I'm not quite sure of the exact scientific method, but if the pool is to shallow than it would be a problem, because you could kick the bottom of the pool. I always find it easier to swim in water around 1.5m deep, because if its too deep it makes me work harder, because i want to stay afloat.
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I would believe so. I don't have the exact science of it, but I did hear the Olympic announcers say that the depth does matter. For example, sometimes you can watch Michael Phelps dive and swim deeper on his initial jump in and on turns. By going deeper, he is able to come to the surface farther ahead by avoiding the drag that the other swimmers create. Essentially, he's able to swim beneath the chaos and come out the other side.
A shallow pool will not allow large swim strokes and will not require you to tread water. Deeper water will not let you swim underwater for long distances without stopping to tread water.
It is said that you do - because there is less turbulance reaching the swimmers on the surface.