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00:00 The side. So we're almost on this morning. And uh uh and

00:11 has uh kindly brought me a cup coffee. So I am fixing the

00:18 just now and then we're gonna get the questions. I see that each

00:23 you as a sent in a I haven't looked at them yet but

00:30 I was pleased to see them all . OK. So um let me

00:44 here. OK. So uh here's first question from uh uh le le

01:13 OK. So I don't know if can see this. Uh uh have

01:21 uh Yeah. Right. Yeah, , I need to uh oh it

01:46 I'm screen sharing but it's not showing . Why is that? Mm mm

02:05 , I have coffee. Yes. you. Right. OK. Uh

02:11 uh uh maybe just dash share your . So it was Yeah, so

02:21 changed 22 to 21 to your uh , I don't think I'm properly.

03:10 . Can they, how come I say it here? So doing

03:23 it does not allow me. So were having trouble again. Audiovisual we

03:29 here in the classroom. We have a large screen and my screen sharing

03:34 not showing on that screen. Mhm one. Mhm I want to make

03:49 that you played your screen by or it is, maybe it's not showing

03:57 screen. It's extend this curse and back and forth between two screens.

04:02 extends the bladder or uh if you I think I know how to uh

04:09 it. Uh um Yeah. So display settings. So um so we

04:20 it to do play. Keep is UK. Yeah. Yes, you're

04:29 . And now shows good. I share with students. Yeah.

04:39 Um So, uh you know, funny that uh we've been uh doing

04:46 uh the Zoom meeting now for two , but we're still learning how to

04:50 it. OK. So um uh are good questions here. Look and

05:00 look here how she has uh uh me four questions, not just

05:05 So she's getting four for the price one. So, uh that's

05:11 So let us bring up uh uh slides and I think that um I

05:21 you uh says slide 36. I that that's referring to the uh uh

05:31 math 101. Am I correct? . So, uh that is very

05:38 . You see here, she was to tell me which slide number because

05:43 put the slide numbers. Oops. I, I've got to um uh

05:49 get my can you see my, um Yeah, you, you can

05:54 my pointer. So I, I the slide numbers here. So that's

05:58 important thing. It's, it's almost . But whenever you make a presentation

06:03 presenting some of your work, always slide numbers on it so that people

06:08 say go to slide 36. So she took advantage of that because

06:14 uh have done this before and I how useful it is. So let's

06:18 down here to slide uh 36. uh too far. Oh,

06:35 34. Um Oh, ok. that's it. I'm in the wrong

06:54 . Ok. I'm in the wrong . So uh uh let me uh

06:59 out of this. Uh oops. . Yeah. Elasticity. Yeah.

07:11 . So, ok. Ok. , uh we didn't get a chance

07:24 do that uh uh to do this , but I do, I do

07:28 to, um, say a few about this uh plot so I'm gonna

07:36 it out here. Um Yeah. . So whenever you're working for a

07:44 , normally the company is going to every meeting with uh an emergency notification

07:52 because it's good practice. And so should do that here. And

07:57 the emergency notification usually says there's no today, but it also gives instructions

08:04 what to do in case there's an . So, uh uh what you

08:09 to do is update your emergency contact uh with the university so that you

08:18 uh uh so that the university always how to find you in an

08:23 So there might be a weather emergency there might be a gun violence uh

08:28 emergency. Uh And the, the has techniques to contact you to warn

08:35 about all these things in advance. um uh so you need to make

08:40 that the uh university knows how to in touch with you. So,

08:45 that's true for everybody and including I, I think uh Carlos won't

08:50 to worry about any uh uh emergen any weather emergencies in Houston. But

08:56 , it's a good idea for the to have your phone number and your

09:00 address and so on just to make that they can contact you.

09:07 So do you, do you? , I do that. That's very

09:13 . It's ok. It's not the look at your screen. So they

09:17 looking, looking at this, aren't ? You know, I, I

09:20 it because I want to show the before, but I need to.

09:27 , it's very, very common. . Ok. Ok. Ok.

09:35 then that's uh uh slide one and my uh uh um lecture two,

09:40 one. So here is uh uh into the, the rest of the

09:46 , which we already did. So now I want to um uh

09:51 um move to slide uh 36 and see, I think the best way

10:00 do this is to catch it. , OK, Honduras when his

10:22 well, I'm not seeing the slide here. It is this side that

10:42 . So I'm not sure why I see any slide numbers on this.

10:49 , one more. Ok. Uh So I don't, I'm not sure

10:56 I'm not seeing slide numbers here, um I'll look into that later.

11:03 should be slide numbers showing on the . I know that I have them

11:08 the f it might be that they're down here in this, in

11:15 they're hidden behind this um uh uh frame here. Hey, but uh

11:23 , you found it uh in your , right? Yeah, they're good

11:28 you. Now, the question is do you do uh um uh a

11:37 subtract? And so the way you that is you just uh uh multiply

11:42 one vector you wanna subtract uh by one. So you reverse it and

11:47 you do an addition, right? . So uh uh that's, that's

11:53 easy. Uh OK. So now let us see here. So um

12:06 , I'm having a problem here. Look, I can't move my mouse

12:12 of this screen, see I can't my mouse outside that screen.

12:19 Yeah. What, what did you ? I just pressed the oh

12:25 Thank you. So, now what wanna do is uh get out of

12:29 um presentation. OK. And I want to share the screen again

12:43 um questions. OK. Slide Then the new distance vector has

12:53 Who? And that's magnitude square, is the two means the definition of

13:02 . Uh um uh in that uh let's, we don't have to

13:06 it up, up, up Uh um In that slide,

13:12 the uh um the magnitude was given the symbol L and uh uh so

13:26 the two means to uh uh I showing you the square of, of

13:31 , not the square of L square distance. Yes. Yeah. So

13:40 why it was a sub. Uh it's uh when you have a superscript

13:44 that, sometimes it's just notation and it means raising it to a

13:50 In this case, it was raising to a power. What I was

13:53 you on that slide was uh the of the distance. OK.

14:02 um OK. Next question is on 40. The strain 10 is defined

14:10 divided by two. There's a one in that definition. And so um

14:16 see. Uh OK. OK. I think it's important to bring that

14:26 . Um stop sharing and then I'm start sharing. OK. So uh

14:44 everybody see that slide? That's where were before. So now she

14:49 uh uh see uh see thi this is L prime square. OK.

15:13 , what she's asking about is this half here? So, uh um

15:19 because we put in a 1.5 that means we have to put in

15:23 two here to cancel out the one . And so, uh uh very

15:28 , uh uh you will see, you will see the reason why we

15:31 that. Um Let's see here. I think, II I think uh

15:42 , uh this is not a good for me to explain why uh I

15:47 in a two but very shortly you find uh that was a clever thing

15:52 do. OK. OK. uh the next uh slide 45 and

16:09 how do you find the uh the of a vector? Is it this

16:24 ? OK. So uh this is good flight. Can I uh uh

16:31 everybody sees this? Uh Let me sure that everybody is seeing this.

16:40 Yeah. So uh Carlos uh you're that I had that uh right.

16:45 , um yeah, but not Not anymore. OK. Um

16:55 Is that good? Yes. So, um um here we see

17:05 a square which was uh uh the dash line is the original um

17:11 orientation of the square. And after deformation, this uh the square has

17:16 um changed into this shape which is square. Uh We call that parallel

17:22 p it, I'm not sure if know that word in uh from your

17:27 , but that's the, the uh uh um you remember back when you

17:33 plane geometry as a sophomore, you that, that the, the name

17:38 a shape like this is a parallel pip it. And of course,

17:42 mean this line is parallel to this , this line is parallel to this

17:46 . OK. So the square in , the dash square has been deformed

17:52 this solid parallel piping. And so deformation is in the one direction.

18:00 uh Can you see that uh uh in the one direction here and also

18:06 here, it's in the one So you see as a function of

18:10 X three, it changes from nothing something. So uh uh you

18:18 I, I think this is a picture which shows that the uh the

18:24 of deformation is in the one direction the one direction and it varies with

18:33 three directions. So did, did answer your question lately? Uh

18:45 it's not uh it, it's not uh excuse me. Uh uh it's

18:50 the, the displacement you is only displacement you want, it's only a

18:57 one component of the displacement. this number is zero. In this

19:05 , uh uh The displacement in the direction is zero because it sees nothing

19:10 this um um shape has uh moved the three direction, it's only moved

19:17 the one direction. So this one uh that's what it's showing here.

19:22 this part is zero, but the three direction is this one. And

19:28 can see that it changes uh uh your new one, um uh changes

19:35 uh from zero. And as you to larger X three, it's getting

19:41 and bigger. And so this term is non zero. So then in

19:48 case, the epsilon 13 is equal one half times this term here

19:55 OK. OK. So uh I was thinking when I um um

20:06 , I was thinking when I was this yesterday that this slide needs more

20:14 and you just proved it, Lili uh your question. OK. So

20:25 gonna stop sharing this screen and I'm go back and share the uh the

20:36 . Yeah. OK. So thanks those questions. Let's uh look at

20:43 next my question from yesterday's lecture. you see this? You know?

20:50 , is about the stress tensor. is not clear to me why applying

20:55 sheer stress would cause infinite spinning? . Um So that's not what I

21:06 . I uh I, I didn't that when you apply sheer stress that

21:10 cause infinite spinning. I can, can I rephrase that? Maybe I

21:16 to II I miss to put that they are not the same. Hm

21:23 . So uh uh uh if they not the, so, so uh

21:27 we see that uh if the two of, for example Tau 12 and

21:32 11, et cetera, wrong tau and tau 21. If those two

21:38 not the same then, um, , it would cause, uh,

21:43 spinning. Well, so, um, I think that,

22:00 uh, for me to explain um, clearly it's gonna take too

22:06 time. So, uh, um, well, what I,

22:13 , so what I'll do is I'm gonna defer that question and,

22:19 , let's see how I can help anyway. Um uh Any textbook will

22:29 a statement like that in, in . If you uh uh um uh

22:36 you have a textbook on uh uh um geophysics, there will be a

22:42 on stress and strength and uh in chapter, there will be a statement

22:49 to this. I think it should . If not, you can uh

22:55 um um look online and uh mm OK. Um uh So, so

23:15 the suggestion Rosada, look online in . Do, do you know uh

23:21 about Wikipedia? So just search online in Wikipedia using the search stress and

23:29 will be an article about stress And I think that in that article

23:34 be a discussion of this point. is pretty good. Wikipedia. I

23:40 very skeptical when Wikipedia began to uh be uh built because I thought there

23:48 so many opportunities here for misinformation that it's gonna be unreliable. But,

23:55 know, there is a whole army people who volunteer their time to um

24:04 uh uh to ensure the validity of , of, of the information on

24:10 . And uh um the way it is that you can join the army

24:19 a volunteer at any time, you sign up with Wikipedia. And then

24:23 you can have uh the uh the to go into any page a Wikipedia

24:32 make a change. And as soon you make that change, it's visible

24:36 everybody worldwide. OK. So, um that is why I thought that

24:50 system was so prone to misinformation because can edit it, they can edit

24:59 uh uh uh out of and, they, they can uh uh put

25:04 their wrong information. Uh And uh instantly visible worldwide, however, in

25:13 background, uh there is a record um uh what you did and you

25:21 find that record by uh when you into uh uh Wikipedia, click

25:26 on, on any page, there's tab for called talk, talk

25:32 And then you can see the history all the changes that have been made

25:37 that um uh page. And um uh normally when you make a change

25:45 that, it is noticed AAA flag hoisted somewhere and there'll be a member

25:52 this army who's an expert in things stress and you'll see that uh uh

25:58 that somebody named Leon Thompson made a and what the change is, and

26:03 he's an expert and he looks at and if he decides that's a,

26:06 good change, he leaves it and he, uh, uh, decides

26:10 a bad change, he'll take it , take it away again, just

26:15 that change. And if Leon Thompson a similar thing, uh,

26:19 twice more then he gets banned from army. Uh, so,

26:25 uh, in that way they ensure , um, the integrity of the

26:32 on, uh, uh, on wiki, on Wikipedia.

26:37 normally for a mathematical subject or a subject like this, there would be

26:41 , um, only a small number people watching but, um, there

26:45 many pages, uh, in Wikipedia are about, um, controversial

26:52 for example, uh, I guarantee that if you go on to the

26:56 today and search for Donald Trump, will find, uh, uh,

27:02 notification that yesterday he was fined $355 . Uh, they are very quick

27:10 put news like that into Wikipedia. so you can go in there

27:16 uh, you can edit the page , uh, Donald Trump and you

27:21 put in there. Um, Donald Trump is an idiot and a

27:26 . So within seconds that one will , uh, um, um,

27:32 because it's, uh, political and and if you do it again,

27:38 , uh, uh, you will kicked out of the system and you

27:40 be able to uh edit at So that's the way they maintain the

27:46 of Wikipedia. So I encourage you actually um become a member of the

27:51 to sign yourself up. And uh when you see uh a change that

27:58 to be done, go ahead and it. And uh um uh you

28:05 get hooked. You know, there thousands of people who uh spend hours

28:10 day doing this. Now, I tell you that the seg has its

28:15 SCG version of Wikipedia, which is in um uh only. And so

28:22 can find that on seg.org. And uh if you look around, uh

28:28 be easy to find, it's called seg wiki and it has stuff about

28:34 uh geophysics and you might find that in this course. Uh uh If

28:41 uh um if you go into the wiki and search for stress, you

28:48 find um uh a description and I it will not be as um um

28:55 a definition as you will find on worldwide Wikipedia because uh it will be

29:02 by a geophysicist who will define things very simply and won't concern himself with

29:08 like uh the symmetry of the stress . That's what I think.

29:17 So let me go on with the uh for Bria, she says,

29:21 is this then related to the fact some of the elements of compliance and

29:25 are the same. Yes, it . Uh that symmetry is very important

29:30 uh that is, that really helps uh uh reduce the number of uh

29:35 elements from 81 which is, you , three by three by three by

29:41 uh down to 21. And that in the number of independent elements comes

29:51 of uh symmetries. And it's true uh any um anybody uh you

30:01 a, a piece of glass, , a piece of rock,

30:06 Uh And then further uh for the uh if the, if the body

30:12 to be isotropic, then the uh list of, of uh independent elements

30:20 down to two out of the 81 two are independent. And um um

30:30 some of those are repeated. So , there's uh doesn't mean there's only

30:34 nonzero terms. That means that there's two independent terms. So we,

30:40 are gonna return to this topic of the number of independent uh elements for

30:46 isotropic body uh later this morning. . So, thanks for that

30:53 Next question is, and you explain physical concept of elastic stiffness and elastic

31:01 . Well, now that's a very question. Um uh um let's just

31:07 about stiffness first. Uh uh uh remember that uh stiffness is uh um

31:16 a tensor with four indices and 81 , three times three times three times

31:27 elements. What is the physical Well, that set of numbers is

31:33 set of proportionality coefficients between stress and . So H Hook's law says that

31:43 is proportional to strain and those uh 81 elements uh give the proportionality

31:52 So when you have uh a large , uh uh a, a large

31:56 of stiffness, that means that a um um oh uh uh given

32:05 a small strain can uh be uh with a large stress because you have

32:13 small strain epsilon times a, a stiffness coefficient on the right side of

32:21 equation. And then on the left , you have the stress. So

32:25 says the stress uh corresponding to that is a large number. So um

32:33 compliance is just the uh the inverse that. If you have a large

32:38 , it means that the uh the is, is uh weak, it's

32:44 and you can uh deform it a with a small stress. So in

32:52 , for example, you have a hard rocks, for example, think

32:56 a granite. So a granite has , has a hard stiff rock and

33:02 has large values of the stiffness of . It has small values of the

33:11 , think of a sponge piece of and just think of a, of

33:16 sponge which you can deform in your and that is highly compliant and it

33:22 a lot when you give it just little bit of stress. So that

33:27 high values of compliance. So does answer the the, is that a

33:33 physical uh explanation for you, Yes. Thank you very much.

33:42 . Well, thanks for these I, I like these very much

33:48 great institution that more. Uh So don't know here uh some three late

34:04 about the dot product when the vector itself is the result still a

34:09 That's a very uh uh important So let us go back, let

34:26 go back to the is a I'm gonna stop sharing this.

35:35 OK. So I think I'm sharing screen with slide number M-16. Does

35:41 see that? Uh uh Rosada? you see this? Yes, I

35:49 ? OK. So this is from math 101 file. Let me see

35:54 . So uh uh uh it and , in answer to Li's question,

36:00 uh you can multiply two vectors together two different ways. So this is

36:06 first one, the dot pilot is scaler not a vector. And let

36:11 just go forward here. Next slide uh uh the second way and the

36:21 product of two vectors is a So you see the difference, there's

36:26 different ways to multiply vectors together. I'm gonna go back to the first

36:33 . So that's the dot product. so uh the, the dot product

36:38 defined this way you take the uh the, the first component of the

36:44 times the first component of the other X two, Y two plus X

36:49 Y three. That's the definitions This right here means def uh um

36:58 . And uh remember when we, can write, uh we can write

37:03 AAA complicated expression like this very simply by saying X of I times Y

37:10 I remembering that when we have a index, we got a sum from

37:16 to 3. So separately uh uh think of this as the definition and

37:25 you can also show and this is to find in any book on plane

37:30 . Uh that uh this uh some three terms is equal to the me

37:37 length of one times the length of other times the cosine of the angle

37:42 them. So I think that's fairly . Uh uh This first way of

37:49 the dot product is fairly simple. second way is more complicated. This

37:55 the cross product of two vectors and a vector. And uh uh here

38:00 here is the result of that for three component vector. You can see

38:05 has three components and each component is difference between two terms. And it's

38:10 of complicated. And then furthermore, uh uh plane geometry book will show

38:17 that this thing is, this thing out to be the uh uh the

38:22 of X times the length of Y the sign of the angle between them

38:28 a unit vector, which is perpendicular both of these. So if you

38:34 X and Y in the plane of screen, then X, then the

38:39 uh vector is gonna be pointing out the screen. And uh well,

38:45 cross product X cross Y is gonna pointing a vector pointing out on the

38:52 . So um uh um I know did a lot of mathematics yesterday and

39:02 lot of notation and I think you're um uh unhappy with that, but

39:09 will be a payoff because the notation we develop is gonna make things easy

39:15 us uh in the future. So um let me stop sharing

39:30 Oh Start sharing. Oops. Um see what I want to have

41:06 OK. I think you can all this, these thumbnails of the

41:12 Uh We talked about yesterday and I'm go back to this one and I'm

41:19 to uh OK. OK. I I'll just remind you that uh uh

41:26 what we found yesterday by considering uh different stresses on um a cylinder of

41:35 rock in the laboratory. We, figured an isotropic. Uh We figured

41:42 that the uh the matrix, the by six matrix which contains all the

41:49 of the uh um compliance tensor that these expressions only. So, uh

41:58 it has a lot uh remember the triangle is the same as the upper

42:03 has a lot of zeros. And the, the main axis here,

42:07 has one over the young motos that's the one direction that's for squeezing it

42:11 the ends. And because it's it's the same in the two direction

42:17 the three direction. And uh when squeeze it like that on the uh

42:23 the ends, it also uh gets , it expands in the perpendicular

42:30 And that's described by these terms here this is protons ratio divided by uh

42:38 young models and the minus signs are here in order. So that pros

42:43 will be a positive number and epsilon be positive number. And of

42:48 uh what uh the minus sign comes when you squeeze a cylinder, it

42:53 shorter and it, but it gets sadder also. So uh obviously,

42:59 strain is the negative of the other , is uh the, when you

43:04 it, it gets shorter. So a negative strain. Um uh And

43:09 we uh it gets fatter, so positive strain. So that's why I

43:16 a, a minus sign here. then uh uh if you uh uh

43:21 it, it uh it responds according the sheer modules like that. And

43:26 uh that's easy to, to show with uh one cylinder, yes,

43:33 putting on there uh a 12 And then because it's iso it's the

43:39 in the other directions. So, in that way, we constructed in

43:44 very intuitive way, this uh compliance which shows all the information necessary in

43:52 compliance tension. Uh But uh uh bad. This is not what we

44:02 for wave propagation uh uh as Uh Well, uh you know,

44:08 have here at the University of we have a world class laboratory,

44:12 uh uh rock physics where we do , all kinds of squeezing of

44:18 And uh uh we, we uh yeah, it's being run by Professor

44:25 and he's got uh a number of in there who are all becoming experts

44:30 experimental rock physics. And so these the sorts of things they do,

44:36 also they do wave propagation experiments. they will um uh send a wave

44:45 these required samples. And we are shortly learn that this compliance matrix is

44:53 what we need for wave propagation before pass on here. I just want

45:00 remind you that we have here uh current three independent numbers, Young's models

45:08 ratio and she models. However, it's isotropic, you can show that

45:14 this relationship between the two. So um uh in fact, there's uh

45:21 uh excuse me, there's this, relationship between uh uh sigma and the

45:27 two. So in fact that there's two independent numbers which characterize the isotropic

45:34 and eventually that's gonna lead to our a conclusion that two types of waves

45:41 in those crops, uh uh B and sheer waves. So that's

45:48 If, if, if we didn't this uh relationship here, there would

45:52 three kinds of a way which And uh remember that I said we

46:02 find this relationship because the material is . So most of our rock materials

46:09 not isotropic, most of them are , which means that this is not

46:16 for most rocks. Most rocks have kind of wave which is propagating in

46:23 rocks. So that's a AAA big . And so we're gonna ignore that

46:33 for a while and we're going to uh soon that the rocks are

46:41 And then later on, once we to be smarter, then we'll uh

46:47 say, OK, uh Back at beginning, we have made an unrealistic

46:53 , we assume that the rocks are . Now, let's think about uh

46:58 rocks. OK. So, uh that's gonna happen at the end of

47:04 , towards the end of the Now, we went on to discuss

47:11 happens, not uh uh uh what when you squeeze the wrong equally from

47:16 sides. And you can imagine doing in the laboratory uh and squeezing it

47:20 all uh um uh equally from all . And from that, we decided

47:27 uh uh when you do that, uh uh uh uh design the,

47:34 book mogul is K and it turns to be uh this number here,

47:39 comes from summing all these others. I encourage you to um um to

47:46 up on this uh in the uh refresh your memory so that you are

47:53 you understand where that come comes Uh Then we, we had a

48:04 quiz here and I think we did quiz and uh uh you all uh

48:09 did well on the quiz. And , and now we came to elastic

48:15 . Uh This is uh uh what gonna need for wave propagation. Remember

48:21 we had Hooks law in two forms where the strain is proportional to the

48:27 and one where the stress is proportional the strain. And so um uh

48:32 this is the compliance matrix and this the stiffness matrix and they must be

48:39 to each other, right? Because uh this uh there's just two ways

48:43 saying the same thing. And so learned that the uh um uh we

48:55 that the stiffness ma the stiffness matrix we're gonna need for wave propagation is

49:02 inverse of the compliance matrix, which just figured out by thought experiments involving

49:08 rocks. So I think what at point, what I want to do

49:14 change into the uh slideshow mode. . So is everybody in the slideshow

49:23 ? And here's the other form And so now what we wanted to

49:27 since we know this uh these things uh thought experiments. We're gonna use

49:33 , uh use this expression for um , I take it to do

49:41 the set of ST stiffness coefficients. . So this is the inverse

49:48 So, so, so, oh . OK. So we're gonna stop

50:00 and start sharing again. OK. that good? We're gonna put

50:16 So it's not Sharon and um then sharing screen two. Sure that

50:36 Are we good? OK. So we decided uh uh uh yesterday that

50:45 complicated product is equal to the um I know the identity tensor,

50:55 the identity tensor with four indices. let's look inside here. We got

51:00 a repeated M oop second, I a pointer comes Mike Porter repeated M

51:10 repeated N. So we're summing over and NS and that comes out to

51:15 the uh uh the uh the identity uh And in fourth rank uh um

51:23 four indices, what does that In terms of things that, you

51:27 , intuitively, it's this combination of delta functions and remember this delta function

51:34 equal to zero if J is um is not equal to P and it's

51:40 to one if J equals P. it looks like um a three by

51:47 tensor with ones on the diagonal and off diagonal. And the same sort

51:52 thing cohere with different indices and so here. And So that's the

51:57 So let's show how to implement that . Let's first look at the 66

52:04 . Uh uh we, we're gonna the 66 component of the stiffness out

52:10 remember that 66 comes from the four notation 1212. So that means we're

52:19 put in here um uh One here two here and one here and two

52:26 . That's this. And then we're sum over all the MS and N

52:31 so when you do that, um you find uh that, that is

52:37 to the fourth, the rank four matrix uh uh putting in the,

52:43 ones and the twos here on, the deltas just like it in here

52:47 that uh uh we have J equals . So we put in here,

52:54 equals one right here and we put equals one right here. So putting

52:59 that together, you see that we this term and so these are both

53:04 . So that's a zero and this one times one. So uh uh

53:08 all of that, that comes together make one half. That is this

53:15 um oh uh this product here. uh we know what all these

53:25 And so uh uh when we, when we stick in here, all

53:34 compliance elements that we just learned about a, a lot of them are

53:42 . So uh the only terms surviving the sum are these two terms.

53:47 we have ac 12, that's the here and we have M equals one

53:52 N equal A and N equals And uh uh again here N equals

53:59 and NN equals two. See, this one comes from those values and

54:06 , uh and there's another term like , these are the only ones that

54:11 all these sums because a lot of uh compliance elements or zero, look

54:20 over for yourself. You will see the only terms out of this

54:24 which are non zero are these OK. The C 1212 and the

54:30 66. And so you might think S 1212 is an uh an S

54:37 . But no, that's not We also, when we convert the

54:43 to from a four index notation to index notation, we need to have

54:49 , a quantity of 1/4 in The reason for that I skipped over

54:54 and I don't wanna go back to today. But if you're concerned about

54:58 , you should go back and look the uh slides that I skipped over

55:02 , which explains why we need AAA here. And so for this term

55:09 again, and we have C Oops, sorry about that.

55:28 Ok. Ok. The C 1221 also AC 66 because of the symmetries

55:46 we talked about earlier. And so , uh when we put in uh

55:52 when we convert from the four index to the two index notation for compliance

55:57 we need, we need 1/4. combining these two terms were left with

56:02 . And so uh that's the, left side of this equation, the

56:07 side is one half. And so the left side with the right

56:11 we deduce that C 66 is the of S 66, which is um

56:22 the sheer marvelous mute. OK. what that means is that we have

56:33 uh uh the uh uh from the a stiffness tensor distance matrix will deduce

56:41 M equal zero. And that these terms are all mute. And just

56:51 with the compliance makers, we have lot of zeros off of uh off

56:56 the side, you can, you convince yourself of that just by following

57:00 previous logic. And now let's consider 11 component. So we put in

57:06 11, put in here, make all this sum. And on

57:09 right hand side, we're gonna get one. And when we carry out

57:15 sum with all the uh the zeros in here from the compliance make we

57:21 with these three terms and converting from indices to two indices were left with

57:27 two terms. And in, in of, of uh young smarts and

57:34 ratio. We uh uh we get of the compliance matrix using these uh

57:44 English words and using the expression that had before we uh uh excuse

57:51 um uh uh th this is simply uh uh multiplying both sides of this

57:58 by young's models. And so we have f finally, young's models is

58:05 by this difference here and we don't want the uh Young's model. We

58:11 modules appears in the, in the matrix. What we want is C

58:17 and C 12. And so, let us say, OK, this

58:25 a useful result to begin with. , what are we gonna do

58:28 Well, we're gonna do the we're gonna do it with, with

58:32 ones and two twos and um go a similar logic and we find this

58:39 relationship between C 11 and C And so you solve those two expressions

58:48 , separately for C 11 and C . And so we get here for

58:52 11, we get an L uh gonna call the result Emma and um

59:00 it's the same uh for 22 direction for 33 direction and the C

59:07 we uh get by solving those two uh uh is equal to M minus

59:13 mu where the view is here. what is M, you don't know

59:18 M is until I show you that is given by this. And suddenly

59:25 think, oh, now I see familiar because I know that K plus

59:31 thirds mu governs the P velocity. furthermore, I know that the sheer

59:38 M governs the sheer modules. And uh uh where did the density come

59:44 ? Well, I, and you know yet, but uh we will

59:48 that in the next lecture to show uh uh uh uh these things come

59:55 the wave equation. So I'm gonna up. Now, we know what

59:58 that these three govern heat wave these three govern uh shwa population and

60:08 three it's not quite clear, is what those three are, are good

60:14 ? But at least we know it in terms that we are familiar with

60:18 other courses where we have the U the K. Now this, this

60:25 a good uh uh uh and go it. This is a good want

60:36 remember that when we say a P as a P wave goes through a

60:42 , it does not make your uh on all sides of the rock.

60:47 makes longitudinal stress. So the stress from uh isotropic compression because there's some

60:54 in there. And that's why you the sheer module syndrome. And now

61:01 way of propagation, the young smokes personalization do not appear this off diagonal

61:09 M minus two new is called the parameter. And you can look up

61:13 lema parameter um uh in the Uh you can do that on your

61:19 . LeMay was actually a priest. you imagine that a priest during elasticity

61:25 in the 19th century? Uh you , all his buddies were uh uh

61:32 chanting uh uh mhm Gregorian chants in choir and some of them were out

61:40 the fields um uh growing grapes. And then uh LeMay was sitting up

61:48 his little cubicle in the monastery during . And to think about that,

61:57 you will find out that uh when look at the wave propagation equation,

62:02 does not appear anywhere even. So can uh uh if we put that

62:08 hit in here, it makes the element look uh simpler and remember that

62:15 are all zeros and these are all same as the upper tri and as

62:23 spreadsheet. Well, we don't have board anymore. We have canvas.

62:29 um uh uh I gave you a uh uh in the com canvas module

62:37 that works as an Excel spreadsheet. can download it and uh look at

62:41 . And the first worksheet in that you to calculate any of these parameters

62:48 from the seismic parameter. So you uh you probably have a good intuitive

62:55 for AAA different type of rock, VP should be and vs should be

63:02 you can just put it in there it will uh calculate for you.

63:07 I actually forgot whether uh the input in terms of feet per second or

63:12 per second. Um There might be way to switch between the two.

63:20 . So here's a little um uh . So how many independent components does

63:30 stiff stiffness tensor have? The charges 81 36 21 3 or two,

63:39 ? How many independent components? So , le uh it says for isotropic

63:50 . So your answer is correct for isotropic rock. But that's not what

63:56 question is. So how many independent components are? Um le let's go

64:05 and look at this. OK. it is. Yeah. So two

64:14 and in this matrix, uh it's um nine of them to be

64:21 Nine of them are nonzero. But uh we've got these repetitions because it's

64:27 . When we get to an these will all be different. But

64:30 , uh you were correct. Uh um nine of them are nonzero.

64:36 uh three different uh symbols appear but we have this uh relationship between

64:43 two. So only two of them independent. So uh um when I

64:50 a question like this, uh uh need to read it carefully because it

64:54 be a trick question. By the , this is a good time for

64:57 to stop and tell you what the uh uh the, the grading,

65:05 , process is gonna be like in course, I'm gonna have to show

65:10 a lot of these quizzes, uh, uh, every day I'll

65:13 showing you quizzes. But, uh, they don't contribute to your

65:18 . The only thing that's gonna contribute your grade is the final test.

65:24 . And the final test will be test of that, uh, of

65:28 sort that you might not have ever before. It's gonna be uh um

65:34 book. So you'll be here when sit down and take the test,

65:38 can have all of this stuff in of you, everything that I gave

65:43 and whatever uh you check out from library or whatever you buy from

65:47 you can have anything is open to . So, uh uh uh the

65:54 is gonna be a test of your , not of your memory,

66:00 Because if you're uncertain about some you can always look it up because

66:06 your books are gonna be open it's gonna be unlimited time. So

66:13 can take three days to take the . So, uh uh the only

66:23 uh uh there's only two conditions, , uh uh there's only two

66:29 one, you have to do it yourself. You, you can't do

66:32 together. And by the way, encourage you all to get together uh

66:37 outside of class, maybe by zoom talk these things over, you have

66:43 uh uh you have uh you know to get in touch with each other

66:48 I think you are able to um together and if you can't zoom

66:53 Utah is gonna help you zoom together or you can meet uh somewhere and

66:59 a, a AAA drink while you're talking. Um Probably that would be

67:05 for Carlos but uh uh I, recommend that you zoom together the three

67:11 you and get you t to, , if you have a question,

67:15 call him up. Uh, uh he's gonna be somewhere else, but

67:19 be happy to uh advise you. , uh I talk about these things

67:24 also, uh at the end of lecture at, at uh, we're

67:29 go today till one o'clock. So the end of this lecture, uh

67:33 you're gonna have, uh, submit a question and, um, uh

67:41 at one o'clock we're gonna be breaking lunch uh until two o'clock and then

67:46 gonna resume at two and we'll go six. So you might not have

67:51 to submit a question between one and . But, uh this evening,

67:56 after uh class, you're gonna email with a question from the morning lecture

68:02 a question from the afternoon lecture. . So back to the final

68:08 it's gonna be open book unlimited do it by yourself only when you

68:16 a test only. And uh, , uh, here's one more

68:20 You have to do it in one . So you can't do half of

68:24 now and half of it the next do it all in one sitting.

68:29 so you, I'm gonna hand it on the last day of class and

68:33 give you about five days before you to turn it in. So,

68:37 , uh, you choose a time included in there will be a

68:42 And so you choose a time when have several hours of, uh,

68:49 , and of no interruptions. So you have a AAA family, uh

68:56 find a place where you can be and concentrate with your books open and

69:02 computer open, everything is open and do the, uh, the exam

69:07 M Juan City uh on your on our system. Hello, I'm

69:17 to give you a number of It'll be, you know, about

69:22 20 questions with, uh, some of them will have several parts

69:27 that question. And so I'm not good at, um, making

69:38 So, what I try to do I try to make exams,

69:42 which if you're, uh, if you understand everything, you can

69:48 the exam easily in two hours. if you have to consult with your

69:54 and so on, it might take three hours but you might want

70:00 So choose a time when, you have a, a all the

70:06 you can devote to it and, , uh, uh, uh,

70:10 normally I find that people who spend time do better. So, but

70:17 people spend hours and hours and they do well at all. And so

70:21 people have not understood the material. , uh, I'm, uh,

70:27 gonna be writing questions, which some them will be easy and some will

70:33 hard and I really don't expect that is gonna get everything right.

70:41 I'm gonna expect that you're gonna be , uh, uh, uh,

70:46 , uh, out of a maximum of 100 I think that the scores

70:50 probably be, be between 8050. , uh, uh, so

70:59 uh, I will decide how uh, uh, assign letter grades

71:05 that. So, uh, if, uh, recognizing that I'm

71:12 very good at writing an exam, if we have two of you

71:17 you know, uh, uh, and 80 the third one gets 40

71:22 that's clearly a difference in learning. if, uh, if two gets

71:28 and 80 the other one gets, , uh, 70 that's pretty

71:32 And so I will make a judgment , uh, uh, how to

71:38 a letter break that if we had large class, if we had 50

71:44 in the class, I would be on a curve and I would say

71:49 average grade is gonna be a B so that it's gonna mean a,

71:53 certain number of A's and a certain of CS and a lot of BS

71:58 a small class like this doesn't, does, doesn't make sense. So

72:03 will uh uh be making um a about how to assign a letter grade

72:12 you turn in the exact. Mhm. My experience is that when

72:22 have an exam like this, which open book, unlimited time. You

72:28 a lot during the exam. Oh was my experience. The first time

72:34 took an exam like this, I lost for the entire course. But

72:38 the exam, uh uh I finally because it was open book unlimited

72:46 And finally, I figured it So I hope the same is true

72:49 you know. So that's the way grading is gonna go in this

72:55 So let's look at the next uh quiz questionnaire. So here's a statement

73:02 isotropic rocks, the normal stiffness component controls the uh the heat wave uh

73:09 velocity. Is that equal to Is that true or false? So

73:15 me turn to uh uh Mesa Can you hear me? Can you

73:26 ? I'm thinking, I think it's . Uh So that uh uh that

73:36 controls the p velocity. And so uh we know that K measures the

73:42 to pressure. So I think that's you're thinking of here. Uh But

73:47 go back we uh uh uh if had a question like this on the

73:51 test and say, oh, let's , let's go back to the uh

73:55 the book is open. And so it is. And here is the

73:59 position and that's not A K, an M, I remember the M

74:05 related to uh image related to K this way, K plus four

74:13 You, so you uh uh uh answer was wrong. You, you

74:17 sort of guessing, but I, you were doing this on the

74:21 you wouldn't be guessing, you would looking it up. Uh But as

74:26 look it up, uh and that's some time. And so if you

74:30 that on every question, you after two or three or four or

74:34 hours, you would sort of run of time and run out of

74:39 And so it, it's really important you to understand these things. So

74:44 don't have to look it up. PC three is equal to M not

74:55 to K. That would be OK. So we're almost finished with

75:02 first lecture and here we are uh hour and a half into the second

75:09 . That's OK. We're uh we're gonna be fine. So let's

75:12 about anisotropy. So, uh if have um uh a crystal, if

75:19 look at any crystal, it has , a, an uh uh uh

75:22 exterior shape with uh uh all his on the, on the side.

75:28 so, obviously, that is gonna to uh uh that's obviously due to

75:35 internal arrangement of the atoms and those uh arrangements are going to make for

75:42 isotropic wave propagation. And in that , we need all of these 21

75:47 . So that's mm clearly, it's nothing we can deal with in the

75:55 . We uh uh when we do experiments, normally, we measure only

76:00 waves, we might also uh uh shear waves. And Utah is measuring

76:07 surface waves which uh uh a combination those uh uh uh you just can't

76:17 in the field ever measuring 21 Uh So uh uh lucky for

76:25 uh they don't have to, even the uh uh uh uh rocks are

76:31 out of these kinds of crystals which the crystals require this kind of analysis

76:40 . So, for uh uh for rocks, uh the uh the grains

76:46 not all aligned. So the grains a lot are uh oriented in a

76:54 way. And if it's perfectly it comes out to be isotropic,

76:59 um uh it's not always perfectly So for the simplest geophysical case,

77:05 gonna find out that the stiffness sensor not 21 elements but five different

77:13 And you can see here the bunch zeros out here. You see

77:17 there's a 11 C 11 is different C 33. So what that means

77:23 that the P wave velocity in the direction and the horizontal direction is different

77:29 P wave velocity in the uh vertical . And similarly, there are two

77:37 cheer mod line here. So we'll more about that later. And then

77:42 1/5 parameter. C 13, which off to the side here. And

77:46 there's a calculated parameter right here. this is the simplest geophysical case.

77:53 this is corresponding to shales un fractured . And it's also um applies to

78:04 bedded sequences. Think about this. you have a bunch of layers where

78:08 long wavelength seismic waves traveling through those many thin layers with a long wavelengths

78:15 many bed thicknesses, it's gonna travel a different velocity uh depending on the

78:24 it's gonna travel with a different velocity and horizontally in between. So we

78:31 that uh in such a situation, uh long wavelength cy waves going through

78:39 thin sedimentary beds, that wave is as though it were a uniform anisotropic

78:49 . So you know that's a pretty um description of most sedimentary rocks uh

78:54 thin layers compared to the seismic So that's why we say that in

79:00 , in almost all seismic situations, the waves are propagating and isotropic.

79:08 that means that everything we learned uh and today. And for the next

79:12 lectures about isotropic rocks. That's only beginning and real locks are mostly an

79:23 . Yeah. Um I know a about that. My uh uh uh

79:29 an expert in such matters. And uh I will share some of my

79:35 with you on the in the 10th . But for now I'm just gonna

79:39 you this glimpse, to show you it's gonna be complicated and to warn

79:45 that riddle walks are more complicated than um uh are learning about in these

79:53 lectures of S Boys and Race. as a summary of that, uh

79:59 we learned at the very beginning is is the study of the deformation of

80:04 materials under stress. And that simple means materials like and glass are like

80:12 doesn't mean rocks. So everything we out here is going to be uh

80:18 directly applicable to rocks, but I that you have been applying it uh

80:24 rocks uh in your entire career. that should uh bother you. Uh

80:30 if Hook was doing stuff only for materials, how can we get away

80:36 applying it to complicated materials like rocks if they're isotropic, consider a sandstone

80:44 . Uh it's got grains and pores uh uh so that's not a simple

80:51 like uh Hook was thinking of. , the grains are little crystals,

80:58 don't have polished faces, but they have internal uh arrangement of the

81:03 which means that every single grain on smallest scale is an isotropic and a

81:09 , those are all gonna be oriented . So the sandstone itself is gonna

81:14 out to be isotropic. But there's situation that uh hope never dreamed

81:21 So we're gonna deal with those issues in the course. So what is

81:29 and what is strength stress is the bringing an area applied to material since

81:36 forces of the vector and since the is specified by a vector, which

81:42 normal to itself, that specifies the of the area. Because of

81:47 the stress is a symmetric three by tension. As we learn that strain

81:55 a measure of deformation, a non measure of deformation. It's defined independently

82:02 the coordinate system. So we talked links, we didn't talk about uh

82:06 uh deformation in the one direction and so on. We combine those

82:11 different into uh showing how the length . It's also a symmetric three by

82:17 tensor. And Hook's law is crucial us. Uh back in the uh

82:24 the 17th century. Imagine that hundreds years ago, h had some

82:30 did some thinking, sitting in his , I think in London and his

82:37 have come down to us hundreds of later, enable us enabling us to

82:42 oil and gas. That's kind of to think about that. So what

82:45 he say? He said he, he made the assumption that stress and

82:49 are proportional to each other in a way. And he did not specify

82:56 one is either cause or effect. that should bother you. Uh uh

83:01 had some disagreement in the, in small class, we had some disagreement

83:06 whether stress causes strain or strain causes . So we did not resolve those

83:13 . You should continue to think about on your own. And we'll resolve

83:18 question by the end of the So then we de define elastic compliance

83:27 the ratio of strain distress. It out to be a complicated tensor with

83:32 different indices. And we can't think that in our minds because we can't

83:38 it down in a simple way and visualize it. But lucky for

83:42 , we can write all that information in the sensor as a six by

83:47 matrix. Now, for isotropic these various components are have contained only

83:54 independent parameters, Young's models and sure and also some of the elements are

84:01 of these two. But that's not we need for wave fag. We

84:06 don't know this yet, but we're need for wave fabrication, the inverse

84:10 compliance, which is the stiffness also complicated tensor. But uh for isotopic

84:18 , we uh I need to only two independent uh parameters, the longitudinal

84:25 m and the sheer modulus m And then uh uh there are functions

84:32 these which uh uh appear in the in the uh the six by six

84:39 , but we don't need them And then we saw briefly that elastic

84:46 is more complicated. So that's the of the first lecture. So,

84:53 I wanna do is um uh I what I wanna do is stop sharing

84:59 and then I want to um stop presentation, art another presentation, but

85:28 gotta find it. So, um . So it's not here. And

86:00 well, I'm gonna have to browse thoroughly. Uh don't uh despair

86:07 I'm gonna find me. Oh, . OK. So, um now

86:35 want to um share this. Um Can people see that now?

86:59 three, the wave equation? You OK. Um Carlos, do

87:08 see it? Yes professor. So I'm gonna go into uh uh

87:15 mode here. OK. So now know about stress and strain and how

87:26 are related. So now we're gonna that into, put them together and

87:33 uh um how they make waves. , I have to uh reshare

87:48 The screen two. OK. Are we good? So uh

88:04 so um it's not responding here. we go. OK. So here

88:16 uh uh um list of course OK. So, um the previous

88:24 are, are gonna lead to what call the scalar wave equation for wave

88:29 in fluids like the ocean. And then making more realistic assumptions.

88:38 going to then find out how uh uh um uh a generalization of the

88:45 WAV ation. We're gonna find this WAV ation with second. And then

88:49 get my, we're gonna generalize this wave equation to the vector wave equation

88:57 is gonna apply to solids. And gonna be uh uh isotropic solids and

89:05 isotropic solids. So, you that, that's still not what we

89:09 is it for applying for our field ? Uh But it's a big step

89:17 from here. We're gonna only deal uh uh uh waves and fluids.

89:23 here, at least we got solids , you know, we're, we're

89:28 more realistic uh as we go. so then uh these equations are,

89:38 not gonna have any source in They're gonna explain how waves it um

89:47 know, propagate without even thinking about source. But we do have to

89:52 about the source. So that's what gonna show next. And then uh

89:58 gonna uh uh think about in homogeneity good because uh uh rocks are in

90:05 , rock formations are in homogeneous. those equations are gonna have lots of

90:11 solutions. And when we do AAA survey, we're gonna find rays which

90:19 to our um uh receivers uh uh lots of different pathways, lots of

90:28 pathways. Uh uh So uh it can get really complicated. And

90:38 our geophysicists have learned over the last years how to deal with all.

90:44 so we are going to see some the, uh, some of the

90:48 of dealing with it. Now as , um, as a further

91:01 we're gonna talk about the concept of reciprocity. And so I suppose you

91:09 have heard of that and you might you understand it, but I

91:15 uh I'm gonna show you something which gonna shock you, which will think

91:20 make you uh think. Uh maybe don't understand it after all. But

91:25 that's later, uh that will become in the afternoon. So let's begin

91:31 , let's begin with the uh the year we're gonna find the scalar wave

91:36 now. So I Newton was the first one who wrote down this expression

91:43 uh uh how particles react to So, if you put a force

91:48 a particle, it's gonna accelerate uh on the mass. Now, that

91:58 was um uh uh a revolution in name because uh before Newton wrote that

92:11 thought that when you have a force makes for a velocity. So you

92:16 you uh you're pushing a, pushing , a um a block of wood

92:21 the table, you're applying a force you make a um uh velocity knock

92:28 acceleration. But uh Newton realized that situation of pushing, pushing a block

92:38 the table means that the uh the forces on the block from the table

92:45 canceling out the, the force that pushing on the side. So there

92:50 uh constant velocity comes from zero zero, total force including the resistance

93:03 on the uh uh from the uh on the bottom of the block.

93:11 you don't have that resistance from, you get an acceleration instead of uh

93:17 velocity. So that was really quite revelation back in those days. But

93:22 not what we need for our We need to uh recast this for

93:26 bodies. And I want you to about uh uh in the first

93:32 we're gonna think about uh water. . So first we're gonna do

93:39 So imagine inside of a fluid, have AAA volume element called a

93:46 You might have uh you might not familiar with this term, but here's

93:51 picture of a Voxel and it's a cube. And of course,

93:57 since we're inside a fluid, it doesn't matter how the cube is

94:04 Uh This word Voxel is uh a of a word, you might be

94:10 familiar with pixel. So when you a two D image uh into its

94:16 pieces, those smallest pieces are called . And so the 3D generalization is

94:22 a Voxel, OK. Now, Voxel has uh uh uh it's a

94:28 uh uh with the edge length that gonna call D and its mass is

94:34 by uh uh uh D cube, is the volume times the density.

94:43 , operating on this Vauxhall, there both body forces and surface forces.

94:48 , uh the, the body forces , are like gravity. And so

94:55 uh uh I was sitting here uh pulled down by gravity but uh canceled

95:01 uh uh uh pressure from below surface from below. So we're gonna ignore

95:07 body force and we're gonna consider only uh transmitted across the surface.

95:17 yeah, where is this Voxel? , let's imagine an or a co

95:22 quarter system here. We're gonna agree a, a right handed uh Cartesian

95:28 system that has an origin over And um uh where is this Voxel

95:37 with respect to the origin of Well, according to some of vector

95:42 this vector stretches from the origin to center of the vox and the pressure

95:50 the center of the Voxel here, gonna call E as a function of

95:54 vector and we're gonna presume that the uh is uh may be variable in

96:00 places. Yeah, it's gonna be at other places. For example,

96:06 uh uh Here is the place uh is um as a um it differs

96:13 the or from the center of the by this vector here, which is

96:20 by uh three components, 00 and over two. You see the,

96:25 full length of the edge here is . So this difference must be D

96:30 two and it's in the three direction . Uh this uh uh uh delta

96:38 is a three dimensions, the three in the three direction only. And

96:45 doesn't have any com any component in one direction or in the two

96:50 And so what is the pressure up ? The pressure is denoted by the

96:57 at X plus this different fact. similarly, the pressure at the bottom

97:06 given by this, where we have minus D over two here. And

97:10 pressure on the side is given by uh I distance vector. In the

97:20 direction. See this is in the direction, this is the one direction

97:24 in the backside, it's a minus and then on the front uh uh

97:29 the back uh uh we have uh different sections are in the two

97:36 This is minus D and this one is D. So this one

97:41 that's in the back and this one in the front. And remember that's

97:46 the center of the Vauxhall center. Vauxhall is uh not showing in this

97:53 . And what are the corresponding Well, they are uh uh uh

98:00 time is the square of the um of the length. So this square

98:07 the surface area and that this is pressure, this is the force pit

98:12 . And here is the amount of units of the unit area. And

98:17 see that we, we put in ad squared in front of all of

98:21 things. So these are the So uh uh this is the force

98:27 the one direction at the uh at , at the left side. And

98:32 here is the force in the one on the right side. Seeing how

98:37 notation works here is the force direction at the top. And here's the

98:44 at the bottom of this one over . So that this has a minus

98:49 here, this has a plus. adding up all these forces, we

98:56 this expression. This is not a . This is just a comp this

98:59 a sum with uh three components and is just adding up all those contributions

99:06 we did before and look here, have some minus signs here. Let's

99:10 sure where that comes from. See , we have a minus sign here

99:15 it's pushing to the left, no sign here. It's pushing to the

99:20 in the same way, we've got minus sign here, but we do

99:23 a minus sign here. So uh when I put together this slide,

99:29 was careful to get all of these signs in the right place. So

99:33 have some minus signs here and then have places like this with a minus

99:38 here. So all of that is correctly. Uh step by step,

99:42 can check me out by going over uh uh afterwards. So here's all

99:47 list left and right is here, and back is here, top and

99:50 is here. Now, that's uh um it's for any pressure distribution.

99:59 now let's figure out the sound way traveling vertically. So traveling vertically uh

100:06 uh a vertical, either top to or bottom to top, we haven't

100:11 yet. But in those cases, pressure did not, does not vary

100:16 the horizontal direction. So that this cancels this one. You see there's

100:21 minus sign here cancels this one. we have only these two terms and

100:30 the pressure at the top pressure at bottom that comes from a sound wave

100:35 vertical. Now, since this V is gonna be small, we're assuming

100:39 gonna be small compared to the seismic we're going uh to approximate this

100:47 which is what we had right here terms of use the tailor expansion,

100:53 we talked about before. Remember the expansion says that when you have functions

100:59 P uh and you know the value a certain position, you can find

101:06 value at a nearby position by taking amount of the displacement, that's the

101:11 two that's coming from here. And by the uh derivative of the function

101:21 with respect to X three because this uh in the three direction. And

101:26 gonna evaluate this derivative here back here New York or using that Taylor expansion

101:36 uh the previous expression we have the forces given by this. Uh

101:42 um right. Mm in the seg of this coalition, this is an

101:51 slide and you can click here and yourself back to the previous slide.

101:56 can't do that here. And we're put in here the Taylor expansion that

102:02 just arrived and then notice that this cancels this one. And so um

102:08 we're left with this and these two are the same. So we're gonna

102:12 able to simplify further. And we that the force in the three direction

102:17 given by minus the cube of the edge times this derivative, where does

102:25 tube comes from? I see we these D squares here and then we're

102:29 uh another factor D here and, uh we have twos uh uh

102:36 So we get uh oh de cubed the derivative with a minus sign.

102:44 you can see where that minus sign from. You can cancel all that

102:48 follow and see where that minus sign from. And um so putting all

103:01 in the way into the equation of , we have that force and the

103:07 that we just arrived equals mass times acceleration, imagining this uh Voxel as

103:13 particle and it's gonna be accelerating in three directions. So we rearrange

103:21 So we uh uh find that the is equal to minus one over the

103:26 times this derivative. Now, because this notation, we uh uh because

103:36 the um time we spent yesterday to about uh vector notation, we can

103:44 generalize this for motion in any direction by changing these threes into eyes.

103:52 we have uh uh uh I uh and we can use any I

103:59 we, we were restricted to IOS . But now because we're so

104:03 we spent the time yesterday to, develop this notation. Uh We can

104:11 it is for a P wave traveling any of the three directions. And

104:18 have a name for this here. called the gradient operator. OK.

104:25 this is a good time to um to follow this link and we're gonna

104:37 this link back to uh uh the 101. So I'm gonna stop sharing

104:45 . Yes, ma'am. And I'm share my screen, let's see

105:00 Um Not sharing the screen yet because want to bring up, I know

105:15 math zero and um oh you Sure. Um OK. Yeah.

105:38 when we, when we did math yesterday, we didn't do the whole

105:51 um because we didn't quite need it . I thought I had myself set

106:11 there but I did not mm You stop present the old one. Shut

106:24 . So that, and what was one? Those three things?

106:41 That sounds good. Yeah. yes. I, I was gonna

106:53 you if we can take a five break. Yeah. So, um

107:00 uh that's a good idea. Let's a five minute break right now.

107:03 by the time you come back, will figure this out. She was

107:11 . So resuming now, uh really you see this on your screen?

107:22 . So, uh what we have now, uh during the lecture,

107:27 encountered a new notation which we uh have not seen before. I actually

107:34 you have seen it before, but maybe some time ago. And so

107:40 we need to do is discuss what notation means. So let's go back

107:45 the um uh math 101. And uh uh I'm on slide 65.

107:54 Well, just a second. I a pointer. So uh on slide

108:01 of the math 101 file and there begins, we did all this other

108:06 earlier. And so now we want talk about vector calculus. So we

108:12 gonna define a symbol. Uh It's Dell. It's like an upside down

108:19 . And uh um it's, it's as a vector and w we didn't

108:26 a uh um uh an error on because uh nobody does the, the

108:32 notation for Dell is without the So we're gonna stick with the standard

108:38 , but it is a vector and , the different components of that vector

108:42 partial derivatives with respect to the different um directions, either two directions or

108:51 directions. And this operator operates on and tensors only from the left

108:57 So we're gonna go here next. uh um if it's applied to a

109:05 , I hear is Dell operating on scaler phi then um uh um that

109:15 uh uh a vector which has these three components, let's go

109:21 And you can see that if you this quantity uh uh uh like a

109:28 of a vector and you're operating on scr, then that vector looks like

109:36 partial derivatives in each of the three directions that can also be applied to

109:44 vector. And uh uh uh uh that happens, it makes a scaler

109:52 uh because uh uh according to the product which we defined yesterday, uh

110:00 we're summing over uh uh equals And so that's uh uh there are

110:06 unpaired uh indices on the right side that. So this is a scaler

110:12 dot vector. And also we can del cross a vector. In which

110:20 , it, it has this more expression as a as a vector can

110:28 applied. Uh uh uh It can be applied to a vector uh to

110:34 uh this gradient tensor. So you here's del with uh uh uh Here's

110:46 dot U, here's Del Cross here here is Dell times you in which

110:56 oh we have a um a This is a two by 210.

111:05 I probably should have made this a by 310. And it can be

111:14 to a tensor to make a, gradient operation on the tensor. And

111:18 leads up to a vector. You how that's a vector we're summing from

111:23 over eyes. And we have here loose J on the side. So

111:28 is has a single, un And furthermore, it can be applied

111:39 to a scaler to make another So if you work through our previous

111:45 uh definitions of del dot and del the scalar, you, you'll see

111:54 uh it's equal to the uh the equals 123 of the second derivative

112:00 of this uh of this, of scaler. And that's also a

112:07 Uh And so this is named after , that's a picture of him and

112:16 can be applied twice to a So you see how um um this

112:23 as we defined it um uh six slides ago can actually be uh uh

112:32 in a variety of different ways. uh just before we go back to

112:37 lecture, I will show you that are various calculus identities uh which uh

112:45 will not prove, but we uh uh universally true. So I'll just

112:51 show here that if you have del a gradient that's a Z that always

112:59 out to be a zero. And del dot uh uh uh this operation

113:05 called the curl operation del cross that's also a zero. And that's

113:14 scaler. And then here's a, vector quantity here. And so,

113:20 words, we say that this one the curl of a gradient. So

113:26 del cross is called a curl This is called a gradient operation.

113:32 what the equation says, the curl a gradient is zero. And this

113:37 uh here's another curl because we're at dell cross uh a vector. And

113:45 we're taking the dot product of And so that's called a di the

113:51 , a divergence of a curl and . And why would we introduce this

113:58 notation for uh uh for vector calculus dealt with all its different manifest

114:10 Well, here's a good reason, fair. Uh uh We don't always

114:20 , um we don't always use a coordinate. Sometimes we use spherical

114:26 And in that case to a an operator looks like this in terms

114:31 various derivatives with respect to R in to angles and respect to the other

114:38 . And uh so why would we to do this? Why would we

114:44 to use F coordinates? We were fine with um uh Cartesian coordinates.

114:51 was easy let me just back up . I'm gonna back up.

114:56 So uh the laplacian operator L squared the scalar has a simple expression like

115:04 . Like so um in um uh co system, just the sum equals

115:14 of the second derivative of that same three terms expressed in one expression.

115:20 like, so that's for Cartesian um um Lalas operator and trust me,

115:28 will uh this Laplace operator will show in our equations very soon. And

115:34 has a simple expression and artesian But in TRL coordinates look how complicated

115:45 , it still has two derivatives but complicated. So why would we want

115:51 do that? Previously, we talked using spherical coordinates for uh uh uh

115:59 academic geophysicists, friends who study earthquakes throughout the sphere of the earth.

116:06 there's an, an example, there's AAA better example more applicable to us

116:12 aspiration. Ge thinks right here when have a, a source say uh

116:20 uh a a dynamite source think of dynamite source in AAA conventional survey.

116:28 that uh is um a source where waves are spreading out in all directions

116:35 like spherical waves. Uh maybe uh uh is a little bit better.

116:46 think instead of dynamite on land, think of dynamite in the ocean.

116:52 he put a, put a charge dynamite in the ocean, making a

116:56 source and it sends out waves in directions, uh And some of them

117:01 down and some of them go up , and uh hit the, um

117:06 surface of the water and get reflected down. So that's a complication that

117:09 need to deal with later. But uh uh aside from that, you

117:14 see the waves are expanding from uh point source in all directions. And

117:22 it's gonna be convenient for us to that in terms of spherical coordinate

117:28 And we can do it trivially, we uh uh uh once we find

117:34 laplacian operator, you're gonna find that up in the wave equation. And

117:39 can trivially apply that either to partition or to circle coordinates if we use

117:46 symbol here. OK. Now, of course, uh you, you

117:52 that we don't typically use dynamite for in the ocean. We use

117:57 but it's the same stuff. Um , um in the borehole, we're

118:05 be uh using uh we're gonna imagine traveling in a borehole. And

118:09 in that case, the, the and operator has this form in terms

118:13 derivatives with respect to axial direction and to radial direction and with respect to

118:20 direction. And so, uh I pose a question and of course,

118:26 uh could be useful for um uh the bar. Now, I'm gonna

118:35 show you here on the uh in math 101 file there is a discussion

118:41 compliance and you might have seen this and you thought, oh, that's

118:45 . This is strictly elasticity whereas all other stuff has nothing to do.

118:51 , uh, much more broadly applicable is elasticity. And right here is

118:56 explanation of why we had those, , strange factors. 1/4 coming up

119:03 the earlier discussion, this lecture and gonna leave that here for you to

119:10 on your own. Not gonna do . And uh in front of you

119:16 , I just point out that here's it is. OK. So what

119:20 gonna do now is stop sharing And I'm gonna uh start sharing singer

119:50 I want insulin. Wait. So folks can you see this?

120:06 , that's presenting our progress. Mhm mhm This one. No uh

120:26 one and OK. And I can . Yeah. OK. I've got

120:39 learn how to do this better. uh this is where uh uh

120:44 we switched back to uh uh the file. We uh switch when we

120:57 this. And can you see Uh This uh Dell symbol is now

121:04 uh the same as we showed before we showed it, Dell operating

121:08 a scale of five here, Dell operating on a scale of P for

121:13 . And so this expression here, I need to order this expression here

121:23 uh expressed in different ways using the uh gradient operator. Dell and that's

121:33 the definition of, of DELL times scr it's a vector and the components

121:40 the vector are given by this. , all of this discussion is valid

121:47 for fluid media. Why is Because we assume that the pressure is

121:51 same on all sides of the how I have a um quiz

122:00 So um uh Carlos uh is this statement true or false? So the

122:14 of motion from Isaac Newton, is the starting point for this derivation of

122:21 scale of ever? Let me, me just back up one slide.

122:25 here, here we have um uh no. Um the scalar wave equation

122:33 not in its final form, but almost final form. And so now

122:38 question is, did we uh get from Isaac Newton? Is that true

122:44 false? I would say it's Yeah, that's true. Of

122:48 Yeah, that question. No sound . Next question is this true or

122:55 ? Sound waves are driven by So they would be slower on the

122:59 where gravity is less? Is that or false? Uh uh Lily,

123:05 one's false. Of course. Next is uh on the equation the

123:14 the pressure at the left center of voxel is given by which for these

123:19 um here's our coordinate system and imagine voxel in here. And uh you

123:26 answer this question by going back to the original pictures, uh, that

123:33 , uh, showed or maybe you just examine it. Um,

123:41 um, verser, let me call you for the answer. Is

123:45 is it a ABC or D talking the left center of the fox?

123:54 the center and a, and here , uh, the, uh,

124:00 , place. Uh, it's, a so, no, so left

124:07 no, it's C, yeah, because of the minus here.

124:11 uh uh A is the right center the Vauxhall and the uh B is

124:17 top of the Vauxhall and D is , the back side of the box

124:23 the rock. So, because this in the true position, very

124:28 Now, let me see here. question is um now, I'm gonna

124:43 on Carlos for this and there's this , true or false. The equation

124:47 for a wave traveling right to left that given by this expression here.

124:54 , uh this is a, a challenge for Carlos because we didn't,

124:59 didn't present this. He can't find answer to this by looking back in

125:05 past uh derivation because in the past , we only considered waves traveling

125:12 Now, this is a question about traveling right to left. So um

125:18 uh uh to get this right, is gonna have to understand what we

125:23 about before and apply that to the uh question of traveling right to

125:30 So, uh uh Carlos what do think? I, I would say

125:34 true. Yeah. Yeah. The difference between this expression and the previous

125:40 is we got ones here instead of . So you see how easy it

125:44 we uh uh we tr we jumped vertical uh propagation to um uh hm

125:58 horizontal propagation just by changing index. , no sweat throat. No.

126:09 here's our expression for propagation in any . Now, it's a vector

126:16 See it's got vectors on the left , and vectors on the right.

126:19 uh uh uh let's try this, us operate from uh uh let's take

126:26 derivative of each of these things by from the left on the, on

126:33 left side and on the right side the operator partially with respect to X

126:38 I. So on the right, expression is spread in terms of

126:46 of the Dell operator. Um in this way, go back and

126:51 what we uh said about Dell dot vector is exactly what this is.

126:59 so the same thing here del dot and that's in vector notation. So

127:05 left side of this equation is uh we just did is a derivative with

127:11 spec to X I of this derivative respect to time of the displacement,

127:17 can interchange the order of these. so now we have the derivative of

127:22 , of the uh with the spec X I is operating directly on here

127:28 , on the displacement, the I of displacement. And all of

127:33 we're taking the second rib. this quantity is the strain. If

127:46 look back at how we define uh uh uh the uh uh this

127:52 uh the, the sum of the the sum of the derivatives uh uh

128:01 uh a 12 and three that expressed terms of strain like this. And

128:05 had a separate um um word for , this some of the diagonal string

128:12 is called the dilatation. And we found the in compressibility was related

128:22 the dilation in this way. And the left side of the wave equation

128:28 looks like this and that's really uh because we don't have the displacement

128:37 We have the, the pressure and don't have any uh uh uh uh

128:44 any special derivatives at all. On left hand side, all we have

128:48 is uh time derivatives. And we the material properties right here. Now

128:56 that back together where the right hand let's go back here. Here is

129:04 , the right hand side here that question index notation, it looked

129:08 this. And so by, by that, uh uh by doing this

129:14 , we convert it, we we transform this displacement here into a

129:21 term. And now we have only . Now we have the uh the

129:26 the unknown here is clearly pressure. , one more thing, let's assume

129:35 the uh the medium is uniform. in the ocean, maybe that's not

129:40 bad uh uh uh uh approximation. that, that means that uh this

129:46 is the same um uh at the of the Vauxhall, the bottom of

129:52 voxel left and right. And this does not change on the small

129:59 And let's assume the same thing about material property K. So we can

130:03 these material properties outside of the Like we show here. And you

130:10 how this is getting to be pretty uh uh uh uh much, much

130:15 simple. We find after this we find that the, the uh

130:21 derivative of the pressure, this time proportional to the second derivative of uh

130:28 the pressure with respect to these three of strain, three components of

130:34 three components of position. That's what trying to say. And what is

130:40 uh what is the proportionality constant is which is K called K over

130:49 So the unknown here is the pressure width, position and temperature. We

130:56 two derivatives. We spend the time derivative of two derivatives with spec to

131:02 . And here is the proportionality And so this is the actual equation

131:08 governs the waves which arrive at a in marine acquisition. So imagine marine

131:18 , you have a, a boat through the ocean. It's got it's

131:22 a source behind it. An air source, that air gun source is

131:26 out waves and the uh uh propagating through the ocean also up through the

131:33 of the surface reflecting back down, it goes into the rocks where reflects

131:38 the rocks comes back through the water the receivers which are towed behind the

131:46 source boat in a long string of may be as long as 10 kilometers

131:54 . And since the way, since uh uh receivers, we call them

131:59 , they're sitting in the water being through the water. And um,

132:09 the waves are arriving through the water in the propagation history, they've traveled

132:15 , through the rocks, but as hit the waves as they hit the

132:20 , they're traveling through the water. this is all we need to

132:25 This equation governs waves arriving at a in ma in standard marine acquisition.

132:33 , uh there's a complication here which the, uh the boat is

132:38 It's not just sitting there, it's uh uh through the ocean, uh

132:44 out uh uh uh energizing the source few seconds, maybe every 30

132:51 Uh And so the receivers are also . And so we've ignored the fact

133:00 the receivers are moving here. But the, the velocity of the,

133:05 the receivers through the water is slow to the um uh velocity of the

133:13 coming through the water. So we ignore that fact. Now, so

133:22 the wave equation or in more compact notation, we can write it here

133:28 terms of the Laplace operator. Remember , I warned you that the Laplace

133:33 was gonna show up and right it's all the complications hidden inside the

133:45 Pasian operator are hidden inside the notation we call it DEL square and everybody

133:52 that's the Lalan operator and in our coordinates, it's this sum here.

134:00 as you saw previously for in other systems, it can be a lot

134:05 complicated. So uh we are clever to hide all those complications uh inside

134:13 little plus an operator. And by way, the waves don't know anything

134:19 our Cartesian Corridor system. That's all our imagination. So the waves don't

134:26 or care about these indices here. the laplacian operator is designed in such

134:32 way that it doesn't care either. gonna be the same. Uh whether

134:38 have uh a Cartesian coordinate system with vertical axis, vertical or horizontal or

134:45 in between all those complications arising from choice of coordinate system. That's all

134:54 inside this laplacian operator. So that's very clever thing we did to uh

135:01 make sure that the wave equation is dependent on our choice of court

135:11 analyzing the waves. Uh It's gonna uh easier um in one coordinate system

135:18 the other. But remember the waves know or care about what we think

135:23 the origin of this cord system. on the left side, we have

135:27 second r respective time. So you know, corresponding to the

135:34 But here we don't have acceleration of , we have the change of pressure

135:39 divided by time. I look at uh um this proportionality constant here,

135:51 got to have the di the dimensions length over times squared. Why?

135:57 because we have uh uh this has have the same physical dimensions on both

136:02 . So we have the pressure on sides. But here we have times

136:06 and here we have distance squared. this thing must have the dimensions physical

136:11 of length over times square. So the same as velocity square. And

136:16 we can just give that in your V squared. And here's the definition

136:22 V square, it's trail a And who knows what that velocity

136:27 All we know is that it's a with the dimensions of uh velocity

136:36 What does that mean? Does that the velocity of the wave, the

136:39 of the particles within the wave, velocity of the boat as it goes

136:44 the water? What does that We do not know at this

136:49 All we know is that we've derived expression and we've given the proportionality constant

136:55 you name the square. So we up with this scalar wave equation.

137:05 , we can do this in another . Uh where the unknown is pressure

137:09 . That kind of makes uh the for us, doesn't it? But

137:14 could have done it another way so we had derived a wave equation with

137:19 same form except the unknown is, the dilatation. And uh it has

137:27 , it has exactly the same exactly the same coefficient here. But

137:32 the unknown is dilatation instead of That's interesting, isn't it? You

137:37 do it one way or the other out the same. But we're gonna

137:41 to analyze the pressure version. Now, let's consider waves only traveling

137:53 . So if it's only traveling this uh vector operator simplifies and we

138:00 uh uh show it this way everything in this equation is the same.

138:05 this is your only way of traveling . Now, the, the solution

138:13 gonna be a function which varies only uh uh uh uh with Z and

138:18 T for this situation where we're failing , we're not gonna have any variation

138:26 the spect action. Why? And this equation says is that whatever the

138:32 is, it's gonna depend only on difference between a time term and a

138:39 term. And uh uh the uh can have either a plus or a

138:46 here and the time is gonna be by um parameter which we call omega

138:52 the depth uh is gonna be uh by a term that we call K

138:58 . OK. So what, what this mean? That means I'll say

139:03 again. The solution is gonna only variations in this quantity phase. No

139:13 combination of T and Z are gonna only this combination. And that's

139:23 Yeah. Uh So that's true for way of traveling vertically. So we

139:30 names for this, this omega is the angular frequency. And uh the

139:36 um we don't say angular and we frequency, that means frequency, the

139:42 per second and the angular frequency is the cycles per second by a two

139:49 which means as uh you can think it as radiance per second. Oh

139:57 , this uh arrows uh should be over here to K three, not

140:03 time. Uh This is point in center. So this box here,

140:08 call out box is referring to K . That's the vertical wave number.

140:13 if you want to know more about wave number, you can go to

140:16 glossary, look up wave number and do that on your own time.

140:23 , let's just verify that this um function actually solves the wave equation.

140:30 uh um uh it's uh I'm gonna here this solution uh with respect to

140:40 and space, we can only have in terms of five. That's what

140:45 said back there. So let's just that. So uh on the left

140:49 of that equation, we've got um uh uh two derivatives respect to

140:56 Let's separate them out, dr them at a time. And so,

141:00 this uh question, um um really pressure with the strength of time can

141:07 expressed in this way, uh using rule calculus. So all we do

141:14 put in a, a partial inspector here, partial inspector Phi here.

141:21 um, now let's do the same of thing. Um um Let's uh

141:28 me. What is the partial R the Phi where Phi with respect to

141:34 , I'll back up. And here's definition of Phi, so that a

141:39 of Phi with respect to time is omega. So that's why we put

141:44 the omega here for this. And we do that same sort of thing

141:50 , uh uh uh uh we're gonna a, another d of the spec

141:55 time, we do it again and end up with the left side of

141:58 wave equation is omega square times the repeat with respect to thought. Uh

142:09 converted a con uh we converted a of the spec of time into a

142:14 with respect to five. Likewise, right side of the equation uh uh

142:20 to a similar thing and we get uh on the right side, it's

142:25 same V squared as we start out plus we have an additional K three

142:30 because uh and it's gonna be good either a positive K three or a

142:35 K three. And the same derived respect to uh uh phi of

142:41 And so these two sides, the side and the left side are gonna

142:48 equal for any function pressure as any of five if and only if we

142:56 this expression here relating K three to over V. So we gotta have

143:03 squared equals V squared times K three . And that condition relating the angular

143:12 with the vertical wave number through the . And uh uh uh and you

143:21 a plus or a minus here. what is this solution? No,

143:28 think that you are familiar with this wave solution here. So here uh

143:36 uh uh this, what I've written here is called a the plane wave

143:43 for the pressure. And it's expressed a constant time is an exponential with

143:50 in the exponent we have I times recognize this right here is the five

143:58 this um uh uh this wave, uh plane wave solution is good for

144:06 value of fry here. Now, just pause here and I talk about

144:14 number E. That's, this is first time we've encountered E in this

144:20 . Are you familiar Lily with Oiler ? E? Have you seen it

144:26 ? It has very special properties invented this guy Oiler? Yeah.

144:36 So uh uh I think your thanks for uh being uh direct and

144:42 here. You've seen it before, you're not quite sure. So Oiler

144:47 is a special number like pi, think all of, you know,

144:51 pi is 3.14159 blah, blah, . It's a very special number.

144:58 Oiler number is another very special It has a value of 2.715 blah

145:05 blah. And those uh it's a i it's an irrational number and those

145:11 go on forever. And it it's a special number which was discovered

145:16 Oiler uh which was a, a genius. He made the smartest guy

145:22 ever lived in the 19th century in . And he discovered this uh uh

145:29 number which happens to be close to , you know, uh that's kind

145:34 um uh uh interesting. Why should be close to P it, it

145:38 very special properties and um um you , why, why this is very

145:46 number? Why is it not uh know, uh 37 0.251 you

145:55 it happens to be close to pi I would say nobody knows why it

145:59 to be close to pi but it's very special number. And so it

146:04 very special properties which you can look in the glossary. So I encourage

146:09 to look up that uh uh not , but uh uh uh look it

146:14 later and has lots of magical properties we will be uh showing some of

146:20 properties uh later in this course. uh it's such a special number that

146:30 we remember uh Oiler most, mostly remember Oiler by because he discovered this

146:37 and we call it e in his . Now, e is raised to

146:46 power I to the five. And you should know about I, which

146:51 the square root of minus one. let me ask, uh um uh

146:56 . Carlos, are you familiar with concept of square root of minus

147:04 Um I think I have seen it but uh no, OK.

147:10 yeah, so uh uh um uh . Have you, are you familiar

147:14 Enn? I? Yes, I . OK. Well, that's

147:19 Uh uh because uh you are working slumber, you will have uh encountered

147:25 many times. Uh Carlos is a . So he uh is not so

147:30 with this, but I is called imaginary number. And so if you

147:36 a square of I, that gives to be minus one. And so

147:41 can it possibly be true that the of any number is negative?

147:47 there are, uh there are a bunch of people who have wondered about

147:54 for centuries. So let's sort of through the um uh the elements of

148:04 theory. So let's start off with idea, one plus one equals

148:11 So uh uh let me say one two equals three. So what are

148:16 numbers? 12 and three? Those abstract ideas, abstract ideas.

148:25 we apply them in physics by attaching to that. So we can say

148:34 apple plus two apples makes three OK. So that suddenly removed from

148:42 mathematics into physics says if we have apple and you add together two apples

148:50 three apples. But what if one is uh uh uh uh what if

148:55 one, the, the, the apple is a big apple, a

148:59 delicious apple, a type of Uh and uh uh the, the

149:04 other apples, suppose those are um honey crisp apples. So what's the

149:11 of one delicious apple and two honey apples? Is that still three apples

149:18 not? What if the honey crisp are small and the diligence apple is

149:23 ? Do we still get three apples the right side of that equation?

149:27 see that gets you into all sorts complicated discussions and mathematics. You don't

149:33 any of that, you say one two equals three because we didn't associate

149:39 was counting. OK. Now what that, so, but it was

149:48 uh Greeks back in Pha Pythagoras thought about these issues and then they

149:55 about what happens if you have. there any numbers between one and

150:02 Well, sure, there's 1.5, one and two thirds. And so

150:07 , they invented fractions and those are rational numbers. And there's an infinite

150:15 of rational numbers like that between one two, an infinite number.

150:22 And how about, if you think , uh uh uh are there any

150:28 between these rational numbers? Suppose you a rational number? One divided by

150:36 and right next to it, you one divided by 213. Are there

150:40 numbers in between there? Well, , there's an infinite, it turns

150:47 there's an infinite number of rational numbers those two numbers. And also there

150:53 uh a, an infinite number of which we call irrational numbers between the

151:01 numbers. And those are numbers like and E where you uh when you

151:06 them out in a, in a form, the digits go on forever

151:11 repeating. Uh You can uh define value of P uh to a million

151:25 significant figures. And you can uh can look that up. Uh um

151:31 If you go to Google scholar, can find a paper where they calculate

151:36 , the uh the digital representation of up to a million significant figures and

151:44 doesn't repeat anywhere. So those are generalization of the counting number. The

151:52 we can call those in integers or natural numbers. And uh uh it's

151:59 of mind boggling to think of how of them are there between one and

152:04 ? And, you know, uh more between two and three, there's

152:07 infinite number of them wherever you So, are we done, are

152:14 , uh this is all part of theory? And are there any other

152:19 of numbers? You take all of , if you, um, take

152:24 of the numbers that we just described take a square of that, you're

152:27 get a positive number. Are there numbers at all? Can you even

152:34 any numbers at all? Or you the square of it? And you

152:38 minus numbers? Wow, you can't that with apples and oranges, but

152:45 can do it mathematically. And all say is let's just assume that there

152:51 such a number and we'll call it , where you uh uh it's,

152:57 it's defined as the square root of one. Oh, we didn't talk

153:02 minus numbers. Everything I said about numbers. You can think about ne

153:07 minus numbers. And I still remember my mind's eye, my, when

153:14 was playing with my nephew who's now uh uh a young man uh uh

153:21 in a high tech uh um uh in California when he was a boy

153:27 four, I was playing with him the rug and I was playing with

153:33 uh uh uh teaching him mathematics, know, I was teaching him how

153:37 count. Uh uh uh what happens you have uh one apple uh plus

153:42 apples and so on. And that fun. Uh We plan for and

153:46 I said, what happens when you uh uh three apples, take away

153:52 apples and he figured out you get apple, pretty smart kid. And

153:58 I said, what happens when you three apples, take away four apples

154:03 the spot. This kid four years invented negative numbers. I was amazed

154:10 a smart kid he has and he out to be smart as well.

154:13 So that's negative numbers. Everything we about positive number, it has a

154:19 thing about negative number. And you see that it gets amazingly complicated.

154:25 it gets even more amazing when you that there's uh you can generalize these

154:30 in another way by imagining that there a number called I and when you

154:38 that number, you get a negative . Wow, a whole new class

154:43 numbers, imaginary numbers and any imaginary uh can be represented as a,

154:51 AAA real number times this odd. you can have imaginary numbers of any

154:58 . Wow, a whole new class um how the numbers. And so

155:06 next thing you say, oh What happens if you uh if,

155:10 you consider something where the, the the square root is um uh the

155:19 is consider number, call it. OK, we say that call it

155:31 so imagine a number. And the root of that number is I

155:36 the square root is not a minus . But this, if you square

155:40 number you get I, is that new class of numbers? Well,

155:43 turns out no, it turns out I is as far as you can

155:48 in this. Um And in this process, you can only go to

155:55 numbers. And there's nothing you can't anything more complicated than imaginary numbers until

156:04 get to vectors and matrices and things that. But I thinking about

156:11 there's nothing more complicated than imaginary to . That's quite uh amazing. Uh

156:18 would think that when you take the root of an imaginary number, it

156:23 be another faster number. No, , it's oh it's another imaginary

156:30 So that, that's worth thinking about it is that this process of um

156:35 a generalization ends at imaginary numbers. . So uh more discussion about I

156:44 in the um um in the glossary that I is appearing here in our

156:51 equation. And so I'm going to uh uh uh so because I is

157:01 in the exponent up here, that that the pressure is not real.

157:08 pressure is I imagine has an imaginary to it. And so at this

157:13 , you should be saying, how can I be measuring imaginary

157:21 And so the answer to this is gonna ignore that question when we're dealing

157:34 plane waves. And we're only going insist that we get real answers when

157:42 propose an experiment. And uh we're do all of this manipulation using imaginary

157:50 and they're gonna be imaginary uh uh in the uh exponent. But that

157:56 this, this whole thing ha it uh is uh we call it a

158:00 number. It has a real part , and an imaginary part. But

158:05 instruments are real instruments. We can't imaginary quantities with our real instruments.

158:12 I guarantee you whenever we work with equations and uh uh we have complex

158:22 all through with the derivation. When end up at the end with uh

158:28 uh an observable quantity, that one gonna be real. No, this

158:36 is not real because you can see uh the imaginary quantity right there.

158:42 whenever we're actually gonna measure something that's uh be real, I guarantee

158:48 So you should keep that doubt in mind. And uh uh it might

158:55 , you never thought about that, you should think about that. How

158:58 we have this imaginary quantity showing up our equations when we have real

159:05 And so, uh the resolution of is what I said, whenever we

159:10 these complex formulations to um uh account observable quantities, the, the observation

159:20 always gonna be real. Yeah, this plane wave solution, the wave

159:30 is related to the wavelength by this here, the wavelength uh uh uppercase

159:36 with a subscript C because we're traveling vertically. So that's equal to the

159:42 um uh divided by the frequency of it's uh that's um I think,

159:51 know that the, the uh well we're actually looking at a a

159:57 Uh but what you will be aware the velocity is given by the wavelength

160:02 the frequency. So the wavelength is like this. And so it's limited

160:07 the, the wave number by So uh uh uh here you see

160:13 K three as the dimensions of one Z, you can see that right

160:19 K three has the dimensions of one length just like Omega has the dimensions

160:23 one over top. So this plane solution uh uh uh has all these

160:31 features that I just uh just spent minutes discussing. It's got this strange

160:38 . E it's got this strange I it's got new parameters, Omega

160:43 K three, but it's gonna be useful to us like, so trust

160:48 , you, you will see this and over again, you need to

160:52 very familiar, very, um in , highly acquainted with, very friendly

161:01 this wave, this plane wave solution the wave equation. OK. Oh

161:17 . So let's make a guess about before we're we're gonna rewrite the phrase

161:22 , uh the phase in this So we factor out the omega and

161:25 left with uh uh uh uh uh over V is equal to plus or

161:33 K three. That's what we decided . Now, since this is a

161:38 for any omega assume that both uh uh uh uh the, the solution

161:46 P with one frequency. And here have the solution for another frequency.

161:52 uh I assume that uh both of separately are solutions. So now we're

161:59 guess the sun is also a So you can verify for yourself that

162:07 is true. If, if this a solution and this is a

162:12 then the sum is also a This happens only because the wave equation

162:21 linear. What do we mean by ? Let's back up here. Here's

162:30 way we, we say it's linear the unknown appears only to the first

162:36 . We've got squares everywhere through but the unknown appears to the first

162:40 . And so there's no term out and you know, involving P

162:44 Now, because of that we have solution here. If we have one

162:53 at one frequency and another solution at frequency, the sum is also a

162:58 , it is also a solution that like it's uh an amusing mathematical

163:04 But that's essential to us in geophysics all of our rays, all of

163:10 waves are composed of sums like this different comp uh different frequency components in

163:18 wave. Any measure that we ever has some uh uh high frequency components

163:25 some low frequency components. And the itself is a salt of all these

163:30 components. By the same logic, you have a weighted sum, uh

163:37 we have a, a one is big number and a two is a

163:40 number, a smaller number. And also a solution. And of

163:44 you have, if we have a with many terms in here, maybe

163:47 uh is gonna sum uh seem to a different coefficient for each one of

163:52 uh uh different frequency components and sum all up. Um I can have

163:58 a 17 of them, we can 1000 of them. Uh That's also

164:01 solution. And what are these They're gonna be determined by initial conditions

164:09 boundary conditions conditions at the source and uh on the boundary of the

164:16 for example, at the surface of earth. So those kinds of boundary

164:22 are gonna determine these. And so these coefficients are called uh uh the

164:33 of the solution. And I know know this term from before of excellent

164:40 when we uh when we give AAA showing how much uh energy is in

164:47 one frequency and another frequency and so that is this set of coefficients a

164:55 now, all of that was for . So um it's easy to generalize

165:02 uh for uh um uh propagation in 3d directions, all we have to

165:08 is change the Z squared to a, an X I square.

165:14 this arrow here uh cars you back the previous generation. If we're gonna

165:21 that, we have to generalize the of phase. So instead of having

165:25 K three times Z, we have K vector times X vector, they

165:33 the same omega T will have the plus or minus here. So then

165:40 you put this definition of a phase here, we find uh uh uh

165:47 any solution uh where the pressure is only on phase. We have um

165:55 uh uh this uh you know, grind through the uh through the um

166:04 rule calculus. And you'll find this not a bad grinding, it's easy

166:09 do. And so we find then Omega squared is equal to V squared

166:14 the sum of the squares of the of K. And so uh this

166:24 assumption that he depends only on phi for any phi if and only if

166:32 length of K is given by omega the. See, this doesn't say

166:37 about the components. It, the of the, of the, of

166:42 wave vector is uh related to the in this way. So,

166:57 it's still a question. It says scalar wave equation has and all these

167:06 . So let's um let's uh work way down through this. Le le

167:12 does it have two derivatives with respect time? It says yes. Um

167:19 per it do, does it have with respect to position? Yes,

167:26 says yes. Uh It ha does have the unknown function to the first

167:31 ? Only Carlos? Not sure, sure. I would say not.

167:43 . Uh uh uh So that is crucial thing. Uh You've got to

167:48 uh have that clearly in your So we're gonna go back here.

167:52 the wave equation right here. No, exactly. Here's the way

167:57 the unknown is peak that only appeared the first power. Everything we know

168:03 wave propagation would be destroy if there a term in P squared. So

168:10 very crucial that we understand that this has the unknown only to the first

168:16 . So that makes it a linear that's here. Now, back to

168:22 uh uh does it have a single which describes the medium, a single

168:32 ? She says no. Um let's go back here. So here

168:36 have a single parameter. We're back . See there's the parameter. It's

168:42 V squared. I know you were that inside the squared is KO

168:50 But though it only appears in that combination K over row, it never

168:54 , you know, uh K plus thirds mirror row. It uh

168:58 it doesn't appear in any other combination K over row. So we gave

169:04 a name. It's one parameter V . So the answer to this question

169:10 a single parameter, yes, which the medium. And, and we

169:14 describe that we can either call it or rope, but that's not really

169:19 parameters. That's only one parameter. we give it a name.

169:23 we don't yet know what V uh means. But uh uh it,

169:29 know it has the dimensions of So we call it A V and

169:35 it has all the attributes above. uh So the answer to this would

169:40 , wouldn't it? Uh uh And is false. Mm OK. So

169:47 let's go to Carlos uh the scalar equation. Is this true or

169:52 Can it be written either in terms pressure with the unknown function or with

169:57 location of the unknown function? Is true or false? I think

170:04 I didn't hear you. Yeah, think it's true. Yes, that's

170:09 . Uh uh We showed that very uh back about 10 slides. That's

170:14 . OK. So brice that uh or false, uh it's only valid

170:19 propagation in the vertical direction since fluids vary in the vertical direction. Is

170:25 true or false? True? now it's true that um that fluids

170:36 in the vertical direction. Let's think the ocean the ocean at the top

170:41 the ocean, the water is warm the Gulf of Mexico, uh the

170:46 is warm and it gets colder as go down. And furthermore, the

170:51 salinity, the salt content of the changes as you ruled out. So

170:56 is definitely true that in the the water, uh the uh the

171:03 properties of the water very in And by the way, they also

171:08 laterally also ve very laterally. Uh uh in a, a complicated

171:16 whereas it varies vertically in a simple , sort of like layers of

171:21 And we frequently ignore that variation but there. And uh uh uh some

171:29 uh uh can actually do our kind reflection seismology. Um um looking at

171:38 in the water color, think about , think about a reflection of a

171:44 wave from the vertical variation of properties the water. So normally we ignore

171:53 possibility, but I assure you that if you look for it in the

171:58 way, you'll find it. Uh uh in all kinds of experiments,

172:02 that's uh um not a common mode with fabrication. So it is true

172:11 it slid vary in the vertical But the question says, is it

172:19 only for propagation in the vertical Well, let's go back here

172:28 we have uh uh uh a propagation 3D in any direction in 3D and

172:36 we had to do to uh make ation was to um what let a

172:46 uh change from X squared in this to uh X I square. And

172:53 we had to change the direction the the definition of, of uh phase

173:00 to a vector X. And uh had to have a vector X here

173:06 a dot product with, with what call the wave vector. It's gonna

173:10 out folks that this wave vector give direction of the wave and the X

173:16 gives the position inside the um uh uh the ocean. So the answer

173:22 this one is false since it's valid any propagation, not just in the

173:33 direction. And if you go through uh our derivation, you will confirm

173:39 in that last form here. Uh generalize from ver from the vertical to

173:45 direction and it all worked because we clever notation. Next question is um

173:53 see, I think um it's up you uh back to you. Uh

173:56 No. Is it to you le uh uh if you have a 17

174:01 solutions uh uh make a sum of those is that also a solution?

174:11 , we showed that if you had different solutions, that's also a

174:15 right? So when you think about uh uh go uh think about

174:20 And uh um um uh you can that argument to make 17 or,

174:28 39 or whatever you want. And also a solution. If you have

174:33 solutions for uh uh a as many solutions as you want to make a

174:40 , that's also a solution. And really important for us because all these

174:46 solutions are gonna have different frequencies and different wave uh vectors and they all

174:54 together it's still a solution and that's to be a solution which hits our

175:02 . And we got to uh to figure it all out what it means

175:06 you know, basically what we're gonna is a complicated wiggle. And we

175:11 a different wiggle at a nearby hydrophone we got uh uh figure all this

175:16 uh uh as a function of position as a function of uh frequency.

175:25 all of that is uh and by way, these sorts of questions are

175:29 unlike the sort of questions that you find on the final exam. So

175:36 encourage you to go back uh through uh questions and verify that you're confident

175:45 your mind, what the answer And uh if you're not confident,

175:52 send me a question uh tomorrow morning o on this point. Now,

175:58 um here's one thing that we didn't about. Uh it, it's useful

176:05 you all to have a printout of lectures. And as I gave them

176:12 you in canvas, you can print out. Uh and then have them

176:17 in front of you while you're listening me talk. And, uh,

176:25 , uh, as you go through quizzes, you can mark right on

176:29 , uh as we're going through the , if you have a question,

176:32 write your question there uh on the in front of you and it,

176:37 the questions you can, you circle the correct answer. And if

176:42 go back to the, the um and you don't understand why that

176:48 given as the correct answer, then can bring it up with me uh

176:54 your email to me overnight. So why it's useful to have a print

176:59 in front of. OK. So we understand um uh now, now

177:09 understand uh propagation, the wave equations propagation in the ocean. We still

177:22 know anything about sources, right? , we just found out that the

177:26 waves propagate, who knows where they from and once they're there, they

177:32 from here to there following the equations uh that we just said that's called

177:37 scalar wave equation because uh the wave as a scalar wave and the unknown

177:45 the pressure, that's a scalar. . Now, let's think about uh

177:53 that's not gonna be good enough for because we have uh beneath the

178:00 we have rocks, the waves go into the rocks like and come

178:05 So we have to know how waves in solids So that's the next point

178:11 , we have a voxel inside of solid where we're gonna consider the stress

178:19 this voxel, not just the because in a solid we have,

178:24 not common to have equal pressure from sides is very common to have unequal

178:31 on the very side. So the is calculated like before the uh the

178:39 is gonna be different up here than is in the center. And

178:44 in particular, this is gonna be uh uh the Tau, we're gonna

178:49 this the Tau three J stress. what this three means that on the

178:55 face, uh uh uh the fact it's uh uh has the, the

179:02 to this top face is pointed in three direction. So the orientation of

179:07 top face is given by unit factor the three directions. So that's why

179:11 call that A three here. And any um uh uh well, uh

179:19 J component of stress on this top , it's given by um uh uh

179:27 know, it's gonna be a function the central position plus um this distance

179:35 from the sand to the top. here's the same thing at the center

179:39 the bottom with the minus side. the center to the right side where

179:44 distance vector is on the, in one position. Here's on the left

179:48 with A minus D over here, the uh front and the back,

179:53 minus D in front and A A D in the back. OK.

180:00 the corresponding forces are um multiply these by D squared in every case.

180:10 in some cases, we have a . And in other cases, we

180:13 a plus here uh depending on uh uh which face we're talking about and

180:21 all these together like we did Remember, this is not AAA um

180:28 matrix. This is uh uh just single, a single equation with six

180:35 in it. So the index one gives the in direction of the area

180:40 the area. So it's the same here as it is here down

180:46 It's a two instead of a And it's the same two over here

180:50 indicates front and back down here. a three instead of a one and

180:55 same three over here that indicates top bottom. The second index indicates the

181:01 of the fourth. So here we a force in the three direction uh

181:11 at all, let's see. Oh is that force in the three

181:17 Because it says, so here, says that we're only interested in the

181:21 three and the force in the three . So that's the direction of the

181:25 as the three everywhere because we're concentrating the vertical form. And so you

181:33 compare that with a scalar case by back through the, through the

181:39 So managing this is it, this uh this is the scalar case we

181:44 have it's the same force in all in that case. Uh But in

181:48 case, it's uh it's a force the three direction only. OK.

181:55 now we specialize to a sound wave , verify the force does not vary

182:00 the horizontal direction. So we're left these two terms only. So you

182:05 compare with the scalar case here. , I did that for you next

182:09 here is the scalar case is only pressure at the top of the

182:16 As before we're gonna use the tailor to simplify the previous expression. And

182:22 comes down to the force in the directions is given by the cube of

182:29 um uh edge times the X three of the cal 33 source cal 33

182:40 . And that derivative is evaluated at center. So that's for the vertical

182:48 only. So you can do the thing with uh force in the two

182:53 in the one direction and they end with this. But this is what

182:58 just derived here is the two direction the three direction you see the uh

183:03 all cases, since we're talking about propagation, we've got vertical variation down

183:09 and we have index 32 and one . It so we have the variation

183:18 in the X three direction because we've the vertical propagation. And notice here

183:24 the stress is only on the X planes. We can have uh force

183:30 the one in the three direction force the two direction or force in the

183:36 chimi. Uh we can have the are in all three directions.

183:47 it's easy for us to uh um generalize that uh for propagation in a

183:56 because of the clever way that we've up this notation. Uh All we

184:00 to do is replace the three with and then recognize and what we have

184:06 the derivative with respect to J of IJ where the J is uh repeated

184:12 . So that's why we have to uh over Js. And so in

184:23 vector notation, we can say that force divided by the volume is given

184:30 the gradient of the strength for Yeah, grading of the, of

184:35 stress tensor and go back to the 101 section. And you'll see what

184:40 , that would when we have the or tensor, it is just exactly

184:45 we're showing here sir. For being environment is the gradient of the

184:54 So put that in the equation of uh uh F equals ma uh in

184:59 words, a vector equals F vector by M. And we find that

185:04 A vector is given by the gradient the stress divided by the density.

185:12 we're gonna express this good way. uh we're gonna express the acceleration in

185:26 of displacement in this way. And that's the left side of this and

185:30 right side here and the same side the right side here. OK.

185:35 this is the vector equation of motion from uh this is gonna be valid

185:40 the subsurface rocks and with the stress . So what's the, what's the

185:48 OK. The unknown here. Is displacement or is it uh stress?

185:54 , um uh we're gonna have to one or the other and we're gonna

185:59 Hook's Law to eliminate the stress. here's the stress and here's Hook's law

186:04 the stress that this equation is what had on the previous slide. And

186:09 uh here's Hook's law for the Now, what we're gonna do is

186:15 yeah, that's right. Express the as we did before in terms of

186:21 of displacement. And here's the one that we saw before. Uh And

186:28 I dodged the question of why, are we, why did we put

186:33 one half in there or that, one half came in in, in

186:37 , in the definition of epsilon? there it is now because of the

186:45 that we talked about before, in distance element, we can uh we

186:50 convert this expression here which from the slide to this expression and look the

186:58 had disappeared. Why is that because this derivative is obviously different from this

187:05 . You see the MS and the are in opposite places here. But

187:09 of these symmetries, we can uh uh uh um uh combine those together

187:18 doing so the two disappears. let's assume that the, the medium

187:25 uniform. So we take this outside the derivative and we're left with

187:30 And you can see on the left side of the second root of the

187:34 time on the right hand side, ro respective position. But it's not

187:40 the wave equation because it has the form. It has too many different

187:46 here to what we want is the posse to appear. OK.

188:02 now we turn to uh another 19th German mathematician, his name is

188:13 And uh we have an actual photograph Helmholtz. And so Helmholtz's theorem says

188:21 uh for any um uh for any field such as the displacement, we

188:28 express that a sum as a sum a part which has a zero curl

188:34 a sum that has zero divergence. we've skipped uh lightly over this.

188:41 uh But earlier in the day, , I talked about uh uh a

188:47 calculus and uh uh we uh we that if you have a vector,

189:00 has zero curl, that means that can express that as the gradient of

189:07 scalar. Remember at the end of discussion about math, 101 vector um

189:16 uh vector calculus I showed three um identities involving Dalton involving Dell. And

189:34 of those said that if you have , a part here which has zero

189:41 , then that means that it can expressed as the divergence of as the

189:46 of a scr because the greed of scr uh operating with the curl operator

189:52 zero. So this one has curl curl free and it can be expressed

189:58 the gradient of a scaler. And next of those three vector identities was

190:06 if you have AAA if we have um a vector field like displacement,

190:16 if you ha if it has zero , then that can be expressed as

190:21 curl of a of another factor. so this is called, this one

190:29 called the scalar potential and this is the vector potential. So those are

190:35 we can uh separate the uh the into a part described by scalar potential

190:44 a part described by a vector OK. So the displacement I think

190:51 have a good um um a good for it. You can imagine in

190:57 mind, the displacement of um the inside of a solid. And you

191:03 imagine how that displacement is gonna vary position. I doubt if you can

191:08 have a good mental uh picture of scale of displacement, it's gonna be

191:13 scale of potential even. So even , I think you don't have a

191:19 um uh idea in your mind of vector potential. But Mr uh Helm

191:27 has told us that we can always any displacement field. And the sum

191:35 these two parts, one has zero and the other has zero divergence

191:42 always. And so, since this has zero curl, we can describe

191:47 uh in terms of the sca scalar and here we can describe this

191:51 we can describe as a vector Now, why is this important,

191:58 of these is gonna lead to key and the other one is gonna lead

192:01 sheer waves. So what we're going do is, oh so uh if

192:15 follow uh uh if, if you back to uh uh the, the

192:20 101 that we did earlier this morning look up the definition of curl and

192:25 definition of divergence you will see spelled in better detail what I just

192:34 And I'm gonna bring you back right to uh uh the glossary and uh

192:42 uh um where we defined Dell. so we said there in the

192:48 we said Dell can be applied to vector to make a scalar. So

192:53 a vector, it has three components I and we uh uh operate in

192:59 , in a dot product fashion with and we have the partial R

193:03 with X I and the sum of . So my equal 1 to 3

193:07 is, that makes a scalar. we also said that when you operate

193:15 uh uh with dell on a you get this vector and this is

193:19 the curl operation. So applying these to the wave equation, say the

193:30 free part of the wave equation is by the gradient of this scalar

193:37 Why we say it's the curl push The superscript P means that it's a

193:43 wave and it has zero curl to . So imagine now what does that

193:53 curl free as the P wave is watch my hands as the P wave

193:59 going through the rock, it's uh the compressing and um and decompressing the

194:08 as it goes through. It's not any of this. It's not making

194:12 sheer of this sort. It's making a, a long compression. So

194:19 can see that uh when we uh can see that the a good English

194:25 for describing this part without this kind motion to it that has no curliness

194:32 it, it only has longitudinal uh to it. So that's why we

194:37 the curl freak part is a P . And so we'll come to the

194:42 part of the sheer wave in a . So let's designate to designate this

194:51 free part. And we're gonna call A PW. And it's gonna be

194:58 uh you know, uh expressed as , the gradient of a of some

195:07 potential function, which we can see mathematically. But I doubt if any

195:13 us have a good mental picture of that scale of potential is. In

195:19 same way, the divergence free part an S way indicating it here.

195:24 I'm gonna go back here and this here has zero divergence that would be

195:47 . The part which has zero divergence an S wave. So let's think

195:51 an S wave going through a solid this. It's not squeezing wa watch

195:58 hand. It's, it's going this horizontally and it's not as it's going

196:02 a saddle like this and it's not the rock at all. So

196:07 it's not converging or diverging the rock it goes through. It's only sharing

196:13 rock. So, uh uh so why a weak site. So that's

196:18 free, only a shear wave. so we give, uh we just

196:22 the notation here. So instead of zero divergence, we say superscript

196:28 for she and that's given by the of some scalar potential vector field or

196:37 potential, which again, I think don't have a good physical intuition for

196:43 but you do have good f physical intuition for these displacement fields on the

196:50 side. So here's the displacement inside P way and here's the displacement inside

196:54 share way. Now, um all of this is uh valid for

197:06 isotropic rocks. For an isotropic rocks a complication. So we're gonna be

197:11 something different or analyze the trapping Not now, but later. And

197:19 look for heat wave solutions. And uh uh uh uh we're looking for

197:25 , here's our wave equation, these terms. And you see we have

197:32 second group of respective time. And just put in here the uh the

197:37 of uh of the scalar potential phi that's gonna be the gradient of the

197:51 uh OK. Yeah. Oh So is the second route of the spec

198:04 time. On the right side, have uh uh uh the green of

198:10 curl free part of um of, the stress. Let's, let's uh

198:19 back up here. OK. If gonna go back earlier right here.

198:31 , uh on that. So on left hand side, we have the

198:34 ro in respect to time. On right hand, we have gradient of

198:39 gradient of stress. So now let's forward again. So right here we

198:50 the gr uh we're, we're specifically that uh uh uh we're taking the

198:57 of the curl free part of the . So that means that uh uh

199:01 the part of the stress which depends the scale of potential F. So

199:08 index notation, um uh this looks this where you see here,

199:15 the index eye here, this, shows only the I component. This

199:20 a vector equation which shows all But for the I component we have

199:24 the left side, this and in right hand side, we have this

199:29 of this stress tensor uh uh uh in this way. Now we're gonna

199:37 the stress using hook flop. So not take the stress and put in

199:44 flaw right here. So again, have the factor of one half.

199:49 have the strain here. We have stiffness tensor here. And uh

199:57 on the right hand side, we're to um uh put in here for

200:03 uh the C component of the B . We're gonna put in uh uh

200:11 a gradient with a spec to K K as we have here uh of

200:18 scale of displacement. And this, uh here is the uh uh

200:25 I think the, the end component the G displacement is given by the

200:32 of the uh with respect to XM the same uh scale of potential.

200:38 , look here um uh we have really with respect to mxmnxk here,

200:48 have XK and XM is the same . And so we can combine those

200:54 and when we combine those two, can get rid of the two.

200:57 now you see how we get rid the one half. And so now

201:01 see why it was clever for us put that one half in there.

201:05 that it comes out this point. , what we're gonna do is is

201:11 that the uh uh the uniform is , it, the medium is

201:17 So this thing does not depend upon , bring that outside the derivative,

201:23 it over here. And uh uh we have um a, a simpler

201:29 involving multiple um uh components with all xjs and X MS and XK.

201:39 looks like it's pretty complicated. I no fear it's gonna simplify shortly because

201:46 this scalar potential fly. But it complicated three equations. One for each

201:54 of I and with 27 terms in , all these sums, you

201:59 we got a sum over J some M, some over K 27 terms

202:04 , in each. So, oh see because of the clever notation,

202:14 of this is gonna simplify very Let's look first at one of these

202:23 for third component only. And so we have uh uh uh a three

202:31 , a three here. And so have three here and, and we're

202:36 have a sum from JK and M um 27 terms like this.

202:45 So um if we show this term uh over J explicitly, so we'll

202:53 the derivatives of spec to XM and put them out here, we're showing

202:57 only with respect to uh only variations respect to um uh from the sum

203:05 sum over XJ explicitly here's X one two and X three. And uh

203:12 see no more Jays here so that j um one equals 12 and

203:24 So now we're going to uh uh uh isotropic elasticity. So remember we

203:32 uh we found this before that uh for isotropic media, the uh the

203:40 uh are quite simple because of all zeros out here. And so,

203:45 of all these zeros, most of terms in this previous equation are

203:50 So we had here, all these in here, most of them are

203:55 because of the properties of the uh stiffness sensor for isotropic media. And

204:03 work only these only these terms So 1234567 terms, I have no

204:16 . It's gonna get simpler still. . Next, we're gonna uh uh

204:21 transfer to two index notation. So example, this 3131 becomes a

204:30 its 3232 becomes a firefox. And similarly down here, so we're left

204:38 with these only these five terms. uh and, and if you look

204:46 here uh uh uh up there, we can collect all these terms.

204:51 so um we uh after we collect the terms, we um uh simplified

204:59 expression to this and using the common using the common names for the elements

205:06 uh uh C 31 and so Uh C 31 is equal to Lambda

205:11 two mu, again, Lambda plus mu and this one is AMP so

205:17 are simply the common name for these components. And because lambda equals M

205:26 two new, all of these terms proportional to M. And so look

205:31 we have everything uh uh uh comes uh uh M and we're left with

205:39 M over on the right side times lobos and of the vertical component of

205:45 displacement. Remember that we're talking about traveling P waves. And on the

205:50 side, we have the secondary respective of the vertical component of uh of

205:58 the displacement. And if we repeat for the other two components, we

206:03 the vector wave equations from P waves nicely like this very simply. And

206:10 remember that M is given by K four thirds mu. And so

206:18 III I said that wrong, let define it is defined uh uh um

206:28 Emma Moreau, let's define that as square of something we call VP.

206:35 finally, we get this expression. after all that um struggle, it

206:40 out to be very easy. Um you go back over this next

206:52 you will find that the struggle wasn't that struggling. It, it's very

206:56 . It's very straightforward because of the we used now. So this is

207:05 be propagation for P waves in any and it's gonna be easy to convince

207:13 that all of these uh uh uh solutions depend upon this phase factor with

207:19 this uh frequency. And this, dot X so long as the length

207:26 K, never mind the components of , the length of K is related

207:31 to uh VP, which we just uh defined. Still, we don't

207:36 what VP is, right? We know whether it's the VP of

207:42 of, of the wave or the or what we don't know yet.

207:46 gonna find out soon. We don't right now. It's just a notation

208:04 . Now, we could do the sort of thing instead of uh uh

208:08 of having the displacement here, we , we could get a wave equation

208:12 the scalar potential. Let me just up here. So here's the equation

208:20 just came up with, we could this in the uh well, the

208:25 of potential back in here right here for the, for the uh uh

208:31 after we got rid of it some ago, but let's put it back

208:35 and now let's take it outside this . So the, the gradient operation

208:42 be uh uh interchanged with the, time derivative like so, and uh

208:48 a similar way, this green operation be uh interchanged with the low cross

208:54 . So we have the uh the operator outside of everything. And so

209:00 means that uh well, the equation good even without the grading operation.

209:07 uh uh we have the same wave for the scale of potential as we

209:12 here for the displacement. And you see there's a big advantage here in

209:18 computation because this is three equations in . And this is one equation.

209:24 is three equations for the three components the displacement. I know you have

209:28 good intuition about the displacement. You have a good intuition about the uh

209:35 the there are potential. But the uh uh the same solution is gonna

209:42 given much simpler by this scalar equation opposed to this vector equation. And

209:54 we uh uh uh uh I remember uh he instruments, our receivers are

210:00 gonna be receiving the potential, they're be receiving the displacement. So uh

210:08 you are operating in the computer uh the uh Gaylor Wave equation, in

210:14 of the potential at the end, got to find out uh you've got

210:19 take the gradient of that potential solution find the displacement to find what the

210:26 is gonna look like. That's what says here. The gradient of the

210:30 here yields the observable displacement. So that's for P waves. And so

210:40 most of what we do in exploration physics is with P waves, but

210:45 shear waves are important. So let's at the same sort of thing with

210:50 shear waves, it's easy to show for the vector potential you get a

210:55 or you get an equation where instead VP here you get something we call

211:01 and that is shorthand for New over , you can um uh convince yourself

211:12 . Now have we have a vector . Uh you know that in um

211:18 , the displacement is perpendicular to the uh direction of propagation. Well,

211:25 because you know that uh it follows this scale of potentials this vector potential

211:32 in in the direction of the And this thing has uh uh solutions

211:41 depend upon the phase defined in this . So long as the wave vector

211:46 related to the frequency by um the velocity instead of the P velocity.

211:56 uh take the curl of this expression , the curl of this expression here

212:02 get the vector wave equation for sheer where now we have the shear wave

212:10 . And uh so here it says uh uh uh uh and it says

212:18 the deice is perpend color to that . And uh um it's a transverse

212:28 no, we defined and we have vector wave equation and previously had a

212:51 wave equation for the P waves. , it says that the solutions to

212:55 vector wave equations have the same properties those of the scalar wave equation,

213:00 are the solutions do have three parameters be fixed by initial or boundary

213:06 So we're gonna be able to um represent the solution as a sum of

213:13 waves. And uh uh I in sum there's gonna be three parameters which

213:19 specify using the initial conditions and the conditions, the functions of the

213:27 But we have either VP or vs here depending on what we're talking

213:33 PR S and a sum of solutions a solution and a weighted sum of

213:38 is a solution. So these are properties of the vector wave equation,

213:44 is the way the waves are propagating the earth. And if we're doing

213:48 land survey, not a marine but a land survey, you

213:53 we're gonna be putting out um uh on the land on the surface

213:59 of the land and uh um coupled the ground. As a matter of

214:04 , my first job in Geophysics was that job, uh uh deploying um

214:11 receivers Trump a quick quiz here. Let me see here. Uh Let

214:23 do the following. I missed a from my wife just now. If

214:35 uh um if you will permit I'm going to send her a

215:03 I'm here. Uh 104, send that note. So yesterday after

215:11 I was uh uh maybe two days , I was uh sitting on the

215:16 just outside the building here waiting for to come pick me up and a

215:21 came along and he said, you're waiting for somebody. And I

215:24 , yeah, I'm waiting for my . He said, you, you're

215:27 that you have a wife to come you up. And I agree that

215:33 am a very lucky guy. uh, um, not everybody has

215:38 wife to come pick them up or husband to pick them up.

215:42 but, uh, I'm very I've had the same wife for 58

215:47 . And uh so I think she's trying to pick me up. Um

215:53 I think about that 58 year longer you all have lived, maybe longer

215:58 your parents have lived. So I've a wife who comes to pick me

216:03 all that time. I'm a very guy. So uh I just sent

216:09 that note. So, so she'll here at one. So we need

216:13 finish up here at one. um let's uh uh look at this

216:20 . Uh question says, is this false? The equation of motion is

216:24 starting point for this derivation of the wave equation. So this is a

216:31 equation, a similar question to what had for the scalar wave equation if

216:36 remember. But this is about the wave equation and that's it, is

216:41 true that the starting point for the wave equation is, is new or

216:47 ? So le le le le, that true? That's true. She

216:51 that's true. And she's correct. . So uh turning to Carlos,

216:57 uh here it exists to or A principal difference between the scalar wave

217:03 and the vector wave equation is that former ie the scalar wave patient analyzes

217:09 on six faces of the voxel. the latter vector wave patient analyzes the

217:15 stresses on the six faces of the . Is that true Carlos or false

217:28 ? We didn't hear you. Yeah. Yeah. Yeah, I

217:30 thinking and then yeah, so what he's doing right now, he's

217:36 over the uh uh uh statement very . It might be a trick

217:43 And so he's reading over it very . Yeah, I think it's

217:47 Yeah. And, and he came the right conclusion. Uh good uh

217:51 good answer. Carlos. OK. versa. This one is for

217:55 the helm theorem separates out the divergence part of this placement leading in the

218:02 case to pure P waves. Is true or false? I think it's

218:09 . Uh Yes. Very good. separates out the con the curl free

218:15 . Yeah. The P waves come the curl free part. So uh

218:19 was a trick question. You it looks like it, the an

218:23 uh it's uh uh you could have fooled. But uh uh you read

218:27 carefully and you say that uh the waves do have convergence and divergence.

218:33 it's the uh it has zero curl not zero divergence. So that one

218:38 false very good. OK. Back you Lili, the scalar wave equations

218:44 the pressure P in a fluid is from the scalar R equation for the

218:51 um fry in a solid for uh uh uh uh these answers here.

218:59 let, let's uh uh think about . I'm gonna uh uh uh Lily

219:05 uh tell me about a uh uh know, is this true that the

219:09 called pressure in a P wave in solid is really launch stress, not

219:15 pure pressure. Is, is this true understanding by itself. Yeah,

219:29 statement is true. The so-called pressure a P wave in a solid is

219:35 a long stress. It's not pure . That's true, but it doesn't

219:39 the question. So we uh uh so we don't wanna pick a,

219:44 uh Carlos, the next one is uh mhm No. B we program

219:51 computers with the equation for five for uh uh for increased efficiency. Number

220:02 , tell me just thinking about this uh uh statement by itself be,

220:07 that true or false? Yes, true because it's easier to program computers

220:14 uh uh scalar equations than for vector . So we do do this.

220:20 , the question for colors is, this answer the question? OK.

220:42 . And while Carlos was thinking about and uh you know, you're coming

220:47 with number C. So you'd be about number C while he's thinking about

220:53 B, I think. Yeah, think about number B, um,

221:02 , let's concentrate on B is uh, uh, uh, answer

221:07 question? Yeah, I think Number B, um, well,

221:17 , I would, uh, I'm, uh, I'm gonna say

221:21 because, uh, um, this business of computers, the,

221:27 equations don't know anything about computers, ? All right. So,

221:34 so the equations are on their own of where they were solving it by

221:39 or paper and pencil or whatever. I'm gonna say that, uh,

221:43 is a true statement. B but doesn't answer the question. So now

221:48 turn to c, it says the P is the observable whereas the potential

221:53 is not an observable. Now, . And, uh, uh,

221:59 , uh, thinking about this see by itself, is that true

222:04 false? Uh, uh, it's . Yeah, that one is

222:09 Now doesn't answer the question. And now we, we, we've already

222:14 A and B and so if this , uh, uh, if we

222:19 C, we're left with D. so I think about that. Do

222:23 really want this? But you gave the answer that yes, this

222:27 um, uh, answer the uh uh, that, uh,

222:31 , uh, this is, a, a fact, a true

222:35 which explains the difference between, the scalar wave equation and uh uh

222:41 heat and the scalar wave equation for very good. So now the next

222:47 in here um uh back to you le le uh vector equations for the

222:56 wave differs from the vector wave equation the P wave because A B is

223:03 all of the above. So um A is true. Yeah. Uh

223:09 , um uh uh the velocity depends the sheer marginist view, not the

223:15 margins. And I, is that ? A statement by itself? And

223:21 think, does uh does that answer question? Mhm I didn't hear

223:35 So uh uh uh let's uh uh this one at a time. I'm

223:39 you about A MM Number one is statement true. And does it answer

223:46 question um is, is the statement ? OK. She says the statement

223:57 true. And now uh uh the is it, you say it does

224:02 answer the question, tell me why doesn't answer the question. I,

224:13 didn't hear you DC. Well, uh uh uh before we get to

224:19 let's think about A. OK. uh so what I'm gonna say is

224:24 this is true statement and it's uh uh it answers the question, but

224:30 some of the others do too. now let's uh uh uh uh look

224:36 um um uh question B and answer and uh Carlos it says the displacement

224:46 in the sheer wave is perpendicular to wave vector, not parallel to it

224:51 in P waves. Uh um is this uh the statement true?

224:56 one and number two, that it answer the question? Yeah, the

225:00 B is also true also true. . So we got two are

225:05 So we're suspecting that we got all the above. But uh uh uh

225:09 we do that, uh um let's to Brisa and ask her about uh

225:15 point C uh is uh the displacement , does the displacement feel of the

225:23 wave? Does it have divergence or ? It's, it's true. It's

225:29 . It has, it has no . And so that's the difference here

225:33 sways and p. So we got of the above. Very good.

225:36 would say folks, this is not easy question, but that's the sort

225:40 question which you might find on the exam. So you, you wanna

225:45 sure you understand all that. here's a good place to stop for

225:50 morning. Um uh So we're gonna up, we're gonna pick up at

225:54 o'clock and go until six. And , um let us stop at this

226:02 . You can stop the

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