How do two-dimensional surfaces vibrate under forced oscillations?

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Chen
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Hi,

I'm looking into the subject of "Chladni plates":
http://www.physics.montana.edu/demonstrations/video/3_oscillationandwaves/demos/chladniplates.html
For a lecture I'm supposed to prepare, and I'm looking for information on how exactly a two-dimensional surface vibrates under forced oscillations. It's no secret that the motion of a 1D string is governed by the simple wave equation. So which equation governs, for example, a thin square plate?

Thanks,
Chen
 
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Chen said:
Hi,

I'm looking into the subject of "Chladni plates":
http://www.physics.montana.edu/demonstrations/video/3_oscillationandwaves/demos/chladniplates.html
For a lecture I'm supposed to prepare, and I'm looking for information on how exactly a two-dimensional surface vibrates under forced oscillations. It's no secret that the motion of a 1D string is governed by the simple wave equation. So which equation governs, for example, a thin square plate?

Thanks,
Chen

This is more of a mathematics problem, so I recommend you look in mathematics books dealing with partial differential equation. In Mary Boas's text "Mathematical Methods in the Physical Science (2nd Ed)", she has a treatment on 2D vibrating circular membrane in Chapter 13 on PDE, giving you all those Bessel function solutions.

Zz.
 
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I have the 1982 version of "Fundamentals of Acoustics" by Kinsler, Coppens, Frey, and Sanders. The index reads:

CHAPTER 4
The Two Dimensional Wave Equation: Vibrations of Membranes and Plates

4.1 Vibrations of a Plane Surface

4.2 The Wave Equation for a Stretched Membrane

4.3 Free Vibrations of a Rectangular Membrane with a Fixed Rim

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4.7 Forced Vibrations of a Membrane

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Thank you both, I'll try looking at those books when I get the chance.
If anyone knows of an online resource for this information, even if it's a bit simplified at first...

Thanks :)
 
That would be the case for membranes, but not for thin plates (much like the difference between the vibration of strings and that of solid bars).
 
These may help a bit. I also picked up a copy of this (used) which is, IMO, a good book for PDEs:

https://www.amazon.com/gp/product/0139586202/?tag=pfamazon01-20

http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-306Spring2004/7CE34382-EFED-4C3E-9AAA-E3EB930C2AFC/0/hristinas_lec14.pdf

http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-306Spring2004/F4CA6117-8F21-486A-809A-5BC3998AA38E/0/hristinas_lec15.pdf
 
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