Waveguides as transmission lines

In the previous chapter, it was shown that waves may travel in hollow pipes in many modes of transmission and that for each of these modes there is a corresponding cutoff frequency. For frequencies below the cutoff frequency the energy in the mode is quickly attenuated; above the cutoff frequency it is freely transmitted. The most important condition in practice is that in which the frequency lies above the cutoff frequency for the lowest mode but below the cutoff frequency for the next higher mode. The lowest, or dominant, mode will then propagate energy in the waveguide. Let us take as the direction of propagation the positive z-direction. Let us suppose that this traveling wave Electromagnetic encounters an obstacle in the waveguide, as a conducting wire placed across the guide, fleeted from an obstacle in waveguide. In the neighborhood of this wire, the solution of Maxwell's equations that corresponds to the dominant mode will no longer suffice to satisfy the boundary conditions.