 Today, we will address the magnitudes of surface compressional moduli and begin an introduction to a new topic on surface waves and ripples. Before we do that, like always we will quickly see the idea behind the surface compressional modulus. You would be able to recall from yesterday that the surface compressibility is defined in analogous fashion to the bulk fluid compressibility or a related expansion coefficient in the context of buoyancy driven convection or natural convection. So, the compressibility is expressed as minus 1 by a dou a by dou pi at constant temperature. Clearly, the units of compressibility would be inverse of the surface pressure. Areas would cancel out in terms of the dimensions living out reciprocal pi for reducing the dimensions or units of the surface compressibility. And clearly, if we have force area curves, we should be able to calculate from the slope this coefficient and then evaluate the compressibility. It is inverse of this compressibility which is more convenient in practice that would then have the same dimensions as of surface pressure. So, in the limited ideal case where pi is constant, C s would be calculated from here expressing A as this constant by pi. Then upon completing the differentiation here, we would be able to find C s as 1 by pi or C s inverse as pi. For a clean surface, because the surface tension is gamma 0, gamma and gamma 0 are identical, pi is 0, the surface compression modulus is equal to 0. And therefore, one may say that for these ideal monolayers, we expect minimum attribute of a property which may contribute to stability of films around bubbles. In general, the the compressional modulus we expect would also depend on the state of the film because the force area curves will be different for different states of films. Expanded or condensed films will have comparatively greater surface compressional modulus with reference to let us say the datum of gaseous monolayers. So, with this we should be able to figure out the values for the surface compressional modulus provided we could ignore the interactions among molecules adsorbed at an interface. And if we could work in the domain where the areas are large compared to limiting areas, permitting the use of the ideal equation of state ideal surface equation of state. But let us look at some magnitudes of the cohesion C S inverse for clean surface 0 ideal whatever value of pi for more complex molecules like a proteins where the area may be given in meter square per milligram. If that is of the order of 1, then the surface compressional modulus you would find is in the range of about 1 to 20. Compare this against the liquid expanded and liquid condensed films and you have C S inverse in the range 12.5 to 50 for liquid expanded films about 100 to 250 for liquid condensed films. And for solid condensed films the magnitudes can be as large as couple of thousands between 1000 and 2000. As I remarked earlier this is a measure of the rigidity of the film or compressional elasticity of film. And this property is important in determining the kinetic properties of monolayers which play important role in determining stability of foam or froth and in retardation of the toroidal circulation which occurs in falling drops. As we will see later the surface compressional modulus is also important in the area which is of generic significance that of waves and ripples. This is the topic that we would be getting introduced to today. So, let us start with a few simple observations. Let us talk about water waves. We have all seen the water waves and often this is a chosen example of waves in elementary courses. It turns out this could be the worst possible example in respect, no way like sound or light. There are all kinds of complications here. So, let us look into some of those. We begin with long water waves in deep waters. Now oceans could be regarded as deep. If there is a disturbance on the surface, that would generate waves. But have you looked at these waves closely? Maybe before I move on to the next slide, let me ask you a quick question. How many of you know swimming? Very few. How many of you have the experience of swimming in an ocean? Alright. Maybe we will have at least one to think about what is mentioned on the next slide. For the rest you can leave it to imagination. Let us say you have a life jacket or a simpler thing, a floating tire in which you can take the comfort of making no effort and staying afloat. They are bound to be some floating objects around you. What do you expect to observe from that reference frame? You are floating. There are other objects nearby and you can observe their movement. Think about it. The question is not completed. This is only the background. So, let me provide some more information. We expect to see all kinds of irregular motions, but among those we will see a pattern. The first character of waves which comes to your mind is the sinusoidal. That type of motion looks the most common. Water on the average is standing still, but the wave moves on. So, the question is, is the motion transverse or longitudinal? That is the question. Water at any given place you would see or expect would be alternatively troughs and hills, but could it be that water simply moves up and down? That is not possible. If water where to only move up and down will be violating something, conservation of volume of water. If water comes down somewhere, where will it go? And for all practical purposes in such systems, you can take water to be incompressible. And we are not talking about the speed of compression of waves. There is a velocity of sound, speed of sound. That is not something we are discussing. So, it must go somewhere else. If water comes down in a portion where currently there is a hill, it must go somewhere around. What happens is actually particles of water near the surface move approximately in little circles. That is what you would see in your frame of reference where you are floating on such water. You would see nearby objects going in circles. So, we return to the question, is the motion transverse or longitudinal? It is a combination. It is a mixture of two. That would add more to the confusion. So, really speaking, these kind of waves will be a bad example, especially in an elementary course on physics. The question is, what would be the velocity of such waves? That should be an interesting problem. Since we are not going to the fundamental theory of physics of waves, but rather take the view of engineers more to your comfort, one might wonder what are the factors or variables or parameters on which the velocity would depend. Think about it. You have these waves passing and you want to know what would be the relevance list when it comes to velocity. What would govern the velocity? And when you compile this relevance list, as you do in all dimensional analysis problems, you would think of the obvious ones, probable ones, even the improbable ones and then rationalize which ones must be retained. That will take some thinking. This is a simpler problem, but in other problems that may get trickier. But you expect now in the relevance list here, a few obvious variables and parameters and I have to tell you of some caution here, especially in compilation of the relevance list for a given problem. When you are trying to analyze the essential physics of the problem, you have to keep in mind one particular historical and most often repeated blunder and that is if you were to look at one single parameter which has very frequently appeared in dimensional analysis reported in literature, for all wrong reasons it is going to be any guess, time, any other acceleration due to gravity. Where it is important, you find often it is neglected and where it is not important it surfaces. We do not know the reasons why acceleration due to gravity has played this game with researchers, but market acceleration due to gravity wherever you come across in your probable list, relevance list, be careful. So, over here we are talking the velocity of the surface wave. So, what could influence this? One, very likely density of water. Secondly, the gravity is the restoring force. If there is a disturbance eventually the disturbance dies out and gravity plays a role there. So, acceleration due to gravity you suspect should be here and apart from that the wavelength, wavelength of this wave or ripple and perhaps the depth of liquid. Now, that is not hard to imagine. You expect the waves to be different in shallow waters maybe the water you can have in a Langmuir trough a few centimeters deep compared to the deep waters like in ocean. You expect that will influence the speed of waves. So, these are the 4 factors density of water, acceleration due to gravity, the wavelength lambda and the depth of the liquid. Now, we want to keep our keep our discussion as simple as possible in this initial stage. So, let us take recourse to one of the limiting circumstances. You know something about your variables. One of the simplest ways to take out a variable is to focus on systems which correspond to very large values. So, in this case this little you can do with density of water or acceleration due to gravity, wavelengths perhaps, but then you may be limited by the kind of waves or wavelengths you are dealing with. So, there is a possibility in our analysis we can think of the depth being the parameter variable which is very large. If it is so large like in deep oceans we could say it is practically infinity. And then once you attribute this infinite character to a variable it goes out of your problem. So, it now boils down to only these 3 density of water, acceleration due to gravity and wavelength. So, let us say the depth of liquid is taken out of consideration, infinitely large depth of liquid. Now, you can write down the dimensions of the quantities you have and see which combination will give you the dimensions of velocity, density of liquid, acceleration due to gravity and wavelength. Let us say there is this ocean. There is ocean. Now, something happens inside and the waves suddenly start. That is what happens. You are asking what causes the waves. Now, obviously there are I am going to come to that there are different ways in which waves could be generated. Here let us say we are looking at some surface disturbance. So, it could be a motor boat speeding past the shore. And so, that means about cavities and the storing force. That when wave moves upwards then gravity takes it down. That is a disturbance which is created in the surface. There is something which creates a ripple which moves on. The energy supplied by that surface disturbance that is why the waves are created. That is what is done. You have let us say a smooth water surface before a motor boat passes over it. So, when it passes over it, its weight will displace some amount of water and that displace water being incompressible has to push water beyond that. So, you have now locally a wave created and then the gravity pulls it down. In turn, this volume has to be distributed in surrounding areas. So, that is the way these waves will be created. Do not be afraid of coming up with conclusions which are your own. It is important. I just told you that depth is out of picture. I gave you three variables or parameters and I am asking you to synthesize a combination which will give you velocity or a quantity with dimensions of velocity. Some constant will do square root of g lambda. So, you are saying square root of g lambda and this means you have done something beyond that. You have taken a decision to leave out the density of liquid. Is that right? The density only has the value of mass in it. The density does not have it. In dimensional analysis and these three variables we cannot have density. Right. That is correct. There is no way we can have density in our picture here to yield a quantity with dimensions of velocity. What is the basically if we see it then the probability in that plane g that is actually saying due to gravity due to the mass that is. No. I think you have to go back to your JEDs. Incidentally acceleration due to gravity is independent of the mass. What is that? How your small close to 0 is the same gravity. Otherwise you come to the same historical conclusion. If you drop a table from top of a tower that historical question was how much time will it take and if I drop a feather how much time it should take. Apart from fluid dynamics it should take the same time. So, do not be afraid of leaving a parameter you suspect originally to be there but does not play a part. So it appears now at least in this picture when you are talking of gravity coming to picture as a dominating quantity we have to leave the density of liquid out. So that is the combination square root g lambda does not contain density. You might wonder if there would be circumstances wherein density might make a legitimate return in our relevance list. I would like you to push you to think about this. Do you think there could be any circumstances related to waves or ripples where density might reckon as an essential relevant parameter? That answer will come from what kind of waves we have, what created those waves and what are the characteristics of the waves. What I am trying to do is I am opening up a problem for you to think of any general situation wherein you may come to a later conclusion I am not saying what exactly that density of the liquid will come back into the velocity expression. Do you think there is a possibility? We just need to think of yes or no here. What you are saying is neither yes nor no it is a maybe situation like maybe perhaps probably the reason why we included the density of liquid in our relevance list at all in the first instance. We left it once we talked only of gravity. So I will come to this question just in a minute or two. But let us complete this particular problem at hand. If we do the complete analysis of dynamics in this given problem then the phase velocity comes out to be exactly this g lambda by 2 pi under root. And now we are going on record here saying that this is for gravity waves. Clearly we are indicating here that there could be other or more general family of waves of which gravity waves are only a subset. Let us look into this little expression in some greater detail. You might be able to relate this to your on shore experience. You do not have to always be able to swim to experience the full beauty of an ocean. There is a lot on the shore. So if you have seen the waves crashing on to shore and recollect the sound these waves make upon crashing you probably will recall some pattern. You would see even around a moderate size lake if there is a motor boat which passes create waves you will have these sloshings at the shore which appear to be first slow and then very very quick. Quicker as time goes. Can you reconcile that with this expression for velocity? Sir this expression comes after infinite depth but since the depth is reducing. Let us say that for all practical purposes the water table is pretty steep. So I am reducing the set of conditions. If the water table is very steep it means that for most part the travel of the waves is dealt by infinite depth assumption. What happens is at the shore is only what you hear but focus on this expression for velocity and see whether you can relate it to this observation. First let me know whether everyone agrees this is true. I am sure you have all heard the waves crashing at the shore. Yes or no? You agree? Now how will you explain in terms of this velocity given in terms of g and lambda? Wave length increases at the shore. Rethink increases. See you are getting elements of that picture but not the right combination. Let me explain this unless you want to think for 20 seconds more. Frictional losses. Let it increase your velocity at this temperature and pressure it so that is why they say sound. Yes, one message I want to give you through these questions is you must try and learn to think directly. Keep things as simple as possible. Understand what is happening. You would hear only corresponding to the waves which are there. Do you agree with this? If there is no particular wave you want here of its existence there will be no crashing of that wave here. So, what is crashing at the shore is certainly a wave which is there. It cannot be that there is no wave and there is a crashing sound here. Second your empirical observation tells you that there is this different sounds which are coming first slowly at small large intervals then at shorter and shorter intervals. What it means is that we got a set of waves of different wavelengths to begin with. Somewhere they originate they originate with different wavelengths and if these waves are with different wavelengths according to this expression you can see that the larger wavelengths largest wavelength will correspond to largest velocity. So, that will be the quickest one to get to the shore that is the one view first here above it crashes then come the shorter and shorter. Once these long waves are gone relatively large wavelengths are gone then come the much smaller waves which obviously come late, but then they are coming in close proximity. So, you hear this increasing frequency that is the picture. So, indication is that this spectrum of sounds you are getting at different shorter and shorter intervals indicates that their waves of different wavelengths largest wavelengths are the ones which arrive first quickest the shortest wavelengths arrive late. Therefore, they come in close proximity well wavelengths may be less than a few centimeters. They will be between them and 0 the 0 wavelength never comes. So, those short wavelengths will come in a close proximity of each other is that clear. So, that is the explanation for these simple observations we have on the seashore wave. The question is does the density return and since we have been specifically addressing gravity waves over here it is indicating that they probably are some other wave wavelengths shorter wavelengths which might have other dominating force, but before I go into that let me quickly address the gravity waves from a slightly different angle. It was a few years ago that while I was going through these lectures on waves and ripples we had that mishap. The tsunami had hit the coast of Tamil Nadu claiming 100 supplies that was an example of a gravity wave and to connect with your question this was a wave not created by surface disturbance, but a disturbance from the sea floor ocean floor itself. The earthquake under the ocean created these large waves it is like thinking that the ocean is like a bowl of water and you tap the bottom that creates the surface waves, but the system is so big the waves you create are very long wavelengths. So, this is the reason why you have those unusual observations the tsunami waves are generally very long wavelengths they could be as long as 150 kilometers. So, the subject of destruction is the shore at the same time when so many lives were lost on shore the vessels floating in the deep waters had no damage they simply went up and down riding on these very long wavelengths it was the destructive crashing at the shore which caused all the disaster. So, some of the tsunamis are gravity waves as deeply touched by that particular instance and ever since I dispel the thoughts of eliminating the discussion on the surface waves and ripples in this course not that you can do much about the gravity waves you cannot prevent these earthquake generated tsunami waves, but perhaps they might be some solutions to elevate the problem will touch upon one of those a little later. We go back into history and take a little account of what were the observations from earliest days about the waves and ripples practices of fishermen people going on long expeditions and so on. So, we addressing this question about elimination of waves and ripples it were in the historical writings of Pliny who recorded the practice of seaman of his times who poured oil on turbulent sea around these vessels to calm down the sea. It looks at first thought ridiculous, but there is lot of wisdom in the experience of these old people here. It does help now of course you would not take recourse to such solutions because you already have disasters striking us on frequent basis with the oil spills. You would not like to pour out oil on sea those issues have been taken care of by advances in mechanical engineering or ship building, but the essential fact remains that pouring of oil can actually help reduce the effect of these stormy conditions and we must understand something about it. It was in 1762 that Benjamin Franklin was told by an old sea captain that Bermudas had this interesting practice. They would pour some oil on top of the sea to calm the waves because when they would dive to spear a fish under net they did not want to be distracted by the patterns of some light. As long as you have waves there will be patterns of light under net. They would not be able to aim accurately at the fish they are trying to capture, but they realize that if they pour some oil on top the job gets easier. So, there is a merit in this observation pouring oil on top of water under these conditions of surface disturbance can actually help eliminate the waves. The question is how? I think the same sea captain also told Benjamin Franklin that there is another practice in the water at a bar on the river. The water which is polluted produces surf and it was a common practice there to pour oil to eliminate this surf. So, Franklin repeated these observations through his experiments first to calm ripples on a small pond and later in 1773 at Devon Point water which is a moderate size lake in England. So, we will talk about this later, but presence of a surface layer or a thick layer of oil can help eliminate the surface waves. Question is how it possibly helped the early sea main survive those stormy conditions that still remains unanswered, but this brings me back to the first question I started out with gravity waves that is where the danger lies, but we admit there would be probably another class of waves which would also become important in some way and together they might actually make a compelling picture. So, we eliminate this, there are three phenomena related to the waves. First the formation, second once they have formed they are damping and lastly the tendency of large waves to break and these are all affected by monolayers on water. When you pour oil on top of water we have more than a monolayer, but the essential effect is same. It seems the presence of oil on surface of water eliminates waves which are of shorter wavelengths and therefore, the disturbance which is created on the surface of larger waves is eliminated, which means these larger waves will have a lesser tendency to break. Let me explain this here, think of can I have the visualizer, think of water surface on a large water body and you think of a large wave, are we likely to have such a wave in practice is the question. You just had to think a little in terms of whatever you might have observed of waves and you would immediately disagree that this would be the correct picture. We do not expect large waves to be so smooth, is not it. Actually the picture would be we have super impose on these waves smaller waves. This is likely to be a realistic situation. So, what is this adding to as far as the winds are concerned this is not a smooth surface rather it is a rough surface, which means whatever momentum is transferred from the wind to this large wave will be actually lot higher than what you expect for a smooth surface. So, that much energy is going from wind into this big crest. If you are putting so much energy this is liable to break. If it breaks it will release energy that is what is causing destruction. But if you have on top of water a layer of oil then what I have drawn earlier that is true. These now are eliminated. These are now smoothened. Once the surface is smoothened less of this energy goes in, less will come out if it breaks or it will not come out at all. This may not break unless it hits the shore. So, that is the picture the small waves are playing an integral part along with the large waves. So, we need to consider these small waves. Observations of this kind lead to some experiments conducted by Fraulein Pockels in 1891, which were conducted in a trough like Langmu trough and she used a monolayer to damp out these surface ripples. In the laboratory what you could do is at one end of the Langmu trough you could create the surface disturbance. The way Pockels did her experiments she inserted at one end a reed which was connected to a loud speaker coil that was actuated by an electronic oscillator. Once you do that you are able to create waves and then you can study the wavelengths and so on. We will look into details of this later. Let me answer that question. So, we complete this discussion today. The way shorter wavelengths come into picture is through this second term over here. First term is identical v square is g lambda by 2 pi that is the gravity part. The second part has surface tension it is 2 pi gamma by rho L lambda density of liquid rho L returns. Surface tension comes into picture. So, we look at some numbers for clean water surface. If the waves are so short as to have wavelength less than half a centimeter you would get velocity about 31.5 centimeters per second and which corresponds to frequency of v by lambda that is 31.5 by 0.5 of 63 seconds inverse. If the liquid is not very deep we may need a correction factor, but let us for the moment not worry about it. So, the two situations that you can have is the second term when it is negligible you have the gravity waves. Gravity is what controls the velocity of waves. If you have second term large and you could neglect the first term then you have surface tension controlling the velocity of waves that will happen for lambda value small. If lambda is small the first term will be small, second term will be large and if the second term is so much larger than the first that you need to only worry about that second term. You will have the velocity of the waves controlled by surface tension and as this customary we call phenomena controlled by surface tension as capillary phenomena. So, we will call v square equal to this second term as expression for capillary waves. So, that is what is indicated here capillary waves are waves which are very small wave lengths gravity waves are the large wave lengths. So, in these asymptotic cases the capillary waves will be given by 2 pi gamma by rho and lambda and generally we may say the waves to be capillary waves if the wave length is less than half a centimeter. What it means now in the context of monolayer is the last thing I want to talk about today. If I have a monolayer on top of water I would have reduced gamma surface tension drops. Therefore, correspondingly the velocity is less. So, what it means clean water will have higher velocities monolayer covered water will display lesser velocities. I would like you to keep that in mind and think about it before we meet next time after the exams. We need to think about how the monolayer is affecting the speed of the wave and the hint I will give is the surface compression modulus C S inverse. We will stop here for today.