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GEOL7333-Seismic Wave and Ray Theory - Day2 08-27-2022-Part1
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00:00 And so here's Stephanie's question. We you might be interested to hear the
00:09 question. Um It's not about the issues, it's uh geophysics question,
00:19 is your advice to someone who wants be successful in this field? Are
00:23 scared to pursue it because of the of oil? Or is this a
00:28 field? So um um actually I asked a few months ago to give
00:36 lecture to the department on uh what takes to be successful. You tie
00:42 by any chance for you in that , I gave a friday seminar.
00:47 so my advice to the students was lucky. So that's not so easy
00:54 be lucky, but it's not true luck just happens. Uh luck happens
00:59 those who are prepared. And so corollary to that is be prepared so
01:06 when luck happens you're ready for And the way to be prepared to
01:10 a good foundational education. And uh a hedge against the possibility that this
01:18 might be dying. What you want a broad education so that you are
01:24 to uh switch fields later. We we have a lot of students in
01:32 , we're now in computer science because they had the skills and they got
01:38 good job offers from companies wanting those skills and math skills. So you
01:45 sure that you uh acquire those I think you already have pretty good
01:51 skills, but you need to have skills as well and familiarity with how
01:56 handle big data sets. That's one we do in geophysics. We have
02:01 big data sets, but normally they're uh you know, a seismic data
02:08 uh as zillions and zillions of lights it, but it's all structured into
02:15 and receivers and so on. And that's a specialized siphon big data.
02:26 more common our datasets, which are so structured and have a lot of
02:35 in them. Uh just for uh if you want to apply artificial
02:45 to solve a problem, uh So basically what you're looking for is patterns
02:50 data and the human mind is really at detecting patterns and data even when
02:57 lots of distractions for example. Uh , uh here's an example um facial
03:10 . So if I had a Cameron it around the room, the
03:13 uh software and the system will be to recognize two human faces out there
03:21 they're very different faces. But they recognize that you have the characteristics of
03:25 human face, which they and so they can zoom in on you and
03:34 get a detailed look at your face then compare it with their database and
03:38 out who you are. And if a terrorist, uh then they can
03:43 the bell and the cops come and you and don't let you get on
03:46 airplane. That sort of thing happens days. And you know what the
03:53 pattern recognition capabilities really not much advanced when I was your age. We
04:01 knew how to do that sort of . But we didn't have the data
04:07 we didn't have the computers, computers small and slow and expensive. And
04:12 we didn't have the computers to do . So it was not a well
04:16 field. Now it's extremely well developed developing further all the time. Sometimes
04:23 a little bit scary. Uh, intelligence taken over. But remember,
04:31 intelligence is not really that it's a name. It's really a pattern recognition
04:37 large data sets with lots of And uh, those two, those
04:44 two things are new since I was age. Um, uh, but
04:51 can acquire and you should acquire while young, the capability to deal with
04:57 kind of, um, uh, , uh, through machine learning
05:06 uh, machine learning how to recognize . So the first part of your
05:16 , uh, advice be lucky and prepared And that's particularly have a good
05:23 , fundamental background in particular math and you can count on during your
05:32 no matter what, where you there's gonna be lots of changes during
05:37 lifetime, in your profession, whatever is. And so you need to
05:43 , keep up with those changes. just tell you a story that my
05:49 was a geophysicist before me, probably why I'm a, physicist. And
05:54 his day, um, uh, recorded the seismic signals on photographic film
06:04 on paper and they analyzed them uh, by spreading those,
06:11 those films, large, large sheets film, they develop them overnight and
06:17 out on the table and uh, to move out things like that with
06:25 drawing tools. And uh, they very low faux, very poor quality
06:31 , but they found a lot of because it was easy to find.
06:34 uh, my father was instrumental in that east texas oilfield, which is
06:40 of the greatest oil fields in the . And um, so you,
06:48 , most of his career, there no computers at all and came into
06:52 field late in his career. And , a lot of his colleagues had
07:00 , you have nothing to do with , but he did. And he
07:05 , he embraced the change and learn to use computers. Of course,
07:11 early computers were nothing compared to over today, but it was a big
07:16 for him and he made it and of his colleagues did. So he
07:21 his job a long time and that he was a very successful oil
07:31 for him for his day. Uh standards were lower than he was highly
07:38 inside his company because when he made recommendation to drill a hole, There
07:44 a 25% chance that it would be success. That was, that was
07:50 success rate and today that would be , uh, terrible if that's what
07:58 did, you would be out of job in two years. Um But
08:05 tools are so much better now that had then and our data is so
08:11 better. So I'm gonna give him pass for that 25% figure. It
08:17 outstanding at the time. And um have moved on. So you've got
08:26 be prepared to move on. Uh ever uh say, well, I'm
08:33 my mid career now. I think can ignore this new development. I'll
08:37 coast through the retirement. That's a attitude. I embrace the change.
08:44 . 2nd part of the question. students scared to pursue geophysics because of
08:49 fluctuation price of oil? Or is a dying field? So the price
08:57 oil has always fluctuated up and And and so I uh Uh I
09:06 I probably saw six or 8 cycles my career. And um I survived
09:14 all by doing my job well. so that's the thing that you want
09:19 do your job well. But I think that um probably, you
09:25 oil is a commodity and the price upon the balance and supply and
09:31 demand is always increasing supplies sometimes gets , sometimes gets behind so that the
09:37 fluctuates. I think we have more than other fields, but all fields
09:43 fluctuations in prospects. So if you stability uh got a government. So
09:55 now, is this a dying Well, it's certainly a changing
09:58 And so the big buzzword this day days energy transition. And uh Professor
10:06 talking about this when he was Um there's a big, there's a
10:11 future for uh solar and for wind and so on. Hydropower, but
10:20 they make up only a small fraction the of need and need is always
10:27 . And so as fast as we convert to uh electric. Uh the
10:34 is good almost as fence. Uh will be demand for oil um for
10:42 for your lifetime. Eventually it's gonna out um probably for your lifetime.
10:55 There will be a large man for while and we simply have to be
11:00 at finding it than we have in past ah over yesterday how, because
11:08 fracking of shales, uh that's led , up until this year we had
11:17 oversupply in the Russian invasion of Ukraine . There's a political decision and suddenly
11:25 uh Russian oil was off the And uh so that meant there was
11:31 shortage and the price went up that's to stabilize in the next couple of
11:37 as other producers step up to make make up the difference. The Saudis
11:43 a lot of spare capacity and we a lot of spare capacity because we
11:50 shut in a lot of coal shell because uh the high supply last
11:57 low price and a lot of wells shut in. So we can uh
12:02 can increase our production also um in short to medium term. But uh
12:09 the long term I think we're gonna to find more oil uh even though
12:18 electric and nuclear are ramping up their from such a low base and the
12:24 is always increasing. So I think demand will be there for a long
12:30 and as it uh finally uh begins find a real shortage. You are
12:40 again to run out of successful discoveries I think the price will go up
12:47 high tech geophysics will be required to the difficult wall remain easy. Oil
12:55 found by by techniques which are really Science and Geophysics has gotten really sophisticated
13:08 the past 30, 40 years. uh like I said yesterday, we
13:16 basically solved the problem of finding oil beneath the complex. Overburden. Haven't
13:22 the problem of how to find the spots of oil production all sales.
13:29 so that um I'm quite confident will solved the next three years and
13:35 you know, by people like us people like you guys, I'm probably
13:40 old for this like in your So, so I don't think it's
13:47 . But you know, there's a that is dying. That's why um
13:53 social students. But what that means that oil companies are going to be
14:00 to find a few students in the , it will bring, oh,
14:09 think it will be high demand for few students in the field, but
14:15 always um always chance to, you , a balance of supply and
14:20 One gets ahead of the other and things happen, the other one gets
14:24 , Bad things happen also. You to be prepared to uh when luck
14:31 your way, be prepared to And that that means getting a good
14:40 education and I think this is part it. Uh This of course is
14:44 a practical course. This is of ideas. And we're about to pick
14:50 the idea of the wave equation, is gonna be implementing what we learned
14:56 about elasticity. So uh this morning will be able to learn how the
15:08 ideas from elasticity leads to the scalar equation for fluids, like uh like
15:15 ocean. And with more realistic assumptions the vector wave equation for uniform,
15:21 should drop solid. Of course the is not like that is not
15:26 it's not ice tropic, but you see that's a step forward. Uh
15:32 as we first derive the equations, not gonna, they're gonna be equations
15:37 how waves propagate, not where they from. And then we have to
15:43 a term in those equations to describe seismic source and that's going to make
15:47 changes. So now we want to that to the earth. So we
15:54 to consider in homogeneous broad formations. then we're gonna look at them uh
16:01 we've developed these equations. Now afterwards gonna look at the solutions. And
16:09 there's a really interesting idea called elastic . And so the last part of
16:17 luxury is about that. Okay, here's the scalar wave equation. So
16:24 have a picture of Isaac Newton and Newton's First law F equals M.
16:36 . This is for and you learned probably when you were a freshman.
16:41 this is for forces on point Well that's not what we need,
16:46 it's the foundation for, we're going recast this for the effects of forces
16:52 continuous bodies. And eventually we're gonna sighting. So we'll start slow from
17:01 for fluids. So consider uh volume inside a fluid. And so when
17:08 see a word like this, uh and in uh royal blue colors,
17:16 means that it's in the glossary. if this were uh a live for
17:21 being taught for the scG, it's on this tab right here that would
17:26 up the glossary. You have to a little bit more work. You
17:30 to go to the glossary file which in the uh blackboard in which I
17:35 you have downloaded. And so they're the glossary. It tells you all
17:41 the boxes which of course is a D generalization, the word pixel.
17:46 you know from uh image analysis. it's got a size which we're gonna
17:53 name d. Lowercase D. And the mass is given by the density
17:58 the tube of the to the Okay, inside the fluid there are
18:05 forces and surface forces. So the forces are like gravity and so they
18:11 everywhere uh on the inside of Um Now uh in Iraq in the
18:21 or let's get that uh in a in the ocean gravity is pointing
18:30 And so it's uh all the But it's constant and it's and it's
18:36 by a constant pressure from below. gravity is pulling down, pressure is
18:43 up from below. That's a surface . Uh And so those cancel.
18:48 then furthermore uh there's pressure on all but those are constant. So we're
18:56 ignore these constant forces. We're going consider only variable forces. Now we're
19:04 locate this rock still at a position is measured from the origin recorded.
19:10 we established an origin over here and we have a vector to the center
19:16 the box in. And so that that that's the vector X. And
19:21 tells where the where the box will the pressure at the center of this
19:28 is P. F. X. it's different at different places for example
19:36 the top surface uh that there is displacement. Uh There is a a
19:46 uh vector uh delta X. Which given by uh 00 at D.
19:56 . R. Two is the uh the difference between this position and this
20:05 . So the pressure is the pressure X. Plus that uh delta
20:14 And on the bottom it's uh corresponding expression except that there's a minus B
20:22 two and on the side uh it's different again because the difference vector is
20:33 zero for two in the one position this thing. Here it is on
20:38 other side with a minus D. then front back. Uh Same.
20:45 the corresponding forces for all these are pressure times the area. Okay?
20:54 here this is a force In the directions at the top and here is
21:07 force right down here is a force the three direction at the bottom,
21:11 see that. And uh correspondent here the force in the one direction from
21:17 right one and here's the one for direction from the left. So we're
21:24 up all these um uh forces on surface of the box and then we
21:32 sum them up. So this looks a make it but it's not it's
21:36 six terms equations. And it's arranged that the top row is the right
21:42 left and the bottom row is the and bottom and the middle row is
21:47 back. And they all have this D squared here given the size of
21:55 Vauxhall. And you see there's there's minus science scattered around uh multiplying by
22:01 pressure and then there's various minus signs indicating the positions. And uh if
22:09 look back previous slides to see where those came from. So let's consider
22:16 sound wave traveling vertical. So the does not vary in the horizontal
22:21 So these top two roads are zero so this is all we have
22:29 So um basically it's constant pressure equal out on now since the box was
22:44 house, what the small means means compared to the wavelength of sound.
22:49 can describe this pressure. This pressure with the taylor expansion. And uh
22:57 pressure at this place here is given the pressure at the center plus this
23:04 series uh change and uh thinking about whether this is the right derivative to
23:19 um go back and and this is fact the and furthermore notice that it's
23:27 at the center. It was evaluated . And this is the difference in
23:35 from um pressure here at the top or two away from the middle.
23:45 then the total force is uh given . Now let me see, let
23:50 check here when a word like that that I'm not not sure whether or
23:55 that's a live link but I'm gonna at. Yeah so it took me
23:59 here so to be frank here, not sure whether uh Back in the
24:10 or not. I'm gonna page forward slide. Yeah. So it took
24:14 back to the uh that place and now here we are back and this
24:20 me that changes colors uh reminds me did that before. So I'm gonna
24:26 forward for the animation. And so we put in the taylor expansions for
24:31 two uh figures top and bottom and see more minus signs coming in and
24:42 the center terms cancel. And we're with only the taylor expansion terms.
24:50 we collect terms and we come out this. So um it's you can
24:55 that it's easy to make mistakes here of skipping out a minus sign.
25:01 I assure you that this is all correctly. And notice that we now
25:07 d cubed here. So we start with the squared, but then we
25:12 a. D. Coming in because the taylor expansion thing. And that's
25:17 now we have the view. Remember this is far more vertically traveling wave
25:24 the ocean. So we're gonna put into the equation of motion. So
25:31 only a force in the three And this is what we just
25:35 And uh so on on the right of Newton's first law says uh mass
25:43 acceleration. So we're gonna call the in the three direction and simply rearranging
25:51 we find that the acceleration is given minus one over road, where did
25:57 road come from? Well, remember the mass is given by road
26:01 Youtube canceled out. So we're left this, that's the acceleration in terms
26:09 the pressure gradient. Now, because the machinery that we set up
26:15 it's easy to uh to change that any uh direction. Always say is
26:24 the acceleration now looks like this secondary respect to time of a displacement vector
26:31 here's the highest component. So we that equal b 12 or three.
26:35 we've been going in any direction and course we have to put on here
26:40 same direction from the gradient. And we um uh we recognized that this
26:52 this derivative is a gradient operator. so um uh we use this notation
27:01 the bill all del symbol to indicate derivative. That's okay. Now remind
27:12 that it is only in the So I wish is this true or
27:22 ? The equation of motion is the point for this derivation. I will
27:27 away reclaiming, raise your emotion. didn't actually say that did I said
27:35 was uh it was first long but the answer to this is true.
27:50 . Sound wave or false. Sound are driven by gravity. So they
27:54 be slower on the moon or gravity less. Say it again.
28:07 uh that's true, but this is in space is on the moon.
28:12 is talking about a sound in wound where the gravity is less because,
28:22 you have to decide, Well it's that the gravity is less on the
28:35 expression. I think this is Yeah, that's fishing. I'm not
28:47 think that. Well is it true sound waves are driven by gravity?
29:00 is a body force, its uh operating throughout the throughout the box ill
29:08 it gets canceled by constant since gravity constant, it's constant in time.
29:14 mean it's canceled by uh by variant pressures. And so that's not
29:22 to lead to waves. So we ignore that. What we found was
29:26 uh um the motion of the particles this box are driven by pressure gradients
29:40 in the center. Okay. Now see if you can figure out the
29:46 um Oh bookkeeping here. Um So have uh we have the coordinates are
29:59 here and uh the question is about left center of the box. So
30:05 means left in the one direction. which of these? Uh is
30:15 Yeah, that's right because this is left center, this would be the
30:19 center and this would be uh the and this would be the bank.
30:27 . Yeah, you're right. So now I didn't I didn't um
30:36 you this but I taught you enough that you can figure this out.
30:41 is uh is this the right equation motion for a way of traveling right
30:47 left, remember what we did was drive up and down? But now
30:51 asking right and left, right, to left. So is this the
30:56 equation of motion for that? Yeah it's accelerating in the one direction and
31:06 depends upon this uh primitive in the . Direction, in the one
31:15 Now. Uh can you tell from ? How about the opposite left to
31:23 ? Let me see here. Uh that that that's not uh on the
31:29 quiz but think about it. Uh this the right equation we already
31:36 Yes, you're correct. That that's pressure right to left. How about
31:40 to right? Is this the right for something traveling left to right?
31:49 . Right. All you do is that case this is gonna be a
31:52 number and uh this is also gonna negative. So uh so that uh
32:02 very good uh you were able to to answer the answer, answer the
32:09 properly even though I didn't teach you answer because you were able to
32:14 but I did teach you to this . So that's the uh um crucial
32:26 skill which not everybody has. I'm to see that you have. Okay
32:32 let's convert these uh vector equations, is affect right, we're gonna take
32:40 divergence of this. So that means we're gonna uh operate on both sides
32:46 the operator D by dx i uh . Now we have repeated indices
32:54 and in vector notation, it looks this and an index notation looks like
33:00 . And you know what we're going do is we are going to um
33:06 move the derivatives around uh through those on the next slot here is here
33:15 the left side of the equation. go back here. So we're gonna
33:18 about this here and the left side like this. It's just put the
33:27 , displacement in its own little square . And then we know in our
33:32 calculus we can interchange the order of , so we bring the derivative with
33:38 to X. Inside here, all way inside here. And then we
33:45 that this is the strain in it's the sum of uh of uh
33:55 stream components epsilon II because we are over all eyes here and we know
34:04 previously that's the dilatation, that biometric . So remember the volumetric changes given
34:13 the in compressibility in this way. let's just see what happens when we
34:20 click here, cannot open the specified . So this only worked for in
34:27 scG environment. So now putting this together. Uh the left side of
34:37 equation now has pressure in here and compressibility has got an extra minus sign
34:45 you didn't have before, that came right here, but it's still got
34:49 acceleration operator on the outside. So we've put together both sides of the
34:57 . And uh now it looks like . And you know what we're gonna
35:02 next? We're going to assume that pressure is uh uniform. So that
35:12 that we can uh bring this I said it wrong that we're going
35:17 assume that the density is uniform so we can bring the density outside of
35:22 operator. You did it. So we have on the left a derivative
35:30 pressure to directors respected time of pressure on the right to directors of pressure
35:37 respect to position and some conscience outside make the units match up and saw
35:47 uh bring the K to the right . Remember we had this minus
35:54 Let's see what we reported this, we minus sign on both sides.
36:22 is now what we call the scalar equation so that the unknown in this
36:28 pressure. And so when we have marine seismic survey, what we do
36:34 measure the incoming uh seismic waves with , not geophones, but hydrophones.
36:42 measure fluctuations of pressure. And it's quite remarkable how they do this.
36:50 to be frank. I am not how those clever engineers have done
36:55 but they've made a little package which can sell uh for just a couple
37:01 $100 makes a hydrophone and they have dozens or hundreds of these on a
37:09 and they have multiple streamers. Maybe geophones in the water at one
37:15 And the boat is pulling it forward the waves are sloshing around and the
37:20 are sloshing around. And even despite those voices. And despite the
37:29 construction of this hydrophone, it's able measure the tiny, tiny fluctuations of
37:33 sensitive waves which come into it. you think that's remarkable? Now,
37:40 seismologist? They don't, uh, , well these days they do,
37:44 in my day they never, when was your age, they never did
37:49 kind of stuff when they were looking earthquake ways. They would always have
37:55 instrument on land in a cave or a tunnel or something like that,
38:01 , protected from the noises and very , many thousands of dollars for each
38:09 . And of course they're measuring three components. But uh, it was
38:15 and it was specialized and it was and there weren't very many of
38:20 So these days they do much better they're looking for earthquake signals.
38:27 and they, but still, they have dedicated systems. Let's measure
38:38 In fact, we have one of uh, stations right here on
38:42 Are you familiar with that one, uh, just across the road
38:48 just across the road, uh, the Georgia building and uh,
38:57 satellite cafeteria, There is a seismic and it's in a lockbox, so
39:06 people don't screw around with it. uh there are people here in our
39:11 who uh the data from that and , make sure it's always working.
39:16 that's a very sensitive instrument which is uh extra equation and Taiwan. Um
39:30 it's it's uh it's expensive several $1000 that. You couldn't possibly do that
39:38 uh exploration seismic because we need to many more receivers. That's secret to
39:46 success is lots of receivers, lots source position, lots of receiver
39:52 So they've got to be cheap and have succeeded in making a very cheap
40:01 and also cheap earphones which nevertheless sensitive to measure these very tiny oscillations.
40:12 uh seismic sources, What more about magnitude of those uh perturbations later.
40:22 in the wave equations we have 22 with respect to time to to produce
40:28 to position and a proportionality concept. uh Stephanie, tell me what,
40:37 is the physical dimensions of that proportionality . K overrun. Well, I
40:51 want to know the magnitude of I want to know the physical dimensions
40:56 it. We have to have the physical dimensions on both sides of the
41:01 . So this is pressure divided by square of time. Right? So
41:06 is pressure divided by the square of and um physical units. And so
41:15 make these work out what, what this got, What got me the
41:19 dimensions of this ratio K. Overall don't know what K is and you
41:25 don't know what row is. I you know that rho is mass per
41:29 mass per unit volume but I'll bet uh uh you're not quite sure what
41:37 is but you know we can figure out. So let's go back
41:41 Okay so uh notation is dimensions and can see that clearly right here.
41:50 Kay must have the dimensions oppression uh order for the left side of
41:59 So the right side has to be . So Kay must have the dimensions
42:03 pressure we're gonna go for. So we have pressure divided by density.
42:08 What does that what what does that out to in terms of time and
42:21 ? You don't really have to uh think you're you're thinking about this in
42:28 wrong way. Uh Caro has to the physical dimension to make the physical
42:37 on the right side the same as on the left side. So uh
42:42 that equal to only have to have squared over T squared? Yeah,
42:49 . And see without working out this pressure. And this is uh densities
42:55 working out uh in detail which each those means. And then taking the
43:01 you can see immediately that the ratio got to be expert over t
43:10 So X squared over two squared is the dimensions of velocity squared.
43:17 so this thing has to have conventions velocity square. Okay, so uh
43:24 notation around, you recognize how changes notice how we slipped in here.
43:30 Del square. And so in Cartesian uh this uh second quarter derivative respect
43:40 X. I. It's the same as a Gaussian operator doll sprague.
43:47 notation which we talked about yesterday, can check it, check it uh
43:52 the files that you have. so just like you said, it
43:57 the dimensions of of exo routine which is velocity squared. So let's
44:04 give that a name. Uh We'll this instead of calling the Contin.
44:11 call it V square. It's a and we don't know yet what is
44:18 velocity of. But uh it's a good guess that that is gonna be
44:22 velocity of the wave. So here's scalar wave equation. Yeah, if
44:34 wanted to we could write it in of facilitation, uh just make this
44:40 and things cancel out and uh get same equation in terms of the rotation
44:48 continue to analyze the pressure version. mention this because when we go to
44:54 vector wave equation, we're gonna be about strains, non stress. And
45:00 mhm. This is a strength. is the volumetric strength and however it's
45:07 in talking about the scalar wave of the discussion in this farm unknown as
45:18 . So, so now let's go to the simple case of wave traveling
45:24 so that so instead of having a operator here, we have only the
45:30 really respect. See now think about , whatever the solution is, it's
45:40 be a solution varying with temperature and and it's gonna whatever it is.
45:49 It's got to be very in this uh combination which we call the phase
45:55 with five written like that. The is a function of time and space
46:01 it depends upon time like this and like this and Parameter Omega and another
46:11 K three. So we will talk about those parameters charts. Okay,
46:19 recognize the uh the parameter omega is notation for the angular frequency. So
46:27 we talk about the cyclical frequency, cycles per second, that's f.
46:34 radiance per second is 25 times. we have a name for this uh
46:43 three. I see the arrows pointed the wrong place to adjust that.
46:49 three is called the vertical wave It's gonna be related to the wavelength
46:58 three itself is vertical way of Now, you might have your doubts
47:04 whether what I said back here. true. It's really true that the
47:12 only uh can't be random combinations of and space, but only through this
47:21 . So let's just test that the side of this. So what we
47:24 is pressure is not really independently not really independently varying with time and
47:33 but only through the phase. So write the pressure depends on the face
47:39 respect to uh Second review of pressure respect to time looks like this since
47:48 uh pressure only uh depends upon We uh we use the rules of
47:56 to uh show the derivative of pressure respect to phase right here and the
48:01 of phase with respect to time Chain rule, cackles, Stephanie,
48:07 you comfortable with that? Okay, . Uh use that kind of stuff
48:17 the time. And so uh the with respect to uh phase respect the
48:26 is just omega from the previous And then we do it again with
48:30 diverted as we get omega squared times second order of P with respect to
48:37 black wise, the right side of equation work through the same kind of
48:41 . And and you find that uh right side is the square times K
48:46 square, where does the K three from comes from these derivatives of phase
48:52 respect to position. And this is for either plus or minus Value for
49:00 three. So now the right side the left side. If and only
49:09 we have this relationship here between a and omega and V. Yeah.
49:21 , so now we are set up consider um solution. Louise. And
49:30 I'm gonna propose that um the solution uh no I'm gonna I'm gonna suggest
49:49 a solution. Is this So uh have pressure as a general function of
49:57 Let's say. Okay so how about function of five? Which is
50:00 oiler number eight uh to uh to power I. So this will
50:11 Uh we're gonna show that this works but any function five will work as
50:18 solution for pressure. So right here making sure you know about oilers number
50:25 . And uh Complex number I is as a square motive -1. So
50:34 talked about that um Good. Yesterday is really crucial for uh first
50:44 Number A is a special number kind like pop and it was, it
50:51 discovered by Oiler And it's a natural . It's a number of a little
50:58 smaller than pie. It's about 2.7 . And it's like it uh is
51:05 irrational number and uh decimal representation. goes on forever Box Mix in
51:17 And it has some special properties which described in the glossary will encounter those
51:24 party player. Now the next idea uh now this is really a complex
51:37 . I think that there's a special like pi which comes which is needed
51:43 describing circles examined. That's a big which you learned in trigonometry I
51:51 Um I suppose in high school that triggered on about pie in high school
51:59 I'll bet you didn't learn about about father's number eight finally learned about
52:05 Number a calculus. Um and uh are there are a few magic numbers
52:18 that, but only a few. kind of a mystery why we don't
52:23 lots of special number like but we only a few. And uh they
52:31 remarkable properties and I encourage you to the glossary number eight tonight.
52:47 I'm sure you have already uh learned those properties in your past forces.
52:55 won't hurt to have a Now we in here complex number in the exponent
53:10 vegan. And what that means is right side of this. Look at
53:17 uh All right. The right this middle party, not the right
53:24 but the middle part is a complex because of this. I and this
53:29 also a complex number. So the has to be a complex number
53:35 But you know that pressure is not complex number. Pressure is a real
53:41 measure. And you can feel it your hands and stuff like that.
53:45 , uh it's always makes a puzzle students think how can we be
53:53 how can we be saying that solution a real property? Like pressure has
54:01 numbers in we're gonna be measuring pressure a real instrument and it's a real
54:08 and it doesn't know anything about complex . And so the answer to this
54:18 is um that uh just at this it's an inconsistency and you have to
54:29 faith that uh we talked about Uh actual thank you your time. Um
54:52 I like. So whenever we have equation like this because and better mind
55:13 with it. Oh. Mhm. the time we get to particular problem
56:05 , your number for the plane wave that we have here, it has
56:20 wavelength which I'm gonna give is uppercase with a subscript three. To remind
56:26 we're talking about vertical propagation here. I think you know that the wavelength
56:36 given by the velocity over the the cyclical frequency. So we put
56:41 here the angular frequency that brings in two pot. And so uh this
56:48 of V over omega is K three or minus. Now I mentioned this
57:02 solution because it's gonna be very useful us. Uh we're gonna be able
57:10 um construct solution for any problem by of terms like this. So we
57:24 to think more about uh about this my solution. So let's make another
57:40 . Of course, you're gonna rewrite face by factoring out the uh omega
57:46 this. And uh then let's think whether uh we have one, let's
57:55 we have a solution for any So, and we don't yet know
58:01 omega is but uh we call it frequency. We know that waves have
58:06 of different frequencies. So let's assume we have one solution with one angular
58:12 and then a separate solutions to to angular a second angular frequency and then
58:19 we're going to guess that the sum also a solution. So you can
58:28 for yourself this is true. Um so this happens because the wave equation
58:34 linear, it means that the unknown appears for the first time ve uh
58:44 there's in they're a v squared, squared is obviously not linear, but
58:47 unknown is linear. It's the pressure of the first power on both sides
58:54 the equation. By the same We can have a weighted something to
59:01 . That's also a solution that's uh obvious to you. But if it's
59:05 obvious, you can work it out just make sure to yourself that perhaps
59:17 a solution. So that's an important . Does not seem like a trivial
59:24 doesn't work because what it implies is we can take a plane wave
59:29 different frequencies and different uh amplitudes and add them up, um weights which
59:40 specially chosen to solve any problem. fact here, of course, is
59:47 we have many times, not just just sums of two, but sums
59:51 many. And these can be an number of frequencies. These coefficients are
59:59 to be determined by initial conditions and boundary conditions, for example, uh
60:05 conditions, uh depending on whether the is a strong source or weak source
60:11 gonna affect these. Uh We decide these A's and uh then we have
60:19 decide about boundary condition. For there might be a free surface at
60:24 surface, might be no boundary at bottom. Uh All those things are
60:33 help us decide These values of Now that's all for a vertical population
60:43 of the mathematical machinery that we set yesterday. It's easy to generalize uh
60:53 to any direction. And all you to do is recognize here that uh
60:59 eye is a subscript I is So we're summing over all eyes.
61:08 then uh since we're gonna be considering directions, we have to generalize the
61:21 of face. So instead of K times e it's a vector dotted with
61:29 and for any solution uh He is function of five on the left
61:35 Uh We have, instead of a , respected time, we have rivers
61:41 to by and uh as a we have these two factors of omega
61:50 out and a similar thing on the side, two factors of K.
61:55 . So now they are multiplying this and you can see this same narrative
62:03 both sides. So that what that is that uh this product K 11
62:12 plus K +22 squared plus K K square as K two square. K
62:18 square. Uh That's the length that's the square of the length of
62:24 vector K. And so um um will be satisfied if K. Is
62:34 to vega Plus or -1 over So, here's a little question.
62:50 We have a list here. Look the bottom of the list that says
62:54 only two and answer is all of attributes above. And so now what
63:00 need to do is you scan through and see which ones those are
63:05 And then the right answer is gonna uh Probably your your f. So
63:14 it's definitely does it have um Remember talking about the scalar wave equation but
63:21 favorite, Not the equation of Um Does it have to do with
63:27 to time? And how about B. Is that true? How
63:32 C. Is that true? Unknown appears to the first power only.
63:41 the unknown function so far? Uh The unknown function is the
63:50 So, she is also true. d a single parameter which describes the
64:00 . That's also true. And that's philosophy square. Uh So it has
64:05 the attributes above. So that's that's . Uh So it can be written
64:12 with either pressure or dilatation as the function false. It's the same
64:31 Question three. It's only valid for in vertical direction since fluids only vary
64:37 the vertical direction. Is that Yeah, that's false. Uh Number
64:44 uh fluids very uh more than the direction. Uh but they do very
64:51 in the vertical direction. Might not aware of this, it's common for
64:55 to think of the ocean as the velocity everywhere, but that's not
65:01 The velocity varies with depth in the because of increasing pressure and decreasing temperature
65:10 also changing salinity. So uh velocity sound and water changes its function of
65:23 , and also it changes laterally. example, in the gulf of
65:27 we have uh it's quite prominent dr , but let me uh introduce that
65:43 idea. You know about the Gulfstream goes up the east coast of
65:48 United States, turns to the east then goes south uh along europe.
65:56 uh um it's pretty obvious, it be intuitively obvious that uh profiles of
66:08 temperature and salinity are different in the stream than they are in the middle
66:15 the atlantic. So uh that means the velocity of sound and water checking
66:22 . Uh picking up on the idea I said before, there are features
66:29 eddies which are circular um uh secular patterns, water which uh move around
66:39 the gulf of Mexico. And in large closed bodies also. And they
66:47 you can have circular motions of the of uh something like, I don't
66:56 1/10 of a meter per second, like that pretty fast folks. And
67:04 generally rotate uh clock wanders around and they affect the fish and they affect
67:13 navigation affect the oil platforms that are there. And uh it's all quite
67:20 complicated things like weather in, in the ocean, whether inside the water
67:29 the ocean, uh and of course velocity of sound inside those eddies is
67:38 than outside. The some cases actually reflections of sound off the boundaries.
67:56 , off the horizontal boundaries of slim and temperature in the ocean. You
68:03 see that the right experiment. You sound reflecting of uh interior boundaries inside
68:15 ocean. And there are people who . And then finally, I want
68:23 say um oh, another point making come back, go on a
68:53 Um Stephanie is this trip? If have 17 different solutions and waves or
69:03 , this is some also solution. . So that's really important. So
69:08 example, if you have a a of coming this way and a way
69:11 coming this way when they cross, just means that there's still a solution
69:16 they don't uh uh it is passed each other. They don't uh
69:23 they don't do this, they pass each other. And of course there
69:28 ways going in all directions all the inside the earth. So that's really
69:35 that as they pass through as they each other, they they don't change
69:45 all. They just add together and pass on. Really important feature comes
69:53 of the linearity. Yeah, all is for uh water waves uh waterways
70:03 that makes you way sound waves in water. I want really. That's
70:17 for us, especially for marine surveys really what we want to know is
70:23 and rocks. So that's called the waving. So go back to the
70:28 salt. Now we have a box the song and we're going to consider
70:34 stress instead of the pressure. Remember is a tensor. Okay, density
70:41 the same way location of the various Hello, various sides is the
70:53 And so here is distress at the of the box. We call it
70:59 component three J. So there's there's Jacobs 123 here, but it's because
71:09 top of it is has a uh the unit vector pointing in in the
71:15 direction. Uh that's gonna that's why have the three here. Remember tao
71:21 two indices. And so the first is gonna be a three on thinking
71:26 the top. But this is a equations, we've got the forces can
71:31 oriented in any direction 12 or but they're all going to be located
71:36 this position. Okay. And here's bottom. And so the only thing
71:44 about this is the location vector Here the side that you see now because
71:52 unit area is pointed in the one we're talking about how one J.
71:58 then on the other side again, won Jae but with a different position
72:03 up in the bottom, see this tau to J here. And so
72:09 corresponding forces are those stresses times the area. And now we have now
72:21 have uh various minus signs here minus here and a minus sign here,
72:28 minus sign here. And that's because force is directed upwards here and the
72:34 is directed downwards here. Oh all forces are carefully designated with minus signs
72:52 in the right places and not in wrong places. And also why on
72:58 I did this right, so that we add them all up and
73:02 this is a simple addition looks like matrix, but it's a simple edition
73:07 you see the minus signs of appearing some places here and not others.
73:11 so uh let's see what happens when compare with the scalar case.
73:17 so here is the scalar case uh it's got pressures instead of stresses but
73:24 um lots of similarities here. Now myself back, that's true. Enter
73:43 not have done that. Okay, um now we have stresses transmitted across
74:02 six faces. So the index one the direction of the, of the
74:11 . These are uh faces right in lap. Now 1, 3 here
74:17 one per year and then front and has uh different direction, direction.
74:28 the threes indicate the direction of the . And so this is all for
74:35 in the three directions. So you uh this is uh like a sheer
74:45 . It's got the forces are pointing the three directions And the unit areas
74:52 in the one direction. All of have forces pointed in the three
74:58 Some of them are shear stresses and of them are uh compression stresses,
75:05 example, here's the compression of stress here. There you go. Second
75:21 . Oh, okay, good news my wife. So this is,
75:41 , so I shouldn't have clicked The comparison with the scalar case is
75:46 here. Now let's specialize this vector for a wave traveling growth. So
75:59 that means that we can uh forced cancel out on the on all the
76:06 . And except for the this is similar to what we had from pressure
76:14 in the ocean as before. We're to use your tailor uh approximation two
76:25 days. And uh so after going the manipulation, same as we did
76:36 for the pressures, we find out the force in the three direction is
76:41 by uh this three greeting Uh down , 3. So that's uh for
76:53 vertically traveling. And then it's trivial generalize that for full force vector and
77:10 is only the force vector and the components of them. Of course,
77:16 the other uh the other components are here. Remember this is still vertical
77:22 only. So these are all vertical Forces in all three directions.
77:34 2, 3 Stress only on the . three planes because this is vertical
77:42 . And uh considering only various in X. So uh further it's easy
77:52 generalize for any direction. That's all do is uh change the threes to
78:00 and uh some over J. And an index notation uh which is simpler
78:10 . So this is a vector equation for three components. Let's look at
78:19 one at a time when we call the ice component and it's equal to
78:24 J uh J derivative of the J the IJ component of town. This
78:33 almost trivial to to uh generalize like because of the machinery we've set up
78:42 . Remember that because jay is We're summing over these. And so
78:50 riding through by the right watch, that. Yeah, force provided by
79:07 . Q. Is given by the of the stress. First per unit
79:17 is equal radiant after stress. Now into the equation of motion. Just
79:23 we did for uh uh motion What we do is you have
79:33 acceleration is equal for over the mass the force we've been working with the
79:38 . And so the acceleration is Given by the green of the stress
79:46 by the density. So the next is we write this out this acceleration
79:56 terms of the second derivative respect Right side is the same and this
80:04 the vector equation or motion with the explicit. However that's not what we
80:10 because uh what's the what's the unknown ? Is it displacement? Or is
80:17 stress? Uh The displacement is hidden the stress. And so that's what
80:22 gonna do is make it less uh hidden by using a hook slow.
80:29 here's what we had before. And put right in here uh Hook's
80:34 And this has got Stiffening the 4th Stiffness Matrix. And because this is
80:45 . J. Here, it's iJ and then Eminem here some of these
80:51 M. S. And ends. now we have um uh huh strain
80:58 the right and placement on the So we expressed the strain in terms
81:06 this. This is our definition of , small stream. And so
81:13 because of the symmetries inside here, can simplify things around. Uh You
81:20 uh all this later for yourself, it works out that this some cancels
81:29 this two. And left for this of this symmetry in the stiffness.
81:39 said. What we're gonna do then we're going to assume the medium is
81:43 . Take this C I. M. N. Outside here.
81:47 you will have, I noticed that trump in the earth. Uh there
81:57 various rock layers and so forth. so uh uniform the beating is not
82:05 in here. So I ask you bear with me for uh and give
82:14 a little slack here and we'll consider case first with the uniform medium and
82:21 we're gonna get to more realistic. this is however, not the wave
82:29 looks like it is because it's got these uh two respected time here to
82:36 respective space here, but it's not wave equation because this is not the
82:42 operator here. It's almost the way black part. We're not there
82:53 Okay, so there are various ways proceed from this point. Standard way
83:05 proceed. It was an argument due uh hair home loans. So we're
83:17 do that here. It's not my way. But uh let's do it
83:27 , because this is the standard way be frank. I don't know whether
83:34 not I spelled the name right? I think there's a T in
83:45 but I'm not sure anyway. He another one of these 19th century german
83:51 . And what he proved this which is described in the glossary uh
83:58 um they're more shadows the following that any vector function of a vector
84:08 You can separate it into two One part has zero curl and the
84:13 part has zero divergence. But let's about what that means. As we
84:22 yesterday we found that um gradient of scalar As zero curl and you remember
84:32 the curl is an operation uh which you, which tells you the curly
84:43 in a in a vector field. here's a vector field here, it's
84:48 vector defined as uh in a field position vectors. And we can define
84:58 curl operation, it's complicated with and talking about yesterday and if we take
85:04 curl, This part of it here said to have zero curl. So
85:12 know from the work we did yesterday uh This part here with zero curl
85:21 be expressed as the gradient of a because we know that the gradient of
85:27 scalar has zero curl. So if operate on this thing, uh if
85:43 take the curl operation on this curl here, in curl operation here,
85:47 one goes away because it has zero and that's what it says here.
85:54 similarly, uh hum host guarantee, that the rest of this unknown displacement
86:05 gonna have zero divergence. So that that uh that we can write it
86:11 the curl of another quantity. We call the vector potential. So cy
86:19 sigh with little air order. That's the vector potential. This is a
86:23 potential. And we know that if take the virgin the divergence of this
86:28 , whole thing, we're going to the divergence of this, that's going
86:31 be non zero. But the divergence this is gonna be zero. The
86:36 of a curl of anything is So when we separated into these two
86:42 , we introduced these two ideas of scalar potential and effective. So this
86:52 what you find in the glacier. operator del can be applied to a
87:00 like this to make a scaler. can also be applied to a vector
87:04 make another vector. And that's the operation. And here is the definition
87:11 , definition of the divergence operator and definition of the curl. Yeah,
87:23 back to what helm whole said had of this thing can be described in
87:31 of a scale of potential. So uh uh so let's concentrate on that
87:36 we're gonna call that because as we about the P wave as it's going
87:43 . It's uh not curly, it's of uh making uh dilatation and compression
87:53 dilatation. It's not doing this. intuitively, we think that uh curl
88:00 part is gonna be the P And so let's concentrate on that.
88:05 it's according to uh if we write this way in terms of a scalar
88:10 , we guarantee that it has no correspondingly the other part uh as a
88:19 waves. So as the shear wave growing along, it has none of
88:23 had no divergence. And so we write it as uh that's what we're
88:29 to call that. The shear wave , the solution has zero divergence.
88:37 , this separation of P waves and ways is uh doesn't work for anti
88:45 rocks. You can still help holds still correct. You can still separate
88:52 displacement into Pearl three part and divergence part. But those two parts are
88:59 P. And S. So we a different set of ideas to uh
89:08 uh P. And S. In psychotic rocks real walks. So uh
89:17 do that later. This is the way. It's good for tropic
89:22 Okay, so um uh what this is that um look for P wave
89:34 by putting in here, the P is part of the displacement field and
89:43 gonna um left side is is We just put it in here.
89:51 uh reading of fun here, we it vague. We just said that
90:00 tao we're gonna take we already know gonna take the gradient of town gradient
90:05 stress, but now we say that stress is dependent only on this scale
90:10 potential five. So in the expectation looks like this and The single derivative
90:18 respect to J. But for all different Jay. Yeah. How is
90:24 going to depend upon the scale of . Okay, so in using hooks
90:32 , like we just did before, here's our stiffness matrix and here's our
90:40 and uh here's our strain right right here. And you see here were
90:45 that it's only the p wave part it here, we're talking about.
91:02 since this is a function only of , we can write it like this
91:07 uh simplify this expression and now it's close to uh something we can work
91:20 but actually it's uh I said it , it's not yet close because on
91:26 left side we have secondary time, what we want. But it also
91:31 um um uh gradient with respect to I skipped over. Where did that
91:43 from? Let us uh Let's back to here. This is uh the
91:51 of motion that we have and now gonna take the uh the gradient of
92:02 . Why did I do that? because uh this curl free part
92:07 uh displacement is the gradient of So that's just a simple substitution here
92:17 displacement. And um but in the of potential in here and also here
92:23 here. You see where it ends here. And so uh working down
92:28 the uh the algebra, uh it's complicated expression and particularly we've got three
92:36 with position down here and uh mixed over here. So we have some
92:42 to do already. You can see this is complicated, It's complicated.
92:49 way this is three equations. Reports I equals 1 to 3. And
92:53 you examine this all these sums sums , em sums over K and sums
93:01 J 27 terms in each of these . So what a mess, let's
93:07 examining one of the these equations, component is uh so we're gonna choose
93:17 equals three right here, here's the component and uh a few. That's
93:24 it looks like on the right I'm gonna go back here and just
93:30 into a three right here and it J K. And M J
93:37 So there they are. So let's the sum over J explicitly. So
93:45 is uh J. P is one really respect. Uh That's one.
93:54 and that's multiplying uh that's operating on here, which is the rest of
94:02 . So take this part of this , stick it over here and and
94:08 show the sum over J explicitly Now at this point we're going to
94:18 the tropic elasticity. So here's our matrix with zero's over here, zeroes
94:31 here and now we stick that into and uh extinct that into this
94:39 And because of all these zeros, lot of those terms disappear. And
94:43 only surviving terms are these here. still a lot but um Start off
94:53 27 and 27 terms and were less one 1234567. Okay, so we
95:04 rid of 20 terms because of, assumed by Satrapi. So now let's
95:11 uh the two index work notation. , so right here, C
95:17 C 3131, sequel to C 44 so on. So now these equations
95:25 simpler and we're gonna collect terms uh , didn't we? We're gonna go
95:46 and do these uh summations. Once assume um linear, once we assume
95:55 psychotropic elasticity, we've got all these out here. And so we get
95:59 lot of zeros. But now these of over M and K are all
96:04 now. Uh and uh we're gonna those by recognizing that this one
96:14 These two are obviously the same Uh and it doesn't matter what's the
96:19 of Differentiation here, so that that shows up on the next page.
96:26 also we go from four uh index to and so this is beginning to
96:33 more simple. And now we're gonna terms and we find uh only these
96:41 terms that we have now sums of , ambulance and using the common
96:51 Now's a good time to change the names. So for example, this
96:55 31, C 31 is uh and C. 44 is view And
97:04 . 3 3 M. And uh here that we have Also uh C32
97:15 Lambda. And because of psychotropic and have two more here. So this
97:21 this some turns out to be the identically as this. Some and you
97:26 from the previous slide that these sums equal to M. And so all
97:33 terms are proportional. So we can out the M. And we get
97:39 the uh we get the little policy operator. So after all this uh
97:47 . We find for this vertically traveling , p way. Uh we find
97:55 simple wave equation like we have but instead of kor role here we
98:00 ammo and we can do the same for the other two components. And
98:07 we find the vector wave equation for waves looks like this. And uh
98:13 like we did in the ocean, can give this ratio a name,
98:18 gonna call it V. P squared remember all of this is about uh
98:23 wave from and you know from previous that aim is related to K and
98:30 in this way. And it's I it's better to give it a name
98:34 is launch launch funeral marshals because in way of inflation never find K appearing
98:42 itself only in this combination. You a name call again. So finally
98:48 get vector wave equation which looks a like the scalar wave equation except that
98:57 different. It has different um components the displacement in here and it's in
99:03 of displacement, not pressure. So like we did before, it's easy
99:12 convince yourself that all the solutions depend on the phase, here's a phase
99:18 in this three dimensional form, but the length of the wave vector is
99:24 in terms of the angular frequency and p wave velocity this way. And
99:31 you gotta remember that, this doesn't the various components of K. Only
99:37 length of stay. Yeah, this what we just uh derived. But
99:50 what we do is we can put say this is the gradient of the
99:55 the scale of potential, put it right here and put it in right
100:03 and then take these uh green operations here, like this one here and
100:09 one comes out here. And because the gradient operator is operating on the
100:17 thing, it comes down to the equation here, but see how this
100:23 is different from this one is this a scalar equation. This is where
100:28 is three equations in one vectors. is scalar wave equation. This is
100:36 scalar equation or p waves always in solid. It looks a lot like
100:45 scalar wave equation for uh waves in ocean. Remember in waves in the
100:52 there are no shear waves, there's uh p waves. So we didn't
100:56 to put on there. The subsequent remind us that it was the waves
101:02 because we know that in the the only sound waves travel the p
101:06 and we call the uh unknown uh we call the pressure or we call
101:14 dilatation. Otherwise this formula looks the . It's a scalar equation, not
101:21 vector equation. And the thing is we cannot measure fine, No way
101:31 us to measure uh this scalar But once we know there's a solution
101:38 like it's supposed we program our computers predict um synthetic science program in this
101:51 looking only for p waves. And we program it to make solutions for
102:04 from the scale of potential. So we end up with the answer in
102:11 computer, okay let's what can we to compare that with actual data?
102:16 we do not have any data for . But then we we can take
102:22 the gradient of the answer in the . The greeting of the solution.
102:29 that is gonna yield us um size with three components. You pete the
102:39 P vector. So why would we to do that? Well if we're
102:44 equations in the computer it's easier to a scalar equation than a vector
102:49 Right? Whatever numerical things you're you're doing it with a scaler instead
102:54 vector components. So it's easier to . Um Pewter should make it a
103:03 science program in terms of potential. then once you have the solution then
103:10 take you take the gradient of that in the here and that's gonna feel
103:16 acceleration for that uh synthetic seismic And then you can compare that.
103:29 , in a similar way, it's to show that if you work with
103:35 vector potential and the vector potential has equation like this for. and the
103:43 difference is there's two differences. Um this is the unknown as a vector
103:50 , not a scale of quality. the parameter of the material parameter is
103:55 shear wave velocity, p wave But otherwise it's the same. It's
103:59 two contributors we expected at the time it's got the little posse operator
104:05 And uh it's it has the same that you can you know, playing
104:13 from the this vector, what how is this vector related to the
104:24 real way? Remember this is a function, not not the displaced.
104:31 for a share way you have two involved, you have the direction of
104:38 of the share way and you also the direction of polarization. Sure.
104:44 . So uh which of those is anything um What is cy cy uh
104:58 gonna find the displacement function by taking curl, I'm sorry. And that's
105:06 give us the displacement of the Sure . And the uh the side victor
105:17 out points in that same direction of of the share wave did not point
105:23 the direction of polarization. So just with the P wave, it depends
105:31 on the face. But the face is is his faith. Is is
105:37 like this where it's related to the with the sheer velocity of the p
105:47 . So I want to know what in this place. Let me just
105:50 the curl of this equation. This us an equation affect your equation for
105:57 rates. This displacement is now perpendicular back here we decided that uh side
106:16 in the direction of propagation but the is sideways to that and it's perpendicular
106:24 the to the operation. So let's about think about the solutions to these
106:40 Mhm. With uh to these vector equations, the solutions have uh you
106:50 have the um represent, represent represent solution any problem as some of a
107:01 solutions like plane waves, plane waves often the best I could use um
107:09 solutions I think about the solution to problem as a sum of plane
107:15 And in that psalm there's it's a sum. There's lots of different uh
107:22 parameters And uh those are gonna be by the initial conditions in the brown
107:30 conditions. And I guarantee you that when we do all that we're gonna
107:35 real quantities on both sides of the . In all cases the solutions are
107:44 of the phase function which is defined this way where the length of the
107:49 vector is given for p waves like , your waist like this because someone
107:57 this is still a solution because these linear equations and waited some still a
108:04 . Yeah, let's have a little and then take a break Stephanie or
108:20 equation of motion is the starting point this derivation of of the venture wave
108:30 false. Okay. So where do think the the starting point is?
108:55 . Well we started off with Equals and uh we start off with
109:00 forces and then we we put them an equation F. Equals M.
109:04 . So that's the equation of And and the only difference is we
109:07 uh looking at all the vector components of the pressures. So the answer
109:13 this one is true. Okay, this true or false? Read
109:30 Yeah that one's true. It's just matter of properly accounting for all the
109:36 on the box. Remember that? got to read it carefully and think
109:46 it because there might be tricky for . That's correct. Very good for
109:54 . Very good. Yes sir. is I have to think about this
110:12 lot. So think out loud as thinking talk. Um Well I can't
111:01 d because you know what pressure, don't know what potential is. I
111:05 it's kind of a fuzzy idea Uh There's gotta be a difference.
111:19 Yeah, so let's talk through Let's start with a the so called
111:24 and a P wave in a silent really launch, you know,
111:27 Not corporate, Is that true or That stable? I'm gonna say that's
111:35 but uh that's not an answer to question. Okay so uh now uh
111:43 two is we program our computers to uh the equation for five for increased
111:50 because it's easier to calculate the scalar vector also true but not in answer
111:55 the question. Um I think you're . I think that the answer is
112:03 because the pressure P. Is the , whereas the potential fire is not
112:08 . Instead we can derive from by observable which is uh displaced.
112:16 So I like that analysis. Look the bottom number D. Is all
112:24 the box. So so you want go through these, talk your way
112:28 this and say does he could start a is that one true or
112:47 Okay that's good. So I'm not to be okay. Now here is
112:59 we got uh so we got two them are true. So we don't
113:04 to just assume that uh C. true. Uh and give and give
113:11 answer D. Because it might be . So let's analyze see as
113:16 is that true? The displacement field a shear wave has no divergence
113:27 This property of the way. And nous is this property. And so
113:33 the sheer way the displacement doesn't do at all. It only does this
113:38 it doesn't have any divergence. So true. Also so uh all of
113:44 above is um correct answer. Okay now uh this is a good place
113:52 break. And so uh let's do . Let's take a quick break and
114:02 back. Let's see um Let's come at 11:00. We'll resume at
114:10 Okay, So we resume the lesson a short break. And so what
114:17 just went through with the potentials and is the standard way of doing
114:22 But I was I've always been uncomfortable that myself because of these potentials.
114:28 me, a potential is sort of fuzzy concept. Uh And uh it
114:33 uh it's not observed, it's not . And uh we already said that
114:40 ahead is not gonna work for an tropic materials uh even though how most
114:46 still works, it doesn't uh the free part is not um it's not
114:54 P wave and uh diversity free part not sure. So let's look at
115:02 same issue um without potentials. So have this equation, here's the equation
115:11 the uh victor way field. And got all these complications and all these
115:22 over the various components. And notice that I uh instead of writing down
115:29 um uh for the component, we're really need to use that symbol I
115:36 it squirt or -1. So so this changed the name there of that
115:46 of that index. And so we to uh So this displacement could
115:53 you know, lots of different p in it from coming from all
115:58 And it could have share waves as . And so um let's think about
116:05 to separate that separate out those parts assuming potential function. No. Um
116:18 since we are two physicists, not , we're gonna solve this in the
116:24 rate. We're gonna guess the answer then we're gonna verify the guests.
116:31 , So we're gonna we're gonna recognize any solution can be expressed as a
116:37 year, a sum of plane So you recognize Stephanie, the plane
116:42 thing here, uh orders number with exponent I to the phase factor.
116:49 then out in front we have uh have a scaling factor awaiting function.
116:57 you see it's a vector just like full displacement is a vector. And
117:06 gonna sum up terms like this. the the important new idea on this
117:13 is I've used the term for So, um you are familiar with
117:21 your analysis of uh wiggles for and any time function. You're familiar
117:30 the fourier analysis of that time. uh and you can read more about
117:36 by the way in the blossom, look up for you. But there
117:46 will find, let me see here the glossary, you will find that
117:53 just like you can do for a a decomposition of a time series,
117:58 can do for a decomposition of a of position. And you can do
118:07 a decomposition of a vector function of and you can um do it
118:16 you can do a joint for your in time and space. And in
118:22 these cases. What fourier proved was uh East fourier components, which we
118:33 deduced using the recipe given by Fourier sufficient to describe anything. Any function
118:43 time and space can be represented as from as a sum of 40
118:49 Like I just described whether uh and kind of puzzling because uh you know
119:02 any fourier component is a wiggle, co sign or a sign that goes
119:08 for about. How can we possibly a localized time signal which has a
119:17 point and an ending point? How we make uh that out of a
119:22 of um oh science and co signs goes on forever. And the answer
119:31 that you can do it and uh choose the uh the aptitudes in that
119:42 and that for you some assume you the the amplitudes of each ah of
119:52 term in a clever way. So reinforced together for short times and then
119:58 a long time they all cancel each out. So you get a localized
120:03 function and the same thing for a space function and the same thing for
120:08 time and space together Now. So call that a basis. Uh we
120:15 that the 48 basis. So uh these signs and co signs with uh
120:21 various uh frequencies and wavelength and so . Uh it's called a basis.
120:29 so So the reason I use that fourier right here is that those basis
120:36 are in fact plan weights. So these plan wave term that I'm showing
120:41 are exactly uh 48 uh basis functions we just talked about. And so
120:48 why it's so useful to talk about waves because although we never see in
120:54 mix, we never see a plane . We know that any solution can
120:59 expressed as a sum of plane wave those. Uh And that's true for
121:07 scalar function like a time series. true for uh for a vector uh
121:14 like uh display shin the J function the J component of dysplasia. And
121:19 it means is that the uh waiting terms are directors and we call these
121:31 that the collection of these uh waiting for the various four year terms.
121:36 called the spectrum. So here uh upper tissue gives the spectrum both in
121:43 and space uh for the collection of waves. So we're gonna make this
121:52 and then put that into um the of motion. And when we uh
122:01 we do that and make the two derivatives from the equation of motion on
122:06 left side, that brings out a omega squared. Why is that?
122:11 need to uh we need to talk that. Let us stop right
122:18 I'm gonna stop right here. Don't the recording. Um So what I'm
122:23 do is get out of this file I'm gonna open up lost three
122:41 Can you see that on zoom? . So I need to stop sharing
122:49 I have to re share. Okay now we're in the glossary. And
123:02 let's go down and find uh oilers . Here's oilers number. It's a
123:19 number named after Leonardo or lor and the special thing about uh oilers number
123:28 you have Oilers number and you raise to a Power. Uh And I'm
123:34 call it power as A. Times . And then take the X.
123:37 of it. What that means. that because of the special properties of
123:43 . That derivative looks like the original . Each of the K.
123:48 R. Now multiplying by K. , now um let's see here.
123:57 This is uh I don't know whether is gonna work. I'm gonna click
124:03 and see uh thanks see where it to take me. It wants to
124:14 me to Wikipedia. So in Wikipedia lots of information about oilers number.
124:23 And so I didn't wanna repeat that . But here's the expense of the
124:28 property. When you take the derivative either the K. X. You
124:32 K. Times either the K. . Okay so um uh Stephanie uh
124:41 . Is that new to you or ? Mhm. Okay. So uh
124:49 now remember that. And and uh e is the only number in the
124:55 which has this property, so that's uh it's named after its discoverer.
125:02 I guess it's a german guy uh and I'm not sure uh where this
125:10 was taken from, I think this is taken from Wikipedia. So uh
125:20 stop sharing this and go back to lecture and I'm gonna share the
125:34 share my screen again. Is that on zoom now? Okay, so
125:49 me put this into presentation mode. , so what we're gonna do now
125:57 uh use that special property of E the wave equation and in the wave
126:04 , you know, we're on the side, we're going to take two
126:07 with respect to time. So when take two derivatives of this, you
126:11 I two of uh this is a e upper case U is a constant
126:17 course. Uh and uh two derivatives you. Uh two tutors of
126:25 the I omega T with inspector T i omega squared squared, omega
126:31 which means minus one is I squared squared. So this on the left
126:36 the result of taking two derivatives of and on the right is to derivatives
126:43 respect to uh directions. Uh L case I'm gonna back up here.
126:55 we have we need to take uh disrespect to K and N. So
127:10 . Yes. Okay, so this a little bit of bad pedagogy for
127:17 . So let me back up And so we have here uh index
127:23 . Obviously we're gonna get confused with when we talk about the wave number
127:31 . The length of the wave vector . So I changed that K.
127:37 L. Right here without telling And I should have told you I
127:45 go back and fix that up. putting this gas into the wave equation
127:54 this and you see this is there's more derivatives here. All the uh
127:59 got taken out and we did the when we uh when we applied the
128:08 we got a minus omega squared here we got a minus K.
128:12 Times K. In here. And what what was left was the original
128:18 function which got canceled out on both . And so this is just an
128:24 . It's actually three equations, algebraic , node privilege anymore. And what
128:29 the unknowns in this equation? it's the uh it's the use so
128:36 you j here and then we're gonna U. M sum with this M
128:43 , 12 and three. So this called the crest awful equations. Another
128:50 think was Um Dutch. That's right , 19th Century Mathematical Physics. We're
128:58 head. I said so this is the equations of motions for a single
129:03 wave for a component. So then have this constraint between omega and
129:12 So we put that in. So gonna divide uh huh by both sides
129:21 by K square. And so on left case grave. Uh on the
129:27 side of omega spirit be right about period, it's gonna be philosophy square
129:32 on the right inside and see a making a ratio with these others.
129:37 you recognize already these uh fractions here give the direction of the wave called
129:46 cosine. So uh polarization, uh know, the displacement is driven by
130:01 uppercase uppercase use and the way the is going is the way the wave
130:07 propagating is hearing by these caves. now let's talk about what kind of
130:15 we can get. Well, the solution is just to assume that uh
130:21 . Is zero on both sides. uh that's that that solves the equation
130:30 matter what is V. But that's useless solution. We call it the
130:35 solution. And so we're gonna seek solutions which are only possible for discrete
130:43 of V. And as we can get uh find certain values of the
130:48 in terms like this Uh for which use or non-0. So let's work
131:05 out explicitly. This is three equations one. So let's write out the
131:10 uh equations separately. And um we're gonna assume my socks. So
131:24 when we do these sums here because so many zeros inside here. Uh
131:31 lot of terms cancel out. You this is some something over M and
131:36 and hell. So there's uh 27 included in here, but a lot
131:44 them are zero because we're going to my socks. And the only ones
131:47 are left are these three equations in here are the three different equations.
131:53 you see what we have here. is the equation for you one.
131:57 on the right hand side it's also you three. So here's the equation
132:02 you three. And on the right side it's also got you one.
132:06 these two terms are a couple of two equations are coupled together, but
132:09 middle equation is not see we got you two on both sides of this
132:18 . So, let's solve this one . This will be easier. So
132:23 the solution for a plane wave propagating at any angle which is polarized in
132:28 two directions. So what are the of, of uh propagation? Or
132:35 given by K. one divided by , and K. Three divided by
132:42 . Whatever you choose for these, are going that's going to determine the
132:46 . And uh so uh intuitively we that uh propagating at any angle in
132:59 13 plans. So look, let's that the 13 plane is the plane
133:03 the screen so this is the one here, in the three direction
133:07 the two direction is out of the and and into the plane. So
133:13 if the wave is propagating in this , but it's polarized if if the
133:19 are moving out of the plane, must be a share wave. And
133:23 , it's what we call an H. Wave because it's polarized always
133:27 the two directions independent of the angle propagation here. So, you know
133:36 SV waves. Those are those are be polarized in this plane. But
133:42 this is a um solution for the the S. H. Way.
133:51 by definition the polarization is like, so here are the other two
134:03 So first we're gonna seek a Certainly polaroid solution. So we're uh
134:10 gonna seek a solution which has uh uh in the direction of you.
134:19 here is you. And so uh . One and K. Three um
134:25 the direction of polarization. Uh So and this is a unit vector for
134:31 you. Uh So let's assume that you is pointing in the same direction
134:40 the wave. So that means that you is gonna have a vector
134:46 uh K. One over K. then uh component in the three direction
134:53 three over K. And so we put that assumption into here, it
135:01 matter which one we uh put in we uh use the velocity, which
135:07 showing right here that is given by . O. Ro And so we're
135:12 to call that the p wave And so our assumption of Washington polarization
135:21 produce um away traveling in the uh the same direction as the polarization,
135:29 direction as the displacement with this And so we call that the,
135:37 our guests for p ways. Now seek a transverse fully polarized. In
135:42 words, one uh we arranged the . Three and the K. One
135:47 this. This vector of uh unit displacement is gonna be perpendicular to the
135:55 of propagation. And why did we this? Well, we choose this
135:59 uh this dot product between you and . Zero if we choose you to
136:08 like that. And so if the product of U. And K.
136:13 , then those two vectors are perpendicular each other. So we're now going
136:17 find a solution where um where the to the obligation. And we insert
136:34 in decoration. And we deduce you follow this through on yourself when you
136:39 the V. Square is uh mu which we're going to call the sheer
136:47 for an S. V. Wave in the same plane. But a
136:54 wave also polarized polarized in the polarized perpendicular to the propagation at at
137:02 points. And it happens to have same shear velocity as the S.
137:07 wave. We didn't know that when started out, but now we deduced
137:12 that's true SV waves and S. waves have the same velocity for extra
137:18 media. And the only difference. The only difference is how those two
137:26 interact with a boundary. So when talk about reflections in next week we
137:33 find out that S. H. reflect differently than sp waves. But
137:38 that's a boundary. That's an issue arises when we have boundaries here.
137:44 haven't mentioned any boundaries or inside of body. And we have an SV
137:48 propagating this direction and S. Waves . Uh Both the SV waves and
137:57 S. H. Waves are propagating this plane, the end of the
138:02 , but the S. H. were polarized perpendicular out of the plane
138:07 the S. P waves are always the plane. So we confirm that
138:12 are um assumption of transverse polarization is . And we found the sp way
138:26 topic. So for as your tropic , we offer this call them S
138:33 an ice tropic bodies that's not going be right for an ice tropic
138:38 It's gonna turn out the same process to two different share wave velocities.
138:45 two different share waves. Two different waves propagating with different velocities in general
138:55 anti psychotropic bodies. And we're gonna the same process, use that um
139:05 a couple of years time. So found separately p waves and share waves
139:12 using potential function. And to me a much better way to think about
139:19 . I've always found potential functions to sort of a vague fuzzy concept.
139:29 call this a degenerate situation when we two different to uh to share waves
139:35 are completely different uh polarized in uh different directions at the same velocity.
139:42 call that a degenerate situation. But we have anisotropy that generously disappears and
139:49 have two different share velocities, Those uh properties for those two waves.
139:55 think about this, I'm gonna go um go back to this and uh
140:05 that gives me a clean screen. I have share waves traveling in this
140:10 with uh polarization in the plane. previously we found share waves traveling also
140:18 this, in this direction or any direction. Let's concentrate on this
140:23 Um with polarization pictures out of the also traveling with the same velocity.
140:31 what does that mean if you have you have uh shear wave traveling with
140:39 polarization which is neither in the plane perpendicular to the plane, but at
140:44 angle to the plane? Well, can be broken down into components parallel
140:49 perpendicular. They both travel with the velocity. So that one that one
140:54 which we just hypothesized with some random perpendicular to the propagation, but a
141:01 polarization in that plan, perpendicular that also works also in ice economic
141:10 . So the polarization can be uh in the perpendicular plane for ice comic
141:18 , but in anti ship topic that's not true. Only those two
141:25 directions that we just found uh work an ice traffic media and for an
141:31 traffic media. Uh If you try I don't know, I have a
141:39 wave with an intermediate polarization, it itself into so much of the uh
141:46 SV wave and so much of the . H. Waves. And those
141:51 each with their own velocity and so get one gets ahead of the
141:56 And so um that's just an example how in real rocks which are an
142:02 topic, everything is different for share . Okay, now let's think about
142:16 a homogeneous wave equation in homogeneous wave . So uh start with the vector
142:27 equations for p waves as we And so I moved the both terms
142:37 the left side of the equation. you will notice that in all terms
142:41 the all the non-0 terms here. unknown appears. And so that's called
142:47 that's called a homogeneous equation homogeneous Now we're gonna put another term on
142:58 right hand side which doesn't have the in it. So that's gonna make
143:05 in homogeneous equation. That's what I said right there. Now we're gonna
143:11 a point source at the origin reading in all directions. And so the
143:18 wave equation comes like this. And here we have um the chronic consumer
143:32 we have the direct delta function Which zero for all Radius R. except
143:43 Article zero. And it's radiating with vector wave which is uh could have
143:51 P waves. No, it's only have P waves in it because we're
143:56 here with p ways. Uh It's um have a time function which the
144:03 decides if he if he used that's one time function, but he
144:11 want to do something else. For , here's an example um um In
144:22 land acquisition for p waves, we don't use dynamite anymore. We normally
144:30 vibrators. So the vibrators are not to be uh expressions like this which
144:37 radiating uh where the uh space function equal in all directions. A vibrator
144:44 going to be focusing its energy So that kind of source is not
144:50 here. This is this is a source where it's radiating equally in all
144:56 . For example. Um If you an air gun in the marine
145:02 we can still use this vector wave in the marine environment. All we
145:07 to do is we set a new zero and suddenly presto change o the
145:14 wave velocity becomes uh the square of care of the world. So this
145:20 this expression is also going to be in the ocean. In the ocean
145:25 typical sources are air guns. And the air gun uh emits um oh
145:36 of compressed air into the water. the pulse expands, bubble expands and
145:43 it over expands and then it like, so it has various false
145:50 of the bubble. So that makes extended time series like this uh that
145:58 out spherical waves through the water in directions, the same in all
146:03 Um but it has a complicated time . So that's why I've left
146:08 This time function to be uh not , but I did specify here that
146:15 gonna be radiating equally in all Just for simplicity. Uh the solutions
146:23 uh vibrator are more complicated, not to be handled in this court.
146:33 will know uh Previously we we ignored term. We just said what happens
146:43 as a wave comes along and hits Vauxhall? What does the Fox will
146:48 with it? Well, the Vauxhall it along according to the material properties
146:53 the box. Uh but now we're , where did this wave come
146:58 Well, it came from the origin with this set of uh time various
147:07 to the source. Now notice here the unknown does not appear on the
147:13 hand side. So the mathematicians call this an in homogeneous wave equation.
147:21 a vector wave equation. And it's p waves because we said it's for
147:24 waves right here. So we're going find the solution in the next election
147:30 now. Uh I know that because origin at r equals zero is a
147:40 place. It's obviously better to use place um operator in spherical coordinates without
147:46 variation, which is given by So psi right, right here we
147:50 the place an operator and we could that in terms of Cartesian coordinates,
147:55 we would get into difficulties because here have the radius here. So we're
148:00 use the uh operator in spherical coordinates is here and right here, you
148:09 immediately see that at the origin where equal zero. That's gonna make a
148:13 because we're dividing by zero squared Also we're multiplying by zero squared over
148:20 . So things are gonna work it's it's not gonna be a
148:25 But we are gonna have to do little bit of work to express.
148:30 Yeah. You know, with that , when we have a source.
148:40 , so is this question for false homogeneous wave equation is the same as
148:48 derived earlier. Except that it applies non home genius, layered subsurface formations
148:55 true or false? That's false. . Right here, right here we
149:01 uh that it's because of the form the equation equation does not have the
149:07 on the right hand side. So what makes it in homogeneous equation.
149:12 the if the if the medium is homogeneous, that's gonna affect uh This
149:19 the material properties are given here. is the wave, this is the
149:24 right here, and this is the . The materialist is crashed here so
149:30 this statement is false. Yeah, are false. Is it the same
149:36 we derived earlier, except that it an extra term describing the source of
149:40 wave. That's true. Okay, now, now we're gonna do the
149:48 closer to the real Earth. Now nine uniform wave equation. Now here
149:54 where we previously assumed uniformity. Back page 63, we were working through
150:00 algebra and we wanted to take this , the stiffness tensor element out of
150:07 derivative. So we just assumed that doesn't vary with X. So we
150:12 put it out here and then we left with this and we eventually turned
150:17 into the wave equation, and actually separate wave equation for P N for
150:24 . So, now, what we to do is consider what happens if
150:27 don't do that. So, using chain rule calculus, we we find
150:32 this same right hand term has two to it. One which describes the
150:41 of this part, that's this, one which describes the variation of this
150:46 , which is this And so here have the gradient respected J. Of
150:52 stiffness element here. Right here, it's assumed to depend upon it.
150:59 the gradient is given right here. previously we assume this term is
151:06 but in the Earth, you that's not zero. So what are
151:11 gonna do about that you term, expresses the non uniformity of meeting.
151:21 , now, so let's suppose now have a layered me and suppose the
151:29 are this thick, we've got another down here, another just uh in
151:35 layers that pick down, suppose I'm ask you, Stephanie, uh are
151:46 gonna need this additional term here for short waves? The wavelength is only
151:53 long and we got a sedimentary layer is uniform within itself. We've got
152:00 wave run through there. Are we need this term here? Yeah,
152:08 right. We're gonna not gonna need for terms like that for terms like
152:13 or for ways like that, this going to go through the media.
152:18 I'm gonna use, my other I'm gonna go through the media thinks
152:20 coming up and it doesn't know about boundary, it doesn't know about the
152:25 until it gets this boundary. And then some of it's gonna reflect someone's
152:30 uh transmit. So that will be boundary problem. We'll talk about
152:38 get reflections and transmissions. But before gets there, it's propagating happily inside
152:44 inside the layer and it doesn't know eventually it's gonna hit a boundary,
152:51 just propagating. And so we don't this term and we can get away
152:55 neglecting it just like we've been neglected suppose that the wavelength is really long
153:00 the wavelength goes from uh the ceiling the floor and it's going through a
153:05 of layers like this. That's gonna a problem. That means we're gonna
153:10 terms like this to describe that And that's the real problem that we
153:16 have in uh in the earth where almost all seismology, we have uh
153:26 which are long compared to the scale of the in home of genetics.
153:34 that's gonna be a problem which we're have to deal with. But
153:42 And the reason we know that is could be in homogeneity on all
153:49 Let me see how we're doing with . Okay, we're doing fine.
153:54 this is a special kind of log is called a more whole image.
153:59 so this is a tool which is inside of a more whole and makes
154:07 uh um uh It acquires data as pulled up through the bar home.
154:15 as it pulled off its spinning rapidly spinning maybe uh Um 100 times a
154:25 or 50 times a second. It's really fast. And meanwhile it's being
154:29 up slowly as it's spinning around. emits high frequency sound uh into the
154:37 fluid, right? Uh You understand the borehole is normally gonna be filled
154:44 borehole mud. Why has it got in there? Well, uh it's
154:51 uh outside the bar hall there might uh fluids in the formation outside the
155:00 hall probably bright, maybe oil, gas, but fluids and they're going
155:06 be under high pressure. We don't those fluids to come into the
155:10 Oh, so there's two ways to that problem. One is to put
155:15 casing on the bar hall. But we do that, we want to
155:21 more about those formations before we hide with the cases. So we send
155:26 tool down, it's called a borehole tool and it's coming up and it's
155:32 in order to keep the fluids from formation out of the borehole, we've
155:38 to have mud in the bar hall enough density to make enough pressure in
155:44 borehole mud so that it's more pressure the mud than in the borehole
155:52 So we don't wanna have it uh don't want to have the mud too
155:58 because if it's too heavy then it's be leaking into the formation. We
156:03 to make it just a little bit . It's the pressure is just a
156:08 bit higher than the formation pressure even it's gonna leak out a bit.
156:14 the way it works is that as leaks out into the forest formation,
156:18 pores screen the mud particles out of mud and it leaves what they call
156:25 mud cake on the um On the of the uh four hole. And
156:35 mud cake uh is actually a good . It uh it decreases the
156:41 it clogs up the pores and decreases permeability in that mud cake so that
156:50 the mud particles stopped flowing out because get something clogged up. Okay?
156:55 it's only occupying the pore space only have to find the four space.
157:02 now this this uh that's why we mud. And so this tool is
157:09 pulled up up the bar whole spinning emitting high frequency sound. And
157:15 and the high frequency sound goes out the mud reflects off the bar hall
157:19 comes back to the instrument. It a receiver directly located at the
157:26 so the the offset between source and zero. And so then it records
157:32 amplitude of the sound which is coming . And so the amplitude of the
157:37 obviously depends upon the kind of rock is in the more whole wall.
157:44 , so now as it's being pulled , you can see that the different
157:48 here give the different amplitudes of reflection the reflection sounds. So they are
157:57 as it's being pulled up there showing different mythology. And so you can
158:03 the layers here. Now it spins and this display is unwrapped. So
158:08 goes from 0 to 360°. And so for some angles there's uh parts of
158:16 tool or in the way. So white lanes here, that means no
158:19 in that as a medical direction. as it's being pulled up, the
158:25 tool rotates slowly as it's being pulled . So that's why these asthma's are
158:29 straight up death. But you can see the layering on this scale.
158:37 20 m showing here and now. let's zoom in zoom in here.
158:44 , you see more layering, Let's in again, more Larry, more
158:48 . This is zooming in uh 10,000 and on all scales blaring.
158:57 maybe this is not true for all for all sequences, but it's extremely
159:02 that you never have this situation where have a locally uniform, we'll always
159:11 that's true. So we neglect that homogeneous term. But so we pretend
159:18 it's homogeneous on the small scale and only have to worry about it at
159:22 risk at this layer boundaries. this says that there's a layer boundary
159:28 , no matter where you're looking. this this is a problem that we
159:32 to deal with. Uh we're not to deal with it. Now,
159:36 gonna deal with it later in the . This is exactly the kind of
159:39 world problem that most courses like uh ignore, but we respect your
159:47 and we uh we know that you've been out of school for a couple
159:53 years. You've been working is you some knowledge of the of the real
159:59 and you know that when I it's homogeneous uh layer like that?
160:05 probably nonsense. Yeah. Yeah. you take a tube sample and uh
160:16 how did how did you collect those samples by the way? Just a
160:26 of. Okay. Yeah. So normally these are fairly shallow.
160:39 Okay. And so the way they it, do you know how they
160:43 it in the field? Yeah. . So I think I know and
160:51 my company made a big um advance this years ago. So what we
160:58 was um uh well I'll suspend this I'll tell you this story because it's
161:05 kind of an interesting story. I at Amoco's Research Center and a lot
161:12 high quality people there and we had geophysics division and geology division and drilling
161:19 and so on and a computing And uh one of the senior guys
161:28 the drilling division had a bright he said you know these silly geophysicists
161:35 around with all their silly uh vibrators services and receivers and everything. Get
161:42 this data and do all those silly and they still don't know where to
161:46 the arm. Why don't we find by drilling? So this is an
161:53 that comes naturally to a griller. There's two answers to that. Uh
161:59 is drilling is too expensive. We just drill random. Uh But what
162:05 could do is we could say well don't we just uh drill in places
162:11 it's likely? And it's and the to that is it's still too
162:15 And so why is it too It's because we drill every hole with
162:21 expectation that we're gonna find oil in . And so the whole that we
162:25 is sufficient size to produce that Why don't we drill our holes in
162:33 different way, Looking for information, for uh not prepared to produce.
162:41 if if we drill the hole for only, it could be a lot
162:47 . Now, you know that if gonna drill down 10,000 ft, what
162:51 do is we start off with a which is maybe 30" in diameter.
162:56 as we go deeper, the the whole gets smaller, we line the
163:04 uh stretches of the hole with steel to keep the fluids out, formation
163:13 out. And then we drill deeper a pipe of smaller diamonds, we
163:18 the smaller diamond down through the big pipe. And so and so the
163:23 gets smaller as it goes deeper and that steel, and then big,
163:32 drilling rig. And so that's what the expense. So we could downsize
163:37 whole thing, Looking only for It could be a lot cheaper.
163:44 we invented at Amoco and then a where we could uh drill a hole
163:50 off with 4" diameter And ending up 10,000 ft 1" time. And the
163:59 we did it was we had a , a 40, a 40 ft
164:06 , just a simple pipe with industrial on the bottom on the pipe.
164:11 we spun that pipe in the hole then every 40 ft we would extract
164:18 easily, bend it and uh extract core, 40 ft long core all
164:25 way to the surface and then uh out the pipe and send it back
164:30 . And uh then eventually we'd have start decreasing diameter. But the whole
164:38 is a lot smaller and you see uh it chewed out um only the
164:46 . Whereas uh standard drilling choose out whole selling. So we should not
164:53 be honest and we uh accumulate all 40 ft course. So the first
164:58 they did this and by the way was done uh very high tech,
165:04 all the results uh monitored by computers I think tele meter back to our
165:13 in Tulsa in real time. Best in the world were monitoring this.
165:18 the bells and whistles. You can . Then after a while somebody noticed
165:23 the rig, of course the rig a small rig, not a big
165:27 , somebody noticed. Hey, look all these 44 cores here. I
165:32 if the geologists might be interested in at these cores. So, since
165:37 were drillers, they had no concept other people could contribute to the
165:43 So they said, well, and course the geologist said of course we're
165:47 . So they put together a a portable geological analysis laugh that they
165:55 bring onto the well site and look these four different courts. And also
166:00 was in the also that they asked uh you'd be interested in looking at
166:07 these parts. Well of course what do is we'll take uh subsections of
166:13 cards and measure the losses, and density and everything else we can
166:18 of. So we made an affordable to go to the well site and
166:23 measurements. So it was quite a system and we drilled maybe five holes
166:30 this, all of them successful. if you solve the pressure of uh
166:38 pressure, mhm. Even solve the . Um What would happen if we
166:48 into overpressure formations? Didn't have any measures for then against the possibility that
166:59 pressure fluids from below can come up this annual list and uh wipe out
167:06 crew. So uh the middle of we were solving that problem, BP
167:13 us and they threw away the whole . So it was not, they
167:22 made a good contribution but it was it was really interesting for us.
167:33 early stages and then we did a thing. We um we took our
167:40 laboratory to London and we put it the parking lot in the head office
167:46 London. He invited all the other companies operating in the North Sea,
167:52 was frontier area in North Bring us samples and we will we will give
167:58 a real sonic log made uh made of vertical while they couldn't do
168:03 See we we could sample every every if we wanted. We could we
168:08 sample every 10 ft. But we to uh every other operator in the
168:14 Sea bringing your conventional course and we'll for measure them for you cheaper than
168:21 and do it collapse offers this We did it for half price.
168:28 piste off core labs but got got lot of business other companies bringing us
168:34 course and we made it basically uh made no profit all for that.
168:41 we did. What we did do we got a complete database all that
168:47 from all over the North Sea. uh just like google is saving your
168:54 whenever you use google. We saved information to good permission. And so
169:00 we accumulate a very extensive database of measured uh from cores. Oh ah
169:18 look very efficient set up uh in efficient lab, it was the best
169:26 physics lab work. That purpose. the interesting thing that I found you
169:41 core out of this kind of rock shells sequence like it's filled with navy
169:51 and it's filled with foods which are high pressure in situ. And then
169:57 you pull it out, the fluids out. We got those of course
170:06 under high pressure within half an hour they came out and measure the
170:12 Then we set the samples on the , come back a month later and
170:16 them again. Guess what they had because it had dried out and made
170:23 changes once when we drive after it dried out and tried to re saturate
170:28 . Uh press it again to the pressures dollars because of uh on elastic
170:44 of permanent effects caused by uh drying of the sample just as it sat
170:52 on the shelf. And so what concluded was that every sample that had
170:56 ever previously been measured was wrong because didn't have it had in it the
171:02 of of desiccation from sitting on the for being transported from wherever it
171:09 Nobody had ever measured a fresh sample we had. And so what this
171:16 that every core measurement that had ever made was wrong for that reason may
171:22 a little bit wrong. Maybe a wrong, depending on pathology and the
171:25 they handled. So naturally core labs not very happy with this. That
171:33 the major finding. Anyway, of we verified that we have um Larry
171:39 homogeneous on all um on all So uh let's now think about what
171:49 that, what is that going to for us when we ignore that in
171:54 in our in our equations. So consider a simple case with laying
172:01 So that means that the derivatives in two directions, horizontal directions are zero
172:06 let's consider only vertically traveling P ways that U. One is zero and
172:11 20. So we're only going to at this equation for you three.
172:15 this is the wave equation term that familiar with now and this is the
172:19 term and it's got in here uh with respect to a vertical position Z
172:30 the launch channel modular. So the is in here uh and variation of
172:36 modular is showing explicitly here and notice this has only a single derivative whereas
172:41 has two deliveries. Okay, so gonna be a problem. Um we're
172:50 gonna solve that problem today. Just it to your attention and I give
172:55 here a little quiz. This that's . So this is very similar to
173:03 question I asked you earlier with only few differences. Yeah, yeah,
173:20 this one is true. Okay, you will come back to that real
173:29 issue later in the course. So let's talk about solutions to this.
173:39 , so this is uh solution. is a a nice graph taken from
173:45 and guild art which I recommend that should buy and um uh it shows
173:52 a function of arrival time here and a number of different lines of uh
174:00 . And so let's uh let's look the first one here is the direct
174:05 which is going from the source at to uh to uh all these different
174:12 . And it's a straight line which that the velocity is um a
174:17 And it's and uh we're measuring in surface of course. And of course
174:21 source the sources of the surface. it's uh propagating horizontally And in this
174:29 it has a velocity of about 650 . That's what they assume in
174:38 And of course they calculated this with model. Now look at branches,
174:47 . And C. Here here is . B. And here is seen
174:53 these are refraction. So you don't yet what a refraction is but I
174:58 tell you that a refraction is where body wave, like we've been talking
175:05 , body is inside the body of hits a boundary at a highly oblique
175:12 and some of it reflects and some the uh refracts. And uh uh
175:20 me say uh differently, some of reflects and some of it transmits and
175:28 amount of reflection and the amount of varies as function incident. So for
175:34 incidence angles that transmitted uh transmitter wave also as a parallel to the boundary
175:43 going horizontal. And that's what we're what we call refraction is here and
175:48 travels with the velocity of the lower . And so here you have to
175:55 layers. And uh B. And . Now look at look at
176:02 see it's a straight line uh far , trace it back closer to neuroscience
176:11 you see it merges tangentially with this reflection which is going to come up
176:17 a second. So this point here uh where uh one obtain agency that
176:25 occurring at an offset. Such that the uh um that offset the incident
176:37 um starts generating two waves, reflected and a and a refracted transmitted
176:46 And that one comes uh he didn't a good enough um description of that
176:54 . So uh what it does is comes down to the boundary at large
177:01 . The transmitted wave is parallel to boundary. And as it goes parallel
177:05 greedy energy upwards every point along And that's what we're that's what we're
177:12 here at the surface. We'll talk about that when we talk about reflections
177:17 transmissions because that happens because of the . So so these are two different
177:23 with uh two different layer boundaries given these two different uh velocities below the
177:32 . And then here are some And these are the ones who were
177:36 interested in E. F. And d. 1st d. Sorry here's
177:46 , here's the writing. So this uh is coming off the same layer
177:53 this. Refraction. They didn't uh didn't Mhm. Show any reflection.
178:05 refraction here. Show one here. show one here, I guess.
178:13 that's because of the properties of the and these are the ones that we're
178:18 interested in. And you will of uh recognized these uh hyperbolic move out
178:29 for E. And for E. F. And what we're measuring here
178:36 the move out velocity for that reflected . So here is a a a
178:46 from a reflection. And uh let's here. How do I know it's
178:53 a from a dipping p reflection. because it's a hyperbole to but the
178:59 the earliest arrival time is not The earliest arrival time is over
179:04 These offsets. So that what that is dipping and furthermore, look
179:11 here is a here's a hyperbole. might have thought that's a reflection.
179:15 here it's identified as a multiple. I think that that wave has gone
179:22 the source down to this reflector back to the surface and then back down
179:29 and uh is recorded at this So that's multiple when it has uh
179:37 reflections. And then we got more here off different reflector. And then
179:43 addition these these are here are called surface ways Sheriff and Gil lark called
179:55 . Ground roll. That's the old uh term my father's generation for surface
180:03 . And the airwave is K. So so here's the ground roll and
180:11 . And the air waving K. you see these are straight lines back
180:15 the surface. Uh And you see um uh they are a little bit
180:29 than the direct way. So the was traveling with a P way p
180:33 . This is slower, this is surface wave velocity. And this is
180:38 yet. This is traveling through the . So um uh Stephanie, have
180:44 ever been on a seismic acquisition Have you ever been on a seismic
180:54 group? And what what were they for Sources Viber side? So when
181:02 virus size was operating, you could in your shoes, you're standing some
181:07 away. You could feel the direct of arriving and you could feel the
181:11 roll arriving. Uh You probably couldn't here an airwave couldn't hear it in
181:17 ears, could you? Yeah. engineering. If they had been
181:25 then you would you would have heard dynamite blast in your ears.
181:32 And then we have the fractions which haven't talked about yet. So we
181:38 talk uh later about um oh. . Refraction and diffraction and reflections and
181:50 . Each of these lines represents an time. It's an arrival time for
181:54 peak of the wave. Yeah. Let's go back here. So for
182:02 , at this time, at this we have all these arrivals,
182:10 1, 2, 3, 4 . And then uh this arrival also
182:16 arriving at this time. And so what makes um and they all have
182:23 own way blitz and their own but we don't know that when we
182:27 look at the size and again they each other and uh causes interpretation,
182:33 confusion. The amplitudes, these amplitudes are not shown on the previous
182:40 only the arrival times are shown. do carry useful information, but the
182:45 shows no indication at all. So , because of all these arriving,
182:59 arriving signals, different pathways, different share waves, P waves,
183:04 everything, reflections or refraction, everything in there. So some of it
183:11 gonna regard as signal and some of is noise. So let me ask
183:19 your tie uh what's the difference between moments. Yeah, yeah, that's
183:35 pretty close to what I'm looking for uh noise is actually signal that we
183:44 care about. That's what you But uh if we think about
183:49 maybe it's uh it's actually a signal we don't understand. So maybe if
183:57 think about this noise and uh more will learn something about the earth that
184:04 ignored first. We don't care about , but it really is signal that
184:10 we should be thinking about, but got to be smarter to understand it
184:16 because of that. The distinction between and noise is very subjective. And
184:23 if you think that this part of data is noise. I might think
184:29 the signal and vice first. back to you Stephanie. Um
184:40 is this statement? How do we compute complete this statement? Uh Beginning
184:48 the statement says that primary reflections from reflectors in the sub service have move
184:53 , which varies often often in which these ways literally literally hyperbolically or
185:00 So uh we'll talk your way through . Okay, we eliminate A.
185:10 B. Well, let's go back look at the cartoon. Okay,
185:25 here here is a reflection right Yeah. The unit. So it
185:31 to me like the earliest arrival as is at uh zero offset Back here
185:38 D zero Offset. Yeah, I to play with sinister rival the roster
185:51 if it's dipping. But here it it's not dipping. So this happens
185:56 dipping reflectors. But your answer is now multiple reflections. Uh and flatline
186:05 how we're gonna yeah, it's gonna the same. And in fact,
186:14 we go back here to the you know, uh this is uh
186:20 a multiple. It says it's a right here age. So that one
186:25 like a hyperbole to and it looks sort of a deep reflector. It
186:30 a lot like this except that you the, can you see this slope
186:34 here is less, this slope here more so this velocity is uh
186:43 This slope is flatter, This philosophy smaller than than it is.
186:47 so what we have is a hyperbolic with a slow velocity arriving late.
186:55 the reason it's arriving late is because spent its time echoing around in the
186:59 parts of the media. So that it. And then it also has
187:03 slow velocity, because as it's going and forth in the upper part of
187:09 media of the medium, those philosophies slow. So this is gonna be
187:15 . Uh but it's it's gonna be hyperbolic and furthermore, it's gonna be
187:21 other high purples in the cartoon. doesn't make any problems. But let's
187:26 down here for uh that one doesn't any. You can guarantee you can
187:33 sure that in the real world uh like that are gonna be lying right
187:39 top of your uh of your You're gonna be interested in a primary
187:46 that looks like this starts here and down here and crosses here somewhere at
187:52 offset. So uh you can be that there will be multiples interfering with
187:59 primary uh reflection of greatest interest. a simple application of Murphy's law.
188:09 gonna be a multiple that bothers Okay, next, uh has moved
188:20 . Which varies how about this? , very literally. And extrapolating back
188:27 the surface. And furthermore, it's to have an extended wave. So
188:38 let's look at here, uh ground is uh call that j that's
188:43 so j So it looks like it's causing much of a problem here.
188:49 uh the wavelet of this means that the peak of the of the ground
188:55 gonna be arriving here. But it's have to be a wave that expands
189:00 all these other times. So it's interfere with everything. And furthermore it's
189:05 have high amplitude. Why does it high active? Well, it's because
189:12 my my definition is a surface So the uh the empathy was contrary
189:19 the surface. Well, guess That's where we have our receivers.
189:24 the amplitude is gonna be on our is gonna be mostly most of the
189:29 is gonna be uh groundwater or surface which we don't want because it's only
189:38 to tell us about the shallow We want to know about the deeper
189:42 . We we want to um uh at reflections. So we're gonna have
189:47 do some clever data processing to see reflections underneath the ground wall. And
189:56 so we have another course in this this sequence. I will give you
190:02 few simple ideas, will get much thorough discussion of solving that problem.
190:10 I think it's Professor joe, is correct given that. Okay, so
190:19 is a fun topic. And so have 15 minutes to discuss.
190:28 so consider this uh seismic survey with places identified here A. And
190:36 Vector source in A. And is a vector source at B.
190:42 Um In both places there are citizen is receiver at a source from
190:52 . Along this vector. And this a receiver at be sourced from a
190:59 . Now the reciprocity theorem, a deep theorem of elasticity uh discovered over
191:09 years ago as applied in the seismic . That says that uh this relationship
191:17 true. So you take the force a. You make a dot
191:22 It's a vector. It force it . Make a dot product with vector
191:27 the data at a source from That dot product is the same as
191:33 corresponding dot product over here. So and the proof in the book by
191:46 , which is almost 100 years Yeah. Huh. For looking only
191:54 P waves, those dot products go because the forces in the same direction
192:01 uh as the data. So we're set the force is equal. And
192:08 what this says in this special case that it says the data measured at
192:15 source from B is actually exactly equal the data measured it be source and
192:24 . So that's called. Uh So that means is if you interchange the
192:28 and receiver position. The data is center. Okay, so before we
192:38 this, we should have known But all along in the bad old
192:43 we used to do split spread surveys so we would have sources and receivers
192:48 all spread all along here in two . And we would sort the data
192:53 that we could get a common midpoint gather and it would be illuminated from
193:02 both directions. So this one which arriving at this receiver would have started
193:07 here, coming down here and up and then uh source from both
193:14 We call it a split spread survey and received on both sides of the
193:21 . Now what we learned from the theorem is that we can obtain the
193:25 information using off end shooting. So only shoot to the left of the
193:30 the midpoint and all the receivers are the right of the midpoint, half
193:39 effort. And so when we realize suddenly we realize that we can do
193:48 streamer acquisition. And so of course off end shooting because the source is
193:53 here, the receivers are always down . And uh so we uh have
194:03 Only raised propagating like this in a environment. So in the early days
194:09 had only single streamers with less than ft long, you know five or
194:16 or 7000 ft long and with hydrophones intervals of about 100 ft. That
194:22 the kind of data that my father acquiring late in his career. And
194:30 reason we can get away with this because uh uh reciprocity theorem says we
194:38 need to have a source back here the same receivers from both from both
194:46 . And so uh now the reciprocity was about uh displacement, but you
194:53 easily extend that to put pressure. . Oh this is really useful for
195:06 when we analyze computers because uh the algorithms have computing costs which depend upon
195:16 number of source position. But in acquisition designs have many more sources than
195:27 . Uh Same old some acquisition designs many more sources than receivers. So
195:35 we interchange the roles of sources and in the computer, these algorithms are
195:40 efficient. Remember that these conclusions only to uh the waves for share waves
195:51 converted ways. We need the full rest process there. The version I
195:56 showed you um any of this Now I know what you're thinking,
196:03 thinking this looks pretty obvious. Let show you a thought experiment there uh
196:12 become very president for you. so imagine a two d problem in
196:18 plane here. And we have a here which elliptical in form and an
196:25 has two special places in it which called the far side and one is
196:31 and one is here. And I'll you about the properties of those two
196:35 in a second. For now let's that we have a source at one
196:41 and a receiver at the other Now here are the here's the special
196:47 of these two foes. It says uh if you if you run the
196:53 between this focus to any point on line and back to the other
197:00 it's the same length of string. matter where this point is including uh
197:06 like here, why this string from source to why to receive is the
197:11 length of string from uh X. a special property of the ellipse.
197:19 furthermore, these angles are the So these lines are like rays of
197:26 . Right? So that means that you fire a source, fire an
197:34 source here, all the rays from from all in points inside this reflector
197:41 end up at the receiver at exactly same time. So imagine an impulsive
197:46 and you get an impulsive reception with everywhere inside the uh the ellipse.
197:57 , those are that's what happens when have sources and receivers at the focus
198:03 of the ellipse. Now let's interchange and resumes. Now the sources here
198:08 the receiver is here and it's very you're going to get exactly the same
198:13 because of symmetry. Okay, that's surprising. Hold on to your
198:18 Now we're going to put uh source on the left and we're gonna remove
198:25 nose of this ellipse so that when source fires half the energy gets lost
198:31 space and the other half collects here at one instant at the receiver now
198:37 onto your teeth because now what the theorem says is that when you enter
198:45 in this situation in charge source and . So now the sources here,
198:49 receiver is here Because this is such narrow opening here are only like 5%
198:57 the energy gets lost to outer 95% of the energy is still inside
199:04 reflector and it all collects here. the reciprocity says that the data is
199:13 exactly exactly the same as in the case where half the energy got
199:20 So this is very counterintuitive, very . So when this was published by
199:35 smart uh um geophysicist at Texaco named , he's deceased now. But uh
199:46 posed this paradox in the pages of Journal. And we had a uh
199:57 discussion in the journal with famous uses coming down on both sides of the
200:05 . Is this the same data as other side? The other case or
200:10 ? And so the issue was finally by Professor Clara Clara Bell and his
200:20 Dillinger who eventually became my colleague at . And they did a computer
200:28 And here here here you can see ellipse with the nose off and here
200:32 can see the sources on the uh can see the sources on the left
200:39 uh most of the energy is staying the ellipse some of it gets past
200:45 and escape to outer space and here the energy and the resulting pressure um
200:53 that computerized experiment here now the interchange and receiver. And here we've got
201:00 source here, half the energy gets to outer space. And uh so
201:05 shows the energy uh arriving for every . And you see it's very different
201:13 this and here's the pressure arriving from angle. When you sum this
201:27 When when you sum this up, total uh total pressure coming from all
201:34 different res uh sums up to this .548 and it's the same sun for
201:41 , even though different contributions from the angles, it all sums up to
201:46 same. And um this gap you know, this is the part
201:51 got got lost. Whereas uh the is very different as you can see
201:58 your eyeball. So uh Claire bow now quite old, He's uh professor
202:09 at stanford and I think he's a about 90 still alive and kicking.
202:16 , very active. I understand it into the office of the day.
202:25 Stephanie, is this statement uh for ? Yeah, it's false because
202:34 the reciprocity theorem enables marine acquisition 1 boat marine acquisition. Uh Is
202:44 uh is this statement true. Is general statement of the reciprocity theorem.
202:52 like that. That true. that's good for you. Uh this
202:59 statement is a special case. In general case, it's false. Not
203:05 , but the special case we now the scale of reciprocity applies only to
203:10 waves and that one is true. by the way it's a very general
203:14 , it includes things like um and includes heterogeneity, it includes everything um
203:23 uh yes, so that's why it's powerful. It's only for p
203:30 So now in this lesson we've learned how the previous lessons on electricity led
203:39 the scalar wave equation for waves in ocean and for rock state leads to
203:46 vector wave equation, not real but for uniform as traffic rocks when
203:57 put in their source, then we how that equation gets modified and how
204:04 gets modified when there's layer in the . And then we looked briefly at
204:10 types of solutions, all of which gonna observe all those are in our
204:16 . And how the concept of elastic leads to significant operation and competition
204:26 So this is a good place uh to break for lunch Utah. This
204:31 a good place to cut the To to what your question is your
204:44 is about what or the ellipse. , so are we still recording?
204:52 , so this is good. keep it recorded. So and so
204:55 me your question. That's correct. go back here. Okay, share
205:24 pressure is proportional to the amplitude and energy is proportional to the square of
205:29 amplitude. Okay, so uh so one and what you measure is the
205:34 of all these angular contributions, The the the angle of arrival of
205:39 the ways. And so um this so what's the same is the uh
205:48 area under this which is the sum all the contributions from all the
205:55 And you can see that the the under this. So the part that
206:00 lost out here is out here. that uh that part got lost and
206:07 part that gets lost here is only the middle. So it has the
206:11 all these tales all the way So uh the area underneath this curve
206:18 black is the same as the area . Uh even though the energy is
206:23 as much. And so remember that energy is the square. So you
206:27 this uh take this curve and square and adjust the scale. And you
206:34 this curve. Same here, take uh and square it and adjust the
206:40 . And you get this curve. so uh it's the uh huh pressure
206:48 equivalent the amplitude which is uh the in these two situations, not the
206:59 . Okay, so let's uh break . And uh we will reconvene at
207:06 PM. Talk about um or we're talk about the solutions. So why
207:17 wave solutions to these
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