I've got a small upstairs bathroom (tub w/a shower). The bathroom has a
bedroom on either side with no access panels to the plumbing. The third
wall is to the exterior. It's a newish house (three years old).

Directly below the bathroom, on the first floor, is a dining room. Evidence
of a leak is showing on the ceiling, more or less directly below the
upstairs bathtub. It strikes me that there are three most likely sources
for the leak. (Note: the bathroom serves as a very infrequently-used guest
bath. Probably not more than ten or twelve showers have been taken there in
the three years it's existed.)

Possible source 1: plumbing, either the drain or the supply.

Possible source 2: leaks in the tile wall or joint between tile wall and
tub, or (less likely) joint between tub outside bottom and floor tiles.

Possible source 3: the roof joint where the exterior wall of the second
floor joins the top of the extended downstairs roof (the upstairs portion of
the house sits atop only half of the downstairs house, and the exterior
upstairs wall joins the lower roof almost above the leak spot on the dining
room ceiling.)

With no ready access to the underside of the tub without pulling out the
drywall, how can I determine which of these is the actual source of the
leak? I'm hesitant to just spray water all over to see which possible
source ends up on the downstairs ceiling, partly because I'm hesitant to
further wet down that ceiling and partly because I don't know how I can be
sure that water from testing one area isn't just taking a delayed trip,
fooling me into thinking it's water from testing the next area. Am I
overthinking this? How do plumbers do it -- just wet things down and see
what leaks?

The matter would be a little simpler to solve if it weren't for the fact
that the rare use of the tub for showers is pretty well matched by the rare
rains. I've never seen the leak when it was actually wet; I only noticed it
recently when it hadn't rained in months and no one had showered there in
months.

Thanks for tolerating (most of you) my simplistic questions over the years.

Jim Beaver