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I've seen a few youtube videos where an aircraft bounces once/twice while landing on the runway. Why does this happen?

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The FAA covers this in the airplane flying manual quite nicely:

Bouncing During Touchdown

When the airplane contacts the ground with a sharp impact as the result of an improper attitude or an excessive rate of sink, it tends to bounce back into the air. Though the airplane’s tires and shock struts provide some springing action, the airplane does not bounce like a rubber ball. Instead, it rebounds into the air because the wing’s AOA was abruptly increased, producing a sudden addition of lift...

There is some more info in the doc worth reading as well.

In the simplest terms this can be one of the outcomes of an unstable approach; oftentimes, when the approach speed is too high, bouncing will result. Touching down on the nosewheel first can also cause a very dangerous bounce case known as a porpoise.

In some cases wind shear can cause bouncing. In this case the shear may cause an unexpected downdraft, forcing the plane down onto the runway vertically and inducing a bounce, or it may cause an updraft leading to an overcorrection by the pilot to descend; when the updraft stops, the plane falls rapidly.

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    $\begingroup$ By the way, a bouncing landing is referred to as "N landings, one arrival" where N = the number of bounces. $\endgroup$ – niels nielsen Nov 7 at 19:33
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    $\begingroup$ This one is quite well known youtube.com/watch?v=lhUO4pdGCf8 and nicely shows the amplification and landing on the front wheel. Some aircraft would not survive that. $\endgroup$ – Vladimir F Nov 8 at 13:26
  • $\begingroup$ @VladimirF I'm not sure that one did survive. I mean, it landed, but I'll bet it took a good bit of work before that thing could fly again... $\endgroup$ – Darrel Hoffman Nov 8 at 14:37
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    $\begingroup$ @DarrelHoffman the locals claim in the comments that this thing still (in 2019 - since 2006) is in that airport with the front chassis removed and unrepaired since that incident, and used as a learning illustration for other student pilots. $\endgroup$ – Peteris Nov 8 at 17:12
  • $\begingroup$ Seems to me it happens more often when there is a strong crosswind, I'm guessing because of a higher approach speed used. $\endgroup$ – Anthony X Nov 9 at 3:20
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Tail draggers are especially prone to bouncing when landed badly, if the main wheels touch down first.

Just before touchdown, the wing is at the right angle of attack to generate enough lift to support the weight of the plane. If the main wheels touch down first, the tailwheel descends a bit further, which increase the angle of attack of the wing. Lift is then greater than weight, so the plane takes off again - albeit somewhat temporarily.

It's similar to landing tricycle undercarriage on the nose wheel first, except a much easier mistake to make.

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