Heave is the up and down motion of a ship or other structure in the ocean. Sometimes a problem can occur where a structure can oscillate up and down in the ocean. If waves are in resonance with the structures natural up and down period these motions can be large. To compensate for this problem some designs require "heave plates". These are underwater and increase the structures drag for movements in the up and down directions. These can be seen in the underwater portion of spars.
The heave problem seems to be when the waterline area is small relative to the total mass of the structure. So spars, multi-spars and such. If such a structure is moving up or down the buoyancy changes slowly, because of the small waterline area, so the motion can continue. Something like a ship or a ball house does not exhibit heave oscillations or require heave plates.
Here is a video with a good demonstration of the heave issue:
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