NavList:

John, you wrote:
" If everything is in a moving frame of reference together, I wouldn't expect any effect on the shape of waves. "

True enough. If you drop a big rock into the water in the middle of the Gulf Stream, the waves from the splash won't look any different from the waves you would get if you dropped that same rock in Boston Harbor at the turn of the tide (or whenever the current has gone to zero).

I'm fairly sure that waves look different in ocean currents because of a refraction effect as the waves originating outside the current cross the boundary of the current entering regions of steadily increasing current speed. And... hmmm... the rest of the physics is trivial and I leave it as an exercise for the reader.

By the way, major oceans currents like the Gulf Stream extend hundreds of feet, even more than a thousand feet below the surface. Basically you have one great moving block of water too deep to give any "roots" to the ocean waves.

Speaking of navigation ;-), it was commonly believed in the 18th century that ocean currents existed only at the surface and many people suggested that dead reckoning could be made much more accurate by dragging deep "sea anchors" of a sort. The deflection of the cable to such an anchor would have been a direct measure of true speed over the bottom if it had really been true that deep waters lacked currents. At least one legitimate proposal for winning the Longitude Prize was based on this idea.

-FER
PS: For those unfamiliar with the above joke, saying that "the physics is trivial" and will be left as "an exercise for the reader" is one way of saying that the physics is very messy and probably involves lots of real-world details and great complexity despite the simplicity of the principle. The devil is in the details.