I’m amused this now being “discovered”. Not only have I personally observed at least “rouge” wave (amongst many very big ones), we also demonstrated the mathematics and probabilities of them in the late 1970’s. Only thing is we didn’t publish for some silly reason.
The wave in question was while aboard the “Shanghai Trader” in the middle of the South China Sea. I was working in the oil industry at the time and I was a young engineer leading a marine and site survey near an island where there was potential for building a marine terminal. We had to sail to a rig in the a few hundred miles away to pick up some equipment. While underway, we passed through a storm. Very uncomfortable. Every so often a “big” wave would appear. Bigger than the rest. That day cured me. To this day I avoid going on ships into the open sea.
The open ocean is composed of a mixture of waves with different fequencies and amplitudes. They add together to form the pattern of waves that one sees. You can do spectrum analysis of the wave forms to find the different components.
When the spectrum is highly “peaked”, e.g. wave energy is focused at a few frequencies with high amplitude, the waves with slightly different frequencies interact to form a very long wave (low frequency) and high amplitude. Actually, it causes a “group” of ways to move this way. We called it “wave grouping”. It’s this wave caused by the interaction which moves along and appears to a stationary observer (e.g. on a boat) to happen less frequently than the “normal” waves. These waves are bigger than normal, and they move in “packs”.
The mathematics of adding waves like this is learned in high school, I think, when one adds together in time two sine waves of slightly different freqency. A more advanced topic would be the Fast Fourier Transforms (FFT) of the wave time series to compute their spectrum, and from the spectrum synthesise a time series. Doing this synthesis for lots of waves of different amplitude frequency can simulate the actual sea surface.
I think I’ll challenge my son to try to work this all out as an excercise. He’s bored this summer and needs a little mental stimulation.