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On an Earthlike planet with one large moon (just like Earth's), I have a large landlocked sea with one narrow exit (like the Black Sea, but much larger, something like six or eight Mediterraneans, with an exit about 10-20km wide).
How extreme will the behaviour of the water be with the changing tides?
For example, will it be possible for a shipping port to operate on a decent-sized island in the exit channel? Will there be whirlpools most of the time, or only when the tide is running strongest?
You are right that the tidal rip through the channel will be extremely fierce, in fact the island and edges of the channel will experience substantial erosion as a result so expect the channel to widen rapidly in geological terms. Even hundreds of years would make a difference, thousands of years certainly would.
The thing is though there is always a period where the water levels on each side are equalized, "slack tide". This will normally last for 1 to 2 hours on earth, in your case it would depend on the length of your day and the orbit of the moon but I would expect it to be similar.
This image illustrates it for you. You have periods of high and low water with the current also ranging between a flow in one direction and a flow in the other. At the point in the center of the flow curve you have "slack water" and at that point ships are safe to manoeuvre. You would need a sheltered harbour to keep ships out of the currents and as you say rips, whirlpools, etc would be fierce in the area.
So ships would time their passage with their tides, using the current to carry them close but arriving during slack water. They would come into dock and make sure they were secure before the tides picked up again.
To leave again they would head out during slack water but timed for when the tides are going in the right direction. As they head out the increasing tidal flow would push them along.
It depends on the shape of the sea. Generally speaking, tidal range is highly dependent on the actual geography. This being said, if the sea extends in a direction roughly aligned with the orbit of the satellite then in good places the tidal amplitude will likely be similar to what we have on Earth on the shores of the Pacific, that is, about 2 meters (7 American feet). If the sea extends mostly in the direction perpendicular to the orbit of the satellite the tidal range is likely to be smaller.
Remember that ports can operate will quite large tidal ranges; for example, Cardiff in the United Kingdom has a tidal range of about 15 meters (50 feet).
A narrow channel (or a pair of narrow channels around the island) can experience very strong tidal currents even if the tidal range in general is small. Look up Euripus Strait, https://en.wikipedia.org/wiki/Euripus_Strait.
Actually, I believe tides will be very limited there, because the narrow exit will limit the flow and the large volume will dissipate it. The Mediterranean sea has very limited tides compared to some other bodies of water. Depending on shape, the tide could be internal to the water body.
