Modeling sound resonance in an arbitrary cavity

I am trying to solve a challenging problem, and I'm hoping for some advice on how to proceed. I want to model sound waves in a cavity for the purpose of determining resonance. The plan is to answer the question numerically, but I need some better bearings regarding the physics involved.

Here is a summary of my main questions:

1. How should I model this situation for a closed cavity?
2. What does resonance in a closed cavity even look like? (as opposed to a "whistling" kind of cavity)
3. What wave equation should I use? I'm assuming spherical for an arbitrary cavity.
4. I don't even know if using diff eq's (wave equation) is the best approach here. Would it be possible, or even advisable, to try modeling noise in the cavity and see what kind of response I get?
5. I don't want my solution to be dependent on the location of the source inside the cavity. Rather, I want to understand what the natural frequency of the space is. Is that even possible, or am I living in a fantasy here?

I want to start by solving this problem for the cube, so that I understand all the parameters involved, and solve for general cavities from there. My naive guess at the solution for cubes (or rectangular prisms) is to look at standing waves in the 3 dimensions (though I don't know which harmonic would be dominant). The problem with this approach is that it does not translate to arbitrary cavities very well.

• A downvote without an explanation doesn't help me make my question better. Dec 3, 2013 at 22:34
• Try to make your physics/conceptual questions clear - at the moment I see a mix of some physics with a lot of information about your Matlab implementation. My advice is remove the coding parts, and state your physics questions clearly. Dec 3, 2013 at 22:54
• While the physics parts of this question are indeed on-topic here (after suitable rephrasing along the lines suggested by innisfree), we'll not be able to help you very much with the numerical computation parts, which are a much better fit for Computational Science. Dec 3, 2013 at 22:58
• Thanks for the advice, I removed the Matlab stuff to focus my questions. I also appreciate the Computational Science link. Dec 4, 2013 at 0:20

How should I model this situation for a closed cavity? Just solve the wave equation with boundary conditions matching the geometry and material properties of the cavity.

What does resonance in a closed cavity even look like? (as opposed to a "whistling" kind of cavity) It is about the same in both cases - a standing wave, with energy leaking out through the walls (if they are imperfect and absorb energy) and through the holes in the cavity

What wave equation should I use? I'm assuming spherical for an arbitrary cavity. The wave equation cannot be spherical, cylindrical etc; the wave equation is just the wave equation. It can be written in various coordinates. But using spherical coordinates for an arbitrary shape cavity is not justified unless the cavity is close to being spherical.

I don't even know if using diff eq's (wave equation) is the best approach here. I am afraid there is no other way to solve this problem other than using the wave equation. Only using the wave equation we can have a clear statement of this problem; otherwise we don't really know what we are talking about.

Would it be possible, or even advisable, to try modeling noise in the cavity and see what kind of response I get? Not sure what you mean by this. Noise is a collection of harmonics with different frequencies and wavenumbers. You need to be able to solve for a simple single-frequency mode to model noise in the cavity. Modeling of noise in the cavity is certainly possible but it is advisable to solve first the simpler problem with single frequency.

I don't want my solution to be dependent on the location of the source inside the cavity. I am afraid the solution has to depend on the location of the source. We know that the sound response of a drum or a bell depends on where you hit it.

Rather, I want to understand what the natural frequency of the space is. Is that even possible, or am I living in a fantasy here? Not sure what you mean but probably this is something to do with solving for eigenmodes of the wave equation in this geometry; this can be done and is often a worthwhile thing to do.