Also, surely you're not suggesting that, when a primary coil is energized, one secondary takes a half-cycle longer than the other to energize.
No, of course not.
Both energize at the same time. With the voltage rise selected to be positive away from center, one rises and one falls. If you want to see a rise in the second wave like you have at the start with the first wave, wait a half cycle and that is what I took "predict the other wave" to mean.
There have been posts suggesting one MUST consider the voltages both to rise from one end of the windings to the other or that polarity must be taken the same way. That simply is not true.
The field increases in all parts of the winding as a whole, not in any particular direction. The one defining the voltage defines the direction.
The point about a half cycle is pick any polarity you want and a half cycle later the world is reversed. An AC wave continuously reverses polarity so the choice can be either way.
I'm trying to get people to think about the system differently than a stack of batteries because that loses sight of a polarity that is constantly changing. Batteries have a fixed polarity so you can't define positive away from the center of the stack (I guess you could but it would stray from normal convention).
One standard convention is to define the positive terminal such that current leaves the source from that terminal when the voltage rises in that direction. You won't see many circuits using a single-phase transformer that care because they are polarity agnostic.
However, a 2-diode full-wave bridge rectifier does care. Current only flows in each winding half for 1/2 cycle. If we define the + terminal using the prior convention, then positive is directed away from the center tap and is actually correct by standard convention. The winding halves never produce during the other polarity cycle available so the transformer is being used as a source of two different phases.
This would be the same as driving two pulse circuits from two sources with a 180d displacement. The cyclical nature is why the single-phase transformer can substitute as a supply for two 180d displaced signals. Two flux events, two windings, four flux results, time-varying. You can't get that with a fixed-polarity stack of batteries.
