We have this at my house too, it has to do with friction and vibration I believe. With our faucet it always goes away when we turn up the hot water so it comes out faster.

Basically, as the water moves through the pipe there is friction, this friction causes vibration. There is a certain point where the vibration is at just the right frequency where they kind of line up with themselves, and it makes them stronger. (it is like pshing a kid on a swing, if you pust randomly, it is counterproductive, but if you push each time the kid starts forward again, he will go higher and higher. . . ) So the vibrations build quickly into an audible whine. The problem is most likely in your pipes and not the faucet.

As far as a solution goes, the permanent answer is to change the pipes at the location of the vibration (could be a bend, or specific section) this isn’t usually practical. I recommend trying to change the speed of the water through the pipes by running it faster or slower. (that is what we do at my house). You may find that at a slightly higher or lower speed, the whine goes away.