 Welcome back everyone, so we will start with the second session today and here we are going to talk about a little bit of introduction to compressible flow. My name is Balchandar Puranek, so now we will start with the discussion on compressible flow, most of you would remember or know that compressible flow as the name suggests is not necessarily restricted only to thermodynamics. In fact, you can imagine that both the concepts from fluid mechanics and thermodynamics are going to be required in order to discuss this particular topic. Many times this compressible flow is not really included in thermodynamics course as well, some places do include it in thermodynamics, some people some places do not. In case it is not included in thermodynamics normally it is included in fluid mechanics course and even many times this compressible flow being a little bit of a specialized topic in fluid mechanics. In the first course of fluid mechanics also many times it is not included, usually a special course itself on compressible flow is offered in many places. And one way or the other what we thought is at least let us get on with some introductory material covered, so that those places which include this topic in the thermodynamics course can utilize it as they say fit in their course. So, what I am going to go through is between today and tomorrow the following list of topics. The first one is conservation of momentum which actually is a fluid mechanics concept or it is basically a mechanics concept here it is going to be borrowed from fluid mechanics. Then we will talk about an important parameter in incompressible flow which is called as a speed of sound. Later on I will talk about what is called as a normal shock phenomenon and finally I will talk about what is called as a steady quasi one dimensional isentropic flow. So, as far as topics number 3 and 4 are concerned the normal shock and the steady quasi one d isentropic flow it is not necessary that you have to go in this particular order. In fact it is quite likely that if you open books it is possible that authors prefer to discuss the steady quasi one d isentropic flow before the discussion of normal shock. It is a personal choice really and in that sense what I would like to convey is that you people should feel free in which sequence you would like to pursue these topics. For some reason right now I am going as what I have listed on the board here after speed of sound I will talk about the normal shock phenomenon and then I will talk about steady quasi one d isentropic flow analysis. So, with this background knowing that we have to combine concepts of fluid mechanics and thermodynamics in order to analyze this compressible flow. Let me first talk about what is meant by our conservation of momentum principle. So, if you remember from the last week's discussions that you must have gone through you have analyzed what is called as an open system analysis. And in the open system analysis you have invariably used some sort of a balance statement and I am almost certain that you have already seen a balance statement for mass as well as energy with respect to what we call an open system or in fluid mechanics we prefer to use term of control volume. So, these two terminologies that is open system or control volume are exactly the same people in thermodynamics prefer to use it open system and people in fluid mechanics prefer to call it as control volume. And what I have written on the slide here is what we call a balance statement for linear momentum again exactly in a manner similar to the balance statement that was written for mass and energy that was contained within the open system. So, in the same line what we can write here is that the rate of change of momentum contained within the open system or the control volume is going to be simply equal to the rate of inflow of momentum minus the rate of outflow of momentum as you can see on the right hand side plus the rate of increase of momentum if there is a source of momentum involved in this. So, let me just quickly go back to the left hand term. So, this rate of change of momentum is basically with respect to time and we may remember that if we are dealing with steady flow situation there is no rate of change of momentum or there is no rate of accumulation of momentum within the control volume or the open system and that is precisely what I have written here that the term on the right left hand side is exactly equal to 0 for a steady flow situation. The rate of inflow and the rate of outflow of momentum from the open system or control volume are because of the mass flow in and mass flow out of the control system. So, when there is a mass flow that is coming into a control volume or an open system it brings along with it a certain amount of momentum. Likewise, when there is a certain mass flow rate leaving the control volume it takes with it certain amount of linear momentum and that is what these two terms on the right hand side at least the first two terms are going to signify. Finally, coming to this rate of increase of momentum due to a source with respect to this balance statement for linear momentum what I would like to interpret this is that the source term is essentially a net force if it is acting on the material that is instantaneously contained within the control volume. So, we know from our basic mechanics that if there is a net force acting on a certain material its momentum is going to increase and that material will experience rate of increase of momentum if it is a net force that is acting on it. In exactly the same fashion for an open system as well you can imagine that there is a certain amount of net force acting on the material that is instantaneously contained within the control volume or the open system. Remember that as far as we are talking about the open system we will always have certain mass flow rate coming in and certain mass flow rate going out. So, it is not the exact same amount of matter that is always within the control system it continuously changing. But as long as there is a net force that is acting on the material that we consider at a given instant in time to be inside the control volume or the open system we will have a non-zero term due to this source. So, this is the way I would like to portray this balance statement for linear momentum as far as a control volume is concerned. If I want to go back a minute and talk about the same balance statement for mass we usually have the rate of change of mass within the control volume equal to the rate of inflow of mass minus the rate of outflow of mass. We do not talk about the source term for a mass. The reason is because unless and until you have situations which will require dealing with creation or destruction of mass which essentially means that we are talking about nuclear sort of reactions we have to not bother with this mass source or massing type term. However, when it comes to the balance statement for momentum we have to include this source term in the form of a net force that is acting on the material that is instantaneously contained inside the control volume or the open system. So, this is what I have written here in words or in plain English language. In fluid mechanics many of you would know that you utilize these balance statements to solve several types of problems. What we are going to do here is that we will utilize this balance statement immediately for the purpose of finding out the velocity of speed of velocity of sound wave directly as an application of the equation. So, let us move on to this important parameter which is called the speed of sound that is required to be considered in compressible flow analysis. And before we actually get on to the calculation or determination of the speed of sound let me just give you a brief background of why we are getting into this kind of analysis. So, the overall picture here is that we are talking about propagation of waves in compressible fluid medium. And what we mean by wave is essentially some sort of a disturbance that you can introduce in a compressible fluid medium at say a given location and that disturbance will move from the location where it was introduced to other locations within the fluid with respect to the motion of the fluid itself. And this is what we call formally a wave motion. If you want to really look at the details of such wave motions you have to go down to molecular level analysis where this wave propagation is quite elegantly expressed in terms of molecular collisions. So, very briefly what ends up happening is that if you end up generating a disturbance at a certain location within the fluid. For example, let us say we explored a small firecracker as what we would do in the valley. So, what that does is that as the firecracker explodes at the location where it is exploding it is imparting a certain amount of energy and that certain amount of energy will create a disturbance in terms of raising the local pressure density and temperature for example, at that location. So, with respect to the ambient what we have is where this firecracker has exploded a higher value of pressure temperature density which simply is what we are going to call a disturbance. What ends up happening is that because you have imparted a some certain amount of energy at that location the molecular activity in terms of collisions at that particular location will go up. And sequentially as time progresses these molecular collisions will transfer this energy in the form of these waves away from the location where we exploded the firecracker. So, technically speaking we can think about a three dimensional spherical geometry sort of a situation where this wave or the motion of the disturbance with respect to the fluid will propagate outward in a three dimensional geometry from the source where it was initiated due to the firecracker. This is the formal model that people want to talk about when we want to describe these wave propagation in compressible fluid media. The simplistic model that we are going to utilize in our course is what we will call a wave front model which is simply a wave front as has been shown here on the slide representing the disturbance such that across this disturbance or across this wave front the fluid properties will change by whatever amount. And therefore in one dimensional situation what we can imagine is that we have a wave front which is propagating with a velocity of c in general let us say in region of fluid which has properties p rho t etcetera. And what we are imagining here is that the fluid into which this wave is propagating with a velocity c is stationary. So, essentially the entire domain here ahead of the propagating wave is at the velocity v equal to 0. Now across the wave across this wave front there will be a change in pressure density temperature etcetera. And I am simply denoting those changes with delta p delta rho delta t etcetera. So, this is the usual model that we will utilize in order to find out the speed of a sound wave. Now what I am showing here on the screen is a general situation where I am talking about a general wave front. If I want to now specialize to what we will call a sound wave front then these changes across the wave front delta p delta rho delta t etcetera are all infinitesimally small or tending to 0. If this is the situation then by definition we call the wave front a sound wave front. And the reason for this is based on actual experimental observation. So, for example, when I am speaking here now I am continuously imparting energy at the location where I am speaking. And that is creating these series of sound waves each of which can be represented using a wave front kind of a model that I just talked about here. It so turns out that the sound waves that we know and the ones that I am creating right now are indeed associated with minute changes in pressure, density and temperature across these wave front. And therefore, the model has a physical basis in the sense that in reality this is what is found to happen when it comes to sound wave. And therefore, just to again summarize if we are talking about infinitesimal changes or very small changes in pressure, density and temperature across the wave front by definition we call that wave front a sound wave front I should say. So, with respect to this discussion now we are in a position to try and calculate the speed of this sound wave front. So, let me go to the next slide which is going to be talking about specifically the speed of sound. Now here I am talking about only the sound wave front and therefore I have changed my nomenclature for the speed of this wave front from that general letter C to A. So, I am going to use the letter A for the speed of sound wave front. So, from now on I am not going to call always sound wave front I am simply going to call sound wave. I suspect that in one of the exercises in your earlier topics Professor Daitonday probably use the letter C for the speed of sound. So, it is just a matter of choice I suppose, but I am going to go ahead with the letter A when it comes to the speed of sound just going back again for a second. If I am talking about a general wave front I am going to use the letter C, but when it comes to a sound wave I am going to talk about the speed as A. So, here it is again what we have is a sound wave moving with a speed of A into stagnant or still fluid medium such as a gas which is at pressure P density rho temperature T and because this wave front we have properties changed by infinitesimally small amounts delta P delta rho delta T etcetera. So, the most standard way of analyzing these problems is to make this system steady in the sense that the top picture on the slide is seen from a so called laboratory coordinate system. So, what I am seeing is that I am attaching myself to the laboratory coordinate system and I see that a sound wave with a speed of A is moving in air which is stagnant and at temperature T pressure P and density rho etcetera. When it comes to formal calculations to figure out the speed of sound the most standard analysis is done in the following manner. So, going back to the slide we make the system steady in the sense that we attach the coordinate system to the moving sound wave itself. So, that with respect to the coordinate system which is shown by this cartoon here the wave is steady whereas it will appear that the fluid ahead of the wave is approaching the wave with a speed of A and the conditions ahead of the wave are P rho and T as they were. The conditions behind the wave now are exactly the same as they were P plus delta P rho plus delta rho P plus delta T and for the purpose of this analysis I am assuming that this wave front is such that delta P and correspondingly delta rho and delta T namely the changes in pressure density and temperature are positive. In other words this sound wave front is increasing the pressure by amount delta P increasing the density by amount delta rho and increasing the temperature by amount delta T and correspondingly I anticipate that with respect to the coordinate system that is attached to the sound wave front I see that the fluid that is ahead of the wave front is approaching the wave front with the speed A. In anticipation of the use of conservation of mass statement I am going to write that the speed with which this coordinate system will see fluid going away from the wave front as A minus delta V. So, some small decrease in the approaching velocity is to be expected as the speed with which the fluid is going to be appearing to leave the sound wave front. And this minus sign is essentially because we are talking about increase in pressure increase in density across the wave front. So, correspondingly we will see that when it comes to conservation of mass statement across the wave front we expect that with the increase in density that is positive change in the density we expect a negative change in the velocity and that is why I am already writing this as A minus delta V as the speed of the flow with respect to the coordinate system that is attached to the sound wave front. So, this is the setting with which we can now analyze and find out the speed of sound. So, going to the next slide I am repeating the previous picture and now in fact what I am doing is I am drawing an open system or a control volume which is supposed to be basically hugging the sound wave front. So, there is an inlet to this control volume and then there is an outlet to this control volume. The inlet is on one side of the sound wave front the outlet is on the other side of the sound wave front. With respect to again this coordinate system which is attached to the control volume which is attached to the sound wave front I again see that the fluid is approaching the control volume or the open system with a speed A. The conditions ahead of the control volume are p rho and t conditions behind the control volume are p plus delta p etcetera and the velocity that this coordinate system sees as the fluid is leaving the control volume is A minus delta V. Now, with respect to this open system or the control volume now we will employ our conservation of mass statement or a mass balance statement and the momentum balance statement with which we began our discussion here. So, if you see with respect to the coordinate system here we have steady flow as it is written on the slide here. So, with respect to the steady flow situation what we have is whatever mass flow rate that is coming into the control volume is the same as what is leaving the control volume. What we will end up doing here is that in such one-dimensional calculations we will assume that the area of cross section is essentially unity perpendicular to the stream wise or the flow direction and therefore on a per unit cross sectional area basis all these calculations are performed. Going ahead then if you employ your mass balance you say that whatever mass flow rate is coming in must be equal to whatever is going out as far as this control volume is concerned. The mass flow rate that is coming in is simply on a per unit area basis again remember the density times the velocity with which the fluid is coming into this control volume and therefore it is rho multiplied by a. The mass flow rate which is leaving the control volume is again density now on the other side of the control volume which is then rho plus delta rho times the velocity with which the fluid is leaving the control volume which is now a minus delta v. This is the principle and here what I am interested in really is making sure that the basic principles are conveyed in reasonably good clarity. There are certain amount of algebra that have to be performed in order to arrive at the final result and some of those algebra I do not want to do here and I am hoping that when you go back you will be able to perform these algebraic steps to arrive at the final results which I am mentioning on the slide. So going back then on to the slide what we see is rho times a equal to rho plus delta rho times a minus delta v what you end up doing here is on the right hand side you open up these brackets and in doing so what you will see is that there will be one term which will involve the product of delta rho times delta v now remember that as far as the sound wave is concerned by definition we said that delta p delta rho delta t and correspondingly delta v are all infinitesimally small. So therefore the product of these infinitesimally small terms is going to be even smaller and therefore we end up neglecting terms like delta rho times delta v. If you do that you can simplify the mass balance to obtain an intermediate result which is given by the first box here it simply says that rho times delta v is equal to a times delta rho and I really request and encourage all of you to go back and perform these algebraic steps to make sure that these final results are obtained correctly. So the left hand side of the slide here showed the conservation of mass statement employed for this control volume on the right hand side what I have is a momentum balance statement which again for the steady flow situation that we are dealing with simply says that as far as this control volume or the open system is concerned the rate at which the momentum is leaving the control volume minus the rate at which the momentum is coming into the control volume must be equal to the net force that is acting on the control volume. So what is the rate of momentum then that we are talking about here? It is simply the mass flow rate times the velocity component in whatever direction that we are talking about. Here is only one-dimensional so it is the axial direction that we are talking about and therefore we have only one component of velocity to speak of. So here that is not really a trouble therefore going back to the momentum balance statement if you want to look at the rate at which momentum is leaving the control volume it is simply going to be the mass flow rate that is leaving the control volume which is if you go to the left hand side of the slide rho plus delta rho times a minus delta v multiplied by the component of the velocity with which the mass is going out in the direction of interest which is again a minus delta v and therefore you see that I have rho plus delta rho multiplying a minus delta v whole bracket raise to 2. This is what we are going to call the momentum linear momentum leaving the control volume. Likewise I have linear momentum coming into the control volume which is simply rho times a which is your mass flow rate coming in multiplied by the velocity with which the mass flow rate is coming in which is again a so I have rho times a squared. And the only forces that we are going to consider here are because of the pressure that is acting on the control volume and let me go to the whiteboard for a minute and draw this quickly. So this is what I have the forces that are acting on the control volume will be due to pressure and we know that pressure is a compressive force. So on the inlet side this is what I will show on the outlet side the force due to pressure will be acting in the opposite direction and this will be p plus delta p. So remember that the flow is going by the way this way. So we can call this as the plus x direction if you want and you have to take into account the signs accurately when it comes to the computation of forces. So when it comes to the force due to this pressure the force is going to be acting in the positive x direction and we are performing these calculations on the per unit area across the perpendicular to the flow direction. So that is not a problem. When it comes to the force due to this p plus delta p on the other side of the control volume that force will be acting in the negative x direction. So with this if I go back to my slide the right hand side which simply says the summation of forces has been evaluated as p minus the whole thing into bracket p plus delta p. Again what you can do is you can open up these products, discard the terms which will involve products of delta rho and delta v or sometimes even delta v squared and if you end up simplifying you will obtain an intermediate result which is at the right bottom. So what you will see is that there is a term here in the intermediate result when you simplify the momentum balance which will involve delta v multiplied by rho the whole thing multiplied by minus 2 times a. And what is now done is you can eliminate this delta v here using the intermediate result that we had obtained from the conservation of mass. So delta v from the conservation of mass is simply a times delta rho over rho. You can substitute that for the delta v in the momentum balance expression. Carry out the simplification and if you do that you will finally end up with an expression for the speed of sound it turns out to be the square of the speed of sound will be equal to delta p divided by delta rho. So this is what the speed of sound expression is and we are still going to talk about this little more but as far as the present analysis is concerned what we have done is again quickly going back. We have simply employed mass balance and momentum balance in a steady situation for our control volume and using these results we have come up with an expression for the square of speed of sound which is going to be equal to delta p which is the change in the pressure across the control volume divided by delta rho which is the change in the density across the control volume. So this is one intermediate result which I will box right now as an important result. One more thing that we can do here is for the open system or the control volume that we have chosen here we can employ what you have already seen in the last week so called one dimensional steady flow energy equation. What we have here is an open system or a control volume into which there is no external heat transfer occurring in or out neither there is any work transfer occurring. So when you want to write the one dimensional steady flow energy equation the work transfer term and the heat transfer term are going to be 0 and therefore I have directly written then the one dimensional steady flow energy equation already discarding the work term and the heat transfer term as the enthalpy ahead of the sound wave front or ahead of the control volume plus a squared over 2 which is simply with respect to the coordinate system a is the speed of the fluid that will be appearing as if it is coming toward the wave front. So h plus v squared over 2 is equal to the enthalpy behind the wave or wave front plus the square of the velocity behind the wave divided by 2. Again in exactly the same manner as we simplified although I did not show the details you can go ahead and simplify this expression also getting rid of the higher order terms which will involve delta v squared etc. and an intermediate result that you can obtain from the simplification of the 1D steady flow energy equation is that the change in the enthalpy across this sound wave front is simply going to be equal to the speed with which it is moving a times the change in the velocity delta v. So this is something that comes up as a result of simplification from here. Now let us see how we can utilize this to qualify the process that is occurring across this sound wave front for which what I have done is I have taken one of your TDS relations only thing is that I am writing it as T times delta s across the wave front equal to delta h minus delta p over rho. So 1 over rho is simply the specific volume and I am writing it as rather than v times delta p I am simply writing this as delta p over rho. For this delta h I am going to substitute the expression that I just obtained from this 1D steady flow energy equation and therefore I have delta h replaced as a times delta v the second term remains exactly the same. What I am going to do now is for this delta v I am going to utilize my intermediate expression from the conservation of mass statement and eliminate that delta v here. So it is simply the a as it is times a over rho times delta rho minus the second term as it is and you can see that 1 over rho then will be coming out as a common factor times a squared times delta rho minus delta p but then a squared times delta rho is simply delta p itself from the derivation that we have completed. In other words what we end up finding is that T times the change in the entropy across the wave front turns out to be 0 which essentially means that the change in the entropy that we can expect across this wave front is going to be equal to 0. In other words the process that is occurring across this wave front so one way you can imagine what is occurring across the wave front is that here is the sound wave front and fluid particles are moving across these wave front and in doing so they are experiencing a change in their pressures, temperatures, density etc by the amount delta p delta rho delta t these are very small amounts by definition. If it so happens then we expect that the entropy of a fluid particle which is moving across a sound wave front essentially does not change and hence we end up saying that the process across the sound wave front is basically isentropic. One other way of coming to this conclusion is simply going back to our control volume we have already said that there is no external heat transfer and you can see that by definition the changes in these properties delta p delta rho delta t etc are all infinitesimally small. So simply using these two facts that there is no external heat transfer and the changes in the properties across the wave front are infinitesimally small we are essentially in a position to say that we are dealing with a reversible adiabatic process which I would call an isentropic process. So there are couple of ways of arguing that the process across the sound wave front is essentially isentropic one way is something that I have outlined on the slide the other way is simply observing that there is no heat transfer external heat transfer and the changes in the properties are infinitesimally small and therefore the process has to be reversible and adiabatic which is going to call which I am going to call as isentropic. So finally with this understanding or physical understanding the square of speed of sound which was simply written as delta p over delta rho is given a qualifier and we write it as the change in the pressure with respect to the change in density at constant entropy and formally we want to write the square of speed of sound as the partial derivative of p with respect to rho at constant entropy. Now for an ideal gas you will recall that if it is undergoing an isentropic process it will follow a relation which is given by p equal to some constant times rho raise to gamma where this gamma is the ratio of specific heats. So let me go back to my board for a second and I will write this that gamma is basically Cp over Cv some authors use k instead of gamma I am using gamma but basically it is nothing but the ratios of specific heat at constant pressure to that at constant volume. Isentropic relation going back to our slide is p is equal to some constant times rho raise to gamma. So if you end up using the recently derived expression for the speed of sound the square of speed of sound and this p equal to C rho raise to gamma you can readily show that a squared is then simply equal to gamma times p over rho and p over rho for an ideal gas is nothing but r times t. So the familiar expression finally that we obtain is a squared equal to gamma times r times t. So going back to my board for a second I have a squared equal to gamma times r times t and this r is the specific gas constant. So this is the most famous result that many of you are probably familiar with also. The formal derivation for this is what we have just gone through. Going back to the slide again the square of the speed of sound then is gamma times r times t and with respect to the setting that we have going back again for a minute here. T is essentially the temperature of the fluid medium into which this sound wave front is propagating. So that is what I have written on the slide here that this t is the temperature of the gas into which the wave is moving. So going back to my board again just to sort of get an idea of what we are dealing with for air let us say at standard conditions I have gamma equal to 1.4 the specific gas constant is roughly 287 joules per kg kelvin and let us say that we are talking about 300 kelvin for the sake of choosing a reasonable number and the square root of all this is going to be the speed of sound in air at a temperature of 300 kelvin. So this roughly comes out to 350 meters per second or thereabouts and this is something that I think normally we would like to mention to our students that if we generate a sound wave in a room which is filled with air and has a temperature of roughly 300 kelvin or 27 degree Celsius then those sound waves will move at the speed of roughly 350 meters per second. So this is what we formally talk about when it comes to the calculation of speed of sound. Immediately what I am going to do now is once we have defined this speed of sound using this speed of sound I am going to define the next important parameter which we are going to utilize a lot in compressible flow analysis and that is what is called as the Mach number. So going back to my slide I define Mach number which I am going to denote with the letter m is equal to the flow velocity divided by the speed of sound in that flow. So imagine that in general the fluid does not have to be at rest. The fluid can be in general moving. The gas in particular that we are talking about here compressible medium will in general be moving with a bulk velocity of v. If you are able to generate a sound wave at the conditions given at a present location within this flow let us say that that sound wave is going to move with respect to the fluid velocity at the speed of a and if you then call the ratio or if you form the ratio I am sorry v over a that ratio is what we will call the Mach number. So the Mach number going back to my slide is then the ratio of the fluid velocity to the speed of sound in that fluid. And immediately you will recall that once you define it in this fashion we have this flow classification the subsonic flow, the supersonic flow and the sonic flow if you remember. If I have the Mach number less than 1 I call that situation subsonic flow situation. If I have a Mach number greater than 1 then I am calling it as supersonic and if it turns out that m is exactly equal to 1 then we call that as a sonic flow where since m is equal to 1 the bulk velocity with which the fluid or the gas is moving is exactly equal to the speed of sound in that gas. So if it turns out that v is equal to a then we have m equal to 1 and we have a so called sonic flow. So just one more point that I would like to convey although I have not written it down here is that many times students do ask and they will ask that suddenly why are we bothered with this speed of sound when it comes to compressible flow situation. Why are we not bothered about this in an incompressible flow situation? And the answer is really that speed of sound will exist whether the flow is incompressible or incompressible. It so happens that if you are dealing with an incompressible flow like for example flow of liquids like water etc. The speed with which sound moves in nearly incompressible medium is extremely high. So right now we just did a quick estimation of what is going to be the speed of sound in air if it is at 300 Kelvin and we saw that it is roughly 350 meters per second. If you were to calculate this speed of sound in water it turns out to be at least something like 1500 meters per second. So physically what ends up happening is that because of these wave propagation whatever changes are going to be introduced in the fluid properties will take place. So formally these are the carriers of information change as we call. So waves when they are moving through fluid will change properties of the fluid at successive locations as they go from one location to the other. If it turns out that the wave speed is extremely high as in case of speed of sound in nearly incompressible fluid these changes in the domain of interest occur essentially instantaneously. Whereas if we are dealing with a compressible fluid medium many times the domain length is such that there is a finite time over which these changes are occurring because the speed with which the sound wave moves for example here is slower and because of these finite times involved when we are dealing with wave propagation in compressible fluid media we actually have to include the speed of sound and the wave propagation phenomena in compressible flow analysis. Usually within the scope of thermodynamics you do not really talk about these things. When you talk about fluid mechanics such details are brought in but in case you know students do ask that why are we bothered about the speed of sound when it comes to compressible flow situations. Roughly and very briefly this is the answer that one can provide that the speeds with which these waves will move incompressible media are relatively low when it comes to those speeds in incompressible situations and because of these relatively low speeds of waves incompressible media interesting things happen and therefore to analyze these things properly you have to include it in a compressible flow analysis. Again you know from a thermodynamics point of view you do not really have to mention these details normally when we go to analysis of compressible flow in fluid mechanics we end up dealing with these kinds of details and here in case there is an interest from the student side you may want to just inform these kinds of things. So what I want to wrap up here is the first part of our discussion which is the speed of sound. Again just to summarize what we are talking about when it comes to speed of sound is a wave front across which changes in pressure density temperature etcetera are infinitesimally small then the process that is occurring across the wave front is going to be argued and we sort of demonstrated that as well is going to be isentropic and the expression for then the speed of sound is that is going to be square root of gamma times the specific gas constant times the temperature of the fluid medium into which this sound wave front is moving is going to give you the speed of sound.