Some fundamental transmission properties of impedance transitions

Reflection and transmission of waves by impedance transitions from a constant input to a constant output characteristic impedence are considered. Several fundamental properties are explored, primarily for impedance transitions with piece-wise constant characteristic impedance in an arbitrary number N of intervals of equal length. For example, the following properties are shown: (i) The relative momentum transmission depends only on the ratio of output to input characteristic impedance. (ii) For a given impedance transition there are at most, and generally exactly, 2^N different transitions, including the original one, with identical transmission properties. (iii) For monotoneous impedance transitions the efficiency of energy transmission is minimized by one with an abrupt change in characteristic impedance. (iv) There exists an optimal impedance transition, with a certain antisymmetry, which maximizes the efficiency of energy transmission for a given incident wave of finite duration and energy. Several of the results can be extended to more general classes of impedence transitions. Simple illustrative examples are given.