 Good morning. So we have been looking at linear wave equation so far. So yesterday in fact we finished with the demos. So to be precise we have been looking at linear first order one dimensional wave equation. Of course as I noted somewhere in between this is not quite it looks similar to but is not quite identical to the class of equations that we are looking at in the fluid mechanics as such. Though I can give an interpretation as of some stream traveling at a constant speed A carrying the property u with it. Another typical equation that we run into is this equation. We will spend a little while try to spend like a part of today or the whole of this class talking about it. I could of course spend the semester talking about it but I do not think I am going to do it. We have spent enough time on the linear one. So this is linear, this is said to be quasi-linear. So if you had a dou u by dou x, dou u dou x whole squared then it would become non-linear or dou u dou t whole squared then it would become non-linear. So this is quasi-linear. If we look at this equation and see whether we can figure out any properties of this equation from the some analytic solution point of view just like we did earlier and see where we can take it from there. And after that I will leave this alone and then we will go on looking at other equations of interest like Euler's equation and so on. So you will hear this being called either the quasi-linear one-dimensional first order wave equation. You may even see it sometimes in literature or in books as the Inviscid-Berger's equation okay which anyway. So it could also be called the Inviscid-Berger's equation but you may see it, you may even see it being called the Inviscid-Berger's equation. Now just to recollect in the beginning when we are talking about this equation we realize the importance of characteristics and in the xt plane we saw that we had characteristics that had slope A that basically means slope 1 over A in this coordinate system that basically means that whatever it is we were doing was propagating at the speed A in a given amount of time right in a unit time. This is basically propagating a distance A in unit time so that the propagation speed was A, that is what we had. In this case the propagation speed is u, is that fine? In this case the propagation speed is u. So the first question is the argument that I used for this, does this argument still hold here? That is can I still say that some s dot gradient of u equals 0 where s equals, I do not remember what basis vectors I used last time but anyway it is okay, ui plus j where i and j are unit vectors along x and t, am I making sense? So this notion of the directional derivative still works here. So you still have du ds equals 0 where this s is measured along this direction s okay and that argument that we used that time saying that along this line because the right hand side is 0 then the story is different because the right hand side is 0 along this line u is a constant, is that okay? And along this line if u is a constant along that line the propagation speed is a constant, fine? Does that work for you? So now all we are going to do is we are going to say oh this has slope u, 1 over u. You travel a distance u in unit time, remember it is still quasi linear, right? It just so happens that along this line u is a constant, that is very important. So I sort of think that through to make sure that there is no issue. So we will use this just like we did last time, we will use this to see whether we can come up with solutions to this equation just like we did for this case, we will see if we can come up with solutions for this case okay. Clearly we cannot do the exponentials like we did last time because the exponential function had the a in the exponent and then you will be writing u in terms of itself, that does not make sense, okay. I mean it does not help us, I mean it is sort of an implicit form, that does not really help us. I do not say it does not make sense but it does not really help us, right? Let us consider some initial condition. So you consider initial condition, this is u, that is x, okay. We will consider a series of initial conditions and see what we get. You consider an initial condition where u equals and we are going to only look at the, just like we did before, either a unit length or a length l or whatever, it does not matter, okay. If the initial condition is 1.0, this is at, all of this is at t equals 0, all of this is at t equals 0. If the initial condition is 1.0 everywhere, what is the solution? The initial condition is 1.0 everywhere. On the xt plane what do you expect? Actually it is a rather boring problem. It is the same as the dou u, dou x plus, right? Dou u, dou t plus dou u, dou x equals 0, that is what it degenerates to. So the initial condition then basically says that all of these characteristics are 45 degree lines, right? With propagation speed 1 and if my boundary condition also continues to be 1, then you will have characteristics coming out of that, is that fine? Okay. So this is in that sense, it is sort of degenerated just because u is a constant, in this half the case it happens here. So if u had been a, it would be the same as that equation, is that fine? Okay, let us consider another one. So if you have something that starts off from 0 and goes to 1 and then is a constant and the boundary condition on the left hand side is 0, it is going to continue to be 0. The boundary condition that I am going to apply for all time at x equals 0, at x equals 0, u equals 0 is about, x equals 0, u equals 0 is the boundary condition that I am going to apply, okay. What do we get for the corresponding characteristics? You do the easy one first. So this is a 45 degree line, it goes to 1, so clearly this goes to 1, right? So let us do that first. So this is 1, right? Let us do the easy part which is to the right hand side. They are all 45 degree lines like this, same D. So they are all 45 degree lines like this. So you get, okay, is that fine? What happens in between? What is it here at the leftmost point? The slope is infinite basically. The slope is infinite, right? So you get the characteristic here is a nice line there. And what happens in between? For instance what happens at 0.5? The slope is 2, it is a steeper line, right? So at the slope is 2, so at 0.5 you would basically get the steeper line. And as you go towards this, lines get steeper, as you go towards this, the lines get shallower till you get to 45 degrees, okay, right? This is called an expansion fan. This is an expansion fan, okay. This is an expansion fan that you would have seen in gas dynamics, right? So this is an expansion fan. So this behaviour is very different from this behaviour. So now we are starting to see the U, D, U, DX sort of kick in, right? The effect of the U, D, U, DX term. Is that okay? A third example of course is a more interesting example. Third example is going to start at 1, it is a 45 degree line going down to 0 here. So this is also 1.0, X equals 1, that is X equals L. And then it is 0 afterwards, right? And for all t, X equals 0 tells you that U equals 0. U equals, I am sorry, U equals 1. Is that fine? Okay. So as a consequence, what are you going to get? Again we will do the easy one first. So between 1.0 and L, you are basically going to get vertical lines because stagnation, right? It is not this, U is 0. This is propagation speed, U is 0. The flow there is stagnated. So you will get vertical lines. Well, if I can draw vertical straight lines, write up to that. And of course at the characteristic at 0, you look at that again, you will get a 45 degree line. So now something bizarre happens, characteristics intersect, right? So I will continue this through. And then what happens to the in between points? They are all the slope is going to increase. And in this particular case, it will turn out that they will all pass through this point. Okay. So I will not draw anymore because it will become a mess. Is that okay? So how do we interpret these two, right? This is very clear. This is just pure simple translation. First one is very clear. How do we interpret the other two? How do we interpret the other two? So as the second one for instance, as T evolves, what do you expect will happen? How does this graph evolve from looking at this? How does it evolve? See, you know at least, you know the propagation speed of this, right? You can actually work out the propagation speed of each of these individual quantities, okay. So effectively what you are going to get as time progresses, you will get a series of functions. So if this is one, this is what you started off with. Let me draw it with a white line like this. Or shall I draw it there? Maybe I can draw it back there. I will draw it back here. So as time progresses at different times, this point is going to move to the right, right? So this point is going to move to the right and you will get a function that looks like this. Am I making sense? Okay. That is why it is an expansion fan. So if you waited for a second time unit, that is twice the first time unit, it is going, it is translating at a constant speed of 1, okay. Then you will end up with, maybe I choose a different color. You end up with a function like that, okay. Is that fine? And you can see that yes indeed we do have a, we do have a, so at different time intervals, at different time intervals, so if I got this right, just say this is at, this is at one time unit, if you want, if you use seconds, 1 second, you use meters per second and second. So this is at t equals 0.25, right? Or one-third, two-thirds, whatever. And by the time you get to one time unit, you are here, right? Equal intervals. They are equal intervals. It will propagate equal, equal distances here. On this end, it does not propagate at all. In between, it propagates at a proportional speed, okay. Is that fine? And indeed this line, this line is getting stretched. Line is getting stretched. Is that okay? So this is at, this is at different time, time intervals, okay. Maybe I will add one more just for the, one of it. I think somewhere there. So this is at t equals 0. This is at 0.25, one-quarter, one-half, three-fourths, one time unit. So by the time you propagate one time unit, it has traveled the distance of one. Fine? Everybody, what happens in this case? What happens in this case? If I were to just follow through, so after a small time interval, right, the leftmost point is propagating at one unit per second, right? And I would get a solution that is like this. If I waited a little more time, then I would get a solution like this. It is going to go through 0 because these are all stagnation points. They are not moving. They are basically those points, points to the right are not moving. And if you waited a little more time, you would get that. And what is the final one? By the time you came to one unit, you have now come to a point where you have traveled the full distance, right? And in fact, at this point, all of these characteristics meet. Can we go beyond that? Does it make sense? Does it make sense? Can we go beyond that? Well, that really depends on, now it depends on what is the actual application. Whether we are able to go beyond that really depends on what application. For example, if you were to look at these as trains travelling on a railroad track, if you were to look at these as trains travelling on a railroad track, there is possibly a railway station here. Ignore all of this stuff. There is a slow passenger train, right? A reasonably fast train and an express train. And if they make it at the same time to the railway station, at the railway station, you have something called findings, right? So, they can even pull a train over and they can actually cross each other. At the station, it is actually possible that they cross. It depends on the application. Am I making sense? All this basically says that this train is fast travelling faster than the other train. And if there is a way for you to shunt the trains from one track to another track, so that they can actually overtake each other. See, that is the key. That they can actually overtake each other. It is actually possible that they do propagate this way. Am I making sense? Is that fine? Right? So, it depends on what is the interpretation that we give to this. What governs this equation? What is the interpretation that we give to these characteristics? Right? The other possibility of course is and these kinds of things happen, the solution can become multivalued. So, the solution can become multivalued here. You look at these characteristics and say, wait a minute, u is constant along this, u is constant along this, u is constant, each train has its own speed. Each u is constant along and yes, at the railway station, the trains can be multivalued. You are there, your train is stationary and you have seen other trains on other tracks going by, right, at a reasonable clip. Fine. So, the u can be multivalued at a given point, right? We are so used to thinking of this possibility that they are multivalued, right? I mean, you do not encounter it very often. We do not encounter it very often. The other possibility of course is that you do have a function even in a regular physical setting which is multivalued. So, if you think of, I think the standard classic example that given as a wave breaking, so you can have, if you go to the beach, I have removed all the froth and all of that stuff, but you could actually have, you could actually have a wave breaking, right? And you could, the wave, it can become multivalued. If your u indicates the depth, right? It is a measure of the depth, okay? Measure of the depth or a function of the square root of the depth, because it is tight to the propagation speed, right? Then it is possible that it actually becomes, it actually becomes multivalued. It depends on the context that you are talking about, right? If you are, on the other hand, there is a traffic light here and all of these cars are stopped, then all of you, if you do not stop in time, yes, this is going to happen, right? So, you can have a pile up, okay? So, there can be situations where something like this, something like this actually happens, fine, right? And in gas dynamics, what we are used to, which is why you have the response that, no, no, it is not possible. So, if you have a pipe, right, which is why I gave you the, if you have a single lane traffic and you cannot get around, you cannot get around, then there is a problem. Then you are going to end up with a situation that I have shown here, where you start off with something that looks like, you can end up with a function that ends up like this. You start off with a function that looks like that and you end up with a function that looks like this, okay? So, what is the big deal? Why am I harping about this? I sort of spent, I could have just said, do you do TA, do you do X equals 0, just try it out for different things, but why am I harping on it? Why am I making a big deal about it? If you think about yesterday's demo, right, when we were doing, when we had dissipation or when we use heat equation, if you have a step, the step tends to smoothen out, okay? And when we had oscillations, everything was smooth, right? Everything that we had was actually relatively smooth. This is a strange situation, right? And it is also possible you will hear people say, oh, nature does not like discontinuities and so on, right? I mean, you can make this broad statement, nature does not like discontinuity. But here you have a situation, right? Here you have a situation where this governing equation actually generates a discontinuity where none existed, okay, right? And for us it is important, for us it is important because we saw that the dissipation term on the right-hand side of the linear wave equation was smoothing. That resembles viscous term in the Navier-Stokes equation, okay? That very closely resembles the viscous term in the Navier-Stokes equation, and it is smoothing, right? But the left-hand side, that is the dou u dou t plus u dou u dou x term, this kind of a term, it is also there in the Navier-Stokes equation, on the left-hand side of the Navier-Stokes equation, has a tendency to make it steeper, okay? Is that fine? You understand what I mean by making it steeper, what is happening to the high wave numbers here, when you go from here to here, what is happening to the high wave numbers? They are decaying, their amplitudes are decaying. That means that if you are going from something like this to this, so imagine you draw the characteristics for this, you expect that this is going to become a shock like this. Why? Because if this is a propagation speed, this is travelling faster than that, right? So if you just have a smooth initial condition, the demo we went from here to here, but instead of going from if you had started off with this and you were solving this equation, what we see now is this is going to become that. It is a competing, opposing effect. Am I making sense? Which is interesting, right? And where does it come from? What is the source? That means I am adding somehow from this low frequency, I am generating high frequency terms. Where does that come from? Okay. The term is here. The term is here. Just imagine that u is like sin theta, sin n delta x, right? u is like sin theta. What happens to this? This becomes sin theta cos theta which is sin 2 theta. And that is where the frequency doubling occurs. You are taking, you are adding to the sin 2 theta, you add to the sin, you understand, you are going to add to the sin 2 theta term. If you were to do a decomposition of some kind, you are going to add to the sin 2 theta term. You are generating a sin 2 theta term where you only started off with the sin theta term, okay. So this is even a function, a smooth function like this is going to get steeper. So in the actual equations where you have both a viscous term and this quasi-linear term, both these effects are competing and therefore what you have studied in gas dynamics that is the thickness of any shock that forms. So in gas dynamics of course these are called shocks, right? Occasionally in mathematical literature you may hear them referred to it as an internal boundary layer, sort of an oxymoron but if you know boundary layer theory you will know what they are talking about, right? Okay, right? So you have, there are shocks, right? These are called shocks and you know that the thickness of the shock depends on the viscosity. Am I making sense? Viscosity is trying to do this. Viscosity is trying to do this. The shock is the quasi-linear term is trying to push it back here and they come to some kind of an equilibrium, right? Left to its own devices the quasi-linear term will actually give you a discontinuity. So viscosity is sort of has the opposite effect which is why that balance is what gives you the thickness of that shock, right? And it changes, it depends on what the viscosity of the fluid is, is that fine? Okay, are there any questions? Right? So this is, this is as far as what should I say as far as this quasi-linear equation is concerned. We will see whether, let me, what happens, can we say anything beyond the shock? So obviously, there are different possibilities. On the one hand, on the one hand you can have, you can have this effect. On the one hand you can have this effect. On the other hand I will redraw this with a shock. I will redraw this with a shock there. On the other hand, before I, so you would get, you expect to get this. I should not have drawn these vertical lines first but it is okay. I will change it, right? And then you have these lines and of course you have conditions coming from the left which I have not drawn there, right? Or you have all of those. You have all of these lines. So all at 45 degrees, they are all parallel to this, right? And they form a shock. So I need a different color for the shock. These characteristics merge and they form a shock. And basically, the shock sort of consumes the characteristics. It eats the characteristics. The characteristics disappear into the shock, right? Is there a way for us to find the propagation speed of the shock? Right? I mean, see, now is the shock just located there? Is it going to travel? Is it going to be stationary? The way I have drawn it, it is going to propagate. The shock is going to travel. That is the way I have drawn it. Is there a way for us to find the propagation speed of the shock? Okay. Should be, is there a way for us to find the propagation speed of the shock? So we will zoom in on this, we will zoom in on this area. We will choose a little more. So now I am not going to, I will just say T. I will not show you the origin. I will say X. We do not know where this is, right? Where here, there is a shock. Our objective is to find the, so of course, the shock. I do not care where it originates. I do not care where it is going. From here, there is a shock. And I will choose a convenient control volume. I will choose convenient control volume with points A and B, okay. It is going to be a rectangle. I will tell you what I am proposing to do, okay. I choose this rectangle. I will rewrite my equation in a fashion that I am able to relate the conditions on the left hand side of the shock to the right hand side of the shock. And I propose to take the limit as A and B approach each other, right? And from that I am hoping that I will be able to infer something about how fast the shock is propagating. Is that fine? For instance, if this is, if this point is XA, in this point is XB, right? So this is XA TA in the XT plane, XB TB. We already know that the propagation speed of the shock, U shock, here U shock is XB-XA divided by TB-TA. It seem to have implicitly assumed that the shock is propagating at a constant speed, okay. Let us see where this takes us.