Ukuleles and Physics

Sorry if you guys can't see the letters on the picture, but I sure hope you can. Well, I just wrote out the equations for velocity. I'm actually kind of scared to write this because I'm not sure if I'm right of wrong. I hope I'm right because if I'm wrong then I would just sound foolish. But what I wanted to say is that we all know from the sound of music lab that a change in frequency changes the note which is being played, so if you play a c note, then a d note, the frequency changes. In an instrument like an ukulele, you can see how this happens. When you change the note, you change the length of the string which you are playing, because you press down on frets with your fingers. This change in L changes the velocity that the string is moving. Then you go down to the second equation. Since the v changes, the frequency and wavelength have to change proportionately. Because the frequency changes, you hear the changed note. I'm not so sure about this way of thinking though, so don't take my word for it!