Why does the angle of attack remain nearly constant as a phugoid mode oscillation progresses? I've seen the plots showing this to be true, but don't have a good way of explaining analytically or physically why this is the case, other than that "it's what the equations say will happen." I understand that, in steady state/equilibrium, a fixed elevator position corresponds to a given angle of attack. But why should the AoA also stay constant during the oscillatory part of the phugoid before steady state is reached?
P.S. The only progress I've made so far is to try to identify the ways that AoA can be changed (which I understand to be via pitch and flight path angle) and to understand how these change during the phugoid. The pitch certainly oscillates during a phugoid, and so does the flight path angle due to the oscillating vertical speed. But why should these effects necessarily cancel each other out, as would seem to be required for us to observe the constant AoA behavior in the phugoid?
(I ask about this in order to more generally understand why AoA and power control airspeed and vertical velocity, respectively, in the way they do.)
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