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

[veneficus5]

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!

 

[Unrest]

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.

 

[veneficus5]

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.

 

[Unrest]

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?

 

[veneficus5]

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.

 

[veneficus5]

actually Unrest I am now interested in your numerical method. My professor said that doing it numerically would suffice.