out on a limb here ---- the (energy, as Tim Wescott writes) differential equations are used, with boundary conditions that require the sudden appearance of waves traveling in the reverse direction IF that boundary is not exactly Zo; the energy is preserved in the new mix of voltage/current values for each of the forward and (now) reverse waves.
regarding "series terminations" --- a series termination, best used with the resistor installed at the SOURCE end of the transmission line, exploits reflections in its operation. Initially the line voltage is only half the Source voltage because of the voltage division of the lumped resistor driving the Z of the line; any circuit monitoring the line will see only Vin/2 and that often is a FORBIDDEN VALUE for logic circuitry; however, at the far end, the receiving end, the math tells us that unterminated receiving end HAS A REFLECTION, and the math tells us the voltage doubles as part of preserving the energy. Thus ONLY at the far end, the receiving end, will a useful full amplitude waveform be created.
At all other points along the line, the voltage will be HALF for some time, and then the reflected energy doubles the voltage. This doubling occurs, eventually, at all points. In general, trying to extract data from this 50% then 100% waveform is a bad idea.
Only at the far/receiving end does a safe-to-use waveform exist.
On the other hand, the use of series-at-the-source termination will reduce overall power consumption.