Oscillation of air particles and speed of sound wave

A sound wave is essentially air particles oscillating parallel to the direction of travel of the wave.

We learnt that $v = f\lambda$, where $v$ is the speed of the wave, $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.

Suppose the air particles oscillate with a frequency of $f_{particle}$. Is there any relationship between $f_{particle}$ and $v, f, \lambda$ of the entire sound wave?

I thought about this because intuitively, if the air particles vibrated faster, then the wave should travel faster, but I am unable to come up with any formula that describe this relationship (if it is even true!).

• How did you arrive at the idea, that the air particles should be oscillating with a different frequency than the wave itself? Why is that "intuitive"? – CuriousOne Sep 18 '14 at 9:45
• In school the only thing we were ever taught about waves is to draw then as $\sin$ and $\cos$ curves. Sound waves were a $\sin$ curve, so were light waves and every other wave in our syllabus. If each point on the curve moved up and down faster, then the entire curve should travel faster, I think. – Yiyuan Lee Sep 18 '14 at 9:48
• I see. Try this: imagine a wave on a lake. Imagine that there are leaves on the water that bob up and down with the wave. Are the leaves moving along with the wave, or are they standing still? What is the frequency of the movement of the leaves? Is it faster than the frequency of the wave, or the same? (If you don't have a lake and leaves, a sink and a bit of flower sprinkled on the water might do for an experiment!) – CuriousOne Sep 18 '14 at 9:54
• Would that mean that the period of the individual particles is exactly the period of the wave? – Yiyuan Lee Sep 18 '14 at 9:55
• Yep. If you don't believe me, you can do the experiment. By analogy, the same is true for sound waves, but it's harder to prove that experimentally, because the movement of air particles in a sound wave is much, much smaller (unless the volume of the sound is beyond deafening). – CuriousOne Sep 18 '14 at 9:57

So your analogy cannot be followed for these reasons, and this is also the cause that we use different formulations to describe groups of 10 or 100 particles, than when we describe media (made of at least ~$10^{23}$).