Rising or falling intonation?
How on earth did you find out?
We should use falling intonation when asking wh questions, but we should use rising intonation to express high-energy emotions like anger, excitement or shock.
And here we do have a wh question and we also have a high-energy emotion, now what?
I can imagine How on earth did you find out? with either rising or falling on the end.
It depends what the focus is of the surprise or shock (indicated by "How on earth").
If it is on the finding out (i.e. "I’m surprised that you managed to find it out. How did you do it?") then I would expect rising tone on find out (or possibly a rise before find, and staying high on out)
If it is on the information you found (i.e. "That’s shocking! How did you find that out?") then the peak is on earth, and find out is falling.