in the previous video we have concluded this equation which was nothing but d square V by DX square is equal to gamma square V right we'll continue from the same equation now if I solve this equation we know that what we're going to get right so let me say that I'm going to get something in terms of V X and T let me call that voltage in terms of X and T which is nothing but space and time I'm going to get V + e to the power minus gamma X plus V minus e to the power plus gamma X let us understand this equation so what I'm saying is if I solve this right you have already solved the particular integrals in machs right so if I solve this I'm going to get a voltage right because this was voltage right so the voltage in terms of space and time X is nothing but your space and T is nothing but in a time what I'm going to get is V Plus e to the power minus gamma X V plus is some constant okay V minus is again some constant e to the power minus gamma X is some quantity we'll see what is that and e to the power plus gamma X now let us say that let me say this is the voltage in terms of space and time but if I want the instantaneous value what I need to do that because the voltage will be sinusoidal in nature right so what I need to do is nothing but if I have to just multiply the home equation by e to the power J Omega T right this will be nothing but your instantaneous value right now if you are dealing with only the RMS and the peak value we can use this equation but if you want the instantaneous value you have to multiply with e to the power J Omega T which is nothing but the sinusoidal reach now what is this gamma let us talk about this gamma gamma is actually called as the propagation constant it is called as the propagation constant we have already seen when we have been studying the plain wave equation right there was also propagation constant right so this gamma would be nothing but alpha plus J beta right where your alpha would be nothing but attenuation constant it will be nothing but your attenuation constant where beta is nothing but your phase constant so it is a constant now if I substitute here okay which is nothing but alpha plus J beta now what I am going to do is I am going to assume it is a lossless medium we know that what is a lossless medium where the free charge carrier in that medium would be zero right so if it is a lossless medium the Alpha would be zero right so we are taking a case of loss less medium or we can say that for lossless medium alpha would be zero right so we are very peculiar about the lossless medium and see how the equation will go if I deal with a lossless medium right so what I am going to do is I am going to continue this equation for alpha equal to 0 so V X of T let us deal with only the peak value or the RMS value okay let us take that so it will be V Plus e to the power minus gamma would be nothing but minus alpha plus J beta right T so it would be x-ray so we are dealing with X so it would be X so gamma is nothing but the gamma is nothing but alpha plus J beta which I have substituted here and it would be X for instantaneous value it would be e to the power J Omega T right another quantity would be here which would be nothing but plus V minus again I'll substitute gamma e to the power plus alpha plus J beta X into e to the power J Omega T right so I got this equation after substituting the gamma s alpha plus J beta now what I'm going to do is I'm going to take this 1 by 1 part for this part and the this part let us understand what is this actually ok so for time being will rub this and we'll continue with that equation now as I said for a lossless medium I'm going to take alpha is equal to 0 so that equation will become so I am only considering this part of the equation later we'll look into B this part of the equation so it would be V Plus e to the power minus J beta X e to the power J Omega T same way if you write that another part it would be plus V minus e to the power J beta X e to the power J Omega T right now if I continue writing this equation e to the power minus J beta X can be written like this can be written like v plus e to the power J Omega t minus beta X right you can see that I can combine that equation plus V minus e to the power J Omega T plus beta X right I can also write this equation in the form of V plus cos Omega t minus beta X and this would be nothing but plus P minus cos Omega T plus beta X remember what is this this is nothing but your V X comma T or nothing but the voltage in space and time right so we got this equation now what is this equation actually telling me now if you recall the basic equation of an electric field which we have discussed in the plane wave right what was that equation let us recall that equation drug this so that equation was if you recall II was given by an electric field a plane wave equation is e naught e to the power minus alpha Z cos Omega t minus beta Z some X cap this was the basic equation remember what it was the meaning is that it's nothing but this is nothing but your attenuation constant it means that it was telling you then it is an e field moving along the z direction or the propagation along the Z direction where the variation of electric field is along the x axis right if I compare this and this equation let us compare this but what it is saying is it has also some constant right for the lossless medium what will become for the lossless medium this will become e naught cos because alpha would be 0 e to the power 0 is 1 so cos Omega t minus beta Z x cap and this is for lossless condition this is for lossless condition now so this equation and this equation if I compare okay so what I can say is this is the wave equation remember this was the wave equation of an electric field no but I am dealing with a voltage right how can the voltage be the wave right so what actually this equation is saying that the voltage which will be flowing in this circuit or I can say that the voltage which will be flowing in the transmission line will have the nature of wave actually so the conclusion is the voltage will be traveling in the transmission line like a wave this is what it is saying now if this was the wave so this is the wave which is travelling in minus beta h6 means plus X direction and this would be the wave which is traveling in minus X direction so you can see that there are two waves or the voltage is traveling like a wave one is traveling in the plus X direction another is traveling in the minus X direction right so let us say that this is your plus X so let us this is the circuit the voltage will be traveling one voltage will be travelling forward right which is nothing but this part there will be another wave which will be travelling backward right so that will be nothing but given by this wave so the conclusion is the voltages in the transmission line will be travelling like a wave the nature of the voltage will be like a wave and one part of the voltage or the voltage will be traveling one in forward direction another will be traveling in the backward direction right if you conclude about the current if in the same way if you go ahead and solve for the current you will get the same equation as the voltage is also wave nature what you will find it out current will also have the wave nature so one out of the current will be travelling forward another part you will see that it is travelling backward so what we have concluded from here is the voltages and the current in the transmission line will have the wave nature and they will be travelling forward and backward at the same time