This *has* been an interesting discussion. Thanks to the O.P. for posing the original question, and thanks to everyone who participated.
The original question was about whether there was a Doppler-type frequency change when a fog
horn was sounding in the wind
(moving air). We've recently shifted the analysis to waves in the water
, probably because we can see and feel them. Along the way we mentioned other wave effects, but most of the discussion has centered on "ideal" waves, where shallowing, breaking, etc, are conveniently ignored (at least that's what I've been doing). I think this is appropriate, as we can look at these different effects in isolation, which makes it much easier to answer the original (and follow-on) questions.
We should of course remember that waves in the water do *not* behave as classical sinusoidal waves, especially when they start interacting with the wind
, the bottom, or other non-water features. Still, I'm comfortable with the conclusion that the frequency does not change when we have fixed (relative to each other) wave source and detector. This holds true even when we have other "non-ideal" factors at play.