 So, let us consider for the TE case ok. So, here the electric field is along the x direction, in this case the electric field is along the x direction. So, incident wave the incident wave the electric field component of the incident wave I denotes the incident wave can be written as x is the unit vector this denotes that the electric field for this incident wave is in the x direction ok. It is given by this expression this we have already seen and for the reflected wave again the reflected wave also it is along the x direction now the amplitude is simply E0 prime ok and here this is minus k y cross theta ok. So, now by principle of superposition the net electric field component in the region between the conducting planes will simply be the addition of the incident wave and the reflected wave. So, we simply add the two field components. So, the net wave is now simply the superposition of the incident wave and reflected wave. Now according to the boundary condition what does the boundary condition say the boundary condition says that E tangential is equal to 0. So, according to the boundary conditions the tangential component of electric field must be 0 at the surface of the conducting plane. Now electric field is in this direction ok and this. So, that means at this boundary it is which component it is the tangential component ok. So, it has to be 0 at y is equal to 0 ok. Similarly at the surface of this conductor the electric field is again tangential component. So, it has to be 0. So, what is the boundary condition? The boundary condition will be that the electric field is 0 at y is equal to 0 and y is equal to a. So, to this resultant electric field let us now apply the first boundary condition that E is equal to 0 at y is equal to 0. So, we put E is equal to 0 at y if equal to 0 ok. So, we get this is what we get. Now this equation is satisfied for all z and all t if E 0 plus E 0 prime is equal to 0 or in other words E 0 is equal to minus E 0 prime ok. So, this simply implies that at when the incident wave goes and it gets reflected it just suffers a phase change of pi there is no change in the amplitude ok. So, this is just telling you that. So, we can substitute instead of E 0 prime we can substitute E 0 and write the expression for the electric field again. So, this is what we get. Now we can simplify this. So, this we can you know that E to the power of j x is simply sin x plus j cos x. So, expanding by using this expression expanding both for this and this and simplifying we get the unit vector x because the electric field is in the x direction 2 times j E 0 and then sin k y cos theta E j k z sin theta minus omega t. So, this this remains and this is now simplified to sin k y cos theta ok. So, again you can write it in this form k can be written as omega by c as well where omega is the frequency in free space. So, now here we can put this k into cos theta as k c and k into sin theta as k g and we can write this expression here. So, we have the expression for the electric field as. So, this is the amplitude and this is the propagation part ok. So, what does this represent now? This represents a wave now which is travelling in the z direction. So, this wave is now travelling only in the z direction with a propagation constant k c sorry k g ok. The propagation constant has also changed and having a sinusoidal variation in the y direction ok. So, the amplitude is now earlier when it was in free space the amplitude was just E 0. Now the amplitude has a sinusoidal variation in y and remember y is the direction in which the boundary was applied. So, standing wave is now formed in the y direction ok. So, so for this expression this represents a wave propagating in the z direction with a propagation constant k g ok. So, k g is now 2 pi by lambda g. So, lambda g is the new wavelength of the propagating wave ok. The when it was propagating in free space the wavelength was lambda. Now the wavelength has also changed to lambda g and this is equal to omega by c sin theta as we have seen from the previous slide. So, lambda g can be written as lambda 0 by sin theta. So, lambda 0 is the free space wavelength and lambda g is called the guided wavelength. That means, the wavelength which is inside the conducting in the region bounded by the conducting boundaries. And similarly this wave is now it does not have a constant amplitude. It is the amplitude is varying sinusoidally along the y axis with an effective wavelength lambda c which is given by k c is equal to 2 pi by lambda c. So, earlier the amplitude in the amplitude in the free space it was constant and now the amplitude has a sinusoidal variation which is given by 2 pi by lambda c. This is to satisfy the boundary condition. So, lambda c can be written as lambda 0 by cos theta and lambda c is the cut-off wavelength. Now from these two expressions if we eliminate theta. So, sin square theta plus cos square theta is equal to 1 we get an expression between lambda 0 lambda c and lambda g. So, lambda 0 is the wavelength of the wave in free space lambda g is the wavelength in the region between the conducting boundary. So, once it is entering the region bounded by the conducting boundaries it is moving with a different wavelength or a different propagation constant. So, that is lambda g and lambda c is the wavelength of the standing wave that is formed in the direction in which the boundary has been applied. Now let us apply the second boundary condition. The second boundary condition is the electric field is 0 at y is equal to a. So, let us apply the condition this is 0 and y is equal to a. So, this equation is satisfied for all z and t if sin of k a cos theta is equal to 0 or in other words k a cos theta is equal to n pi where n is an integer. So, k can be written as omega by c a cos theta can be written as lambda 0 by lambda c from the previous slide. So, this is equal to n pi. So, here lambda 0 gets cancelled with omega by c and we are left with lambda c is equal to 2 a by n. So, you see that lambda c is the wavelength of the standing wave formed in the y direction that is the direction in which the boundaries have been applied and it depends upon the dimension of the system. So, this is analogous to the case of these string. So, there we saw that the frequency or the wavelength of the fundamental mode depends only on the dimensions of the system and the disturbance of the dependence of the disturbance on y it cannot have any arbitrary wavelength, but only an infinite set of discrete wavelengths. Why? Because n is an integer. So, it cannot have now any arbitrary wavelength, but only an infinite set of discrete wavelengths. So, in between the two infinite parallel conductors in order to satisfy the boundary conditions the field arranges themselves to form certain patterns of the standing wave. So, here if n is equal to 1 this is the Te1 mode. So, this is the y direction in which the boundary has been applied and this is the z direction in which there is no boundary and the wave is propagating in this direction. So, we see that a standing wave is formed in the y direction, the direction in which boundaries applied. So, this is the first mode and if you put n is equal to 1 here we have lambda c is equal to 2a. So, this is the length from 0 to a. So, we get here a is equal to lambda c by 2. So, this wave which is propagating in this direction if we calculate or if we see its wavelength in this direction this corresponds to lambda g which is the wavelength of the guided wave. Similarly, for n is equal to 2 we have two half-period variations in the field in the y direction, the direction in which the fields have been applied. So, you can see from here that there is this field is 0 here again it is 0 here and in between it goes to 0. So, if I plot in y the variation of the field from 0 to a I have 0 here maximum going to maximum and then in the opposite direction. So, at a again it is equal to 0 to satisfy the boundary condition. This wave is again propagating in the z direction and this is the guided wavelength which corresponds to the wavelength of the guided wave. So, you can see this here. So, this is the y direction in the y direction it is a standing wave. So, if you can see at any point here. So, here the electric field amplitude is always 0 and at this point here just like the case of the string. So, this is the first mode. So, the amplitude at these points is 0 and at this point it is oscillating from minus a to a whereas in the z direction. So, this is the y direction whereas in the z direction it is a propagating wave and this wavelength corresponds to lambda g. This is the second mode. So, in the second mode the wavelength. So, we have two half period variations of the wavelength. So, here it is 0 here it is 0 and again in between it is 0. So, analogous to the case of the string. So, now let us see the difference between the electromagnetic wave in free space and the electromagnetic wave in between two parallel conducting planes. In free space this was the form of the electromagnetic wave. The electric field component of the electromagnetic wave the amplitude was constant and it was in this case it was propagating in both the y direction and the z direction. So, the amplitude was constant and it was propagating in both the y direction and z direction. So, it was in the y z plane. Now, when we applied boundaries what happened was that the resultant wave has a form like this. So, now it is propagating with a new propagation constant here the propagation constant was k. Now, it is propagating with a new propagation constant kg and it is propagating only in the z direction. It is no longer propagating in the y direction and y is the direction in which the boundaries were applied. So, in the y direction it has formed a standing wave. So, there so you can see that in the amplitude there is a sinusoidal variation in the y direction. So, wave is no longer propagating in the y direction because it is a direction in which the boundaries have now been applied. Coming back to this expression for lambda 0 which is the wavelength in free space, lambda c which is the wavelength of the standing wave. So, c here denotes the cutoff wavelength and lambda g this is the wavelength of the wave that is now propagating inside the system of infinite parallel conductors. Now, for a given system lambda c is constant because lambda c depends only on the dimensions of the system. So, lambda c is constant. Now, as lambda 0 is increased lambda g also increases and the phase velocity that means the velocity with which the wave is propagating in the region between the parallel conductors this is also going to increase until lambda 0 becomes equal to lambda c. So, as lambda 0 is increased lambda g also increases. Now, when lambda 0 becomes equal to lambda c what happens is that lambda g and v g both become imaginary. So, the wave now gets attenuated. So, any wavelength any lambda 0 greater than lambda c will not be able to be propagated inside the system it will get attenuated it will not enter inside this system. So, the free space wavelength at which the disturbance changes from being propagated to attenuated this is known as the cutoff wavelength. In terms of frequency since frequency and wavelength are inversely related in terms of frequency if you do it. So, you will find that waves below the cutoff frequency are attenuated and above the cutoff frequency are propagated. So, the corresponding frequency here is called the cutoff frequency the subscript c denotes the cutoff. So, here let me just summarize the field patterns in the Te mode. So, this is the Te1 mode this is the Te2 mode. So, here in the y direction standing wave is formed. So, if we see the Te1 mode there is one half period variation in the y direction from 0 to a and if we see the Te2 mode there are two half period variations or one full period variation of the standing wave in the y direction. So, this is from 0 to a and the wave is a propagating wave in the z direction and this corresponds to lambda c this is wavelength here corresponds to lambda c and here n will take value 1 here n will take value 2 and this corresponds to the guided wavelength. So, in the direction in which the wave is propagated. Similarly, we can do the entire calculation for the TM mode and we will get similar results. So, here for the TM mode the magnetic field is in the x direction. So, here at the boundaries this magnetic field is the tangential component in both cases the magnetic field is the tangential component and tangential component of magnetic field is it need not be equal to 0. So, the boundary condition is that the normal component of magnetic field is 0. So, in this case for the first mode we will have. So, we will not have magnetic field equal to 0 here and here. So, we will have but still we need to have one half period variation. So, we can have one half period variation from here to here. So, you will see the field is maximum here the field is maximum here and then it goes to 0 it goes to 0 here and then again it is maximum in the opposite direction here. Okay. So, this is the magnetic field along y this again is the z direction this corresponds to lambda G. This is a TM2 mode here you have one we have two half period variations along y. So, the field is varying from here to here. So, we have the field here as so let us say at this point the field here is maximum then it goes to 0 at this point then it is maximum in the opposite direction here and then again it goes to 0 and then again it goes in the positive direction okay and then similarly we can have TM3, TM4 modes and so on. So, this was like a wave guide in one direction. So, a wave guide is generally a hollow pipe of infinite x 10. So, that means now here we had in this case we had applied the boundary only in one direction. So, if we applied boundaries in both the directions now let us say x and y and the wave is free to propagate in the z direction. So, then we will have a standing wave in both the x and y direction. So, such a system is known as a wave guide. So, wave guide is a hollow pipe of infinite x 10. So, let us consider the propagation of electromagnetic waves along a hollow pipe of arbitrary cross section in the x y plane. So, it has some arbitrary cross section in the x y plane okay and uniform along the length. So, this is a wave guide of arbitrary cross section. Now, each component of E and B the electric field and magnetic field must satisfy the wave equation in vacuum which is given by this. So, the wave equation is del square E minus mu 0 epsilon 0 del 2 E by del t square is equal to 0 or del 2 del 2 E. So, this mu 0 epsilon 0 can be written as 1 by c square. So, you can write it at this as this and similarly for the magnetic field you can write it as this. Now, in addition to satisfying the wave equation inside this it should also satisfy the boundary conditions okay, what are the boundary conditions? The tangential component of electric field is 0 at the boundary and the normal component of magnetic field is 0 at the boundary okay and intuitively from what we have done in the one dimension we can write that now since boundaries are applied in the x and y direction. So, now the amplitude will have a sinusoidal variation in both x and y okay it will have some variation in both x and y direction okay. So, in the one dimension case since the boundary was applied only in the y direction the amplitude had a sinusoidal variation in the y direction. Now, since the boundaries are applied in both x and y direction and free to propagate in the z direction. So, the amplitude will have some variation in x and y. So, it will form a standing wave in the x direction and y direction and it will propagate in the z direction with a propagation constant kg. So, putting the second equation in the first equation in the wave equation we get for electric field del 2 E by del x square plus del 2 E by del y square minus kg square E plus omega square by c square E is equal to 0 we take this quantity in bracket okay and this we know that is equal to kc square okay where kc is what it is the wavelength or the wave number of the standing wave pattern okay formed in the direction in which the boundaries have been applied. So, and this is derivative double derivative with respect to the transverse components. So, we can represent it as del perpendicular square. Similarly for the magnetic field also by putting the form of magnetic field in that expression wave equation we get a similar expression and the two expressions can be combined and written like this. Now Maxwell's equation in free space R curl of E is equal to minus del B by del t okay. So, this can be evaluated we can so we can write this as del by del x del by del y del by del z okay E x E y E z is equal to minus del B by del t and B also you can expand in terms of components B x B y and B z okay. So, if you okay so if you now just equate the terms on the left hand side and right hand side the x component y component and z component you will get 3 sets of equations here okay. So, the x component is del E z by del y minus del E y by del z is equal to minus del B x by del t and similarly for the y and z components okay and also using this expression for curl of B is equal to 1 by c square del E y del t we will get again 3 sets of equations here okay. Now the both electric field and magnetic field have a form like this. So, if we take the partial derivative of this with respect to z we will get i into kj and if we take partial derivative of the electric field or magnetic field with respect to t we will get minus i omega. So, everywhere substituting del by del z with ikg okay so we substitute del by del z with ikg and del by del t with minus i omega also here. So, we arrive at these sets of equations. So, these sets of equations so we can now rearrange and simplify. So, we get E x E y we get expressions for E x E y B x and B y. So, now we notice that all the four transverse components okay. So, E x E y B x B y all the four transverse components they can be written in terms of the z components E z and B z okay. So, notice that in all the four cases E x E y B x B y they can all be written in terms of just E z and B z. So, that means that inside a waveguide we need to know just E z and B z okay. If we know E z and B z then we know all the then we know all the six components of the field because E x E y B x B y can be derived from E z and B z. So, from here we can conclude that it is sufficient to determine E z and B z as the solutions of the two dimensional wave equation other transverse components can then be calculated from the other equations okay. So, now notice from here that TEM waves cannot propagate in a waveguide as if you have E z is equal to zero and B z is equal to zero. So, if you have E z is equal to zero and B z is equal to zero okay. So, what does it mean? So, you have E z equal to zero B z equal to zero. So, since all the other fields depend upon E z and B z we will have E x equal to zero E y equal to zero B x equal to zero and B y equal to zero. So, that means all field components will go to zero and we will not have any field at all okay. So, this implies all fields are zero and no more. So, that means TEM boards cannot propagate in a waveguide. So, electromagnetic waves in free space are TEM waves when they enter inside the waveguide they propagate either as TEM mode or TEM they cannot propagate as TEM mode okay because they have to satisfy the boundary conditions. In TEM mode the magnetic field is transverse to the direction of propagation. So, B z is equal to zero okay and TEM mode electric field is transverse to the direction of propagation. So, E z is equal to zero. So, you can have two modes of propagation the TEM mode and the TEM mode. So, let me summarize whatever we have done in this lecture. So, we have seen that electromagnetic waves in free space are TEM waves propagating with a propagation constant k and velocity c but they cannot be used for acceleration. So, they have a time varying component of electric field but this electric field is always perpendicular to the direction of the velocity of the beam. So, it cannot be used for acceleration. Now when electromagnetic waves propagate in region bounded by conducting boundaries they form a standing wave which is known as modes okay in the direction in which the boundaries are applied. The amplitude has a sinusoidal variation in the direction the boundaries are applied in order to satisfy the boundary conditions. So, the standing wave is formed in order to satisfy the boundary conditions at the boundary. The wavelength of the standing wave depends upon the dimensions of the system. So, in the direction in which there is no boundary the electromagnetic wave is a propagating wave with a new propagation constant k g okay. So, with this we have learned today about the propagation of electromagnetic waves in between conducting boundaries. So, how they rearrange themselves? They change from a TEM wave to a TEM or TE wave in order to satisfy the boundary condition and how they form modes or standing wave field patterns in the direction in which the boundary has been applied. In the next lecture we will see electromagnetic waves in waveguides and cavities. So, in waveguides we will apply the boundaries in two directions and in cavities now finally we will apply the boundaries in all the three directions and then we will see that how modes are formed and how these modes can be used for acceleration of charged particles as proposed by Andres.