we need to see that how we can simulate these wave flow and then how we can take protective measure preventive measure. So, for all those understanding wave is very important and as such we started with what are the different types of wave how we can classify waves and then as I am just saying that the how the wave propagate that is very important for our various requirement practical requirement that is why the speed of wave propagation is also necessary and there comes the term that we call as celerity well what we mean by celerity is that it is the speed of wave with respect to the fluid media well if we just simply say that celerity is the speed of the wave then it is not complete because we need to say that it is in reference to the fluid media if the fluid media is it at rest then of course the speed of the wave absolute speed of the wave and celerity will be same but if the fluid is also moving and then over the fluid the wave is moving then celerity is the speed of the wave with respect to the fluid media and for analyzing these wave say what is the celerity to have an expression for the celerity when we try to analyze it at unsteady flow we face some complexity and this unsteady flow therefore we try to convert to an equivalent steady flow so that it become convenient for our analysis and last class we did so how we can we could see how we can convert this to a equivalent unsteady flow and we are starting from equivalent steady flow and we are starting from that in this class well say this is a wave forming here a solitary wave single wave and the bed is there and by say and let us first consider that this is a steel water maybe in a tank and by moving a paddle we have created a wave a single wave is moving like that and it is moving with a speed c well and this is a unsteady flow this is a unsteady flow now to convert this to a steady flow situation as I did explain in the last class that like a walker as the belt is moving upstream or say when we walk the belt moving the opposite direction in a walker and that is why although our that we are walking on a walker but we are not moving forward and then someone can observe us that how we are behaving when we are walking but we are in a steady position we are in a static position so that advantage we can have here like the belt moving the opposite direction in a walker we can put a opposite direction the velocity c which is the actual velocity of the wave then what will happen that this is also moving in a velocity c wave is also moving in a velocity c in this downstream direction and as now this fluid media is moving in the opposite direction with velocity c and wave which was moving in the downstream direction with a velocity c so relative velocity of this will become 0 and then it will be remaining in this position and now we can observe it carefully well now let us see how we can analyze it say this depth p at this section suppose at this section depth b y and so amplitude of the wave is say a this is the amplitude of the wave and so what will be the depth here depth in the wave portion where we have the wave this will be say y plus a well and when we talk about velocity then c is the velocity at this section say section 1 and if I talk about the section 2 then what will be the velocity say velocity is v because this velocity will not be equal to the velocity c because depth here is change so let us write this velocity as v well now we can write a continuity equation and from that continuity equation we can find that how we can express this v and c how we can relate this v and c well how we can write the continuity equation say if we consider this channel to be say rectangular and suppose we are considering unit width of the channel so this is let me just write here this is a equivalent steady state and now let us see the continuity equation how we can write say continuity equation between section say 1 and 2 between section 1 and 2 how we can write and that we are writing considering unit width of the section unit width of the section well we are considering unit width of the section then what will be the discharge in continuity equation state that here the discharge and here discharge will be equal discharge at section 1 and section 2 will be equal fine now discharge is equal to the velocity at this point 1 section 1 and the area area is equal to y into each width in fact if we consider width then width will be coming but we are considering width to be equal to 1 so it will be c into y so we what we can write that c into y is equal to here we are talking the velocity as v so v and depth here is equal to y plus a and from this what we can write that v is equal to c y divided by y plus a let us keep this expression here and now we can write another expression this is from continuity equation we are getting this relation that is v we can express in this form well but we can have another expression from the consideration of say energy equation now what will be the energy at this point it will be y 1 plus that means at section 1 it will be y 1 plus sorry y we are writing y y plus c square by twice z and here because velocity is c here and here it will be y plus a plus v square by twice c and let us write that expression well and you can write that energy equation energy equation between section 1 and 2 of course when we are writing energy equation between section 1 and 2 and we are saying that energy at this section at that section are equal what we mean that there is no energy loss in between that assumption is inherent there well now let us write that expression say y plus c square by twice g that means v square by twice v and velocity square by twice g this is equal to depth in the section 2 is y plus a plus say v square by twice c let me write one step v square by twice c now what is v is we know that v is equal to c then y by y plus a so we can just replace it in this form that is y plus c square by twice g is equal to y plus a plus write that c square this in place of c square by y square by y plus a whole square so v square we can write that v square by twice g will be and v we can write c square into y square by y plus a whole square well this is the expression and from this let us try to write because our interest is to get an expression for c and here what is not known and what we want to try is the c and what is our unknown thing is y a here the amplitude and at the same time this expression we are writing for a very for a way where the amplitude is very small I mean small amplitude way as we just if we recall our last class then we were talking about one expression that c is equal to root over g y for rectangular channel we obtained that and that was for small amplitude way so here also we are considering that this depth is considerably large than the amplitude a well let us just separate this term and see how we can write it this y and y will of course get cancelled and let me write first a in terms a is equal to then c square by twice g c square by twice c is here so we can bring c square by twice g and then we can write one minus say y by y plus a whole square well and this can be written as say let us break up y c square by twice g our intension is to see that if because in the expression of c in fact we do not have the a term root over say g y that was that that what that is what we got in the last class so let us see by we are starting from the first principle and we are trying that yes if we can get this expression and here we do not have a a term so let us see how we can do that and this we can write say y plus a whole square and then this will be say we can break up y square plus twice a plus twice a y y square plus twice a y plus a square then minus y square well here we are in a position to simplify now well so what we can do that from here let us write one expression for c so c is equal to we can write that c is equal to twice g a will be coming c square let me write c square is equal to twice g a and then this will be just we can write in the reverse direction this will be say y square plus twice a y plus a square this is there and here what we will be having this and this is getting cancelled already and here we have twice a y plus a square so that twice a y plus a square we can write here as a keeping common and we are writing twice y plus a well now this a and this a we can cancel at this point and let us see how we can write this expression well so we can write c square is equal to now root over twice g twice g and entire things we can put under root sign root over twice g and then here it will become say y square plus twice a then twice y plus a let me write c square is equal to now root over twice start from this expression again y square plus twice a y plus a square divided by a twice a plus it is a and a is getting cancelled so what we are writing that twice g and in this part we are writing that expression well plus a square well so now here as we know that a square is say a itself is very small so this a square term we can put equivalent to 0 or say it is negligible so if it is negligible now what we can write that c square now it is c because we have written already in terms of root over so c is equal to say our target is to get this term g y so we are writing g y means from here we are bringing a y common we are bringing a y common here in fact we had a term a y a y plus a square right so y we are bringing common and 2 we are taking to that side so what we can write root over g y then here we can write so as 1 y has come so it will become twice y 2 is going inside and then plus 4 a 2 is going and this is coming so it will be twice y plus a well now this we can further simplify in a way that this is equal to root over g y and let me write it in this form that twice y plus a fine here also let me write twice y plus a then in fact we need to write plus twice a we need to write plus twice a well so this twice y plus a twice y plus a that we can have 1 and this will become twice a by this expression and so let us write c is equal to root over g y then we can write root over 1 plus twice a by twice y plus a now this expression twice a so a is very small and here in the denominator we have twice y so it is quite a large expression and as y is very large as compared to a so this expression will become every small term divided by every large term so this we can now neglect so if we neglect this term say as a is very very small as a is very very small as compared to y therefore 3 a by twice y plus a is say negligible is negligible and now just neglecting this part it will become c is equal to root over g y well so that is the expression that we can derive what we got in our last class again starting from the first principle we could see that for a small amplitude wave we can have the expression for c as root over g y and if it is a suppose non rectangular channel then what we do that y will not represent this things properly so what we write that c is equal to root over g d what d is d is the hydraulic depth d is the hydraulic depth so using this hydraulic depth in fact for rectangular channel this will become y but for trapezoidal channel it will be a by t so using this expression also we can get the value of c so after that let us see that well for a wave of this type for a wave of this solitary wave of this solitary wave of this kind we can have an expression for c but if we have a charge if we have a charge and how to analyze this charge whether the c value will be same as or the same expression whether we can use or not well that first we should know what is charge this is also in unsteady flow this is important I mean important topic that is charge we need to do lot of analysis in fact on charges and but here we will be just doing some limited exercise on charge that you can see that charge is a moving wave front that brings about an abrupt change in the flow depth of the channel so basically what we mean by charge is that say this is the bed and normal flow of charge is suppose moving here and somehow a wave is coming like that it is a moving wave front that is why we call this as a wave front this wave front and then it is so this water is moving like that otherwise also and then we have a moving wave from this wave front is moving with a velocity say v w this wave front is moving with a velocity v w and this can be this we refer as a charge in fact it can be in the other way also say your water level is here water level is here and water itself is moving in this direction suppose and somehow due to closure of gate the you have suddenly closed the gate so water level here is rising and then the wave can be moving in the opposite direction also this is say wave moving upstream and this is a wave moving downstream so whatever may be the situation we call such moving wave front which is moving and creating a abrupt change in the depth so depth earlier was like this and there is a abrupt change if the depth changes gradually like this then we will not call this as a charge but here depth is changing abruptly and that is why we call this as a charge and the charge and so this charge also we can express in the form of say equivalent steady state for its analysis well let us see how we can represent this in the form of a equivalent flow well let us consider this one anyone we can consider but let us consider this one suppose this is say one and this is section two so say one and two why we are considering because this is both in the same direction here it is in the opposite direction that will give us an opportunity to handle a case where the flow and waves are moving in the opposite direction till now we have not discussed that sort of situation so suppose the velocity here is v1 and velocity here will be changing velocity here means in the entire section say v2 well now vw is the velocity which is moving in the opposite direction this is a unsteady situation at the next moment of time this wave will be moving further this wave will be moving further and say we will not find it here so let us see how we can do this to keep in a steady position well what we can do in this we can apply an additional velocity in the opposite direction to the speed of the wave well so please just see we are meaning that opposite direction to the speed of the wave but this speed of the wave is not the celerity celerity will be the relative speed so we are we need to apply a velocity in the opposite direction to the speed of the wave otherwise it is moving with a velocity v1 now if we apply another velocity say vw if we apply another velocity say vw in this part then total velocity at this point will become vw plus v1 and here also it will become v2 plus vw and then velocity of this in this direction will become 0 because it is moving in this direction it is and then that is why the this things is not in a position to move further it will be remaining in the same position well so let us represent this and let us see how we can derive okay I am now having this portion and this equivalent steady state position we can keep here we can draw here so it will be like this now we have here the velocity will be v1 plus vw and here the velocity will be v2 plus vw and of course there will not be any velocity of the wave it is not moving rather so this is basically we can say equivalent steady situation well now again for this search just like we were writing earlier we can write continuity equation and momentum equation so continuity equation we can write here say continuity equation and let us put some depth value here say depth here is y1 and depth here is say y2 well now what is continuity equation so again considering unit width of the channel considering unit width we should mention this one considering unit width this will become say y1 into v1 plus vw plus vw is equal to y2 into v2 plus sorry it is not v3 it is vw v2 plus vw well so that is our continuity equation and this continuity equation can give us or from this continuity equation we can express y1 in terms of y2 and we can do those things and then what is this v1 plus vw that is also interesting in fact if from this point when wave speed or absolute speed of the wave is say it is moving with a speed vw and then the fluid is moving in the opposite direction v1 then what is its relative speed its relative speed will be v1 plus vw v1 plus vw because both are moving in the opposite direction so it will be it is in fact it will appear that it is moving faster when v1 is in this direction vw in this opposite direction so wave speed relative wave speed will be v1 plus vw and this relative wave speed in fact we call as c so this v1 plus vw is nothing but c well that we can write that equation so continuity equation we have seen and then let me write the continuity equation again and let me keep the diagram here quickly say it is v1 plus vw sorry v2 plus vw and this is v1 plus vw that we have and depth here is y1 this is y2 now the continuity equation we have already described so y1 plus y1 into v1 plus vw is equal to y2 into v2 plus vw now we need to write the momentum equation we need to write the momentum equation so for momentum equation for writing the momentum equation we need to see what are the forces acting so the pressure force from this direction and the pressure force from that direction and then the momentum change in momentum between this point and that point so pressure diagram we can draw like that and as we know we have discussed that part a lot so pressure here we can write as half of rho g into y1 square because pressure here is nothing but rho g into y1 or w y1 we can write unit weight as w or we can write rho g so writing in any form we can get it say rho g y1 and then area of this triangle will be half of rho g y1 plus multiplied by the way y so it will become half of rho g y1 square so momentum equation what we can write the force from this side that we can write this is continuity equation and now we are writing momentum equation so momentum equation we can write here say half of rho g y1 square that is the pressure force from this side and minus half of rho g y2 square that is the pressure force from that side that is equal to change in momentum rather rate of change in momentum so what we can write that mass flowing per unit time and change in velocity so mass flowing per unit time how we can write that discharge flowing per unit time well discharge as it is not changing from here to here no other flow is being added nor the flow has been taken out so we can write it in terms of discharge of this force because this is known to us what is depth y2 that is of course not known to us because after coming of the wave this is happening but this y1 will be generally known to us so let us write the discharge in terms of this one that we can write discharge in fact is equal to y1 into v1 plus vw is the velocity and area is actually y1 into width but width is 1 so this is what the volume flowing we can say and then this volume flowing if we multiplied by rho then it become the mass flowing so this is what the mass flowing and then v square or say sorry not v square this is the mass flowing and the change in velocity so change in velocity is equal to in fact v2 plus vw minus v1 plus vw so ultimately the resulting will be v2 minus v1 so that is what the change in flow velocity this is our momentum equation well from this momentum equation we can further derive some of the relation which is required for finding the expression for c in case of charge and well before that let us write the expression for c here this is another equation that is c is equal to we have we can write that c is equal to say v1 plus vw that we have already explained how it is so this is equal to c celerity c that is the expression for celerity c and this expression celerity c we can further say starting from the momentum equation we can derive it in a different form and let us see how we can do that well now we need to mention one point here that we got one expression for c is equal to root of our gy and this equation is called Lagrange's equation of celerity this equation c is equal to root of our gy this equation is called Lagrange's equation of celerity well and now let us see what will be the expression for c in case of charge in terms of y1 and y2 well in general we can have it as v1 plus vw but we then we do not know what is vw and so we need to have it in a different form like that earlier we got in terms of depth y so let us see what we can do starting from this again starting from this momentum equation we can write that this is equal to say rho we can cancel in each part then let me take common here say half of g then y1 square say half of g y1 square if I keep here then it become 1 minus y1 square by y2 square well and on this side we can take say y1 is here and then v2 minus v1 is there so what is the v2 is the velocity so velocity and this expression actually y1 plus v1 plus vw this expression this expression is nothing but this is showing how much is the discharge I am writing this as q this is q means unit discharge because we are talking about discharge through unit width so this is equal to q and what is our v2 or v1 v2 is nothing but q by say y1 because v2 into y1 is equal to y2 is equal to q and v1 is equal to q by y1 because that we know that is again from continuity equation we know that v1 y1 is equal to v2 y1 that is also known to us so in fact this v2 can be written as v2 can be written as q by y2 means q is equal to v1 plus vw q is equal to y1 v1 plus vw divided by y2 so that can be written so well so if we bring this q common here let me write it in this from here say y1 plus v1 plus vw and then this q we can let us keep rho still here because here also rho discharge will be coming so this discharge is nothing but say we can again write this part as y1 into v1 plus vw that is what the discharge we are writing here bringing common and then inside this will be say 1 by that means q we are bringing out this q is nothing but this expression and then here it will be 1 by y2 minus 1 by y1 and as we can see very clearly that this term and this term is same so we can just because we are writing for discharge only so this will be square so this can be written as half of gy1 square and 1 minus y1 square by y2 square is equal to this now we can write y1 square and then we can write here as say v1 plus vw square then within bracket we are putting say 1 by y2 minus 1 by y now let us see how we can further simplify this expression here that is the v1 plus vw is nothing but c so from that we can put the value here as c square say y1 square and this y1 square we can cancel and then we can just rewrite this expression in a simple form like that c square we have already replaced this part that means this we are writing as c this one this is nothing but c so putting that what we can write c square is equal to say half of g half of g then we can write say y square y1 square minus y2 square that we can write y1 plus y2 into y1 minus y2 and this is divided by y1 square and then this the denominator part we can write say y1 minus y2 divided by y1 y2 that means this part this is become y1 y2 then y1 minus y2 so this is coming here and this is becoming y2 square and y2 square then this y1 y2 square minus y1 square that we are writing as this one say y1 y2 and y1 plus y2 now from here what we can do this is equal to half of g then we can write y2 by y1 so this y1 and this y1 is getting cancel and we can write this as y2 by y1 y2 by y1 y2 by y1 into y1 minus y2 by y1 minus y2 by y1 minus y2 by y1 say which will be y1 plus y2 that will be the expression and we can of course further simplify from this part you can see that our target is always to get the expression in terms of root over gy and then whether we can have it in further simplified form so c square this is equal to we can bring one y from here one from one y one from here so this is equal to say gy1 one y1 we are bringing from here and then the other part we are writing in a different way say half we are writing here and this y2 by y1 is there y2 by y1 and then as one y1 we have brought here this will be 1 plus y2 by y1 well so that is the expression we are getting and as you can see that this will be y1 so c square is equal to this part then if we just try to take c then we will be getting c is equal to root over gy1 then root over say half of y2 by y1 and 1 plus y2 by y1 plus y2 by y1 well now in this expression normally if we refer to our earlier diagram this say y2 will not be known to us because it is coming this wave is coming and the y1 will be known to us of course so to get the value of c then say generally we are getting the expression in terms of y2 by y1 and here it is gy1 and this y2 we need to know in many case so this is y2 sorry this should have been y2 square by y1 square yes we are taking common y1 square here and as such it will be y2 square by y1 square and this is y2 square by y1 square right and that is why we are getting this expression in this form like that one this one and so this is coming in this form well fine so that way c is equal to root over gy1 and this expression we are getting in terms of y2 by y1 and this is the expression normally we use for computing charge ok charge means a celerity in charge and then this help us in various in solving various numerical problems small numerical problems we can solve by using this particular expression well at minute for c2 however continuity and momentum equation and that is what we need to know so we are getting this expression in terms of y2 by y1 39 is not it 39 means in other words it takes 20 minutes 19 minutes problem is not going to happen in 19 minutes not at all it is not that easy right? it is not easy it is not that easy it is not that easy let us take a problem typical problem in fact the problem, typical problem, in fact we will not be solving the problem as such, but we will see how the steps need to be followed and how we can get the results, so, the problem suppose he is like that, what are flows in a rectangular channel at a depth of 2 meter we carefully 1 meter per second, well, say this is the channel and here water is ంిితి బానిమి మిడ్చన్చి వటం్సిస్ని పరపటిత్బిమాసి మారందాహి మరనిమలికిసంచె. So, this is say depth is 2 meter and the velocity of flow is 1 meter per second well and if discharge in the channel is suddenly trivial what will be the depth of flow. So, here it was the depth of flow suppose in the discharge if we suddenly trivial then what will be the depth of flow suppose depth of flow is increasing like that and this will in fact continue to increase this will continue to increase this will continue to increase and then wave is moving in this direction and after sometime the depth will be increasing. Now here this is in fact our say we can put as y 2 and this is our y 1 of course we can write either way also and then velocity here is say v 1 velocity here is v 2 and then wave velocity and then wave velocity let us put as v w. Then how we can write continuity equation and momentum equation for this case and what we know here is the depth y 2 and this v 2 is known velocity is 1 meter per second and depth is say 2 meter. So, that part is known and other things are not known and what we are knowing the discharge what was earlier flowing is just trivial. So, this discharge what will be getting here y 1 v 1 say considering unit width what will be getting y 1 v 1 that will be 3 time of that of the say y 2 v 2. So, how we can write the continuity equation again to first make it steady what we can do to make it steady we can apply a velocity opposite to this v w. So, we will be applying here v w and then if we apply v w what will be the velocity here ultimately this velocity will be changing to say v 1 minus v w after applying this one and this will be changing to this velocity will be changing to v 2 minus v w well and then continuity equation we can write say v 1 minus v w into y 1 is equal to v 2 minus v w. So, that is what we can write the continuity equation now here we can put the value of v 2 and y 2. So, what we can write that this is nothing, but say v 2 is equal to 2. So, 2 sorry v 2 is equal to 1 1 minus v w into 2. So, this equation will give us some information then we can write the momentum equation we can write the momentum equation say.  And then this expression also we can simplify means we can put the value like half of say rho g y 1 square minus y 2 square means 4 this is equal to rho we can write this as rho then v 2 minus v w that we can again write as say v 2 was our 1 minus v w and this y 2 is already known to us. So, this will this is 2. So, this 2 depth is we got the depth s sorry we got the depth s 2 meter. So, that we can put here say 2 and then this v 2 minus v 1 again this we can write as 1 minus v 1. Now, this is say equation 2 well that what information we have we have a relationship between what information we have that you can refer to the slide we have one information like that v 1 y 1 which is the discharge which was the discharge after the charge is coming that is v 1 y 1 this is the actual discharge what is coming here and this discharge is triple or three times of that v 2 y 2. So, we can write it like that v 1 y 1 is equal to say three times of v 2 y 2 and this value we can have that is equal to three into say one is the velocity into the depth 2. So, this is equal to six. So, that way we can have I am just writing the numerical value not writing the unit here. So, this is one expression and then we have these expression and solving these expressions we can get ultimately the value of y 1 because our target is to know the value of y 1 and. So, if we know this value of y 1 our problem is solved and to get the value of v 1 we have these equations say this is one equation and then here of course, we have this unknown v w and this is one equation where we have this unknown v w. So, v w and y 1 are the unknown that we can solve by trial and error method and we can get say y 1 is equal to 2.45 meter. This is just a solved problem already in the book of Rangaraju and just I am taking here just to show how the search problem can be solved. Well, then we will be taking up the next topic rather the equation of unsteady flow. Well, well, so equation of unsteady flow that we know that already we have discussed some of the equations of unsteady flow when we were discussing our earlier discussion on say governing equation of a functional flow. For unsteady state we were discussing about the continuity equation. The continuity equation that as we know it is say del a del t plus del q del x is equal to 0 that we could get and this of course, we can write as say del a del t plus this q if it is q is written as a into v this can be written as say a del v del x plus v del a del x is equal to 0. Well, so here I am not taking up again this continuity equation that you can refer or rather we will be knowing by referring to our earlier classes and then the two governing equations are required basically for solving the problem of unsteady flow. And then one of the most popular equation of course continuity equation is there then another equation that we take is the equation of motion and this can be of course, we use momentum principle or energy principle to get this equation of motion and for some case we use say continuity and momentum equation in couple for solving the problem. Sometimes we use continuity and energy equation for solving the problem depending on the situation where suppose energy is energy loss is there there we go for momentum couple continuity momentum couple. So, that way these considerations are there of course, in this class we are not going for all those detail, but we will be just discussing the equation of motion that was given that was first deduced by Barrett Saint-Venant and popularly known this equation as Saint-Venant equation and that is say 1879 long back this equation was derived well and this is a dynamic equation of unsteady flow in open channel and for what condition it is the shallow water wave equation shallow water wave equation. That means why we are giving this a shallow water wave equation because when this equation was derived this vertical acceleration was considered to be negligible. So, vertical acceleration is negligible and then we should know that for what condition we can apply this and derivation of this equation not this equation this is continuity equation derivation of the equation of motion can be done by two approach one is energy approach and another is that we can start from Newton's equation of motion. So, by both these approach we can have the derivation and here we will be discussing how by energy approach we can derive the equation in a simple way and then let us see what is the energy equation say if this is the flow and this is going like that well let me take some space because we will have to write here. Say this is section one and section two so section one and two well in this section one and two let us take a small length let us take a small length and let us write this as say as this small let us write this as say del x this length is and datum is here datum is here. So, this depth is say z one and this depth is say z two that is the elevation head up to the bed and then up to this depth this is y one we are considering slope to be small that is why we are not writing y one cos theta or those term we are writing straight weight is y one and then here this energy is v square by twice g v one square by twice g energy head up to that point and then here in the second section section two this is the y two and the v square by twice g is suppose that much v square by twice g v two square by twice g and then we can draw the energy gradient line as this one energy gradient line as this one and there is definitely some energy loss this we refer as so when we draw it parallel to the datum then we will be finding that there is a head loss and this head loss we can write as h l and in fact this head loss is having two component one is that one is that it is friction loss that is known to us because we did discuss about this friction loss during our discussion of gradually varied flow itself and but here there will be another component because in case of unsteady flow that local acceleration exist local acceleration exist just to recall briefly say when we say v then it is a velocity which is a function of x and t of course in steady flow it become a function of it is not a function of t but in unsteady flow this become a function of t now when we talk about so this change in v say d v that we can write as del v del t into d t plus del v del t into del v del x into d x del v del x into d x now if we write that it is a function of x and t so if we just write it as d v d t d v d t then we are getting del v del t plus say this d x d t so d x d t divide into del v del x and this d x d t is nothing but v so now perhaps you can recall that we did discuss earlier also d x d t is equal to v del v del x and then this is called local acceleration this is called convective acceleration now in case of say steady flow this term is not existing so that problem become little simpler but here we have this term so this local acceleration exist so because acceleration is there in any whether it is fluid or any solid but if acceleration is there that is a mass moving and that mass is having some acceleration means some force is being utilized to have that acceleration otherwise that acceleration will not be there so some force is being utilized to have this acceleration and when some force is being utilized and things are moving then there must be some work done force is applied and it is moving so some distance has been moved so some work done will be there and some work done is there means some energy loss is there so we can say that in this sort of flow there is two type of loss one is h f another is h a so this two loss will be there this h a is say loss due to acceleration well now if we write the energy equation so writing energy equation between say 1 and 2 between 1 and 2 what we can have between 1 and 2 this we can write say z 1 plus y 1 plus v 1 square by twice g is equal to z 2 plus y 2 plus v 2 square by twice g plus h l well plus h l or say we can say loss loss h a loss loss h l okay now this loss as I have explained that let me write here say loss h l is equal to say friction loss friction loss that we write as h f and plus the acceleration loss acceleration loss that is say h a so our expression become now of course in this loss term we need to put this two so this is equivalent to h f plus h l already we have written here now we need to have an expression for this two so h l is equal to we know that it is h f plus h a now how we can get the h f because friction slope we know that as f if friction slope is as f friction slope is say as f then we know that as f that is friction loss divided by delta x delta x is equal to as f or we can say that as f the slope into distance as f into this delta x will give us the h f so friction slope may be this like that and we can have as f up to certain point h f and then this slope is actually as f and then other part of loss is due to say this acceleration so that is clear to us that we have discussed earlier also so in place of as f by delta x also in place of as f we can write that as f is equal to as f into delta x well now that so for our next acceleration loss is there how we can calculate that so acceleration del v del t local acceleration del v del t exist here so if this is the acceleration then if this is the acceleration then the mass suppose if we consider that unit weight of fluid unit weight of fluid then mass is equal to say weight by actually weight by the g so this is here as our weight is equal to 1 so this is equal to 1 by g and force utilized forced utilized to produce this acceleration force utilized to produce to produce this acceleration that is equal to mass into acceleration mass into acceleration so that we can write as 1 by g into del v del t so this is what the force required force utilized then this forces work for how much distance this force is working and then acceleration is being there and then the fluid is moving this mass is moving from here to here so the distance of say delta x it is being moving so what is the work done and distance move distance moved by this unit mass by this unit weight rather by this unit weight of fluid is equal to delta x that is what we are considering therefore the work done how much work is being done work done is equal to 1 by g del v del t into del x del v del t into del x well so this is the work done and what this work done is that is the energy loss energy loss so energy loss is equal to that means 1 by g del v del t into delta x so finally how we can write the head loss so this energy loss is nothing but h a so this energy loss due to acceleration that is the work done due to produce that acceleration and that is the energy loss we can write like that therefore so what we can write that head loss h l is equal to h f what we can write h f plus h a h f plus h a and that h f we can write s f into delta x and then this h a we can write that is the acceleration loss 1 by g and del v del t into delta x so putting this value of h l in our equation of energy this one then we can get a simplification of this equation of this equation of this equation of motion and that we will be doing in our next class and then from that we will be also trying to see how this equation can be solved and then by solving this equation how we can get solution to our real life problem of open channel flow so thank you very much.


What's up everyone? It's your girl, Kenzie here from Layton Sports Cards and I am ripping a personal here for Brett S. We've got an order of two 2022 Panini National Sports Convention VIP exclusive silver packs and they are in lots of five.  We've got one, two, three, four, five on the first bundle and one, two, three, four, five on the second. All right. Good luck. Thanks again for hopping in. Let's see what we can get you. I'm not ripped any of these silver packs yet. Just the um, Thursday ones. We also have a couple of golds in queue. So definitely looking forward to that. We've got a Traylon Burks rookie and foiled rookie of Drake London. Nice one there. Predict that they have a pretty big year. The Falcons won't be half bad this year. We've got Kawhi Leonard and blue foil of Jason Tatum with the foil finish on the uh, packing. Patty Mahomes and we've got a James Harden foil Sixers. Yeah, for sure, Duffy. I've seen some ripped and they're, they're nice packs for sure. Good looking cards. What's up, Mike? Especially the ones with like autos and some of the patches on these. They're, they're nice. Tyree Killen is a Dolphins uniform. That's cool. And we've got a Clayton Kershaw foil. All right. Last one of the first bundle of five. We've got Jalen Duren and foil of Shohei Otani. Nice Showtime foil. All right. Again, second bundle of five, one, two, three, four, five. Here we go, Rhett. Look forward to rippin' some of the gold ones too. Like I said that are in queue. We've got rookie of Garrett Wilson and purple foil of Steph Curry. That is number 20 out of 25 on the purple. Nice splash, bro. Yeah, Mike. I should be able to do that for you. Got Trayon because you just put an order in and we've got nice Christian Polissic out of 99. Color match there with the red and the USA jersey. Got AI and foil of Jonathan Taylor for the Colts. There's foil there out of 199. Couple packs left. Ben Simmons and green foil, KD for the Nets. That is one of five KD. Wow. Nice hit there, Brett. I'm curious to see where he goes. Nice KD to five. Beautiful head. Place about two. Okay, let me just take a look real quick after this for you, Mike, but yeah, that shouldn't be a problem. Should not be a problem. Got Devin Booker and last hit for you. We've got a rookie, Bobby Whitt Jr. on the orange to 199, 11 of 199. Nice wit rookie to end that out. All right, Brett, that'll do it for your silver packs. Thanks again for your order and for your patience. We'll get these right out to you.


This is Dockimer 11062A. We are here at Summit Automotive in Fond du Lac, Wisconsin, your new and used light duty truck. Headquarters today we are checking out this Super Clean 2014 Ford F-150 Super Crew 5.5 foot short box. This truck has the XLT package. It also has the 3.5 liter twin turbo V6 EcoBoost motor which comes out 365 horsepower. This truck has been fully safe and inspected by our service shop. It has a fresh oil and filter change. All the fluids have been checked and topped off. It has four brand new Goodyear tires and this truck is 100% ready to go. Ruby Red Metallic is the color. We shoot all of our videos in 1080p 60 frames per second so if you have HD capabilities on your computer tablet or smartphone device turn them on right now because it is definitely your best way to check out the quality and condition of the truck before seeing it in person. And if you'd like to check out all the photos on this truck in the upper right hand part of your screen as they link to our website click that and check us out there. This one comes with the factory alloy rims with the chrome covers and it has Goodyear Wrangler 4 to 2 to HT tires. These are 275 65 R18s and they are brand new tires. We put those on in our safety inspection. You can still see the little wear knobs there. So brand new tires all the way around. Front fender is in excellent shape. Headlight lenses are nice and clear. Factory fog lights from bumpers in really nice shape as well. I didn't see any major dense dings on that and it does have the chrome tow hooks in front. Chrome trimmed grill as well and the hood of the vehicle is in really nice shape. I didn't see any dense or dings on there and the passenger side front fender is in nice condition as well. Passengers side rim no major scuffs or scrapes on there that tire is also brand new and as you go down this side of the vehicle take note of how clean the body is how reflective and mirror like that paint is. We take these HD videos so if you are far away or even if you're close by and you just cannot make the trip down but you're still interested in purchasing the vehicle you can see the truck hear the truck and have confidence in the vehicle that you're looking at before you even get here so that when you do get here there's absolutely no surprises this one comes with a Lakeland truck cap. That's why we offer the Summit Auto Fastpass option in the upper right hand part of your screen a link right to our website where you can apply for financing get pre-approved see a payment quote even appraise a trade all from the comfort of your own home so when you get here all you have to do is drive the truck love it you'll be all set to go back tires are brand new as well and the frame and underbody for being a 14 and 90,000 miles is in pretty nice shape and from Wisconsin I should say but very nice under here and it's like that all the way underneath bodies in nice shape has the factory chrome step bars and the cab corners actually look really good on this truck that's usually the first spot that goes on these F-150s but this one looks really good on the passenger side as we come around to the back of the vehicle rear bumpers in nice shape no major dents or dings on that has the backup parking sensors full towing package which includes a receiver hitch four pin and seven pin wiring the tailgate is in really nice shape I didn't see any dents or dings on there and that Lakeland cap is in nice condition to inside you get a factory Ford drop-in bed liner and the caps in nice shape to think those windows slide open and they got a screen on them so if you wanted to put pets back there you could as you go down the driver's side just as clean as that passenger side no major dents or dings in the box has the capitalist fuel fill which is a really nice feature and for full disclosure this back rim nice shape and that tire brand new as well cap is a nice condition on this side too coming down the rest of the side truck looks really good cab corners look pretty good on this side as well doors look pretty good driver side door code entry we do have the code for that has the heated mirrors with the built-in directional sign signals driver side blind spot mirror the only thing I saw is one tiny little ding on the door right there but pretty hard to see but that's why they're used inside the XLT package gives you the gray cloth interior there are no rips or tears on those seats power driver's seat we have a set of weathertech floor mats throughout on this truck auto headlamps power pedals power windows power locks and power mirrors and as we hop inside the truck you can see that this one has 91,076 miles get the Ford tough animation instrument clusters very nice and clean we're a little low on the battery right now since I've been doing the videos here and had the lights on leather wrap steering wheel bluetooth and audio controls on the right cruise controls and information center controls on the left has the six-speed automatic transmission with the optional tap shift turned out four-wheel drive factory brake controller you have AM FM and Sirius XM radio capabilities on this radio you have your stability control right there down here your climate controls you also have the sync system so you can connect your cell phone to it and this is also where your backup camera shows up I'll show you that when we start the truck here in just a second carpeting and passenger side floor mat and seat are in excellent condition no rips or tears on them and the headliner is in really nice shape as well you get map lights power sign mirror window and we'll take a quick look at the back seats and then we'll start it up check out under the hood back seats just as clean as the front seats no rips or tears back here it does have the latch child safety system for any child car seats you may have power sliding rear window with the built-in rear defrost get a weather tech floor mat back here as well which keeps your carpeting looking good and a set of factory formats underneath to boot side curtain airbags and child safety locks on the back doors bottoms of the doors look pretty good as well and we'll start and take a quick look under the hood starts right up I would personally like to thank you for checking out the video today and hopefully from this HD video you'll have been able to tell just how clean this truck is all the way around inside now under the hood we have the 3.5 liter V6 twin turbocharged EcoBoost motor and Jinbe is very clean runs very smooth has a die-hard pattern once again this truck has been fully safety to and inspected by our service shop has a fresh oil and filter change all the fluids have been checked and topped off and this truck is 100% ready to go I'd highly recommend this truck from a quality and condition standpoint also for brand-new tires and to see more pictures of this truck or one of our other 450 new and used cars trucks SUVs minivans Wranglers half tons to quarter tons one tons you name it we got it go to our website right there www.summitauto.com full pictures and descriptions of every single vehicle from two locations all at summit auto.com and if you'd like to check out more HD videos you can go to youtube.com slash summit auto remember to like subscribe and share on this video and all the videos that you see there in fact in a second you will see a link to subscribe to our YouTube channel on your left a link to more Ford F-150 videos like this one I'm right if you have not been to our website on the bottom link to this vehicle on our website click those check us out and we're super excited to be offering this very clean 2014 Ford F-150 supercruise shortbox XLT and ruby red metallic thanks again for checking out the video


Hello friends, so we are going to discuss a new topic and this is Limits to the value of t ratios that is trigonometric ratios. So we are going to understand What are the values or the range of values all the six t ratios can take? So start with this for this identity sine square theta plus cos square theta is one, right? so we are first trying to attain or arrive at the range the values of sine theta can take So sine square theta plus cos square theta is one now You know that sine square theta is greater than equal to zero all the time. Why? Because it is a positive point. It's a square term So square of anything if you square anything it was always going to be greater than equal to zero if it is zero Then square of zero is zero But if it is anything negative or positive square of any anything any number will be always positive So I can say sine square theta is greater than equal to zero. Similarly, cos square theta will always be greater than equal to zero Now some of two so two positive quantities are there one is sine square theta another is cos square theta and some of these two is one right that means one Two positive quantities are added up to get one That means each one of them will be less than one so hence sine square theta is less than equal to one and Cos square theta is less than equal to one So this will be one. So let's say when sine square theta is one then definitely cos square theta is zero and When cos square is one then definitely sine square theta is zero. So this is what it means So now sine square theta is less than one So anything which is less than one that means if the square of anything is less than equal to one It always means that the value will be from minus one to plus one Is it it you can take examples for example if it is a zero point two negative So if you square this you will get 0.04 right and if it is let us say 0.3 so square this you will get 0.09. So if you see These all are greater than equal to zero. So till let us say if you take minus 0.99 999 like that. So it's square again will be equal to very near to one again So 0.9 only 99 like that Okay, so you you can check similarly minus one will be less than cos theta and less than equal to one Right. So hence we got the first two ranges. So sine theta is less than equal to one and cos theta is less than equal to One and sine theta is greater than equal to minus one and cos theta is also greater than equal to minus one This is the first Result now Cosic and theta is nothing but one upon sine theta reciprocal of sine theta So hence when sine theta is less than one then one upon sine theta will always be greater than equal to one Right. So if any anything is less than one less than equal to one reciprocal of it will always be greater than that Is it it so hence if you see Cosic theta is greater than equal to one you can check taking value Let us say point two is less than one but one upon point two will always be greater than one Correct. So this is for co-second theta. Similarly, if you if you take secant theta is a one upon cos theta So hence secant theta will always be less than equal to minus one and secant theta will be greater than equal to one So cosecant and secant in terms of range Are similar that means both cosecant and secant have same range that means It will be either less than minus one or it will be greater than one Correct. Now, let's let's talk about tan theta So tan theta is sine upon cos which when which can be written as under root one minus cos square theta by cos How because sine square theta plus cos square theta is equal to one So hence sine theta sine square theta can be written as one minus cos square Theta. So hence sine theta can be then reduced to this one minus cos square theta. Is it it? So hence I have represented sine in terms of cos itself. So one minus cos square theta under root divided by cos theta Now look at this particular term carefully. Now, it means cos theta tends to zero from the positive side What does it mean? So this arrow means tends to so when cos theta value becomes very very close to zero But the value is positive now The value can be either positive or negative. Let us say this is my x-axis and let us say in x-axis I am representing cos theta values So cos theta well, let us say this is my one and this is minus one So cos theta value in this area is less than one But it is positive similarly cos theta value here in this area this region is Negative and but very close to zero, isn't it? So hence cos theta value can be let us say point plus point plus point zero point zero zero zero one and it can also be minus zero point zero zero zero one Right. So when I'm talking about zero plus I mean to say that cos is positive but very close to one and when I say zero minus I say cos is very close to zero, but it is a negative number, right? Now as cos is very close to zero, but not zero mind you. It's very close to zero, but not zero So tan theta takes a very large positive value. Why because I am dividing by a very very small positive value So let us say if you divide one by zero point zero zero zero zero and many zeros Right followed by let's say more lots of zeros and one then this is a very huge number, isn't it? It will be ten to the power number of zeros here, right? It let's say 20. So it's a very huge number So hence if you divide anything by a very very small positive number then the number and You know the quotient or whatever is the result of the division is a very very large positive value, correct? So that means tan theta is taking a very large positive value It can take any, you know positive value and in that sense we say that tan theta tends to positive infinity similarly When cos theta is very very small Value but negative value, right? Then tan theta will take a very negative large negative value Why because let's say the same thing here if I change this changes to minus one then this Item becomes minus 10 to the power. Let's say 20 20 is just a number. I'm saying it's a very large negative value So what do we learn? We learn that tan theta value can be Now tan theta can definitely be zero. Why because if my sine theta becomes zero then tan theta becomes zero Okay, so hence what happens? I can see that tan theta is taking any value between minus infinity to plus infinity, right? Then theta range is between minus infinity to plus infinity similarly cot cot theta also can be explained in the same terms and Again, you can take two values of sine very small positive and very small negative and you will see cot theta like tan is also the same minus infinity to Plus infinity. So what is the summary? So I have summarized the above results if you see So basically it is nothing but sine theta will be varying from minus one to one and it in Alternatively, it can be also said that mod value of sine theta is less than equal to one, right? The modulus whenever you take the mod value of sine theta, it will always be less than equal to One and yes, you can always write this as well. So zero less than equal to Zero less than equal to sine theta mod sine theta is less than one because mod is a positive quantity So it will not go into the negative realm. So hence. This is another way of writing this Right and similarly for cos is this so minus one less than equal to cos theta less than equal to one minus infinity less than tan theta is less than infinity here if you notice I have not put equal to sine because you cannot equate anything to infinity because you know, it's in a way It's you know, I'm not I'm not known not defined number. So hence I am I cannot equate it to tan theta It is it is to it is to show that the infinite is a very large number. It's not equal to tan theta Okay, similarly minus infinity is Is less than equal to quarter theta, sorry, less than quarter theta and less than infinity Seq and theta again less than minus one and see can theta is greater than one Similarly, cosecant theta is less than minus one and cosecant theta is greater than one, which can be again written like this Hope you understood this this information is very vital We'll also have a demonstration video on which we'll show you how the graph of Sine theta versus theta behaves or all the trigonometric ratios with respect to the angle behaves You can check that out in the same Channel


Trade it and upgrade it now at Appalachian Wireless and receive up to $700 off when you trade your old eligible device in and upgrade to a new iPhone 14 Pro Max or Samsung S23 Ultra on an unlimited Appalachian advantage plan because we are you. We are Appalachian Wireless. This summer at the Virginia Kentucky State Fair, one woman won first, second, and third place for best cookies, candy, and savory bread. She also won the blue ribbon for cake, pie, brownie, sweet bread, and best overall baked goods along with eight additional blue ribbon wins. This woman was Linda Skeens, a 74 year old grandmother from Appalachia. My son actually was the thing of that. He was going to enter a model car in the competition when we first moved over here in 83. And I think this was 84, 85. He was going to do that. And I had embroidered pillowcases and he talked to me into entering the fair and I won a ribbon and I got hooked on it and I've been doing it ever since. Despite having over 1,500 awards, Skeens encourages her friends to continue to enter competitions alongside her. A lot of my friends do enter stuff in the fair but they respect me and I do them and one of them actually made the comment down there at Russell County Fair. She said, if you beat Linda you're doing good. So and I feel the same way about her because she's really good but everybody goes in there with the thing you just go in there you do what you can do and you hope for the best if you win be happy and if you don't just try to do better the next time. Despite her success in competition, Skeens taught herself how to cook out of necessity. Well actually when I first got married at 16 I couldn't cook. We snacked a lot and all that and I figured I better learn to cook when they start coming along. But I just a lot of times I would get a recipe from my family, my mom, my mother, my friend or I just make one up or get one on the book and I just start cooking and I decide if I wanted to make it I'd make it. Once I learned how I love it I do I enjoy cooking. Skeens will be publishing her first cookbook, Blue Ribbon Cooking, recipes and tips from America's favorite county fair champion this summer. One of the most important aspects to Skeens when writing her book was ensuring her recipes were accessible. When I first started learning to cook I would watch these tv shows where you'd have to have all these special equipments all these special foods. I hadn't even heard of some of them. Couldn't find them at the local Pigley Wiggly at that time. They didn't know what I was talking about. So I like something simple that's what I told Mike Huckabee. He was talking about my peanut butter fudge. I like it easy simple that you can find ingredients and it don't take you all day to make. Delicious. And it's good too. Skeens stated all of her friends and family have encouraged her to write a cookbook for years and she's excited to finally have the opportunity. For more information and to pre-order Blue Ribbon Cooking recipes and tips from America's favorite county fair champion go to lindeskeensblueribbon.com. Reporting for Mountain Top News, I'm Brianna Robinson.


Welcome. Thank you for joining us for this JSA TV interview. I'm Greg Elliott, VP of Business Development with 1623 Fartum. With us today is Mike Rockwell. He is the Global Head of Solution Architects with Megaport. So thanks for joining us today, Mike. I'm super excited about this conversation, so really appreciate it. Yeah, me too. It's great to be here with yourself and JSA TV. For the viewers and yourself out there, I do use a lot of hand motions and expressions. This is a video production, so I'm not having a medical event. Just forewarning here. I always like to get that out of the way because it can be good to know. Good generation. I speak with my hands too, so I get it. So let's jump into it. So over the last however long, we've had multiple, multiple conversations with customers, central customers, industry experts about cybersecurity and how the network is a key component to that. Can you shed some light on how Megaport could be used as a tool in that fight for better network security? Yeah, now that's a great question. I think anytime we're onboarding a customer, the first thing that comes up is they want to validate our security profile. So we're diving straight into the questionnaires or accident 100 questions. So they always want to validate that we're a secure option for their network. A little just kind of preface about what Megaport does. We're a global network as a service provider. So primarily customers are building on demand private connectivity from data center to the cloud. Certainly, there's other ways that customers utilize the network, but we are a network as a service provider. When it comes to security, it's kind of interesting. I think there's three things that come to mind in our conversations. One is simplicity, another one is control and another is flexibility. So when I talk about simplicity, it is still shocking. The term network sprawl comes to mind quite often. We'll get on the phone with the customer. We'll start talking about their cloud connectivity strategy and they'll pull up a busy diagram. Right. And this thing looks like a plate of spaghetti and meatballs. I want to throw the sauce on the screen and start soaking it up. Right. But one of the issues with that is they start to dig into it deeper. And we're strictly talking about kind of the data center to the cloud connectivity portion and the conversational progress into, Hey, what's what are we using this link for? What are we using this link for? What's this connected to? What resources ultimately are are we trying to reach from this location to this location? And, you know, that's a huge security risk. Right. If we don't know what network links we're using for one, that's an issue. If we don't know what endpoints ultimately they're connecting to, that's an issue as well. So when I talk about simplicity and really in the form of Megaport, a lot of it comes back into you. They want to take control of that network infrastructure and they want to be able to manage it from a single platform. Really a lot of the same conversations around SD-WAN where they're taking their control of their network, they have a management platform, able to push down all their policies through their controller to all their edge devices and have that single platform for management and bringing that into the kind of network connectivity model. That's one of the easy resources that Megaport provides is that management console where they can see all those network connections and see the endpoints ultimately they're connected to. The other piece is control. So Megaport, we operate a private network that any endpoint from A to B that the customer is connecting to. It's over Megaport's private layer to network. So they know exactly from point A to point B that it's routing across a private platform. We're not handing it off to a third party provider or routing it over the internet. So really we're taking the given the customer control to route the traffic, how they like, over a transparent connection. The next piece is flexibility in the world today with the distributed applications, distributed users. We want to have the flexibility to make it make change in a moment's notice, right? So that old traditional architecture of waiting 30, 60 days. You know, that's just doesn't work anymore. So they want to have that flexibility from a pure security standpoint. Some of the key things that that we're seeing. Max sec comes up right of it, right? We're providing a layer to transport across our network. So max sec being a layer to encryption or security protocol. A lot of a couple of the cloud providers, AWS and Azure, specifically come to mind. They offer it over their dedicated connectivity model for their private connections and express route and direct connect. So we're helping customers to facilitate those connections today, and that's definitely a huge topic. It sits outside of kind of that standard partner model, but certainly gives the customer that simplicity and control. The other piece is sassy. I'm sure is. I think there's a recent Nannog event, my guess is that you know, secure access service edge was top of mind. They talked about it quite a bit. I'm sure they did. And, you know, we're getting involved in some of those sassy solutions as well. So we have a mega port virtual edge that allows customers to deploy virtual network functions on mega port compute. And then they have the benefit of building out the private connectivity from the data centers like 1623 Farnum, your data center to the cloud providers, but also integrating that underlay into their SCO and overlay. And where that really comes into play from a security perspective is now they're able to build out these these virtual pops. So maybe you have a customer that's there in 1623 Farnum. They don't have a data center or presidents or a president that sits within Europe. They can deploy an MVE and essentially turn up a sassy pop where they're able to incorporate their network as a service and security as a service on one central platform. And that's integrated with with their SDWAM providers. So Cisco, Fortnight and Versa are some of the folks that we're working with.  That's great. So can we go a little bit deeper into maybe the higher regulated verticals like health care or government or financial? How do you tackle this? Oh, man, well, we work across all verticals, as I know you all do. I think in the data center space, this is probably your daily thing. Keeping up with these certifications, it's got to be quite the the resource. A lot of resources dedicated to that, right? You know, our network itself is built in third party data centers, such as yourselves at 1623. And, you know, so a lot of accreditation and a lot of the security is built in with that in that data center from a network perspective. You know, we're we're taking the customer out of that data center to the cloud provider edge primarily. So when you start thinking about, you know, the verticals and government, for instance, I think is one of the the topics that's in the question. You have to get out of the data center. You need to get to the edge of the cloud. You need to be able to access a government on ramp to reach your your government cloud resources, right? To sit in that regulated environment outside of the commercial cloud environment. So with Megaport, those are the type of connections that we're facilitating to, you know, with Azure, Google, and a Azure Google AWS and Oracle primarily, and they all have different ways that they set up those connectivity models and certainly we're able to provide that for them. You start thinking across the different verticals. And again, we're we're connecting network resources from point A to point B. So we're providing that private layer to connectivity. So it is a secure platform. One of the interesting things or one of the interesting customers that we're working with is Intercontinental Exchange or ICE. They're a NASDAQ Clearinghouse. They utilize our network as part of the underlay for their ICE Global Cloud Connect offering. And essentially through ICE, they have their own dedicated network where part of the fees that they're trying to access within the public clouds. So they're utilizing our network to build private connectivity from the edge of their network, from their data centers to the cloud provider edge and ultimately to access these data fees that they're pulling back down into their network for their customers. So that's one of the financial use cases that we support. And certainly there's a lot more detail around that that customers can look at on our website. But I think, you know, one of the interesting things that we got from their head of their global network services is that you are a Megaport software to find a private network is really purpose built for the capital markets. So I think if someone at ICE is giving us that type of credibility from a security perspective, that our network is considered to be secure across all the verticals that we work with. That's great. Yeah, just back to your point on just all your case studies that Megaport has on it. I spent a lot of time going through Megaport when you guys decided to come to 1623 and pop us with the dual feed. But like it helps to see those use cases out there and how Megaport is helping so many different types of businesses. Yeah. I don't think it's, I mean, any, you know, service provider out there, you know, getting a use case is not. I think you would agree, right? It's not easy, right? It takes time. It takes resources within the business. So, you know, I think if you're lucky enough or you're good enough to get a use case, that really shows the value that your customer is bringing to the business, or that your service is bringing to the business. Sure. And so most of our customers and other companies out there have a hybrid cloud strategy and you kind of touch on this a little bit earlier. But like how can businesses use Megaport as they're trying to educate or execute on their digital transformation strategy? Yeah. Hybrid multi-cloud. I mean, that's our daily conversation, right? Sure. You know, I think if I'm an expert at anything, which I don't know if I am or not, but yeah, I feel like that's probably what I'm an expert at. I think if you talk about it every day, you've probably become somewhat knowledgeable, that there's no way around it. But the biggest thing about when you start thinking about building a hybrid connectivity or building connectivity to the cloud is, as you know, at 1623, you have Google in your data center, right? You've got the Edge, the Google Edge in your data center. Customers can connect directly to 2623, connect right to the Google network. And great for Council Bluffs, that central region with Google, outstanding option, right? But as you know, not the edge of each cloud provider is not going to sit within your particular data center, right? Or any data center provider, it doesn't matter, right? Right. They're not everywhere. So you have to have a strategy to be able to get outside of your data center and get to the edge if you're going to have a hybrid cloud strategy, right? And one of the great things about Megaporton where we fit that need and we assist some of the folks that sit within 1623 Farnham, they can essentially connect to our network. And 1623 is an example. And we have 700 data and enable data centers across the globe. But 1623 customer connects to our network in your data center. It essentially brings the cloud provider edge to the customer. Because once they're connected to the Megaport network, they're able to build private connectivity to the cloud provider edge across the globe. So a great example in this particular use case, you're sitting in Omaha, Nebraska. You connect at 1623. You build a private connection across the Megaport network to, say, AWS or Azure in Chicago if you want to go east. If you want to go west, you can build connectivity into Denver. You can build it all along the east or west coast. But you can also extend your network outside of the United States. You can build connectivity across Europe. You can build connectivity across APAC. So when we start thinking about hybrid cloud connectivity and having a strategy and building your network, one of the things that we like to say is really future-proofing that network, is we all know that in working with our customers that maybe I'm in AWS today, maybe I don't have any intention of going to GCP or anywhere else. But inevitably, someone's going to find a resource in another cloud that they're going to want to access. And they're going to want to build that connectivity. So you have to have a strategy that allows you to really pivot and make change in a moment's notice. And that's really where our network is purpose-built to facilitate those types of connections. Yeah, that's great. So I've been in the business for about 15 years in the carrier hotel space. So I've seen how customers can be successful when they have multiple options. And as their business changes and scales and whatever they have to pivot, they have at a carrier hotel type facility, you have 50 carriers and ISPs. You can now have on-ramps to different clouds. But then I think Megaport just takes it to the next level, just to your point. So we really love that you guys have a diverse deployment at 1623. It's just another builder to the network ecosystem. Do you have any interesting, the ICE was a great example, but do you have some other interesting ways customers or your customers are using Megaport that you can share? Yeah, I think there's many ways we continue to expand. I think what's interesting is our customers sometimes find ways outside of maybe what we had envisioned for the network, which is outstanding. It's incredible, really. I think the other interesting thing is when you talk about hybrid cloud and multi-cloud, when customers do work in a multi-cloud environment, they want to be able to route directly between those clouds but also back into their data center. So as a service type of functionality is another great tool to have in the bag. Because if you are going to develop a multi-cloud strategy, typically those resources, you want to sit in very close proximity, right? So a great example here in the US is the East Coast. I mean, Ashburn, anybody who's in cloud and is in probably anywhere in the world, especially here in the United States, we know that that Northern Virginia tech hub is unbelievable, right? The data center operators that are there, the cloud provider regions, they all have regions that sit there, right? So if I'm building out a multi-cloud strategy, that's probably one of the primary places where I'm going to build it out. One of the great things that tools that we have is a virtual routing appliance that a customer can turn up on demand. It sits on top of our private network. It sits, the appliance itself physically sits right at the edge of the cloud provider networks. So if I'm on a route from say AWS, I have some front-end applications that I'm utilizing in AWS and I want to keep the Oracle databases, people have Oracle databases, I want to integrate those two networks together. One option is doing that over the internet, which the reliability issues there, security issues there, also throughput issues. An easy way to do that is to have a virtual routing type of platform that you have access to. So that's one of the probably unique use cases that we see. The other thing that's great about that is I can still offer that virtual edge. I can still build private connectivity back to my reuse sources at Farnham as well, at 1623 or any data center across the Megaport network. So that's kind of one of those, I don't know if it's a unique use case, but it's certainly one that I think customers aren't always aware of, that they're able to use as a service type of routers as prevalent as they are today. Oh, that's great. Well, this conversation has been amazing and I really appreciate you taking the time. Anything else you'd like to add before we head on out?  That's a risky proposition. But I think the other thing that I would add is that we didn't really touch on is building in that resiliency as well. You mentioned out of 1623 Farnham that we have dual fiber paths from a layer one perspective and then from a layer two perspective, we have protection across the network. The other thing that's a key tool to have in the bag and again, bringing those edge connections into your data center ultimately from far away is being able to build that high availability architecture and that resiliency across the platform. So the only other thing I would say around that hybrid cloud is work with a strategy that does allow you to connect potentially in multiple geographical locations, but also gives you the flexibility to manage it as you see fit. And I think that's it. Maybe there's probably more. I'm sure there's a lot more we could go on forever. Yeah, no, we could, I appreciate that. Thanks for hosting Greg and it's been a great conversation and thanks to JSA TV. Yep, thanks Mike and thank you to JSA TV and we'll see you on the next one.