What is the difference between a puff of air and a sound wave regarding creation and propagation? While watching a Schlieren video of a hand clapping, I noted a very distinct difference between a sound wave and a puff of air, which were both created by a hand clapping.  What is the difference between a puff of air and a sound wave regarding creation and propagation?
In the video, it appears that some of the energy goes into the sound wave and some of the energy goes into the puff.  Is there a principle that governs the distribution of impact energy between oscillation (sound) and pushing (puff?)  
Newton's laws for air imply the wave equation, but are not equivalent to it.  If air was described by the wave equation and nothing else, then any disturbance would travel at the speed of sound.  What set of equations can you write that describes both sound and macroscopic motion at the same time, and what dimensionless parameters specify their relative importance?
For example, one might think that a very low frequency disturbance would be likely to create a "wind".  On the other hand, the "pop" heard when you say "p" into a microphone without a pop shield doesn't sound like a super low frequency, yet I suspect that the purpose of the pop shield is to slow down the puff.
 A: Based off of that video, the differences you're pointing out are the nice wavefronts from the speaker at 2:04 and then the clap shown at the beginning and the end.
It's true that the wave fronts from the speaker (and even the book) give nice "crests" and "troughs" whereas the clap kinda just... is this blob-y thing. There are several potential reasons why these appear differently.


*

*Harmonics: Speakers (and books hitting tables) produce very "pure" sounds. They hit the air very strongly, deforming very little. This produces a nice, single wavefront (per movement of the item), which you see as a dark line in the Schlieren videos. Hands, on the other hand, are floppy things and hit more "softly," jiggling as they come together. This produces a less "pure" sound, so this translates into a very amorphous wave front. (You can also say that hand-claps have harmonics, whereas the speaker and book have little to no harmonics.)

*Shape: the book and the speaker have a nice square or round shape to them. This produces much "nicer" and clearer wavefronts than our oddly-shaped hands. Flat or round objects make for waves we're generally used to seeing while studying physics.

*Perspective: the book and the speaker both had really ideal ways of setting them up to see the obvious waves. Hands, however, present a challenge. How can you show a non-symmetric 3-d wave on a 2-d screen? At best, you could see a wavefront, but you'll likely just see a blob, especially if the shape of the 3-d wave isn't really spherical. The symmetry of the objects allowed for good perspectives. Clapping hands can lack that kind of symmetry, preventing good shots.

*Power: that speaker or that book hitting the ground may have been more forceful than the man's clap. I suspect a more wave produced by more force (with "higher amplitude") appears darker. So, if that speaker produces a louder sound than hand clapping, you'll see darker lines there.

*Camera Tricks: Finally, we should address the fact that, although these images all use the Schlieren technique, they likely have different settings for different shots. If we put the speaker next to that clapping man and filmed them both, maybe the speaker's wavefronts would appear as weak little things, just like the man's clap. This is just a possibility, though, so this reason is much weaker than the others.


The shape of these wavefronts all have to deal with the thing that made them; as far as propagation goes, they all move under the same laws. It's their initial shape, the force with which they were made, and the thing that made them which determines the differences in the waves.
A: Sound waves are air vibrations which means that the particles of air are not displaced permanently (in ideal case), while "puff of air" is a kind of "wind" meaning permanent displacement of air particles.
In practice both these phenomena often go together, like in the case of hand clapping (or lower quality bass loudspeakers for that matter).
EDIT: Concerning the difference in the creation - it's pretty straightforward, and bass loudspeakers are a good example here.
If you take modern bass loudspeakers, which tend to be smaller than they used to be, they make up for the small area with a larger displacement. Effectively, they produce a lot of "wind" in addition to the sound wave (and as a result, the measurements look OK, as the microphone does not really differentiate, but the sound quality is worse, despite advertising claims - my opinion). On the other hand, bigger loudspeakers - 15" or even 18" do not need such a big displacement, and as a result they produce more sound and less wind.
When you clap, your hands, having relatively large area, move quite a long way before they eventually hit each other. This makes a large displacement of air particles before the sound wave is finally produced.
To sum up - if you want to produce sound, and not a puff, you rather provide a strong, and yet short strike/blow.
A: The video shows not just physically displaced air, it can also show temperature gradients.  Part of the "puff" that you are concerned with is due to a physical displacement of air from the colliding hands and part is due to the warmer air near the hands convecting and conducting heat away.  As illustrated later in the video, the visualization technique can show temperature gradients as well as density gradients.  However, temperature gradients alone do not necessarily lead to sound waves.
A sound wave is defined by the following dispersion relation:
$$
\frac{\omega}{k} = C_{s} = \sqrt{ \frac{ \partial P }{ \partial \rho } }
$$
where $C_{s}$ is the speed of sound, P is the gas pressure, and $\rho$ is the mass density.
Temperature gradients do not always lead to density gradients, as illustrated by sun spots, which have a lower temperature relative to the surrounding gas but effectively the same density.  Technically, these structures remain in pressure balance with the surrounding medium due to increased magnetic pressure that compensates for the decreased thermal pressure.  While this is not exactly the same as the "puff" in the video, it illustrates my point that a temperature gradient alone does not necessarily lead to a sound wave.
So to answer your question, the circular pulses seen leaving the hands at a very high speed (compared to the "puff") are sound waves created by finite values of $\partial P / \partial \rho$.  The "puff" is likely created by a combination of temperature gradients and displaced air.
