Physics in Action: Standing Waves on a Metal Sheet

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We've seen what standing waves in a one-dimensional system, like a vibrating string, look like. For example, the ends of the string don't move. Those are called nodes. And there are also nodes at various points in-between, which depend on how fast the string is vibrating. Now you may wonder whether you can have similar things happening in a two-dimensional system. Are there normal modes in a two-dimensional system? Are there regions, which do not vibrate in a two-dimensional system? Can you even have standing waves in a two-dimensional system?
Well, we can answer those questions by showing you what a two-dimensional system really looks like. We've set up here a brass plate which is going to start vibrating in a few seconds, and we've sprinkled on top of this brass plate a bunch of ordinary beach sand to show you how the plate vibrates. Randy is going to bow this brass plate with a violin bow and that's going to cause the plate to vibrate in a somewhat complicated way. Let's see what happens. We'll come back at the end to explain what you've seen.
Notice what happens. As he bows, some of the sand in some of the regions gets thrown off. There are regions over here, like here and here, where there's hardly any sand at all, but there are regions over here and here where there's lots of sand. Now think about it. Why has that happened? If you think about it for a few seconds, you'll realize that where there is no sand, that's a region where the plate vibrated a lot and kicked off the sand. And where sand has remained, those are regions, like nodes on a string, where the plate didn't vibrate at all and that's why the sand remains on the brass plate.
Now Randy can bow this plate in different ways and in different regions, and we could change what the pattern looks like. Notice what happens now. This looks more like a star pattern. And as he continues to bow, the pattern changes even more. Notice the regions now where there is no sand look somewhat different from how they looked before, and in fact these regions are somewhat complicated. As you go on and on, depending on how we bow, where we bow, we can change the pattern from one pattern to another.
Again notice this rather beautiful and symmetric pattern. Again there are regions where there is no sand. Those are regions where the plate vibrated a lot. Regions where sand remains, those are regions where the place hardly vibrated at all.
So what have we learned from all this? We've learned that a two-dimensional plate, like a one-dimensional system, does have normal modes. Instead of standing waves in it, we can find regions where the plate vibrated a lot and other regions of the plate didn't vibrate at all. This system is much more complicated but the general idea of normal modes applies to two-dimensional systems just as it does to a one-dimensional system like a simply vibrating string.
Waves
Standing Waves
Physics in Action: Standing Waves on a Sheet of Metal Page [1 of 1]