Don Hosek - Flute - October 1999
We enter the second octave

Practice continues, with my beginning to start playing notes in the second octave. Despite what my dad had claimed, the fingerings of the second octave are not identical to the first octave, courtesy of a slight variation for D2 and D#2. For these two notes the first finger on the left hand is opened. It actually makes sense as I think about it.

Notes beyond the first octave are sounded using the second harmonic of the tone. I don't know entirely the physics of how this works, but it ends up that the column of air in the flute is effectively cut in half, creating a sound an octave higher. With the long tube as fingered with the D and D# fingerings, this division is somewhat harder (but not impossible to effect). Raising the first finger on the left hand makes it that much easier to generate the desired tone. As we shorten the tube when moving up the scale, the tone becomes progressively harder to maintain with that finger raised.

I read Theobald Boehm's book, The Flute and Flute Playing in Acoustical, Technical and Artistic Aspects hoping to get a formal explanation of this, but didn't get it. I did, however, get an interesting account of the experiments that lead to the design of the modern flute.