Butterfly stroke timing
The following video shows two swimmers performing the butterfly stroke with different timings of the second kick. We will refer to the kick that occurs as the hands enter the water as the first kick, the kick that is generally timed as the hands exit the water as the second kick. You should be able to see that the swimmer in the bottom half of the screen performs the second kick earlier, the legs finish the downbeat of the second kick while his hands are still midway through the pull. Here's the view from above water:The timing can be observed more clearly in underwater video of the same swims:
Here is a still shot at the point where the bottom swimmer has completed the downbeat of the second kick:
Note that the arms are only midway through the pull.
For comparison, here is a shot where the top swimmer has finished the downbeat of his second kick:
Note that his hands are just leaving the water.
Finally, here is a shot of Michael Phelps at the same point in his stroke:
His hands can just be seen passing his swim suit on their way to leaving the water.
There are a few reasons to prefer the hand exit timing.
The top swimmer performs the upward part of the kick cycle with straight legs during the middle third of his pull, the downward part of the cycle starts with the knees descending, followed by the knees rising as his feet descend. This maintains a relatively flat and streamlined body position, minimizing drag resistance.
In contrast, the bottom swimmer's kick is mostly from the knee down, the upper legs are never recovered upward, and the downward kick occurs earlier and ends up deeper. The last third of his pull occurs with the legs are at maximum depth resulting in much more drag resistance. Compare the body positions in the first still image above.
For the top swimmer the second kick is providing extra propulsion just as he is launching forward into the recovery, helping maintain the speed generated by the pull and making it easier to cleanly recover the arms. The bottom swimmer gets the extra forward propulsion early in his pull which does increase his speed at that point but then he loses considerable speed during the recovery and has to expend extra energy in the finish of the pull and the recovery of the arms.
The downward force on the water generated by the downbeat of the top swimmer's second kick should also help maintain a flatter position during arm recovery.
The bottom swimmer would likely benefit from performing the upbeat of the kick with straight legs and then achieving the knee bend from a downward movement of the knee, as he does in his first kick, rather than an upward movement of the lower leg. This will shift the timing and likely reduce the amplitude of his vertical movements, both of which should result in reduced drag and effort.