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A mathematical model for multiple reflections on lossless, distortionless signal transmission lines is developed for step changes in voltage or current at the source. The model is quite general and is valid for a source driving either one or two lines each loaded with an arbitrary number of discontinuities at arbitrary distances. It will give an exact solution for discontinuities and terminations which are purely resistive and allows for resistances of any size, none of which need be alike. Although an exact solution is valid only when the capacitances are negligible, a method is presented which will partially compensate for a capacitively distorted wave if the capacitive-characteristic resistance time constant is reasonably small compared to the transit time of the line. Considerations for digital computer simulation of the mathematical model are discussed and several examples including open backplane wiring comparing actual and computer results are given. The model and simulation are especially useful for lines which are not terminated in the characteristic impedance at the source or load end and also where modification of the model can be made to investigate reactive and nonlinear terminations.