Have you ever noticed how tight the bends are near trumpet valves? Do brasswind lack clarity, or volume? Admittedly, the sound of the flute is more fragile than that of a trumpet (trumpet players may disagree), but it has been proven abundantly that we can make perfect wind instruments with tight and complex bent bores

Cecilia Oomes plays Syrinx by Debussy.
Cecilia Oomes, formerly with the Koninklijk Concertgebouworkest (KCO), now playing solo.
The recording was made October 20th, 2006 in the Nassaukerk in Amsterdam. Oomes played on a silver Jack Moore flute, fitted with a Flutelab headjoint #1105, Sterling silver with 14K gold lipplate and riser. The recording was done with a Shure 16A microphone and a Sony minidisc MZ-R70.

So how does the Swan Neck flute sound compare to a straight headjoint?
The turbulence near the embouchure hole is not dealt with as efficiently as in a straight flute. This makes it somewhat harder to produce a clear sound. The difference is small, but audible. Also, this type of headjoint has a little more resistance in the higher notes.

How does a sound wave travel trough a bend?
Unlike moving air, pressure distributes itself evenly in all directions. This is why a bend does not hinder the air wave. But the wave exhibits a peculiar behaviour: it will try to cut a bend short. This will effectively shorten the path of the wave, so the pitch goes up. Only a little bit, but enough to throw an instrument out of tune. Scientific research has recently proven that the effects can be compensated by minute restrictions of the bore. Instrument makers have known this for centuries. The difficulty is not as much making the bend to work. The difficulty is combining the subtle Boehm parabola in the headjoint with the bends and its acoustical compensations. Our headjoint bores are designed by hand, then the compensations are calculated in a spreadsheet and the result can be seen in CAD file (Computer Assisted Design).

How does a flute work? You may think of a flute as an air duct, but it is not. It is a pressure duct. Waves travel down the tube and impart some of their energy on the surrounding air. But some of the wave bounces back into the tube. At certain frequencies, the waves arrive back at the embouchure hole the same moment a new wave starts. So the up wave coincides with the down wave. By this mechanism, sustainable waves are formed at the harmonic frequencies.

What are the effects of bends on the sound in general?

Animation courtesy of Dr. Dan Russell, Kettering University.