My understanding of longitudinal wave propagation is that some impulse (say, a vibrating speaker membrane) pushes particles into other particles, creating regions of higher density, or compressions. However, the particles don't want to stay together for long (perhaps due to entropy?) so the particle that was bumped into moves forward, while the initial particle moves back in the opposite direction, resulting in a chain of oscillating motion. If this is correct (if it's not, please help me understand in an intuitive manner), how is the momentum of the particles conserved? In other words, how does a particle oscillate backwards with the same momentum if it transferred some of it to the next particle? Does this have to do with why waves dissipate?
Actually, if there is just an initial impulse, the particles should collide with other particles, and then become stationary (in an elastic collision between two particles with the same mass, one moving and one stationary, the moving one will transfer ALL momentum to the stationary particle). Assuming such ideal conditions, one would observe that the compression "wave" (which is actually just an impulse), will move forwards, leaving stationary particles in its wake. This is why we can only hear a sound for a short moment (the wave moves past us after that).
To get a continuous wave, you need the source to continually be generating the compression and rarefactions, which will then be transmitted due to particles moving from the high-pressure regions to low-pressure regions, and so on. When the source stops, the waves will eventually move past the observer, and the observer will no longer hear a sound.
(A similar principle applies for transverse waves.)