A common analogy that even I often use is to compare the bow of a string instrument to the air on the brass. Just as the bow moves across the string to create vibration, the air must move past the lips on a brass instrument. Usually I hear this comparison to make a point about breathing, such as there is no sound without the air.
It’s interesting to view the vibrating bowed string and compare it with the vibrations of the lips. Here’s a video showing the vibrating string is slow motion.
Now compare that with some of the footage from this film by Lloyd Leno, Lip Vibrations of Trombone Embouchures.
The pattern of vibrations from a single string and the lips have some similarities, but the interaction of the two lips vibrating in tandem does make some important fundamental differences with the lip vibration.
One important distinction to think about is that while on a string instrument the sound is produced essentially through the amplification of the string vibrating, on a brass instrument the sound we’re hearing isn’t really amplification of the vibrating lips but rather the sound of the vibrating column of air inside the instrument. A slightly different analogy would be to think of the air inside the instrument as the vibrating string.
On a string instrument the pitch is changed by lengthening or shortening the string, essentially changing the speed of oscillations. For the brass embouchure the closest parallel would be the way we change pitches by increasing or decreasing the surface area of the lips that vibrate. Check out the Leno film and look closely for this. But again, the lips role is to excite the vibrating column of air inside the instrument, so the analogy isn’t quite complete.
Another helpful comparison might be to take a look at how strings vibrate when an overtone or harmonic is produced. This can be done with a vibrating string by lightly touching the string at certain points and exciting the string to vibrate with the bow or by plucking it. What happens is that a “node” forms at certain points and the string vibrates in segments, not traveling from one end of the string to another. I can’t really see that in this video, but you can watch a harmonic on a cello string in slow motion here.
Much like stringed instruments, we can change the size of vibrating column of air inside a brass instrument by adding or subtracting to the length of the air column with the slide or valves of the instrument. For example, when we lengthen the instrument by moving out with the slide or depressing valves, the pitch gets lower. But if that was the only way we could change the pitch we would be limited to a range of only a diminished 5th, not quite enough. Brass players primarily change the pitch by playing along the harmonics of the vibrating air column.
Consider a trombonist playing a pedal B flat, the fundamental pitch on this instrument. The player excites the column of air into vibrating by blowing air against the firmed lips. The vibration starts at the lips and then travels all the way to the bell of the instrument where some of the energy of that vibration escapes. The construction of the instrument is such, though, that some of that energy gets reflected back down the column of air back to the lips where this cycle starts again as long as the player is blowing air. This is sort of like bowing an open string.
When the trombonist plays the next first position note, the B flat a major ninth below middle C, the vibrating column of air inside the instrument forms a node half way between the lips and bell. This essentially creates two oscillating columns of air, vibrating sympathetically. Since the reflection of this energy only travels from the lips to the center of the horn and back again, the speed of this reflection is twice as fast and the lips will vibrate in tandem with this oscillation. This is sort of like playing a harmonic on a string by lightly touching the string exactly in the center while bowing or plucking.
At the next first partial pitch, the F below middle C, the column of air inside the trombone will divide into three parts, with two additional node points between the lips and bell.
The acoustical principles that all come into play here can get very complex very quickly, and I’m not an expert in this subject. If you want to learn more, I recommend resources by Nick Drozdoff, a high school physics teacher and excellent trumpet player. He has a whole series of online lessons that include some excellent discussions of the physics of brass instruments.