First, a bit about where I came from: A page from my graduate program's website.
Brass instruments fascinate me. The interplay of human natural and artificial structures is complicated, physically and mathematically unwieldy, and altogether unusual and beautiful. The fact that the tissues of the lips provide the basic oscillation for this instrument is at the heart of my fascination.
The original modern model of the lips is that of Helmholtz, from the masterwork On the Sensation of Tone (1877). His idea was that the lips were like a valve for a one-dimensional flow of air. This lip valve was controlled by pressures, upstream in the mouth, downstream in the mouthpiece outside the lips, and now we might also add between the lips (like a "Bernoulli" pressure - moving air exerts less pressure). These valves could be categorized and analyzed depending on the action of the valve under various overpressure/underpressure situations. An "outward-striking" valve would tend to open when there was greater pressure upstream and less pressure downstream. The "lip reed" of a brass instrument may be seen in this way: when there's more pressure inside the mouth than outside, the lips can blow open. Alternatively, an "inward-striking" valve would tend to close in that situation, and open when there's greater pressure downstream than upstream.
Diagrams
Current thinking and the state of the art model
Solid model/FEA
While still working on a model of the lip reed
Literature
D. O. Ludwigsen, "Physical modeling of the trombone player's lips," Proceedings of the Stockholm Music Acoustics Conference (SMAC03) , Roberto Bresin, Ed., p. 209 (2003).
D. O. Ludwigsen and W. J. Strong, "Normal modes of a finite element lip reed model," J. Acoust. Soc. Am., 109(5), Pt. 2, 2483(A) (2001)
J. R. Moffit and D. O. Ludwigsen, "Self-sustained oscillation as a function of blowing pressure in a one mass lip reed model," J. Acoust. Soc. Am., 108, 2591 (2000)
D. O. Ludwigsen and W. J. Strong, "Physical modeling of a trombone player's lips," J. Acoust. Soc. Am., 104, 1768 (1998)