how they are formed
According to their tales, rogue waves of 100 feet or more were seen in seas which could not support waves higher than 60 feet by the then known equations. This is an important matter because the design of ships takes into account the highest possible wave in determining the strength of the build. Also, the shape of these waves was unusual. Since they were mid-ocean waves they should be, in theory, 'swells' possessing a gradual slope. However, these rogue waves were 'breaking and had a very steep slope.
Now, clearly a wave does not break unless it encounters an obstacle. What obstacle is there in the open sea upon which such a monster wave could break causing it to rise up as breaking waves do?
The matter was not believed until satellite radar images of waves in a large storm were taken which clearly showed the unusual formations in open ocean just as mariners had said all along. (This is one of the few times that Hynek's Impossibility Theorem has failed. ;o)
Here is the cross-section of such a wave.
Note the deep trough in front of and behind the waves.
What could cause this obvious anomaly?
A scientist noticed a resemblance between the cross-section of the rogue wave and the cross-section of a solution to Schrodinger's wave equation in quantum mechanics. The program left the viewer here with no actual resolution. The matter is under study.
Let us proceed to "study".
First and foremost this is not a quantum mechanical event. The only relevant force here is wind velocity. So how does the wind make the big wave?
Like this ...
First, there is a normal statistical distribution of waves just as is predicted by the aforementioned standard scientific equations. So there is most definitely some probability that a bigger than usual wave will occur.
When such a wave occurs, the wind will naturally blow on it more than the surrounding ones which have their heads down as it were. So it gets pushed more becoming bigger. It can't get bigger without more water so that water comes from the trough immediately behind the following wave (the windward trough). After all the wave is being pushed away from the following wave.
Now, the growing wave catches up to the wave in front of it and ... because of the dynamics of water waves ... breaks on that wave whilst at the same time "eating it" as well making it all the bigger. It's now a monster wave ... breaking ... with steep sides ... and a big trough front and back. It has a big trough in front because it grew bigger and the water was taken from there, i.e. it's minus the front wave that was eaten. Remember, water is an incompressible liquid. Therefore, for every wave there is an equal but opposite trough. So, if you are going to have a big wave you need a big trough where the water came from.
(This is a 5 frame rollover image)