We are all taught about transmission lines and its associates — characteristic impedance, reflection, standing wave, etc. It is well known what happens when a transmission line is terminated with different types of impedances:
1) Open – reflection occurs
2) Shorted – reflection occurs but inverted
3) Equal to Zo – no reflection
4) Between Zo and open/short – partial reflection
So far, so good. Every textbook and other reference will tell you this. But I cannot find a straightforward explanation why it is like that. Some demonstrations show the charging of capacitors through inductors and how the end capacitor doubles voltage due to collapsed magnetic field in the inductor, etc., but I find it very confusing. So maybe a mathematical demonstration would be convincing.
How can we show mathematically, for instance, that when the load impedance Zl is equal to Zo then reflection is zero, and so on?