How do two-dimensional surfaces vibrate under forced oscillations?

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Discussion Overview

The discussion centers around the vibrations of two-dimensional surfaces, specifically in the context of Chladni plates and the governing equations for their forced oscillations. Participants explore the mathematical frameworks applicable to these phenomena, including references to relevant literature and resources.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Chen inquires about the governing equations for the vibrations of a thin square plate under forced oscillations, drawing a comparison to the wave equation for a 1D string.
  • Zz suggests looking into mathematics books on partial differential equations, specifically referencing Mary Boas's text for solutions related to 2D vibrating circular membranes.
  • A participant mentions the 1982 version of "Fundamentals of Acoustics," highlighting its chapters on the two-dimensional wave equation and vibrations of membranes and plates.
  • Another participant notes the distinction between the behavior of membranes and thin plates, indicating that the governing equations may differ significantly.
  • Chen expresses interest in finding online resources that provide simplified information on the topic.
  • A participant shares links to additional resources, including a PDE book and lecture notes from MIT, which may be useful for understanding the topic further.

Areas of Agreement / Disagreement

Participants generally agree on the need for mathematical resources to understand the vibrations of two-dimensional surfaces, but there is a disagreement regarding the applicability of certain equations to membranes versus thin plates.

Contextual Notes

There is an acknowledgment of the complexity involved in distinguishing between the vibrations of different types of surfaces, and the discussion reflects varying levels of familiarity with the mathematical concepts involved.

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.
 
Last edited by a moderator:
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

.
.
.

4.7 Forced Vibrations of a Membrane

.
.
.
 
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).
 
Chen, I just deleted my post after rereading the OP. Hadn't read your last post before I did that - sorry.
 
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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