So I am studying about waveguides and I am going through all this mathematics and then I realized that I do not understand the general concept of how wave guides even work. I can go through the math all I want but I don't understand one thing.. If a wave is radiated in free space and is unrestricted, then it travels and attenuates as an inverse square law. Why is it that once we put the radiation source in a wave guide, it does not decay and can travel long distances? In the beginning of the mathematics, the book starts with the wave equations for E and H derived from Maxwells Equations. How ever it is the wave equations for a lossless condition. Why can we assume a waveguide is lossless? Shouldn't they attenuate as an inverse square law just as if they were radiated in free space un-restricted?