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My question is based around this video:

https://www.youtube.com/watch?v=1yaqUI4b974

The video shows sand settling in vibrational nodes of a plate at certain resonance frequencies. It appears that all the patterns are radially symmetric however at 5:15 I see a pattern that is bilaterally symmetric.

My question is why this pattern occurs horizontally instead of vertically? For a rectangular plate I understand that the two situations would occur at different frequencies, but what about a square plate? Is this a random occurrence?

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  • $\begingroup$ I think I will make my first attempt at answering part of my own question here, as encouraged by stackexchange. $\endgroup$ – cspirou Sep 3 '14 at 14:09
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    $\begingroup$ Related: physics.stackexchange.com/q/90021 $\endgroup$ – Chris Mueller Sep 3 '14 at 18:26
  • $\begingroup$ Good link Chris. Although my question relates to why a specific orientation occurs since another orientation seems just as plausible. $\endgroup$ – cspirou Sep 4 '14 at 9:06
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I believe the horizontal and vertical modes would be "equally likely" - except that if the plate is not perfectly square, one mode will be excited just before the other. And once you "lock in" to one of the modes you cannot excite the orthogonal mode since the plate, being no longer flat, will be significantly stiffer along the other direction.

Even if the plate was perfectly square I would expect that the nonlinearity of the situation (frequency of one mode depends on presence of the other mode - think about how corrugated plates are stiffer along one direction) will quickly cause the situation to fall into one of two degenerate states.

But without careful repeated experiments, that is just speculation on my part.

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In the context of this specific video, I believe that you cannot truly have a perfect square plate no matter how good your tools are. Therefore the pattern actually does settle on a specific frequency and not at two different frequencies.

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  • $\begingroup$ Your own answer neglects much more important effects, like the details of how (and where) the driving force acts on the plate. $\endgroup$ – CuriousOne Sep 3 '14 at 14:12
  • $\begingroup$ You are correct. However I didn't really think it was important because if the driving force was away from the center, then there wouldn't be the large number of radially symmetric patterns that the video displays. $\endgroup$ – cspirou Sep 3 '14 at 14:18

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