The new and old systems work together fairly well for horn- whistle/bell. However they do differ slightly. I have pw that will not respond to new transformers, and new engines that won't respond to post war power.
The reason is likely the shape of the waves. PW and modern each use a slightly different technique to get the offset voltage.
(I tend to keep things simple as the general understanding is more useful to a novice at first. I used to have to give a dozen teens a year or more a 2hr crash course that would allow them to be able to pull of arcade gaming assistance beyond " here is a token" with one hand, and an out of order sign in the other. Call me out if I soar too high, or sink too low explaining).
I just choose what runs where if I really want that sound that day. I also added diodes on a homeade button to work the new engines whistle/bell with my pw transformers 100% reliably. (only 1 new transformer, designated to accessory power since it has no boost effect and I run more postwar anyhow.)
Also note using PW power gives the new engines a boost they don't need. A 5v boost can cause "dangerous" speed if applied to an engine that only needs 6-9v to run normally.
The offset wave of ac is better known as a dc offset because the offset portion can be measured as pulsed dc flowing only one direction. The sine wave final shape mentioned has less of a wave, looking more like a cliff vs a hill (hill is old; "pure sine"). New components tend to chop fast rather than ease into things.