Acoustic levitation: high or low pressure?

[FranzDiCoccio]

TL;DR Summary

I just finished watching the awesome science demonstration by Harward University where a (ultra)sound standing wave is used to levitate small light balls. I'm not sure I understand where (relative to the wave) the levitating objects settles down. Is it the nodes or antinodes of the pressure?

So the video I'm referring to is the second in this webpage. Around time stamp 3:55 mr Wolfgang, the demonstrator, says that the little balls settle down at the high-pressure areas, which are signaled by the bright bands in the Schlieren image. We understand this by noticing that the area near the reflector, where the pressure is high, is bright.
So it would seem that the the balls settle at the antinodes of the (pressure) wave.

For some reasons this does not sound right.
For one, below the diagram under the video in the webpage it says that "The brightness of the schlieren effect is proportional to the magnitude of the change in refraction". If I get it right, refraction is proportional to density which should be proportional to pressure.

So, since the balls settle at the brighter spots, I'd say that they like to be where pressure changes the most. Now, if we're talking about change in time, that would be the antinodes again, right? There pressure rapidly changes from a minimum to a maximum value (about a background, "unperturbed" value).

If instead we are talking about change in space, the greatest (average) change should be at the nodes.
This seems to be the point of view of this discussion, where the time-averaged effective potential for the levitating object is the square of the spatial profile of the standing wave (plus the ramp given by gravity).

Is it possible that what mr Wolfgang meant is that the bright bands are the areas where the pressure changes the most with varying position, i.e. the nodes of the standing wave? If this is the case I guess that saying that those are high-pressure areas is not correct.

I know that atoms can be trapped in optical standing waves, and while some atomic species are "high field seekers", others are "low field seekers". I'm not sure there could be something similar for small objects in standing acoustic waves.

Thanks for any insight
Francesco

 

[scottdave]

The nodes are where the pressure is not changing. At the antinodes, the pressure is oscillating (changing) with time.

 

[FranzDiCoccio]

hi scottdave,
your reply is very short, so I'm not sure I get what you mean to tell me.

Are you suggesting that the bright areas are bright because the pressure is changing a lot with time?
Thus, being antinodes, the bright areas would indeed be the places where pressure reaches its maximum value, as the demonstrator says.

Also, this should mean that the balls are subject to a time-averaged potential whose minima are at the pressure antinodes, right?

The Harvard webpage does not discuss this, but at some point (somewhere below the diagram referring to the optical configuration) it says that the spatial derivative of the refractive index is maximum at the bright spots. Also, it says that one could consider density or pressure in place of the refractive index. The rate of change it refers to is a spatial one:
$\delta = k L \dfrac{d \rho}{d x}$
My doubts came from the fact that the spatial derivative is maximum at the nodes.

The time-averaged effective potential is briefly discussed in this reply. From the look of it, the local minima of such potentials again correspond to nodes. Antinodes seem to be local maxima, i.e. unstable equilibria.

I hope I clarified my doubts. Maybe there is something obvious I am not seeing.

Thanks again for your help.

 

[scottdave]

What I was clarifying is that the pressure is changing at the antinode, rather than being a constant "high pressure".

While not exactly the same, think about a string stretched between 2 fixed ends and vibrating. If vibrating at one of the harmonic frequencies, then the nodes appear not to move and the antinodes move back and forth. The tension in the string would vary, depending on the displacement from the rest position.

My experience is with electrical signals, where the wave oscillates between positive and negative voltage.

I would need to research how the spatial derivative of the refractive index behaves.

 

[FranzDiCoccio]

Hi scottdave,
thanks for your help. I have found an article in the American Journal of Physics that should cast some light on the subject. I have not had the time to read it seriously, but straight from the abstract it challenges the claim that the particles come to rest at (or close to) the antinodes of the standing pressure wave. The article specifically discusses the experiment at Harvard, which is described in Ref [1]. I guess one of the authors of [1], W. Rueckner, is actually the demonstrator in the online videos.