Helmholtz 2D PDE around general shape (eg a figure 8)

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Discussion Overview

The discussion revolves around solving the Helmholtz equation or wave equation for a figure 8 shape, specifically in the context of determining the resonant frequencies of a classical guitar. Participants explore both analytical and numerical approaches, considering the physical properties of the guitar's wooden top as a membrane or a plate.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to understand how to solve the Helmholtz equation for a figure 8 shape, considering it as a Dirichlet boundary condition for the wooden top of a guitar.
  • Another participant notes that the wooden panel behaves more like a plate with bending stiffness rather than a membrane under tension, questioning the applicability of the Helmholtz equation in this context.
  • There is a suggestion to also consider the resonances of the air cavity within the guitar, which may be significant.
  • One participant expresses a preference for non-numerical solutions but acknowledges the need to develop a numerical method for their presentation.
  • Concerns are raised about whether both tensioned membranes and stiff bending plates can be modeled similarly, indicating a potential area of exploration.
  • Participants discuss the purpose of the presentation, emphasizing the desire to present something new and interesting related to PDEs while keeping the topic accessible within a limited timeframe.
  • A later reply expresses interest in the numerical method mentioned by another participant, indicating a shift towards numerical approaches.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to model the problem, with differing views on whether to treat the guitar's top as a membrane or a plate. The discussion remains unresolved regarding the applicability of the Helmholtz equation in this scenario.

Contextual Notes

Participants acknowledge limitations in their knowledge of analytical solutions and express uncertainty about the modeling of the guitar's top as either a membrane or a plate. There are also references to prior knowledge of PDEs limited to simpler geometries like circles and squares.

Who May Find This Useful

This discussion may be useful for students and researchers interested in the application of partial differential equations to physical systems, particularly in acoustics and materials science.

veneficus5
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Hello physics enthusiasts! I was looking for resources, and stumbled upon these awesome forums.

I am looking for how to solve the helmholtz equation / wave equation on a figure 8 type shape. I wanted to find the resonant frequencies of a classical guitar.

Would this work? I am considering the wood top of the guitar to be the membrane and the figure 8 shape to be the dirichlet boundary condition.

I need some help though as i only know how to solve for squares and circles.

Numerically would fine too, although I would prefer nonnumerically.

Thanks a lot!
 
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If you're looking for the resonant frequency of the wooden panel, don't forget that it's not a membrane under tension, it's a plate with bending stiffness. I only know you can use the Helmholtz eqn. for membranes, but I guess you can use it for plates too?? I'd like to know how to formulate that problem.

You could also find the resonances of the air in the cavity. That might be more significant for a guitar.

I'd just do it numerically, using off-the-shelf finite element analysis software.
 
Well if I do it numerically, I'll have to come up with the method. I am doing a presentation on it. I should have stated that.
 
veneficus5 said:
Well if I do it numerically, I'll have to come up with the method. I am doing a presentation on it. I should have stated that.

Hmm, what's the purpose of the presentation? Would you have to actually write your own software rather than just handwaving how an exiting product works? Sorry I don't have any experience (that I remember!) with analytical solutions but I've worked on a numerical solver for this problem.

But what do you think about the tensioned membrane vs. stiff bending plate? Can both be modeled the same way?
 
Unrest said:
Hmm, what's the purpose of the presentation? Would you have to actually write your own software rather than just handwaving how an exiting product works? Sorry I don't have any experience (that I remember!) with analytical solutions but I've worked on a numerical solver for this problem.

But what do you think about the tensioned membrane vs. stiff bending plate? Can both be modeled the same way?

The purpose of the presentation is to present something relatively new and interesting related to PDEs to my class in about 20 minutes. Even though it should be a stiff bending plate, I think I'll just explain that detail at the beginning of the presentation and look at the membrane version anyways. Everyone has done PDEs analytically, but only one other person and I have done PDEs numerically.

The stiff version might be worth looking into, but I want to keep it easy and understandable by only deviating slightly from what we already know (20 minute presentation) edit: and what we know is the wave equation on a circle and square with or without heat sources.
 
actually Unrest I am now interested in your numerical method. My professor said that doing it numerically would suffice.
 

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