Mostly they avoid heavy storms. With good weather forecasting it is possible to delay departures or take a slight detour to avoid storms and then use a bit more fuel to arrive on time. Secondly the modern cruise ships are huge giants so even large swells are quite small compared to the size of the ship. The larger ships do not have the same issues that smaller ships face. It does put more pressure on the hull which needs to be reinforced of course. And for comfort they do have stabelizers that help them stay level in bad weather.
Point your nose in the waves. And then do the rollercoaster dance.
If you don’t have engine power to do that, well you get a big permanent submarine full of non-moving zombies.
Seriously, the hull can take massive waves nose first and shrug them off. It’s designed for that. Waves coming onto the side are another thing. Not much the low frequency tsunami ones, those are very very long waves and the ship will be more like “lifted by an elevator” than actually being on a wave. Tsunami is otherwise devastating to land because the long wave will crash onto land with so much water in it (due to be very long) that it can simply submerge a region before stopping.
Ship wise, the most devastating are storm waves because they are shorter in lenght and more mixed in direction, so you get hit more randomly and frequently, plus you can get “killer lenght” waves that basically are as long as your ship. This means while you pass each wave crest the center of the ship is raised with nose and stern out of the water, potentially snapping a ship in half. Of course the design accounts for it, but you prefer to run away and around the storm instead of live testing the design with some thousand scared people on board. And it’s quite easy to go around a storm.
Cruise ships are also very big and heavy, this gives them a lot of brute force to punch through big waves, the crew point them into the wind and just crest each wave as they come.
As an example, Royal Carribean’s Wonder of the Seas displaces over 230,000t, Titanic was 46,000t, Even the US Nimitz class aircraft carrier only weighs in at 100,000t.
In addition to what everyone is saying, cruise ships are designed for passenger comfort.
All their heavy equipment is low in the boat on the low decks. This make them have lower center of gravity than say a container ship so they rock less.
Cruise ships also have gyro stabilizers that send out fins as the ship rocks to cancel out the rocking some.
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