Particle Velocity vs. Wave Velocity of a Longitudinal Wave I am having some trouble disentangling the concept of particle and wave velocity for acoustic waves. Because the waves are propagating along the same plane as particle oscillations, I would (wrongly) assume that increasing the particle velocity would result in an increase in wave velocity.
Some sources that I have been looking at do not necessarily describe the relationship that I am struggling with. I found that this applet was fantastic in illustrating how an increase in particle velocity will not result in an increased wave speed. Rather, the increased particle velocity manifests as increased particle clustering and greater wave amplitudes.
I was wondering if anyone could explain to me: how does an increase in particle velocity not result in an increase in wave velocity for longitudinal waves?
 A: You can imagine the particles undergoing an ascillatory motion around their equilibrium positions. The simplest model is a 1D chain of particles.
As a particle oscilates around the equilibrium, it makes conatct with the neighbours once per period and so the the wave (the perturbation)  is "handed over" from particle to particle. So the propagation velocity is related to this time interval between two events of neighbouring particles "making contact" and transmitting the wave. (it does not really have to make physical "contact"; they may interact by EM fields).
What we call the particle velocity is the velocity of the oscillation which follows a harmonic (sine or cosine) time dependence. There is an amplitude of the velocity as well as an amplitude of the displacement. When you say that the particle velocity increases it means that the value of the maximum velocity increases.
Now, as for any simple harmonic oscillator, increasing the velocity is related to an increased amplitude ($v_{max}=\omega X_{max}$). So it goes faster but the distance to travel is larger. As a result, the period stays the same, is independent of both amplitude and particle velocity. So, no matter how fast the particle move, it "makes contact" once per period (or twice) and the period does not change with increasing velocity.
The period of the oscillation, as for simple harmonic oscillator,  depends on the stiffness (spring constant)and mass. And so does the propagation velocity, which depends on some parameter measuring the stiffness of the medium and the density which represents the mass factor in the period.
