John, Please actually read some of the document I cited. There's useful information before the heavy maths even starts.
The first page states:
"
...methods for compensating bends in air columns when designing wind instruments have been proposed, as, for example, a reduction of the bore diameter at bends .
However, reducing the bore diameter at bends to compensate for the tuning change supposes that this change is weakly frequency-dependent, or, at least, that the frequency shift, or the length correction, remains the same sign over the musically useful frequency range.
It is shown in the present paper that the frequency shift, or the length correction, in a bend, may vary significantly over frequency and that its sign changes. The curvature not always increases the resonance frequencies, but may even decrease them, resulting in a complex inharmonicity.
"
So this sort of compensation only works for one register. Of course there may be bore perturbations elsewhere in an instrument to compensate but the conclusion has to be that a curved and a straight crook will behave differently on any particular instrument.
The first page states:
"
...methods for compensating bends in air columns when designing wind instruments have been proposed, as, for example, a reduction of the bore diameter at bends .
However, reducing the bore diameter at bends to compensate for the tuning change supposes that this change is weakly frequency-dependent, or, at least, that the frequency shift, or the length correction, remains the same sign over the musically useful frequency range.
It is shown in the present paper that the frequency shift, or the length correction, in a bend, may vary significantly over frequency and that its sign changes. The curvature not always increases the resonance frequencies, but may even decrease them, resulting in a complex inharmonicity.
"
So this sort of compensation only works for one register. Of course there may be bore perturbations elsewhere in an instrument to compensate but the conclusion has to be that a curved and a straight crook will behave differently on any particular instrument.

