one squared, |
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| 59:50 | is X two squared, this is three square. And then, |
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| 60:23 | I think we're back to the, lecture now. So uh uh then |
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| 60:29 | you see, now we have the of the dark po and so then |
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| 60:34 | uh with that definition in hand, we see that uh it's just the |
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| 60:40 | as we had before. So the is, is a good for your |
|
| 60:47 | . And um now, so uh uh uh continue on with a little |
|
| 60:58 | review. So here um we're talking matrices. So in mathematics, a |
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| 61:07 | is a rectangular array of number, , symbols or expressions, the individual |
|
| 61:14 | are called elements. And here is example of a three by two, |
|
| 61:18 | by two main things you see it's rows. So uh right. |
|
| 61:30 | it's this is actually a two by matrix. Yeah, it's got two |
|
| 61:37 | and three columns. So, so is wrong. So I, |
|
| 61:40 | I'll have to fix that up. for pointing that out. Oh, |
|
| 61:46 | sort of the definition of a Now, next question is what is |
|
| 61:52 | sum of these two matrices? So gonna uh um uh turn to you |
|
| 62:00 | since you won the last one. Tell me what's uh what's the uh |
|
| 62:07 | correct answer here? I, I it's D I don't really remember what |
|
| 62:19 | will go through. D Yeah. you, you think it's D, |
|
| 62:24 | that what you're saying? Yes. . OK. So uh uh let's |
|
| 62:28 | uh he, what do you She, she agrees. What do |
|
| 62:33 | think? Carlos? Not sure, sure. Professor I did too. |
|
| 62:44 | . So what I know what you here, you, you, you |
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| 62:47 | the uh uh every one of you the same thing. Hold on a |
|
| 62:50 | . Let me get myself a OK. Yeah. Uh You did |
|
| 62:54 | uh component by component. You took three and added this 10 and it |
|
| 63:00 | a 13 in this upper left So we call this the 11 position |
|
| 63:05 | then you looked around here and say uh OK. So where do we |
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| 63:08 | a 13 knot? Here's one of , th here's one with a |
|
| 63:12 | So the answer is got to be C or D, so then we |
|
| 63:16 | one plus five, that's a Uh uh So that is, I |
|
| 63:21 | , this one is wrong. And then let's just check. Two plus |
|
| 63:24 | is seven and four plus 10 is . So the answer is D |
|
| 63:29 | So that, I think is pretty that's AAA straightforward extension of uh your |
|
| 63:38 | sense. Now, now this gets complicated. What's the matrix product? |
|
| 63:49 | I'm gonna turn to you uh um Carlos. Uh What do you think |
|
| 63:56 | the matrix product here? Not Is it not clear? So, |
|
| 64:10 | suppose we did it by the same we did before, I think it's |
|
| 64:15 | but not, not true. Uh . So uh uh tell me why |
|
| 64:19 | think it's b because the first, position 11 would be the multiplication between |
|
| 64:27 | times 10 and, and then I have to add, they fight |
|
| 64:34 | No, I'm not, I'm not . Professor. OK. Well, |
|
| 64:37 | you are correct. So the rule you first, let me give you |
|
| 64:42 | wrong rule. Uh When you have product like this, you don't do |
|
| 64:47 | uh um um element by element like did in the, in the uh |
|
| 64:53 | . Um You don't just take three uh 10 and, and uh and |
|
| 64:58 | that the uh uh the matrix And in fact, none of these |
|
| 65:03 | here have a 30 in that 11 , right? So what Carlos did |
|
| 65:08 | , he said, uh, that's not what we do when we |
|
| 65:11 | a matrix product. We don't just three times 10. Uh, |
|
| 65:15 | but we also do one times five add those together and makes, |
|
| 65:20 | uh, 30 that makes, um, 35. And here, |
|
| 65:24 | fact, we have a 35. now, um, uh, |
|
| 65:29 | let's, uh, let's look how would we go about this |
|
| 65:45 | This is like to multiply by 10 four, multiplied by five. Oh |
|
| 65:58 | , that is true. No, she is suggesting is we have three |
|
| 66:06 | 10 plus four times five, you , uh uh uh that's not what |
|
| 66:13 | wanted. So I can see here we uh we do have a problem |
|
| 66:19 | understanding um matrices and how you can matrices by multiplication. Combining them by |
|
| 66:30 | was easy. And of course, them by subtraction is also easy, |
|
| 66:34 | combining them by multiplication is complicated. uh well, we, we're boggled |
|
| 66:43 | and um then uh uh I suppose would also be boggled. What happens |
|
| 66:50 | combine them by dividing instead of by . So at, at this |
|
| 66:55 | what we're gonna do is we're going get out of this um uh screen |
|
| 67:03 | and I'm going to um minimize that I'm going to open up another file |
|
| 67:21 | I call math 101. And I'm stop sharing and I'm go and I'm |
|
| 67:34 | start sharing OK, time, I'm a hard time here getting this |
|
| 68:18 | to show the slideshow. OK. , this one here is, what |
|
| 68:36 | was that? Mhm Yeah. Stop , right? OK. So |
|
| 69:03 | so yeah. OK. So uh you, you and I have to |
|
| 69:10 | on this uh after last week and make these changes uh more uh |
|
| 69:17 | OK. So um then uh this , this file is uh available to |
|
| 69:26 | And oh And by the way, all have this file uh available to |
|
| 69:33 | on canvas and you might want to it and print it and it's in |
|
| 69:39 | , it's in a format so that can have uh uh uh three slides |
|
| 69:45 | page and with room on the side make write notes and um uh where |
|
| 69:53 | want to handwrit notes. So uh sometimes students like to do that. |
|
| 69:59 | let us now uh oh hm go the uh uh this mathematics refresher. |
|
| 70:10 | you see we have vectors uh which had no trouble with matrices, which |
|
| 70:16 | did have trouble with sensors, which might not even know what they are |
|
| 70:21 | calculus and then compliance here. So uh you can see there are a |
|
| 70:26 | of topics here and um oh 12 uh pose, pose you the |
|
| 70:37 | um um Is this a vector? so the answer is, oops, |
|
| 70:46 | , this is just an air. . Let me get yourself a |
|
| 70:51 | Is this a vector? And the is no, it's just an arrow |
|
| 70:55 | uh it's a typesetting trick. So uh a lot of people don't know |
|
| 70:59 | but if you type in, in Microsoft product dash dash, right. |
|
| 71:07 | All right, bracket you get Um So this is just, that's |
|
| 71:18 | a vector, that's just a AAA of typesetting. Now, the question |
|
| 71:22 | , is this a vector? And the answer is you see there is |
|
| 71:27 | notation. So if it's notation for apples, seven oranges and four |
|
| 71:33 | that's not a vector. It's, a shopping list. But if it's |
|
| 71:38 | for three units in direction one and directions in direction two and four units |
|
| 71:47 | direction three, then it's effective. it's, it's compact notation. But |
|
| 71:54 | ha we all have to agree if notation for what and uh if it |
|
| 72:00 | this meaning, then it's a OK. So we have to come |
|
| 72:06 | an agreement. What do you mean direction? One? So uh and |
|
| 72:09 | on. So when we have that , then that's we call that a |
|
| 72:15 | system. And so uh usually uh the court system uh is uh has |
|
| 72:21 | orthogonal directions. And um you can that I have only a, a |
|
| 72:27 | dimensional screen to show you here, you can uh uh understand that this |
|
| 72:32 | two vector is pointing out of the . And if you want to know |
|
| 72:39 | orthogonal means that's in the glossary. so we agree when we uh uh |
|
| 72:45 | we, we wanna agree on this system, how it's oriented and also |
|
| 72:51 | the origin is. No, sometimes label these three directions, 12 and |
|
| 73:01 | , or sometimes we label them XY Z uh uh those are trivial |
|
| 73:08 | And usually the order of the directions to the right hand rule. |
|
| 73:14 | So I want you to look look at my hand here, I'm |
|
| 73:20 | up my right hand and I'm showing one, two, three. |
|
| 73:29 | suppose I did this with my left , 123, you see that's |
|
| 73:37 | So, uh uh um is a of the right hand rule in the |
|
| 73:46 | , I'll uh I'll let you look that after class, but you can |
|
| 73:50 | that when we make um uh a about a coordinate system, we have |
|
| 73:59 | agree on all these things, how oriented, how it's uh how it's |
|
| 74:03 | or is it a right handed court , left handed court system or |
|
| 74:08 | where the origin is? And um if the rocks are isotropic, all |
|
| 74:18 | these decisions that we make about the system, the rock doesn't know anything |
|
| 74:26 | that. Right. That's all in mind. The rock does not know |
|
| 74:31 | about that if it's an isotropic Now, it turns out that if |
|
| 74:38 | an anisotropic rock, uh uh um , that's gonna be different. |
|
| 74:45 | uh uh for now, let's realize uh the rock, uh let's, |
|
| 74:51 | , we, we're going to be all these ideas to isotropic rock where |
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| 74:57 | of these um decisions about the court , uh The rock doesn't care. |
|
| 75:05 | are all in our own minds. so we better not come up with |
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| 75:09 | , uh with uh uh a procedure we think the rock knows what we're |
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| 75:18 | about. In other words, uh , we better formulate our uh our |
|
| 75:26 | in such a way that it's independent our choice of coordinate system. I'll |
|
| 75:34 | you more on what that means uh on. OK. Now, it |
|
| 75:49 | seem obvious that this is the kind court system that we want. This |
|
| 75:53 | uh was uh uh designed by uh uh a French uh mathematician whose name |
|
| 76:01 | Descartes in the um uh 18th And so it's called a Cartesian co |
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| 76:07 | . But uh there are uh lots cases where we wanna use something |
|
| 76:12 | So for example, uh the directions different at different positions. For |
|
| 76:19 | you could have let's talk about a cylindrical coordinate system. So here's |
|
| 76:24 | cylindrical coordinate system where uh we have vector is the radial direction, one |
|
| 76:30 | the angular direction and one is the is the accidental direction. So that |
|
| 76:36 | be the coordinate system which would be natural way for us to, to |
|
| 76:41 | the mathematics in a borehole. And that case, the uh the uh |
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| 76:49 | the cylindrical axis would obviously be parallel the borehole. And so you can |
|
| 76:55 | that if you did that, it make a lot of advantages for describing |
|
| 77:00 | propagation in the borehole. Uh if you are uh uh uh interested |
|
| 77:06 | uh uh global seismology and how uh seismic waves travel uh uh uh around |
|
| 77:13 | uh the earth. And you might talk about that in terms of a |
|
| 77:18 | cord system where the, the three are radius, a latitude and |
|
| 77:33 | sometimes the directions are not even See how, how uh this could |
|
| 77:38 | a right-handed quarter system. You can , hold your fingers out pointing |
|
| 77:43 | your uh your index finger in the direction and your uh um uh uh |
|
| 77:53 | and your middle finger in the two . And if you did that, |
|
| 77:56 | would see that the three direction is downwards. So this is actually um |
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| 78:01 | left handed court system. Now, would you want to do that? |
|
| 78:08 | , suppose you uh uh we're analyzing propagation inside of a crystal and the |
|
| 78:15 | of the crystal are not orthogonal to other. So you might want to |
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| 78:21 | the corner system lined up with the axis of the crystal, which is |
|
| 78:27 | by the atom, the arrangement of inside the crystal. So you see |
|
| 78:31 | this discussion of uh choosing a corner is not so trivial as uh Mr |
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| 78:39 | cartes might have thought about in this , we'll, we will use an |
|
| 78:46 | Cartesian chord system right handed, except it's exclusively noted. Otherwise. So |
|
| 78:54 | you see uh uh uh uh notation this, these are gonna refer to |
|
| 79:00 | axes and we can either uh uh write it as a uh a row |
|
| 79:06 | or column vector or write it with arrow or maybe just with a single |
|
| 79:12 | trip I. And if we, we write it like this, we |
|
| 79:15 | understand here that I could be a or it could be a two or |
|
| 79:19 | could be a three. So these all different ways of writing and what's |
|
| 79:24 | the same thing. And of uh there are two D variants of |
|
| 79:32 | these now intuitively, a vector is quantity that has both magnitude and |
|
| 79:38 | So uh for example, position is a vector displacement. So if you |
|
| 79:45 | a AAA point uh identified by a and then you displace it, that's |
|
| 79:53 | a vector velocity is a vector acceleration the vector force is a vector. |
|
| 79:59 | things don't have are not better. example, pressure and temperature geophysical examples |
|
| 80:05 | are not trends. And the magnitude given by this form which we al |
|
| 80:13 | um discussed uh half an hour Um And it was, this is |
|
| 80:21 | , a famous theorem by this guy . Who, whose name was |
|
| 80:26 | So this is the Pythagorean theorem. deep theorem. And I know you |
|
| 80:32 | learned about this in high school. Pythagoras uh um invented this uh thera |
|
| 80:44 | long, long time ago when nothing so obvious as so many things are |
|
| 80:49 | to us today. But it was extraordinary thing and it's uh it's uh |
|
| 80:55 | down to, to us through the , tell us true. Now, |
|
| 81:00 | it was then, now the direction the vector is given by these |
|
| 81:09 | So uh uh uh this is the of the first component divided by the |
|
| 81:17 | . That's this radius here. Let say something more about um um about |
|
| 81:26 | . Uh But when we have these vertical lines here, it means the |
|
| 81:32 | value of this. And uh you , I don't know why I have |
|
| 81:40 | equal, this approximate equal sign I think that's another type of, |
|
| 81:44 | think that should be a definition instead a, of a, in, |
|
| 81:51 | , instead of an approximation. And this is absolutely true. That's |
|
| 81:57 | definition here. And this is a . This is the result of the |
|
| 82:03 | genius of Mr uh Pythagoras now. uh um So that, so that's |
|
| 82:12 | R is. And uh the direction given by these three ratios which are |
|
| 82:16 | direction cosines. And uh uh you convince yourself, I, I think |
|
| 82:24 | I'll leave it to you to convince that these are the cosines, these |
|
| 82:28 | non dimensional numbers which are the co of the angles which uh uh uh |
|
| 82:36 | the direction of the vector. if this is a ve if this |
|
| 82:43 | a vector which has a magnitude of , we call it a unit |
|
| 82:47 | And then we denote that with uh we call this a carrots |
|
| 82:51 | That's a carrot right there. So uh has length of one. So |
|
| 83:00 | unit vectors in each of the three systems are denoted in this way, |
|
| 83:05 | . And so, and now we to the same uh quiz that we |
|
| 83:09 | before and um uh so we can over this because everybody knows that the |
|
| 83:14 | the, the length of this vector given by B and this is quiz |
|
| 83:23 | B and uh um uh uh the answers. And the only difference is |
|
| 83:29 | the notation up here for the vector . I'm gonna go back one. |
|
| 83:34 | this is a, a row This is a column vector, same |
|
| 83:37 | choices, same correct choice, same as we had before for a three |
|
| 83:45 | uh vector. It's uh uh still theorem. I think when you uh |
|
| 83:53 | Pythagoras theorem uh in high school in plane geometry course, you probably only |
|
| 83:58 | with two dimensional vectors. So it be sort of pleasing to you to |
|
| 84:03 | that in the real world with three . It's uh just a, a |
|
| 84:08 | extension of what you learned in plain . And how would we go about |
|
| 84:15 | two vectors together? We talked about before in terms of symbols. Let's |
|
| 84:21 | about this in terms of the pictures . So here we have two |
|
| 84:25 | we have X one X two for vector, we have Y one, |
|
| 84:29 | two for this vector. And so can add the two together by adding |
|
| 84:34 | up, nose to tail like So here's the nose and here's the |
|
| 84:38 | and um like just add them up I forgot who it was. Somebody |
|
| 84:44 | this group said, well, you , the first component is just the |
|
| 84:47 | of X one plus Y one and is one and the sum is for |
|
| 84:52 | other component of X two plus Y . And that is what is implied |
|
| 84:56 | this um picture of a, of moving this one up here, |
|
| 85:05 | to tail like it's showing hair and the sum is given by the black |
|
| 85:10 | . And so we can write that notation this way. Now, do |
|
| 85:17 | see uh uh we uh here's the example where we um a profit from |
|
| 85:26 | index notation. So if we say uh the uh uh the sum of |
|
| 85:31 | is the vector Z, then it for its I component, the sum |
|
| 85:36 | these two I components. And the is true for either I equals one |
|
| 85:41 | I equals two or I equals We have all those three equations summed |
|
| 85:46 | with just one if we, if give the subscript as an I instead |
|
| 85:53 | a one or two or three. uh just by leaving it vague, |
|
| 85:59 | uh uh this to be a a discrete verbal, you can't |
|
| 86:04 | I equals 1.3. You gotta have equals one or two or three. |
|
| 86:10 | then this uh uh is three equations one. I uh I'm gonna show |
|
| 86:21 | immediately some more um some more advantages this index notation. Remember we talked |
|
| 86:32 | the dot product and the dot product of two vectors is this one. |
|
| 86:37 | this product plus this product, this , we can write that compactly in |
|
| 86:43 | way and see what uh it says here when, whenever an index is |
|
| 86:49 | , it means we sum over all values. Well, we could have |
|
| 86:53 | in here a summation sign. But didn't have to if we understand that |
|
| 87:00 | you see a formula with two repeated , that means you, you s |
|
| 87:07 | one plus I equals two plus I three. And this uh um |
|
| 87:14 | this convention was invented by a fellow Albert Einstein, whose name you might |
|
| 87:24 | know from an, from other So uh it's amazing to me that |
|
| 87:32 | worked with a vector algebra and matrix for decades and nobody until Einstein in |
|
| 87:44 | realize that whenever you have two repeated , which are identical, that |
|
| 87:51 | always means that they are uh uh to be summed over. And if |
|
| 87:58 | have an equation which has, you , three repeated indices, you |
|
| 88:04 | you're screwed up somewhere. That's not , a well formed equation. |
|
| 88:13 | I'm not gonna prove it, but can convince yourself that this expression X |
|
| 88:18 | Y is equal to X scalar times scalar times the cosine of the angle |
|
| 88:26 | them. And it's a scalar, made out of vectors, but it's |
|
| 88:35 | scalar. Isn't that interesting? Now, like I said in the |
|
| 88:42 | line here and multiply vectors together in different ways, this is one way |
|
| 88:49 | dot product, the other way is cross product. And so here is |
|
| 88:56 | cross product between two vectors and I'm you only the I component. And |
|
| 89:04 | is the definition of it. It's is X time X of J times |
|
| 89:10 | sub K times epsilon with three indices . And K. Now look |
|
| 89:17 | you can see that the K is and the J is repeated. So |
|
| 89:21 | is an implied sum here. And , who knows what this means? |
|
| 89:26 | , I'm gonna show you what this . Here's the definition of E some |
|
| 89:33 | that's a matrix with uh uh uh uh with three indices here. And |
|
| 89:39 | has uh the value of zero. any two indices are the same and |
|
| 89:45 | has a value of one if the indices are 123 or 231 or 312 |
|
| 89:55 | it has minus one if the indices uh anything else. So let me |
|
| 90:02 | give you uh uh uh uh talk about the, these three choices |
|
| 90:08 | are uh similar. Uh look at the 123 and take the one and |
|
| 90:14 | it around to the end And that's 231, which is what it says |
|
| 90:19 | , take the two and move it to the end and that makes |
|
| 90:23 | So uh uh these are uh related that way. So when you uh |
|
| 90:35 | this definition of epsilon with three indices this equation, you find it's this |
|
| 90:42 | it's uh uh we're gonna get a three component vector where the first |
|
| 90:48 | is given by this difference in these terms. Second com component is given |
|
| 90:54 | and the third component is given And you see it's a vector, |
|
| 91:00 | product, the cross product of two is a vector where the dog product |
|
| 91:08 | two vectors is a scalar. Now Yeah, I'm not gonna show |
|
| 91:17 | I'm not gonna prove this, but can be shown that this is equivalent |
|
| 91:22 | uh X cross Y is equal to times Y scale. Uh uh uh |
|
| 91:29 | of X times length of Y times sine of the angle between the two |
|
| 91:35 | all of that is multiplying a a unit vector which is perpendicular to |
|
| 91:40 | X and Y. So uh I you probably learned this a long time |
|
| 91:46 | right now, I'm gonna ask you accept it as um uh uh sort |
|
| 91:51 | his definitions. So we have another , by the way, you see |
|
| 91:58 | these quizzes that we have here in course. And um the, the |
|
| 92:05 | course offered by the seg is structured such a way that if you uh |
|
| 92:12 | get the uh the little quizzes Exactly. Right. It cycles you |
|
| 92:19 | and so you go through the material . So uh in that way, |
|
| 92:23 | don't need a, a human Um So let's, let's look at |
|
| 92:28 | . Uh this is quiz. Uh I think this is it uh this |
|
| 92:37 | the quiz and the lecture tube and the fourth quiz in the lecture |
|
| 92:41 | So it's given as the d so length of this vector is given by |
|
| 92:46 | . So, is this true or ? Actually, we had this um |
|
| 92:51 | we had this uh same question So Carlos, what's the answer? |
|
| 93:00 | ? You muted. So the answer true. Yeah, that, that's |
|
| 93:06 | . Right. You, you remember . And furthermore, I hope you |
|
| 93:09 | a further understanding of this notation. . So with that introduction, we're |
|
| 93:15 | gonna go through the rest of the um uh uh quiz. Let's move |
|
| 93:22 | to matrices. Yeah, when a talks about a vector, he doesn't |
|
| 93:33 | the words we have here. and he's, remember we said intuitively |
|
| 93:37 | a, a vector is a quantity a magnitude and a direction. So |
|
| 93:42 | kind of makes intuitive sense I But that's not the way a mathematician |
|
| 93:46 | . He says a mathematician mathematically a is the quantity which changes to different |
|
| 93:53 | . If you make a different choice a recording system according to a certain |
|
| 93:58 | , remember we uh uh decided that can decide what the uh quarter system |
|
| 94:04 | . And if we choose a different system, that's up to us. |
|
| 94:09 | why not? Um But uh if want to uh uh uh a |
|
| 94:16 | it's gonna just be described in the coordinate system differently than in the old |
|
| 94:20 | system according to the vector transformation rule I'm going to uh uh uh to |
|
| 94:29 | you, you know. So uh us uh uh choose a coordinate system |
|
| 94:39 | orthogonal or system which is in So we got uh uh the uh |
|
| 94:46 | unit vector in the one direction is the right unit vector in the three |
|
| 94:51 | is down, you know, because geophysicists, if we were physicists, |
|
| 94:56 | would probably have that pointed up. now we have X two is pointed |
|
| 95:01 | into the screen or out of the . Uh uh How are we |
|
| 95:06 | let's make a right handed quarter And so uh Y le tell me |
|
| 95:12 | uh X two pointed into the screen out of the screen. She says |
|
| 95:17 | pointed into the screen. And so want you to hold your hand like |
|
| 95:22 | right to the side, hold your like this with your index finger uh |
|
| 95:28 | in the uh parallel to XY and thumb then pointed uh uh uh uh |
|
| 95:35 | down to me this way, uh finger uh uh pointed um uh to |
|
| 95:43 | right oh index finger pointed to the middle finger now perpendicular to the index |
|
| 95:54 | . And as I've pointed, it's out of the screen and then my |
|
| 95:59 | is the third direction that goes But you see, that's not what |
|
| 96:02 | have. What we have is is uh the other one. So |
|
| 96:06 | do it again. One, three. So she is right. |
|
| 96:15 | uh um uh uh X two, , if we're gonna have X one |
|
| 96:23 | to the right and X three that means that X two has got |
|
| 96:27 | go uh uh into the screen, out of the screen. So uh |
|
| 96:34 | I want you guys to uh a class, go over this again, |
|
| 96:39 | have this file uh uh go over again until you understand. Uh uh |
|
| 96:45 | is the consequence of a right handed system. Yeah. So that's the |
|
| 96:53 | . OK. Now, we had set up the, the physics which |
|
| 96:57 | independent of this choice, right? we have an isotropic rocks, it |
|
| 97:01 | care what kind of important system we , it doesn't know about right-handed or |
|
| 97:06 | or anything. Uh uh So, uh we want to set up the |
|
| 97:11 | , so it's independent for that. . Now, uh go back to |
|
| 97:18 | uh uh uh uh two dimensions and is a vector in two dimensions. |
|
| 97:23 | got X one in this direction, X one component here. This is |
|
| 97:28 | gonna be the X one direction I have had an arrow here and uh |
|
| 97:33 | error ahead here. And so this the X one component, the X |
|
| 97:37 | component and the length of this vector already talked about now consider another choice |
|
| 97:45 | partner system which is rotated. the, the vector itself is the |
|
| 97:50 | watch that black arrow. I'm gonna back, see the black arrow is |
|
| 97:54 | same, but the court system has . OK. So here is as |
|
| 98:02 | have new um uh components, one and X two prime uh uh uh |
|
| 98:09 | to uh uh the um rejections of vector onto the new quarter system. |
|
| 98:22 | um how did we decide this new system? Well, we, we |
|
| 98:28 | rotated the old, old quarter system like this. And we have the |
|
| 98:34 | that the uh uh the coordinate, angle is uh positive. If it's |
|
| 98:40 | clock, I could show it So this is putting all that together |
|
| 98:51 | , the original uh components X one X two, the new components X |
|
| 98:56 | prime and X two prime. Yeah, X one prime and X |
|
| 99:03 | prime. Yeah. What is the then between X one prime and X |
|
| 99:11 | and X two? Well, here can work it out with signs and |
|
| 99:16 | uh uh um from your uh uh understanding of plane geometry that um |
|
| 99:27 | these expressions um come from plane So we're not gonna prove that. |
|
| 99:36 | uh uh you, you can work out for yourself and with paper and |
|
| 99:47 | . So these are the same equations we just had and look at these |
|
| 99:55 | be written in the following way, equations with prime. So here is |
|
| 100:03 | equation with a prime where the I be either a one or a |
|
| 100:07 | And it says that uh uh uh prime some something I is equal to |
|
| 100:16 | sum of two terms. And what the two terms? Well, there's |
|
| 100:21 | uh uh there's the old components J the original components J and you |
|
| 100:29 | we're summing J equals one or two a matrix which is we call R |
|
| 100:35 | IJ. And here is the definition the matrix. Yeah. Um uh |
|
| 100:45 | le let's just uh uh work through a little bit. Uh Let's say |
|
| 100:51 | I equals one. So that means gonna reproduce this equation and showing that's |
|
| 100:57 | special case of this. So uh uh what is it? We got |
|
| 101:02 | sum of two terms. So uh let's take J equals one and uh |
|
| 101:08 | and now we got, I equals , J equals one. So that's |
|
| 101:11 | cosine theta, that's the same as have here multiplying times X one. |
|
| 101:18 | , we also have in the we have uh uh J equals |
|
| 101:23 | So for J equals two, we RIJ which is this one right |
|
| 101:30 | Rijr one two. So that's this right here. And, and that's |
|
| 101:36 | . So you see how in that we have found that we verified that |
|
| 101:43 | is compact notation for these two two equations uh uh in one. |
|
| 101:52 | you can verify the second one for . But uh uh uh in the |
|
| 101:57 | way that we just did by introducing matrix, we have uh found a |
|
| 102:05 | way to uh express this transformation. this is the rule for transformation with |
|
| 102:15 | like a vector. So a mathematician immediately will recognize this is the transformation |
|
| 102:21 | for uh for uh the, the vector components when you change the chord |
|
| 102:30 | . So this array is called a . It can array components. And |
|
| 102:35 | gonna uh we're gonna denote that whole with a tilde like it shows here |
|
| 102:43 | it has two indices, we say is, is a rank two and |
|
| 102:47 | denote the components like this. if it has three indices, we |
|
| 102:56 | say it's rank three, but uh won't see too many indices like |
|
| 103:00 | But uh uh look here, if index counts up to three, we |
|
| 103:04 | it has dimension three. In this , it had uh uh this only |
|
| 103:10 | up from 1 to 2. So has dimension two, but for three |
|
| 103:15 | um vectors, we're gonna use uh dimension three. And yeah, so |
|
| 103:22 | an example. Uh And so uh of these have ranked two because they |
|
| 103:30 | only two indices. But this one uh three dimensions and this one has |
|
| 103:34 | two dimensions. And here's a, a further convention that the first index |
|
| 103:41 | the rows, say first row, row, third row, and the |
|
| 103:45 | index counts the column. So the column, second column there. Now |
|
| 103:52 | easy to imagine matrices are ranked three even more. You'd have to write |
|
| 103:57 | in a complicated way. Uh But can easily imagine that can't you? |
|
| 104:04 | this particular matrix has dimension two oh . But it has three indices. |
|
| 104:18 | we can't write it if it has indices, we can't write it on |
|
| 104:22 | screen, can we? Because the only has uh uh the two |
|
| 104:27 | So we have to write in in a complicated way like this. |
|
| 104:31 | have to show it in a complicated like this. OK. So now |
|
| 104:37 | are ready to start combining matrices. let's define a matrix C which is |
|
| 104:43 | result of adding together matrixes here. this is the definition for the components |
|
| 104:50 | um uh of C composed by just the components of A and corresponding component |
|
| 104:59 | B. And look how we're, really beginning here to get a lot |
|
| 105:06 | advantage from writing the uh uh these with indices because this is true for |
|
| 105:13 | I and any J. And I could be uh you can have |
|
| 105:19 | uh uh two dimensions or three dimensions 17 dimensions or whatever it all looks |
|
| 105:24 | same. Yeah, in this uh uh with this index notation. |
|
| 105:33 | from this definition, it's clear that doesn't matter the order of addition, |
|
| 105:40 | go back here doesn't uh uh you how to add things together and it |
|
| 105:45 | matter. You could, yeah, , you could uh uh uh move |
|
| 105:49 | A over here and the plus it wouldn't make any difference. And |
|
| 105:55 | uh uh the, the mathematicians uh that property community that is uh uh |
|
| 106:02 | , the, the B can commute to a and it makes no |
|
| 106:07 | Now, now we get to the uh complicated, this is where we |
|
| 106:11 | it before. So with the notation we have uh develop now this is |
|
| 106:18 | be easy. So, so now gonna define the matrix which is the |
|
| 106:26 | of these two. And it says uh uh We don't find the products |
|
| 106:34 | the components of C by simply multiplying corresponding components of A and B. |
|
| 106:41 | , this is the definition, the component of C is equal to this |
|
| 106:48 | over K of these components of A B. And notice here that I |
|
| 106:55 | appearing only once on each side of equation J is appearing only once on |
|
| 107:00 | side of the equation, but K repeated. So that means we're gonna |
|
| 107:05 | over K and notice the clever thing that uh uh uh uh the repeated |
|
| 107:15 | , hey, here's a in the position for A and in the first |
|
| 107:22 | for K. So these um these are gonna uh make things easy for |
|
| 107:37 | in the future. That's why I said this in the, the second |
|
| 107:42 | here is repeat of this first And so this is what Carlos uh |
|
| 107:48 | has suggested for the uh for the of two indices. If you want |
|
| 107:54 | one, you take the product of time, this plus the product of |
|
| 107:59 | time this and that gives you the the +11 component of ST written out |
|
| 108:07 | . It's like some as a second of here's the 12 component is the |
|
| 108:12 | of this one times this one plus product of this one times this |
|
| 108:18 | And so what you might wanna think is what I think let's make out |
|
| 108:24 | this row. Let's make a a vector and turn it into a |
|
| 108:30 | vector and put it right here. imagine a column vector here where |
|
| 108:34 | a 11 here and a 12 then you just multiply cross wires like |
|
| 108:41 | and you get this, I realize it's the same first ended and it |
|
| 108:52 | and the same second index here and . And what's summed over is the |
|
| 108:59 | indices one here and two here. . This is what I said |
|
| 109:10 | It always happens in matrix algebra, if an index is repeated in an |
|
| 109:16 | , you must s over that. here is some and be because here's |
|
| 109:23 | repetition, we don't have to show this some explicitly. And this |
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| 109:27 | invented by Alper Dice. Yeah, defined C as A times B and |
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| 109:41 | spell it out here with uh uh uh the definition of what this means |
|
| 109:48 | terms of disease. Now, let's at B times A and give that |
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| 109:52 | new name. So from uh uh did before uh we can write that |
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| 109:59 | that what B times A means is this. And now it's pretty clear |
|
| 110:06 | uh uh C is not equal to because this is a different sum di |
|
| 110:12 | di different quantities in there. And what that means is M matrix multiplication |
|
| 110:20 | not community. Now, what happens you can multiply a vector by a |
|
| 110:31 | , let's define a vector, which a result of multiplying a vector X |
|
| 110:37 | a matrix A from the left. . So we're multiplying from the |
|
| 110:42 | So A is multiplying from the left X. So X is a vector |
|
| 110:47 | is a vector and A is a . So uh this is a matter |
|
| 110:53 | notation. So here's the definition right . So here is the definition uh |
|
| 110:59 | this ve uh uh matrix vector multiplication here. And you see again, |
|
| 111:07 | got AAA bubble, a repeated index right there. So this is how |
|
| 111:19 | can remember matrix multi uh uh uh vector notation, write the uh write |
|
| 111:26 | all as ve as column vectors. here's the Y vector, here's the |
|
| 111:31 | vector, here is the A major then just do what we did before |
|
| 111:36 | the, for the first component uh these two together, then add this |
|
| 111:44 | and similar thing for the second We saw this when we talked about |
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| 111:52 | rotation maker. Yeah, we had uh uh uh uh uh uh an |
|
| 112:02 | vector X expressed in a new coordinate where the prime and what are the |
|
| 112:08 | between the two uh uh between the and the X prime? Well, |
|
| 112:14 | given by this expression here where R the matrix coefficient matrix uh of the |
|
| 112:20 | matrix has got a cosine of the here. Same quantity here, sine |
|
| 112:25 | the angle here at a minus sine here. So this, this right |
|
| 112:31 | is matrix notations for these two And I think it's a lot easier |
|
| 112:36 | remember this than this. I can't this, but I can remember this |
|
| 112:41 | I can remember the form of the matrix. It's got the cosine and |
|
| 112:46 | cosine got a sine and a And so this, this uh equation |
|
| 113:00 | is exactly the same as this We call this, uh um we |
|
| 113:05 | this index notation. We call we can call this um uh uh |
|
| 113:11 | vector notation. And actually, we even need the summation here because of |
|
| 113:16 | summation symbol here because we know that , if it's uh two components here |
|
| 113:22 | uh repeated, it means we're gonna . And by the way, I |
|
| 113:28 | rename this, I could call that K. And if, if |
|
| 113:31 | if I call that A K and A K, then it doesn't |
|
| 113:35 | We call this a dummy index when repeated, it doesn't really matter what |
|
| 113:40 | you give it because you're gonna sum all possibilities. Anyway, it doesn't |
|
| 113:47 | uh uh what value for I, choose, you choose, I equals |
|
| 113:52 | , I have that correspond to this one. And that'll give you a |
|
| 113:57 | answer than if you choose I equals . But the J here is a |
|
| 114:01 | index. OK. So now let's about this for rotation. So we |
|
| 114:12 | this X prime uh components and multiply . And again, by uh rotate |
|
| 114:20 | by another um rotation matrix about AAA uh angle. Well, we know |
|
| 114:26 | to do that because of everything we've so far. This is trivial. |
|
| 114:34 | know that the rotation matrix is gonna like this except it's got a |
|
| 114:38 | it's got a AAA five prime instead a thought. Otherwise it's exactly the |
|
| 114:49 | . So let's now consider these rotations sequence. So this is the one |
|
| 114:54 | just talked about. And I'm gonna a bracket around the X prime square |
|
| 115:00 | . You see this bracket is the difference here. Just put a bracket |
|
| 115:04 | the bracket. And now inside the , I'm gonna use previous um uh |
|
| 115:10 | . Uh X prime came from X by uh uh by the matrix R |
|
| 115:17 | , and see this is a wow by P as this one is rotation |
|
| 115:24 | P five prime. And uh so mathematicians uh have proven that you can |
|
| 115:33 | these um brackets in this way and call that we say that matrix operations |
|
| 115:43 | associative. So you can uh associate one with this one as long as |
|
| 115:49 | don't change the order. And then can use elementary trigonometry to prove that |
|
| 115:58 | thing is, is simply a single matrix multiplied uh where, where the |
|
| 116:05 | of the matrix is the sum of two angles. So if you rotate |
|
| 116:10 | this much and, and then by much, it's just um the same |
|
| 116:15 | rotating all at once together with one . So that's all in two |
|
| 116:23 | And, and it's easy to uh now uh to three dimensions. Uh |
|
| 116:28 | example, a 3d vector can be these ways to rotate uh about the |
|
| 116:33 | the three axis. Uh uh We simply multiply this 3d vector by this |
|
| 116:39 | uh three dimensional rotation matrix. And we've, we've got a subs a |
|
| 116:45 | three here just to remind you that a three dimensional rotation matrix. And |
|
| 116:51 | is the definition. And if you closely in this part of it |
|
| 116:55 | the upper left corner is exactly the as we had before. And now |
|
| 117:00 | got uh uh more indices to So it's got to have uh it's |
|
| 117:05 | a one here and zeros along And this one means we're rotating about |
|
| 117:12 | X three axis. So that's the form as we had before. |
|
| 117:26 | suppose we wanna do what we did uh I have uh rotations followed by |
|
| 117:33 | . So this time I'm gonna uh , the second rotation is gonna be |
|
| 117:37 | the, the X one axis. here is uh my uh rotation matrix |
|
| 117:43 | the one axis I I, and gonna call this the angle theta not |
|
| 117:51 | and I'm gonna call this double prime I did before. This rotation matrix |
|
| 117:56 | uh uh operating on the uh the the new, the new |
|
| 118:04 | And this uh uh is trivial from we did before we know how to |
|
| 118:10 | a rotation. Now, um uh You, you will recognize this sum |
|
| 118:17 | we, you'll recognize, we don't to have uh the summation side because |
|
| 118:22 | see that the K is repeated. this looks so simple. And the |
|
| 118:25 | difference is that since we're rotating about S, the X axis, the |
|
| 118:32 | matrix looks differently. It's got the here in the 11 position. It's |
|
| 118:37 | the zeros here and it's got in uh the angle theta here. Uh |
|
| 118:43 | because it's, it's this data but look, it's, it's the |
|
| 118:47 | form. It's, it's uh uh , cosine, sine and minus sine |
|
| 118:53 | same form as before. Now, remember the trick we did before was |
|
| 119:04 | took this X prime and we put in brackets here here to express in |
|
| 119:09 | of the fir the first rotation. then here we uh uh we uh |
|
| 119:15 | the brackets without changing the order. that's all OK. And so these |
|
| 119:22 | now uh theta and Phi are called angles invented by this guy Oiler. |
|
| 119:31 | so uh uh I gotta remind you name is not Eer in English, |
|
| 119:38 | would maybe say you were, but was German. And so that's called |
|
| 119:46 | . And uh notice here that you can't simply do what we did |
|
| 119:51 | by adding together the angles because uh different actions. And furthermore, you |
|
| 119:57 | interchange the order. OK. So is a lot of uh notation. |
|
| 120:06 | you think about it, there's not many complicated ideas here. It's just |
|
| 120:11 | , a bunch of notation. And might wonder why are we going through |
|
| 120:17 | notation? And uh the answer I'd is um would be not obvious to |
|
| 120:24 | . But uh uh by the time finished with this course, you will |
|
| 120:28 | glad that we uh uh did this uh we'll see lots of vectors and |
|
| 120:36 | of matrices and lots of indices. uh uh in the index notation, |
|
| 120:42 | gonna be all so easy. So have a little quest, uh what |
|
| 120:52 | the matrix su we uh we, did this one before and the answer |
|
| 120:56 | the same as before. Now. we're in a position to uh uh |
|
| 121:02 | look at this matrix product. And what we, you know, we |
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| 121:07 | to do three times 10 is 30 one times five is five. So |
|
| 121:12 | makes it 35. So that means in the +11 position of this |
|
| 121:17 | it's either gonna be this one or one. OK. So now we |
|
| 121:22 | three times five is 15 plus one 10 is uh uh 10 makes the |
|
| 121:29 | sum of 25. So uh theirs 25. So uh it's not this |
|
| 121:38 | . It's just the answer is C . So, uh uh so this |
|
| 121:50 | the one we did before. uh so this is the one which |
|
| 121:54 | have, we haven't done yet So let me, let me turn |
|
| 122:00 | versa and versa. Tell us what the, the answer for this. |
|
| 122:05 | uh It's a matrix times a And so immediately, you know, |
|
| 122:11 | not this answer because the answer for has gotta be a vector. So |
|
| 122:16 | , you know, it's not gonna this. So it's gonna be uh |
|
| 122:19 | of these three. So what do think it is, is to |
|
| 122:24 | it's uh BB. So we yeah, yeah. So, and |
|
| 122:28 | you did here is you took a times 10, makes 30 plus one |
|
| 122:34 | five makes 35. And uh that these other two. So it must |
|
| 122:39 | B yeah, this is what I before. Did you notice it in |
|
| 122:48 | two problems multiplying uh a matrix times vector is just like the first part |
|
| 122:55 | ma multiplying a matrix times of matrix of this vector is the same as |
|
| 123:05 | column of this matrix. You can that this column of this matrix is |
|
| 123:12 | same as this vector. And so uh uh me uh Meader just did |
|
| 123:20 | she implemented this and it's the same we did before in the matrix matrix |
|
| 123:27 | . This vector is the same, column vector is the same as this |
|
| 123:36 | . So another way to think about a mathematicians way is you can say |
|
| 123:41 | a vector is just like a one matrix. It has since it has |
|
| 123:46 | one index mathematicians say it is a of rank one. So we've seen |
|
| 123:54 | of rank one matrices of rank What would you say is a matrix |
|
| 124:02 | rank zero? That would be a that would be a number like pressure |
|
| 124:11 | temperature? Mhm OK. So next come to tensors and I think that |
|
| 124:20 | is a good place for us to with the math already. And uh |
|
| 124:27 | uh so tonight, you might be to go and look at the next |
|
| 124:32 | sections on tensors. It's uh what gonna learn is it uh is it |
|
| 124:38 | tensor is a, is a special of matrix. So with this uh |
|
| 124:45 | so this is a good place for to, to break right now. |
|
| 124:48 | break for 10 minutes break for uh 19 minutes and come back at 30 |
|
| 124:59 | past the hour. And we will resume with the lecture about elasticity having |
|
| 125:08 | this uh common understanding of matrix and notation. So I'll see you all |
|
| 125:16 | um oh 18 minutes. Ok. . So uh let's see here. |
|
| 125:53 | we have people? But I don't if our friends are with us or |
|
| 126:03 | . Um Yeah, here's Carlos and pro said, OK. So we're |
|
| 126:22 | to go. OK. So with mathematical uh diversion uh Let's get back |
|
| 126:33 | um uh elasticity. So first item stress. So the question is what |
|
| 126:42 | stress? And so uh uh here's definition, it's the, the force |
|
| 126:49 | unit area distributed across the unit So um uh we, we typically |
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| 126:59 | the um the uh the stress of uh the notation tau with two subs |
|
| 127:10 | I and J and one is for unit vector and one is for the |
|
| 127:15 | vector. So if you imagine say you know, like a postage stamp |
|
| 127:20 | , uh unit area uh inside of rock oriented according to its normal |
|
| 127:27 | So we don't care about uh uh orientation and we can twist this thing |
|
| 127:32 | the, around this direction. But the unit area is oriented according to |
|
| 127:38 | um the, the perpendicular normal, , the normal vector to that uh |
|
| 127:45 | unit area. And you can imagine on this um uh on this area |
|
| 127:52 | ev at every point in the there's a force and the force vector |
|
| 127:56 | be uh pointed like this or maybe other direction, but that's gonna be |
|
| 128:02 | by the indices. So one index the orientation of this force uh area |
|
| 128:10 | and the other and the other index the direction of the force factor. |
|
| 128:23 | uh we're gonna have uh uh I 123 and J equals 123. So |
|
| 128:30 | we're gonna have a three by three for stress uh nine elements indicating the |
|
| 128:41 | of that particular component of the force unit area. So here's a |
|
| 128:49 | So it's a special matrix whose components made from vectors. So we call |
|
| 128:56 | a 10. Uh So you saw word tensor in the math uh uh |
|
| 129:02 | , you might wanna come back to uh uh later. Uh We are |
|
| 129:07 | gonna turn uh need this concept of tensor later in the course. But |
|
| 129:12 | for now, we're just gonna leave at this, that a matrix is |
|
| 129:17 | uh excuse me, a tensor is matrix whose, whose uh components are |
|
| 129:22 | out of vectors. So you can of this as column vectors or you |
|
| 129:26 | think of this as row vectors, Andes of course refer to court and |
|
| 129:35 | . And so if you have a orientation of the coding system, the |
|
| 129:38 | are all different. So um um uh uh uh imagine a little uh |
|
| 129:50 | well, you know, we, did not cover this uh uh |
|
| 129:55 | So I'm gonna skip over this quiz uh uh we'll come back to this |
|
| 130:02 | question tomorrow. And uh also we'll a little bit more um from mathematics |
|
| 130:10 | on one uh tomorrow. So for now we're gonna get past this. |
|
| 130:17 | so uh I analyze that um uh factor that stress uh matrix. It's |
|
| 130:27 | , a stress tensor. Which is matrix made out of vectors. And |
|
| 130:32 | let's analyze this component by component. the 11 component is gonna have a |
|
| 130:38 | area pointed in the one direction. here's our coordinate system and it's uh |
|
| 130:43 | uh this one here uh uh uh indicates that the unit area is pointed |
|
| 130:50 | the one direction and the other one that the forest is also oriented in |
|
| 130:55 | one direction and it's pressing everywhere on unit area. OK. And so |
|
| 131:04 | is uh uh the uh a picture the 33 component, the unit area |
|
| 131:10 | pointed in the three direction and the is also pointed in the three |
|
| 131:18 | So I think that's pretty clear. So here is now the 13 |
|
| 131:22 | So we got the unit area and , and uh it's pointed in the |
|
| 131:26 | direction and the force is gonna be force pointed in the three directions. |
|
| 131:31 | see it, it's like a, sheer force on that, a sheer |
|
| 131:36 | on that unit area. And here the sheer stress 311, I said |
|
| 131:45 | uh the, the, the stress 31 back up here is the |
|
| 131:51 | the stress tau 13 and this is 31. So here it has the |
|
| 131:57 | area is in the three direction and force is in the um in uh |
|
| 132:03 | the one direction. Now, you be asking yourself who said which an |
|
| 132:10 | switch. And who cares? Remember had the convention that when you have |
|
| 132:17 | , this is the rows and this the columns, rows and column, |
|
| 132:22 | which corresponds to the unit area and corresponds to the fourth area. Anybody |
|
| 132:32 | an answer to, to these which is which and who cares? |
|
| 132:39 | think it's a very deep question And so the answer is that if |
|
| 132:45 | two are not equal to each then whenever you apply the sheer stress |
|
| 132:51 | would cause the uh the body to spinning and spin faster and faster and |
|
| 132:58 | , instantly spinning. And since that happen, you can squeeze a rock |
|
| 133:03 | you can shear a rock and it spin, it may be deformed itself |
|
| 133:08 | sheer, but it doesn't spin. therefore, uh since this doesn't |
|
| 133:13 | these two must be the same. so we don't care uh which nex |
|
| 133:20 | which since these two are the So when you have a symmetric tensor |
|
| 133:29 | this, the T is said to orthogonal. And I suppose that you |
|
| 133:36 | quite uh know why we should say orthogonal. Uh So let's postpone that |
|
| 133:44 | now, uh postpone the answer uh that for now and just say that |
|
| 133:50 | when we uh stress tensors are And so the uh uh yeah, |
|
| 133:58 | we describe that, as we it's Ortho, so why is uh |
|
| 134:08 | all orthogonal tensors are uh are So uh if you have a, |
|
| 134:14 | uh if you write it like And by the way, these symbols |
|
| 134:19 | , that doesn't mean the absolute That means it's, it's it. |
|
| 134:23 | that means it's a uh a tensor these. So a tensor uh uh |
|
| 134:31 | the, the strength sensor has this that uh it looks like it's got |
|
| 134:35 | different components, but uh actually, six of them are independent because this |
|
| 134:40 | is equal to this one and this is equal to this one, et |
|
| 134:44 | . So you count them up and only a six independent quantities and the |
|
| 134:53 | in, in the stress tension. , we skipped over a math quiz |
|
| 135:07 | told us how to rotate uh uh tensor with two indices. And here |
|
| 135:15 | comes up to bite us again. here I'm just gonna say, and |
|
| 135:18 | go, gonna give you the answer when you have AAA tensor like a |
|
| 135:23 | tensor with two in C and you express it in a different court |
|
| 135:30 | then what you need to do is rotations, one from the left and |
|
| 135:34 | from the right. And these two correspond to the two indices. So |
|
| 135:40 | this thing has two indices uh implicit . And uh uh it, we |
|
| 135:47 | to um uh if we rotate it express it in a different quarter |
|
| 135:51 | same quantity in a different quarter we need to rotate it twice, |
|
| 135:55 | on the left and once on the and F rotate from the right. |
|
| 136:03 | uh We do the same sort of multiplication as we had before. Uh |
|
| 136:08 | this rotation matrix is to transpose of you saw before. So whenever we |
|
| 136:16 | AAA rotation matrix down, going to a three by three rotation matrix uh |
|
| 136:24 | and you wanna know the transpose of , then uh the transpose of that |
|
| 136:30 | , then you just uh uh uh uh rows and columns. Now |
|
| 136:40 | Uh so because, because the stress uh uh symmetric because it's orthogonal, |
|
| 136:50 | that means is that you can always if you uh if you examine that |
|
| 136:57 | cancer with its nine components, but know that only six, some of |
|
| 137:03 | are independent because it's symmetric, you rotate it this way and you can |
|
| 137:08 | it that way. And you can find a special rotation, a special |
|
| 137:14 | system whereby it looks like this that off the diagonal and three different components |
|
| 137:21 | the diagonal. And these three stretches called the principal stretches. And the |
|
| 137:31 | magic coordinate system which shows this pattern called um uh the principal coordinate |
|
| 137:42 | No uh in most sedimentary bases where exploring for oil and gas having near |
|
| 137:51 | layers. Mm In cases like the stress sensor is the obvious one |
|
| 138:00 | is the principal uh uh current system 11 axis vertical and the stress sensor |
|
| 138:07 | like this. That is true in deformed basis in the mountains. That |
|
| 138:16 | true in the mountains. The principal system uh could be oriented anywhere |
|
| 138:23 | in any direction. And if you yourself over 10 m in an up |
|
| 138:27 | down or sideways or anything, it be different. Mountains are complicated. |
|
| 138:34 | we're sort of lucky that in the of uh rocks where we normally explore |
|
| 138:43 | the uh principal coordinate system is usually has one axis vertical um uh uh |
|
| 138:53 | . And furthermore, the vertical stress usually the largest. And of |
|
| 138:59 | the reason for that is uh the stress is the one which is uh |
|
| 139:05 | from the weight of the overlying Now, the uh um the, |
|
| 139:16 | two horizontal stresses are usually um uh uh but similar to each other. |
|
| 139:25 | uh uh we can say that Tau H max is, is uh almost |
|
| 139:31 | same as Tau of H men, very different from the vertical, which |
|
| 139:36 | the, the maximum stress. So these numbers are something like 60 or |
|
| 139:42 | of the vertical number and maybe they by two or 3%. And these |
|
| 139:52 | uh we said that normally in our of rocks, we have one axis |
|
| 140:01 | . And so what about the other ? Are, are they oriented east |
|
| 140:05 | , north, south in between? are they oriented? That's not such |
|
| 140:10 | an easy question to answer. And I'm gonna postpone that question for a |
|
| 140:15 | bit. Uh uh uh But um you need to know is that the |
|
| 140:26 | components are usually significantly different stress than vertical components. So that the orientation |
|
| 140:42 | those horizontal axes does depend upon tectonic and it usually varies specially. So |
|
| 140:51 | the orientation of the horizontal stress might relieved by uh might be revealed by |
|
| 140:58 | . So uh this is an outcrop surfer outcrop. As you can |
|
| 141:06 | And can you see these joints in ? Of course, you can see |
|
| 141:10 | layers and you can see these joints all parallel to each other and all |
|
| 141:19 | avertly. So these joints lie in same plane as TV and TH |
|
| 141:28 | So tau V is vertical and tas H max is has landed in this |
|
| 141:34 | trying to show you in three So or in a parallel to uh |
|
| 141:39 | to this joint. And uh uh uh TX men is uh located perpendicular |
|
| 141:46 | the joints. Here's the way for to um uh think about this. |
|
| 141:58 | One more thing here is um let's look at this joint. It's |
|
| 142:13 | is it like a fault plan? do you think is the displacement of |
|
| 142:19 | is the sense of displacement across this ? Uh Do, do you think |
|
| 142:25 | the one side of it is moved relative to the other side in the |
|
| 142:30 | direction or the vertical direction or Well, we can answer that question |
|
| 142:36 | by looking at this picture, just looking at this, we can say |
|
| 142:42 | since this um uh uh since this is vertical, we can say uh |
|
| 142:54 | uh uh geologists have AAA name for , they call it type one fractures |
|
| 143:03 | which the displacement is perpendicular to the face. So this thing had uh |
|
| 143:10 | has uh uh has not sh sheared the, there is no shear in |
|
| 143:15 | plane. Here, it the the is perpendicular to the plane. So |
|
| 143:33 | a rock mass is going to fail fracturing fail by cracking, obviously, |
|
| 143:41 | the failure plane is gonna be such the failure direction is in the direction |
|
| 143:46 | the least stress. So we say this uh uh this perpendicular direction here |
|
| 143:54 | the least horizontal stress and the maximum stress lies in the plane of the |
|
| 144:01 | and then the vertical stress is bigger either one of those. Now, |
|
| 144:07 | is probably a good time to let know that there are other types of |
|
| 144:11 | . And you can see one can you see this fracture here? |
|
| 144:16 | one is not vertical. And so know that this fracture as here, |
|
| 144:25 | a sheer fracture that is the displacement in the plane of this fracture |
|
| 144:30 | the displacement is perpendicular to the uh the fracture we know that because it's |
|
| 144:39 | , this one is sheer. So has shed uh uh the, the |
|
| 144:43 | is lying in the plane of the . And all of that is a |
|
| 144:48 | of uh um structural geology, which will not do much of in this |
|
| 144:55 | , we're gonna be talking about infinitesimal and infinitesimal strains accompanying wave propagation. |
|
| 145:06 | uh uh I brought this up at point so that you will know uh |
|
| 145:12 | principal stresses. Um I had so so here's what you know about principal |
|
| 145:19 | that the, that there's a magic system where the stress tensor looks like |
|
| 145:24 | for our kind of rocks. Uh uh usually has zeros off diagonal and |
|
| 145:29 | different numbers on the diagonal. This is the biggest, that's the vertical |
|
| 145:34 | . These two are significantly less but similar to each other. And so |
|
| 145:39 | can see that here now inside. uh So, so that's the, |
|
| 145:51 | way stress is in uh um in in the ocean. It's a special |
|
| 145:57 | because uh the ocean has zero sheer to it. And so in, |
|
| 146:05 | the ocean, you always have the uh three quantities uh here uh on |
|
| 146:11 | diagonal. And uh uh it's true any orientation. By the way, |
|
| 146:16 | uh uh uh uh if you uh this coordinate system, uh thi this |
|
| 146:22 | tensor in the ocean to any other system, you're gonna get the same |
|
| 146:27 | because the ocean is made out of . And the, the, uh |
|
| 146:31 | three principal stretches are all the same uh uh given by uh minus the |
|
| 146:37 | pressure, mi minus the pressure and mind the minus sign, that's just |
|
| 146:42 | convention. And so immediately you see that in rocks, we have stresses |
|
| 146:49 | in the ocean, we have So in rocks, we have uh |
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| 146:54 | uh different stresses in different directions in the ocean, we have the same |
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| 146:59 | in all directions. And so uh a bit uh uh uh dis disingenuous |
|
| 147:07 | , for us to think about pressure of rocks because uh inside of rocks |
|
| 147:14 | the different components of the stress tensor always different from each other. |
|
| 147:24 | another way to write that is in ocean, the, the uh the |
|
| 147:28 | tensor is given by minus P times IJ. So that this is called |
|
| 147:34 | del the chronicle delta. Look it in your glossary. It's a number |
|
| 147:38 | is equal to zero if I and are not equal to each other and |
|
| 147:42 | they are equal to each other, it's a one. So you |
|
| 147:53 | So let's do a little quiz which of stress is shown here. So |
|
| 147:59 | we have AAA CO system and um bread and we have a, a |
|
| 148:08 | area in blue with uh its um uh unit area vector pointing um into |
|
| 148:17 | screen or out of the screen. you can see the force vectors are |
|
| 148:24 | in the screen pointed in the one . So um what component of stress |
|
| 148:32 | shown here? Since the unit area its vector pointed in the two |
|
| 148:47 | it's got to be this one. mean, because this is the only |
|
| 148:50 | where there's a two. And sure , the uh uh uh the other |
|
| 149:00 | uh uh which gives the force is in the one direction. Was that |
|
| 149:09 | ? Was that an easy quiz? one was, was easy. |
|
| 149:15 | but iiii I don't understand why. mean there must be a mathematical demonstration |
|
| 149:21 | that. But you said that the perpendicular, the two tensors, |
|
| 149:25 | 12 and the two ones are It doesn't because physically, physically uh |
|
| 149:33 | mean, those 10 sources, if add those tensors, uh I |
|
| 149:38 | you don't have any rotation of of the Yeah, I, I |
|
| 149:42 | you can convince yourself that if, those two are not equal, then |
|
| 149:46 | uh rock will spin and it will faster and faster and faster. And |
|
| 149:51 | that doesn't happen, these two must equal on all scales on, on |
|
| 149:57 | scales. So think about uh uh the hand simple scale and think, |
|
| 150:02 | about the uh the molecular scale and uh at, at all scales uh |
|
| 150:08 | we have to have the stress it's uh symmetric otherwise everything will be |
|
| 150:16 | . OK. Yeah. OK. , so much for stress. So |
|
| 150:21 | let's talk about strength. So um is deformation and it's defined in a |
|
| 150:30 | way. So let's consider two nearby . So here is our coordinate system |
|
| 150:35 | here is the origin of the coordinate and these points are near to each |
|
| 150:40 | . So that this uh uh uh is X and this is X plus |
|
| 150:46 | X. So obviously this is the uh the vector delta X and this |
|
| 150:52 | just notation here. Uh uh uh delta XE, just notation for this |
|
| 151:00 | vector stretching between these two. And gonna be assuming that these two points |
|
| 151:06 | nearby to each other in a certain . And also we're talking, we're |
|
| 151:12 | about a continuous medium here. So know that a rock has uh ultimately |
|
| 151:18 | out of atoms. Uh But let's uh uh uh let's describe the rock |
|
| 151:27 | terms of a continuous solid. So distance factor delta X has the magnitude |
|
| 151:36 | the square of the magnitude given by X times delta X. And if |
|
| 151:40 | wanna know what uh uh L itself , you just take uh the square |
|
| 151:44 | of that. So that's the, the, uh this is the square |
|
| 151:51 | the length of this fact. Now these same two points deformed by a |
|
| 152:00 | field. So this one gets deformed this way and this one is deformed |
|
| 152:05 | this way. And you see that deformation is a little bit different. |
|
| 152:09 | uh this is the displacement at the , the displacement U at the point |
|
| 152:14 | the position X and this is the U at the position X plus delta |
|
| 152:22 | . So the new um uh uh factor is given by that. And |
|
| 152:33 | uh that's the, the, the the distance factor given by uh |
|
| 152:38 | the same as uh uh we had . That's this delta X plus the |
|
| 152:44 | of these two, the spice. you see how uh uh uh uh |
|
| 152:52 | easy for us now to talk about uh uh sums of vectors because uh |
|
| 152:59 | talked about mathematics earlier. So what the length of this new uh distance |
|
| 153:05 | ? Well, it's obviously uh the of it is given by delta X |
|
| 153:11 | uh times delta X prime I summing I, I'm gonna back up. |
|
| 153:20 | , uh I should have shown here uh uh delta X I times delta |
|
| 153:28 | I with reported with repeated is uh like I did here with the |
|
| 153:40 | Now, we said they were close by assumption. So in that |
|
| 153:45 | the uh uh form the displacement at second location is equal to the displacement |
|
| 153:54 | the first location plus this correction which depends upon the amount of the |
|
| 154:02 | . And this is an approximation. this is called the Taylor approximation. |
|
| 154:07 | Now, let me ask uh uh Lee, are you familiar with the |
|
| 154:12 | approximation? Yeah. Well, let ask you, Carlos, are you |
|
| 154:17 | with the uh Taylor approximation? No ? Uh How about you, |
|
| 154:25 | Are you familiar with the, with Taylor approximation? No. OK. |
|
| 154:31 | we are going to uh uh have um uh I use this many times |
|
| 154:39 | this course. So I'm gonna uh deviate from uh uh from this uh |
|
| 154:47 | go to uh the uh the So I'm gonna stop sharing and then |
|
| 154:56 | going to uh start sharing. No, I'm gonna uh I'm gonna |
|
| 155:05 | uh do that because I haven't done proper preparation yet. Oh Yes, |
|
| 155:11 | have. Yes, I have. So let me, OK. Now |
|
| 155:28 | gonna share again. I hear the one. OK. Sure. What |
|
| 155:42 | that with 32? See uh that's not your brain. OK. |
|
| 156:09 | got that. OK. So this from the glossary and you can verify |
|
| 156:25 | yourself. Hold on a second. me uh OK. And so it's |
|
| 156:30 | , it's uh slide 73 in the . So here is a uh picture |
|
| 156:34 | Mr Taylor. I uh would not surprised if that's a poor uh photograph |
|
| 156:39 | uh Mr Taylor. He lived a time ago, an Englishman. And |
|
| 156:44 | uh so he uh uh um he the following uh uh uh idea. |
|
| 156:53 | so we call this the tailor expansion the tailor approximation. And he says |
|
| 156:58 | any function of uh of X, very useful approximation connects its value at |
|
| 157:05 | um initial uh position X zero to value at a nearby X with this |
|
| 157:12 | . So let's look and see what have here. Uh So this is |
|
| 157:16 | any function defined at X zero plus X is, is expressed in terms |
|
| 157:24 | its value at the uh at X , same as this, but this |
|
| 157:29 | X zero plus a correction term which upon this uh different delta X, |
|
| 157:36 | is the same as we have here this derivative evaluated at the uh at |
|
| 157:45 | zero. So this is the first tailor expansion valid for small X. |
|
| 157:51 | let's see what's next. Uh Uh There is uh more and more |
|
| 157:59 | more about this. Uh So I'm leave it at this point and I'm |
|
| 158:03 | go back to the uh the previous . You can read all about Mr |
|
| 158:09 | expansion um in uh in the glossary after class. And so now what |
|
| 158:20 | gonna do is I'm gonna stop sharing I'm going to um then start sharing |
|
| 158:37 | . Yes, you don't want to saying that right. Yeah. Mm |
|
| 158:46 | to present the the go back to lecture. OK. OK. So |
|
| 159:11 | to review the displacement at this separated is equal to the displacement at the |
|
| 159:18 | position plus correction term which involves only separation and the uh uh uh and |
|
| 159:26 | derivative and if you look here, see repeated Js. So that means |
|
| 159:32 | gonna have to sum over J equals , the index I is not |
|
| 159:39 | So we don't sum over that at . So using this approximation, which |
|
| 159:47 | due to Mr Taylor, we put approximation into this expression here which we |
|
| 159:54 | uh uh it's the, the definition the, the new separation is the |
|
| 160:00 | separation plus the difference in displacement. we come out at the end, |
|
| 160:05 | separating out, uh we, we out at the end with uh uh |
|
| 160:11 | the displacement of uh um uh at uh the whole position is equal to |
|
| 160:21 | . At, to me, the displacement at X prime is equal |
|
| 160:25 | displacement at X plus this term which depends only on the separation and |
|
| 160:34 | what I've done here in a second . Since we're gonna sum over these |
|
| 160:39 | indexes, we do not, it matter whether we call them Jrxj or |
|
| 160:45 | . See that. Oh I'm here, I uh I didn't get |
|
| 160:50 | laser part. So here we have repeated here. We have Js repeated |
|
| 160:56 | it doesn't matter what we call them we're gonna sum, in either |
|
| 161:01 | we're gonna sum from 1 to So we call those dummy in the |
|
| 161:08 | . So then the new distance vector uh uh this magnitude. Uh uh |
|
| 161:15 | uh this is what we disproved and is what we disproved. Uh, |
|
| 161:20 | , uh, see here it's got and here it's got KS, |
|
| 161:23 | doesn't matter, multiply all this And then we do some tricks |
|
| 161:31 | Uh, um, we're gonna rename repeated indices. Uh, uh, |
|
| 161:40 | see, for this term. we have an I repeated with an |
|
| 161:48 | . So let's, let's repeat uh, change called J, repeat |
|
| 161:52 | to J. That's, and change the KS to is from this |
|
| 162:05 | It's been turned to, I's same and here and is, are gonna |
|
| 162:12 | to MS and when we do all , we can collect terms and we're |
|
| 162:17 | with this, all of that is focus uh uh uh uh taking advantage |
|
| 162:24 | the machinery, the mathematical machinery, we discussed earlier this afternoon. And |
|
| 162:31 | uh recognizing that when you have a quantity like here, I and I |
|
| 162:38 | and K uh uh uh it doesn't what you call it. So in |
|
| 162:42 | controlled way, we uh change ch change index notation and you can go |
|
| 162:48 | this yourself again uh later and make we did it right. And uh |
|
| 162:52 | uh we ended up then uh in way where we can collect terms where |
|
| 162:57 | have a product of terms delta delta X I with all of these |
|
| 163:03 | here. And then that's the new magnitude square. And the um uh |
|
| 163:14 | off the old distance magnitude square and it with only this different term |
|
| 163:22 | So now we're gonna define this strain . So all of this was uh |
|
| 163:27 | uh so that we could define this brain cancer epsilon in this way. |
|
| 163:35 | that the difference in the magnitude is by this expression there. I'm gonna |
|
| 163:41 | up now. And you see this gonna be the strain tensor and multiplying |
|
| 163:46 | these two position tensors. And uh enter it to two here and that |
|
| 163:54 | is to cancel the one half That's gonna turn out to be a |
|
| 163:57 | thing to do. It looks like a stupid notation, but you will |
|
| 164:01 | shortly that it's a clever thing to . Now in uh uh whenever we |
|
| 164:09 | seismic, we have uh uh uh source and then at some distance we're |
|
| 164:14 | receiving the seismic waves, the source be quite strong if you've ever been |
|
| 164:20 | a seismic um uh uh acquisition Let me just ask. Uh have |
|
| 164:26 | uh y le have you been to on a seismic acquisition crew? You |
|
| 164:31 | from Calgary and you've never been on seismic ac acquisition crew? Oh, |
|
| 164:36 | . How about you? All other ? Uh uh uh Yeah. |
|
| 164:41 | I haven't. Uh uh Carlos. you been on a seismic acquisition |
|
| 164:47 | So, uh on, you on, on land, right? |
|
| 164:51 | uh So it's, and a, vibrator crew with a vibrator. |
|
| 164:56 | Yeah. So, in a, a vibrator crew, what they have |
|
| 164:59 | a big truck and he droves, , pulls up to the source point |
|
| 165:04 | he stops and he lowers a pad to the ground and raises the wheels |
|
| 165:09 | the truck off the ground. So the weight of the uh uh of |
|
| 165:15 | truck is bearing on this vibrator And then it's got uh uh um |
|
| 165:21 | motors on the uh truck and it the truck up and down and it |
|
| 165:26 | a seismic wave into the, into earth. And if you're standing |
|
| 165:31 | you can feel your whole legs are , you can feel it. And |
|
| 165:35 | your, your, your, your might shake, your hair, might |
|
| 165:41 | if, if you're standing nearby. those are strong motions. But by |
|
| 165:47 | time it gets, by the time waves uh uh get over to the |
|
| 165:52 | recording instruments, they're weak, why it because all the energy gets, |
|
| 165:57 | spread out over a big shell of uh of, of distant. |
|
| 166:02 | so by the time the wave arrives the recording, it's weak. So |
|
| 166:07 | what we're measuring. So the strains gonna be small booster. And so |
|
| 166:18 | gonna be able to neglect this term here. And so uh uh |
|
| 166:32 | this is our definition of seismic It's a sum of two of two |
|
| 166:40 | uh uh gradients. So this is displacement U and this is the position |
|
| 166:47 | . So this term is a gradient the J component of displacement according to |
|
| 166:53 | K direction of position. And over is another term where the two indices |
|
| 166:59 | interchanged. So then immediately you see uh EJK equals EKJ. So this |
|
| 167:07 | a symmetric tensor and it's or an tensor just like we had for |
|
| 167:13 | So the fact that stress is symmetric strain is also symmetric that's gonna simplify |
|
| 167:20 | lives a lot. So let's look at uh some examples of this here |
|
| 167:28 | the long strain epsilon 33. So epsilon 33 is one half the sum |
|
| 167:36 | these two terms. You can see are both identically the same since these |
|
| 167:40 | indices are the same. So it's the, the change of displacement |
|
| 167:45 | respect to position in the both in three direction. So if you |
|
| 167:50 | then uh uh uh uh a box was uh um uh ha was a |
|
| 167:58 | box following the dash line and it uh deformed a displacement uh And um |
|
| 168:07 | the three direction at the displacement in three direction. And uh uh uh |
|
| 168:16 | that might be different uh uh for box, then for a similar box |
|
| 168:23 | here and a similar box up see as we consider these other boxes |
|
| 168:30 | I I yeah, or looking at values for X three. So uh |
|
| 168:37 | you see the displacement U three, displacement delta U three. And uh |
|
| 168:46 | might not be the same for another up here and another box here, |
|
| 168:50 | what I mean. Uh So that's this gradient is not uh uh uh |
|
| 168:57 | zero. No, when a wave through a rock, maybe the |
|
| 169:15 | uh uh maybe inside the wave, a volumetric strength, the volumetric strain |
|
| 169:21 | gives the local change in the volume the rock due to the passage of |
|
| 169:26 | wave. And we give it this uh uppercase the and in terms of |
|
| 169:33 | , it's the sum of the diagonal uh uh components of the strain. |
|
| 169:40 | this one plus this one plus this . That's the s that here is |
|
| 169:51 | interesting thing about this. If you uh uh any other coordinate system, |
|
| 169:59 | uh uh every one of these things uh as uh every one of these |
|
| 170:05 | refers to a coordinate direction, one , three direction and all that is |
|
| 170:11 | that's referring to the chord system, it is in our brains. The |
|
| 170:16 | doesn't know anything about that. But is a quantity which uh doesn't care |
|
| 170:22 | uh how we imagine the cordon system because this sum is the same for |
|
| 170:30 | coordinate systems. Now, suppose uh as the wave goes through the |
|
| 170:47 | it does something else besides making a change. Maybe it makes a sheer |
|
| 170:53 | as well. No. Wow, course, that's gonna happen if it's |
|
| 170:59 | sheer wave but in a P it um uh is there also some |
|
| 171:06 | in a P wave uh uh in a P wave, I think |
|
| 171:11 | a P wave traveling vertically. we'll use this picture P wave traveling |
|
| 171:19 | . So it's changing the shape of box, this square from a square |
|
| 171:25 | a shorter rectangle. So that's not pure v volumetric change. That's uh |
|
| 171:33 | volumetric change plus a sheer change, it? So we, we probably |
|
| 171:39 | call a P wave A P wave uh W when we say a P |
|
| 171:44 | , maybe it means primary instead of of pressure because the stress is obviously |
|
| 171:51 | longitudinal stress. It's not a compressional , it's not a pure pressure, |
|
| 171:57 | a longitudinal strength. No, not stress. Well, let's, let's |
|
| 172:03 | of another example of strain. This the, the sheer uh the, |
|
| 172:06 | 13 strain. So that's defined in way and it's measured. Uh uh |
|
| 172:12 | , here you see uh uh information in the one direction and there is |
|
| 172:23 | deformation in the three directions. so this is defamation, uh uh |
|
| 172:29 | U one. You can see it . You can see also in this |
|
| 172:34 | , there is no delta U three uh U three there's no uh there's |
|
| 172:42 | change in the uh three direction. how about this one? But I |
|
| 172:53 | OK. OK. Let's figure out is the uh uh the component of |
|
| 173:01 | here. So you can see that a component of displacement in the two |
|
| 173:07 | . So it's gonna be either this or this one, you've eliminated these |
|
| 173:12 | . Does it change as a function ? Oh Yeah. Um uh uh |
|
| 173:22 | direction. Yes, it does because is the uh the displacement is zero |
|
| 173:27 | and it's something else here. So this changes with, with respect to |
|
| 173:32 | three direction. So this is a answer. Does this displacement change with |
|
| 173:38 | one direction? Well, no, uh uh it's shown here um uh |
|
| 173:45 | be uh lying in the 32 But this is the wrong answer. |
|
| 173:50 | this is the right answer over Again, the displacement is in the |
|
| 173:55 | direction that varies in the three This is the answer. OK. |
|
| 174:05 | now we have defined and discussed Strat we have defined and discussed the strength |
|
| 174:14 | now we're gonna put them together. that is first done by Mr Robert |
|
| 174:24 | back in the uh 17th century. are no pictures of Robert Cooke. |
|
| 174:37 | are no paintings of Robert Hook and are no, obviously no photographs of |
|
| 174:42 | Hook. Uh uh But uh so happened to him, he was a |
|
| 174:47 | guy. He was the first um of uh the Royal Philosophical Society. |
|
| 174:56 | uh the uh the, the, major uh uh organization for scientists uh |
|
| 175:04 | uh uh in the UK and he one of the originators, he was |
|
| 175:11 | famous guy. Um but there are images of him in existence. He |
|
| 175:25 | a famous feud with Sir Isaac Newton Newton uh was uh even more famous |
|
| 175:34 | Hook. And so the speculation is uh uh Newton arranged for all of |
|
| 175:40 | images to be destroyed. Interesting Now, books law uh was formulated |
|
| 175:50 | the 17th century to describe the deformation of homogeneous materials like iron or |
|
| 176:01 | That's not a very good description of is Iraq has various grains and it |
|
| 176:06 | pores as well. I want you ignore that um fine point here. |
|
| 176:12 | uh we're going to apply Hooks Law homogeneous materials anyway to rocks. And |
|
| 176:20 | we'll talk in the eighth week of course. We'll talk about the eighth |
|
| 176:25 | of this course. We'll talk about we have to modify H's Law to |
|
| 176:32 | the rocks for. Now, I you to ignore this distinction. The |
|
| 176:40 | Law says that stress and strain are to each other. So written uh |
|
| 176:47 | uh uh uh we can write it way. This is uh a component |
|
| 176:51 | stress and a component of strain and uh AAA bunch of proportionality constants. |
|
| 177:02 | so let's choose J equals one and equals two, for example. And |
|
| 177:07 | we're gonna uh have J equals one K equals two here. And then |
|
| 177:10 | gonna sum over all these other components these other indices are repeat it. |
|
| 177:20 | this quantity, this set of coefficients called the elastic compliance tensor. There's |
|
| 177:27 | word again, tensor. We're, gonna uh talk more about tensors tomorrow |
|
| 177:35 | . This set of uh this collection um uh proponents is uh all the |
|
| 177:45 | tensor and it has rank four. uh uh uh uh before we |
|
| 177:50 | we looked at matrices of uh uh only two indices, they were ranked |
|
| 177:56 | and this is rank four what they . But you can see how you |
|
| 178:03 | to have it that way. And this one has rank two and this |
|
| 178:06 | has rank two. So in order have each component of stress here portion |
|
| 178:12 | every component of, of each component strain here portion to every component of |
|
| 178:18 | . Here, you're gonna need four of uh to describe the collection of |
|
| 178:25 | . So putting those all together, call it the elastic compliance tensor. |
|
| 178:30 | we can write it in tensor notation way. See that's got one squiggle |
|
| 178:34 | over the rank two tensor strength, squiggle over the rank two tensor |
|
| 178:42 | So two squiggles over the rank four 10. Now another way to uh |
|
| 178:54 | books law is this way which says stress is proportional to strain. So |
|
| 179:01 | we have stress and strain and a set of coefficients. And this is |
|
| 179:06 | the elastic stiffness tensor. And in notation, it looks like this, |
|
| 179:13 | this with this, you see very structure. Now, what is the |
|
| 179:23 | between this set of coefficients and going this set of car fish? |
|
| 179:31 | there one is the inverse of the . Let me show you how this |
|
| 179:36 | here are the two equations that we saw. We can combine those by |
|
| 179:41 | taking this and putting this expression for . That's, that's this here. |
|
| 179:50 | that right in here rearranging the um the parentheses we can do that because |
|
| 179:56 | uh uh uh mathematicians say these quantities associative. And now we have stress |
|
| 180:04 | the left and stress on the right thing better be the identity matrix. |
|
| 180:12 | that is the f the rank four matrix, which is uh uh defined |
|
| 180:17 | terms of the rank two identity matrix called the chronic or delta in this |
|
| 180:26 | . No, in the general this is really complicated. For |
|
| 180:35 | um uh Books law says that the, the 11 stress is equal |
|
| 180:40 | the sum of these nine components What a mess. But the isotropic |
|
| 180:50 | um case is uh uh is gonna out to be simpler for reasons which |
|
| 180:55 | come to. And the way it's simplify is uh these uh all these |
|
| 181:06 | different components that they're not all gonna independent that that's the way we're gonna |
|
| 181:11 | that. So here's a question for . The stress cause strain or the |
|
| 181:18 | cause stress. Now, Hook doesn't or care. Hook says that uh |
|
| 181:29 | is proportional to strain or we can it the other way, strain is |
|
| 181:34 | to stress. This is uh uh linear relationship which Hook uh described uh |
|
| 181:46 | uh relating stress to strain but who the did not know or care about |
|
| 181:54 | ? So he doesn't have an answer this question here. The stress cause |
|
| 181:59 | or the strain cause stress. So me uh uh um uh uh pose |
|
| 182:04 | to you. Um Let me uh up your pictures again. OK. |
|
| 182:10 | uh Brisa, uh how would you that the stress cause strain or the |
|
| 182:15 | cause stress? I would say that stress causes strain. OK. So |
|
| 182:25 | that's your answer. So what's your here? Why did that, that |
|
| 182:34 | may or may not be correct? tell me your thinking. Well, |
|
| 182:38 | , I'm, I'm thinking that but maybe it's going to, I'm |
|
| 182:43 | to repeat the same but it's in to have a strength, you need |
|
| 182:48 | have a stress that caused. so uh uh I know what you're |
|
| 182:55 | . Uh Look at my hands, , you're um you're putting a stress |
|
| 182:59 | a rock and so it strains fair . But can't you say the same |
|
| 183:04 | ? I'm putting a strain on a and so that pushes stress back, |
|
| 183:09 | resists if you put a AAA strain it and it's re resisting with uh |
|
| 183:14 | stress. So uh uh uh those descriptions are about the same, it |
|
| 183:22 | to me. So, Carlos, do you think? Does stress cause |
|
| 183:28 | or the strain cause stress? I say that it could be both. |
|
| 183:36 | mean, you know, there there is a relationship. So, |
|
| 183:40 | it, but I think in the stress is what it, but |
|
| 183:44 | is causing the strain. OK. uh can you give me a better |
|
| 183:50 | than that's the same answer as Persada and maybe you're right. But, |
|
| 183:54 | give me a AAA uh give me better reason. Mhm It's hard, |
|
| 184:07 | it? Yeah. Uh So you , can you give uh do you |
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| 184:11 | with these or do you have a answer? Strain causes stress? Thanks |
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| 184:20 | doing OK. So uh uh Yee is giving me the other answer. |
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| 184:26 | says that strain causes strength. So uh uh uh yeah. |
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| 184:35 | so she's doing uh she's showing me her hands, she's applying a strain |
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| 184:42 | the uh the rock in between my here is pushing back with the |
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| 184:46 | So uh uh um we have a here. Uh We're about 22 to |
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| 184:52 | . Uh uh Utah. What do think the, the, the stress |
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| 184:56 | strain or vice versa? Remember, is not gonna tell us the |
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| 185:09 | Those risk. A change of strength stress. OK. So we have |
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| 185:16 | strain to begin with. And now saying uh uh uh strain uh uh |
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| 185:23 | strain causes stress. So now uh have 2 to 2. And so |
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| 185:28 | , I get to uh uh I the uh uh but I, |
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| 185:37 | sorry, I have an idea but the stress, it's something |
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| 185:46 | right? That will cause the strain is internal. I don't like |
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| 185:54 | I don't think like how the strain from inside. OK. So, |
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| 185:59 | let me see here here. We uh uh um a rock and I'm |
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| 186:04 | squeeze it from the outside. But you think that there's stress on the |
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| 186:10 | as well when I'm squeezing this? the, the stress is maybe uh |
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| 186:16 | is both on the inside and on outside. So, uh I like |
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| 186:23 | actually. Um uh we have a vote and um uh so uh Hook |
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| 186:33 | help us. And I, I don't know whether Hook asked himself |
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| 186:36 | question or not. Uh So, the uh uh uh so let's um |
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| 186:49 | let's think more about this. We're go away from elasticity second here. |
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| 187:16 | I'm thinking that uh iiii I don't to um uh answer this question at |
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| 187:24 | point because the next six lectures are gonna be applying Hook's Law and Hook |
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| 187:33 | know or care about the answer to question. And we're only gonna return |
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| 187:40 | this question at the end of the where we go beyond Hooks Law. |
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| 187:50 | so I want you to um uh that in your mind that who does |
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| 187:57 | have an answer to this question and uh uh you will come up with |
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| 188:02 | answer when you're thinking about this uh and maybe we can uh discuss it |
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| 188:08 | more tomorrow. Oh But uh for , let me uh uh uh step |
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| 188:20 | from elasticity and talk a little bit thermodynamics and you'll see what the connection |
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| 188:26 | uh uh uh immediately. So the law of thermodynamics says that is |
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| 188:33 | a law of conservation of energy. so it says that when you uh |
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| 188:39 | uh a change in internal energy or volume is given by the amount of |
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| 188:44 | done minus the amount of heat So I know you all took a |
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| 188:51 | in thermodynamics some time ago. So is uh the same as they taught |
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| 188:56 | there. Now, the second law thermodynamics says that this amount of heat |
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| 189:02 | is given by the temperature and the in entropy. Now, if the |
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| 189:11 | done is infinitesimal, then that work equal to the stress times the |
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| 189:19 | And we'll put in here for the let's put in here this expression from |
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| 189:24 | law and then rearrange the uh uh the uh the brackets and uh um |
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| 189:33 | then the, the change in internal density is then given by this expression |
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| 189:39 | the deformation. What is the term uh of the heat injected? And |
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| 189:45 | reason I'm showing you this is that uh uh uh you can see that |
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| 189:52 | s uh on the left hand hand is a scalar. And so all |
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| 189:58 | these indices are repeated. There are single in indices here J is repeated |
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| 190:05 | , J is repeated here, et . So there's lots of summing going |
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| 190:08 | in here and to end up with scaler and then we subtract off the |
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| 190:13 | injected. Now, during wave In the first instance, we're going |
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| 190:19 | assume that the deformation is idio And that's a thermodynamic term, which |
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| 190:26 | that uh the change in entropy is . So that means that as the |
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| 190:32 | is going through the rock, no escapes from the rock and no heat |
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| 190:37 | injected into the rock. And so that this term goes away and the |
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| 190:43 | simplifies because of the uh of the . Um uh uh uh because of |
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| 190:57 | this summing going on here, it be that if we interchange the order |
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| 191:03 | um uh the pairs see here, has JK at the front here, |
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| 191:07 | has JK at the end. it has MN at the front and |
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| 191:11 | at the uh MN at the back MN at the front. And so |
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| 191:17 | uh uh we can uh uh re these multiplications any way we want. |
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| 191:23 | very clear that uh uh that, this argument about uh um energy since |
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| 191:31 | is quadratic in the strain, it be that the uh stiffness tensor is |
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| 191:39 | in this way. And furthermore, can make a similar argument about the |
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| 191:44 | tensor. Because of this argument, rank four tensors are not so complicated |
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| 191:55 | they appear. That's gonna be very news for us, not as complicated |
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| 192:01 | it appeared. We there's a further because the stress and strain are both |
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| 192:09 | tensors must be that you can interchange J with A K like we do |
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| 192:14 | . It must be, you can the M with the N like we |
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| 192:17 | here. And it's got to be same for the compliance sensor. Because |
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| 192:22 | these symmetries, we have a wonderful simplification which is the this stiffness sensor |
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| 192:31 | we're gonna be needing for wave propagation be mapped to a stiffness matrix with |
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| 192:38 | two indices in the following way. here we have 1/4 rank tensor being |
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| 192:45 | to a second rank matrix. And does the ma matrix go? |
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| 192:50 | it goes pairwise. So every pair and K is gonna map into one |
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| 192:56 | index alpha or beta as the case be. And 11 is gonna map |
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| 193:02 | a, one that's pretty obvious. maps to a 233 maps to A |
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| 193:07 | . OK. But look here 23 gonna map to a four and the |
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| 193:12 | is gonna be for a 32 because of the symmetries of stress and strength |
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| 193:18 | 13 maps to a five and 12 to a six. And uh uh |
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| 193:23 | that means that we can write all information which is hidden inside here in |
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| 193:28 | simple way. Just think about If I, if I want to |
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| 193:32 | a rock in terms of these book tensor uh E elements, I, |
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| 193:38 | could write down uh the I's and Js uh three by three on this |
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| 193:43 | . But then what would I do the MS and the ends? I |
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| 193:46 | simply couldn't show it to you. you can't see it, you can't |
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| 193:49 | about it. So because of the that I just showed all the information |
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| 193:54 | that uh uh uh uh rank four is contained in this rank two |
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| 194:03 | which happens to be six by And furthermore, we're not finished simplifying |
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| 194:11 | . Furthermore, this uh uh is because of the previous symmetries. So |
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| 194:18 | can just omit that lower triangle. now we have uh uh the compliance |
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| 194:25 | expressed as a compliance to me, stiffness tensor ex expressed as a tensor |
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| 194:31 | , as a stiffness matrix can be by these 21 constants. You can |
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| 194:38 | them up six along the diagonal 15 the upper triangle. And all these |
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| 194:43 | here are the, are the same the upper triangle. And did you |
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| 194:46 | the mistake I just made, I this c and I call that uh |
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| 194:54 | a, a compliance matrix. C compliance. No, that's the stiffness |
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| 195:01 | . The compliance matrix of this with S. Isn't that terrible that the |
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| 195:07 | who set this mo uh notation in motion, uh 100 and 50 |
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| 195:14 | ago, they used S for compliance C for, well, for |
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| 195:21 | stiffness, constant, terrible thing, we have to go with it. |
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| 195:25 | We try to use our creativity We'll get in big trouble, but |
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| 195:29 | is the compliance matrix and a similar . OK. So I'm gonna skip |
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| 195:37 | this. Uh uh And I have little quiz. Is this true or |
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| 195:44 | Elie? That's true. Good enough be, can it be formulated |
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| 195:55 | And equivalently in terms of stiffness or Carlos. Is that true or |
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| 196:02 | I think it's true, professor. , that's also true. Right. |
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| 196:08 | . Oops, sorry about that. . And so this three by three |
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| 196:16 | three by three stiffness tensor has in most general anisotropic case, how many |
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| 196:23 | components? Uh three oh yeah, . Um You say that uh uh |
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| 196:40 | for you. Uh You, I was struggling to uh uh uh |
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| 196:44 | your name properly. So now uh did you say two confused with |
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| 196:53 | the indexes index? Uh No, , it's got four indices. Each |
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| 196:59 | of them goes from 1 to So if you multiply this up, |
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| 197:02 | looks like it comes to uh OK. But remember we said that |
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| 197:09 | uh because of all the symmetries in , uh many of these are the |
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| 197:14 | . And so it can be represented a six by six matrix. |
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| 197:18 | So that makes you think that 36 be the answer. But remember we |
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| 197:23 | that this six by six matrix is . So 21 is independent answer. |
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| 197:29 | where does the two come from? turns out and you'll be very happy |
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| 197:33 | see to hear this. It turns that for isotropic rocks, all these |
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| 197:39 | come down to two P waves and waves, right? So that's why |
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| 197:48 | the uh uh uh uh the anisotropic has 21 and the isotropic case has |
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| 197:57 | . So that's the uh the reason we like to deal with isotropic rocks |
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| 198:02 | though the rocks are not isotropic, ? Because we have a hard time |
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| 198:07 | our minds around 21 different uh uh components. So that is an issue |
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| 198:15 | we will deal with in the 10th . So for uh uh uh for |
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| 198:19 | , for the first seven lectures, are going to be dealing with isotropic |
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| 198:25 | where there's only two independent components. . So let's figure out what these |
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| 198:37 | like. Uh Then for uh uh uh for an isotropic rock here are |
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| 198:43 | , here is the I isotropic uh compliance tensor. Uh And we're gonna |
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| 198:49 | all these and it's not gonna be hard as you think. Uh um |
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| 198:54 | gonna do the isotropic case one at time and remember we only have to |
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| 198:58 | this top uh track. OK. , so here is the isotropic, |
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| 199:04 | the, the, the uh the strain uh uh uh and in |
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| 199:13 | of Hook's law, it's expressed in way and we got a sum over |
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| 199:17 | in M. And so we have these nine terms on the left side |
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| 199:22 | this expression for the 11 strength. using the wort notation, we can |
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| 199:29 | um uh uh compress that uh to . Uh So here, this |
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| 199:37 | the s 1111, remember that maps an S 11 and uh uh uh |
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| 199:43 | a similar way, applying the void . That's another German name fit. |
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| 199:49 | don't pronounce the G, it's a notation. And so this has turned |
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| 199:56 | to be this. And so you wonder here what happened to the uh |
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| 200:00 | to the twos that you might Uh there's uh uh uh these two |
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| 200:07 | , but uh these two stretches here the same. So, uh that |
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| 200:14 | um uh shouldn't we have uh uh two here somewhere. Well, it |
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| 200:20 | out uh that the, the answer no, for a reason I skipped |
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| 200:24 | because we're behind time here. You uh go back and uh uh read |
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| 200:30 | , uh the lecture slides for yourself you'll see where I skipped over the |
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| 200:35 | of uh of uh appliances. So, so now let's uh uh |
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| 200:41 | at it, the special case we have a horizontal isotopic cylinder and |
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| 200:46 | gonna squeeze it from the ends uh AAA stress 11. And we're gonna |
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| 200:52 | get out uh uh what's gonna happen this? It's gonna have a 11 |
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| 200:58 | and it's also gonna have AAA radial also, you know that when you |
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| 201:03 | the cylinder like this, it gets and it also gets fatter. But |
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| 201:08 | uh now we're talking about only the strength. And so then the previous |
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| 201:15 | comes down to this. Why is , I'm gonna back up, gonna |
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| 201:26 | up. So uh we have uh uh the, the there's no 32 |
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| 201:33 | only uh uh only uh AAA 11 . So this is zero, this |
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| 201:39 | zero, this is all of these zero because the only nonzero stress is |
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| 201:44 | 11 stress. And so that's what have. So the ratio of that |
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| 201:53 | and uh that stress and that strain given by one over the young's |
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| 201:59 | That's the definition of Young's modules. , that is not a modulus which |
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| 202:06 | which uh my um it happens, , which occurs in, in uh |
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| 202:13 | our expressions for wave propagation. But an easy one to visualize you. |
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| 202:19 | where young's mars comes from squeezing as isotropic cylinder like that. And |
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| 202:26 | in this way, we have uh defined the 11 component of compliance. |
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| 202:32 | furthermore, by uh uh because it isotropic, uh all these are the |
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| 202:37 | , this is the 22 direction and 333 direction. And so um uh |
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| 202:45 | right away, we've got uh three of the um 21 countries. |
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| 202:51 | for 11 stress, it will also a different to two strain. So |
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| 202:56 | is the 22 strain. And you here uh uh on the right hand |
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| 203:01 | , we have all these terms with the different stresses. Um Yeah. |
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| 203:18 | so in our case, all of , uh uh uh all those other |
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| 203:22 | are zero. So the only stress this one right here and uh this |
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| 203:27 | right here to the uh all these disappear except for this one. So |
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| 203:33 | this. Now, uh uh is a common name for S 12? |
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| 203:44 | . Uh uh Back in the 19th , there was another um uh physicist |
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| 203:50 | , his name was poison. And the French way to pronounce this is |
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| 203:56 | poison, it's poison. And uh he defined this ratio uh uh in |
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| 204:02 | case here, uh and we call B Poisson's ratio is the ratio of |
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| 204:08 | two strains, the ratio of this and this strain. And so, |
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| 204:15 | uh uh since we have the uh 11 strain is given uh in terms |
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| 204:20 | Young's models, this way, this ratio is equal to um uh um |
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| 204:28 | Young models times minus one times the 120 by the way, the minus |
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| 204:33 | here is so that pass on ratio will turn out to be positive because |
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| 204:40 | uh um epsilon is positive, uh 11 is positive epsilon 22 is gonna |
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| 204:47 | negative going the other way. So need to have the minus sign to |
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| 204:52 | the ar ratio positive. So that that s 12 is equal to minus |
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| 204:59 | ratio divided by young smart. And bang, we've gotten this component and |
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| 205:06 | bang these two are the same because isotropic, that's I think there will |
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| 205:15 | no 12 strain in response to 11 because all of these terms are |
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| 205:22 | every, every one of these terms zero except for this one. So |
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| 205:28 | must mean that that uh uh uh 61 must be zero. So we |
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| 205:33 | this to be zero and bang all others are zero for a similar way |
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| 205:38 | a similar reason. Now, if lay on there a 12 stress, |
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| 205:48 | will get a 12 strength and that given by one over the sheer |
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| 205:53 | That's this. And in a similar because it's isotropic uh uh uh there |
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| 206:01 | a uh uh she marks they are same and in the same way, |
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| 206:07 | uh uh these others are zero. we've analyzed the complete compliance tensor and |
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| 206:14 | only took us about five or 10 . And so you see in |
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| 206:18 | sheer modulus that you're familiar with from wave propagation. You see here Young's |
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| 206:24 | , which does not appear in the for seismic wave propagation and proton |
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| 206:30 | And uh remind you the uh the lower triangle is the same. |
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| 206:36 | , uh there's a further simplification because material is isotropic, we have this |
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| 206:43 | between the three components. And this from the requirement. If we rotate |
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| 206:48 | sample in any direction, the compliance has to be the same. So |
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| 206:53 | enforces this relationship among these three Yeah, we've seen all these uh |
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| 207:11 | uh all these um uh module I shear modulus Young's modules, et |
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| 207:18 | whatever happened to the uh the, bulk modulus, the in compressibility that |
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| 207:24 | a quantity that does show up in uh he wave uh propagation. So |
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| 207:32 | let's think about that. And next consider uh the case of isotopic |
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| 207:38 | So this is the case of isotropic , whether it's in the ocean or |
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| 207:44 | uh rocks or anything, we can imagine that we can apply an isotropic |
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| 207:51 | um onto uh any sample. And volumetric change is given, like we |
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| 207:57 | before is the sum of the strain using Hooks law or uh we uh |
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| 208:03 | this in terms of the compliance And uh uh if the pressure is |
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| 208:10 | is, is isotopic and this one a special case of uh uh diagonal |
|
| 208:17 | component. So carrying out all these uh sums here, uh uh we |
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| 208:24 | that the, the change in volume equal to the uh it's portion to |
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| 208:29 | change in to the pressure with all sums of compliance, matrix compliance elements |
|
| 208:36 | here. And so that means that OK, that, that, that |
|
| 208:44 | here is meaning summing over these components . And if you sum them all |
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| 208:52 | , it's uh it's three over E sig signal over E, you can |
|
| 208:57 | that if we sum those up and gives the, the bulk modules, |
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| 209:03 | we divide uh uh the volume change the pressure that gives one over the |
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| 209:10 | modules, that is uh the same we derive from that sum. So |
|
| 209:17 | found just earlier this other expression. uh uh what we just learned uh |
|
| 209:23 | expression for K, we can find sigma in terms of K and |
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| 209:34 | Now, this, we're almost finished today. Um I want to uh |
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| 209:41 | uh I ask you uh uh I out to you, there are two |
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| 209:47 | cases of interest. If the sheer is zero, that is in the |
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| 209:53 | , then well know ratio is one . You can see that right here |
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| 209:58 | just put zero for here and here you get one half. If |
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| 210:04 | on the other hand, K is pars ratio comes out to be minus |
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| 210:10 | . Can you see that K equals here and here we get a minus |
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| 210:13 | for K and so that means the, the, the maximum value |
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| 210:19 | P ratio is one half, the value is minus one. You might |
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| 210:26 | that the minimum value for cross ratio be zero, but it has showed |
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| 210:30 | that it's minus one. Never happens rocks that uh uh uh people |
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| 210:36 | have manufactured some artificial sponges which has negative cross ratio. So, so |
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| 210:51 | isotropic rocks, uh ratio or ratio upon all these uh uh rock |
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| 210:58 | Usually rock lies between 0.1 and Ok? So no, uh uh |
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| 211:08 | is a little quiz coming up Uh uh And I'm gonna leave you |
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| 211:13 | take that quiz on your own tonight write down a question for me and |
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| 211:20 | it to me by email tonight. all have my email address and uh |
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| 211:25 | it to me tonight and we will off at nine o'clock tomorrow morning uh |
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| 211:31 | uh uh discussion of your questions from lecture and then we'll continue uh on |
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| 211:40 | tomorrow's lecture. So uh this is good place for us to stop. |
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| 211:46 | so that's what we're gonna do. right now. You can turn off |
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| 211:49 | , uh, um, um, off the |
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