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Premise: a massive organic structure sits somewhere in the Atlantic Ocean giving of an intermittent telekinetic pulse. This would presumably create a constant chain of tsunamis.
What sort of abnormal (if any) geographic features would such a scenario create and how would it otherwise affect the coastlines?
Really big tides and really low tides. With the average exactly where it is right now. There might be multiple high and low tides (vs the two a day of each that we have now).
Because the Earth rotates through two tidal “bulges” every lunar day, coastal areas experience two high and two low tides every 24 hours and 50 minutes. High tides occur 12 hours and 25 minutes apart. (ref)
Since the pulses of the organic structure probably don't match the rhythms of the existing tides, it would make predicting them difficult (easy enough with computers, if the structure's pulses are very regular) but not something you can just figure out in your head, like people can now.
Many ecosystems depend on marshlands near the coast. These would be inundated with more seawater than they had ever had to handle in the past. This would potentially wipe out many species locally. The problem is both the level of the water (too much and too little changes what can survive) and the increased levels of salt due to changing the mix of seawater and fresh water).
In theory, some of the marshes, tidal estuaries, etc, could reestablish themselves further inland. Except for the fact that most coastlines are prime real estate and are covered in asphalt, concrete, and buildings.
Most harbors will be unnavigable. The low tides will mean a lot of boats can't pass at all and larger boats could be damaged if the tides are so low that they're holding weight through their hulls sitting on the ground. Engines will get filled with silt, etc.
The high tides will mean the boats that are there will slip off their moorings. Docks float with the tides and being on a boat is the safest place to be in heavy rains that raise the water level. A friend and her family rode out Hurricane Harvey in Houston last year on their boat (their home) and were fine...except the tidal rise was a lot more than expected and they came within a couple feet of the docks physically rising so much that they would have gone over the pillars that hold them in place. Docks are supposed to go up and down but not up and over. Higher pillars would fix this part of the problem, but are expensive and time consuming to install.
So you'd basically eliminate most of the houseboating population of the affected part of the world. They'd be left in just a few naturally deep harbors with enhanced docks. Not that big a deal you say? Maybe not. Except...
You'd also eliminate most shipping by boat. And that is a big deal. Those shipping boats are very large and can only go in a few harbors to begin with. In 1000 miles of coastline, California only has 11 major ports. Losing even one creates economic problems. Losing, say, half would be catastrophic.
There is no way a huge ship could navigate a harbor, dock, setup, unload its cargo, break down, and get back out to sea in however many hours it has where the water level is high enough to navigate and not cause the ship to hit bottom. Moving cargo on to smaller ships to bring into the docks is possible but more expensive and still something you'd need to be very careful in timing for.
This answer falls under "how does it otherwise affect the coastlines" but you also asked about geographic changes. Cliffs would crumble and fall along the coasts. This happens anyway but these tidal changes would speed up the process tremendously. And that's even if the tides are gentle. On some coasts, like California, that would wipe out most of the railroads.
Lakes that are currently slightly inland would get bigger. Rivers would change size and possibly even course. Sand would move further inland and current sandy beaches would be stripped of sand. Silt in rivers near the coast would increase. If these rivers are not dredged regularly, the rivers will become more and more shallow and could disappear and the area turn into marsh. If human habitation doesn't prevent it.
