If you are obtaining resonances, I find it a bit weird that you went for dispersion relations. Physically, it does not quite make sense, since there is no propagation along a direction. Mathematically, it depends strongly the boundary conditions, and perhaps you are wrong about the Dirichlet one. My first thought wold be to look at the density of states, i.e. $D(\omega)$ such that the number of resonances below $\Omega$ is $$N_<(\Omega)=\int_{-\infty}^\Omega D(\omega)d\omega$$ Is your fit good for large resonances, do you see a clear asymptote?

Thanks! I should have added "I'm not a physicist". To me it seemed to make sense to assume the resonances are standing waves, induced purely by the boundary conditions, and to ask what wave equation could lead to this. The boundary seems to go to zero amplitude for the state variable we're measuring, but who knows what's actually happening. (Sorry for not being concrete, I don't want to disclose unpublished results from my experimental collaborators).

I also deleted the bit where I can only see something that looks like 1s, 2p, 3d, and 2s modes in that order. The experiment didn't/couldn't probe frequencies beyond this and/or there is a cutoff where travelling waves decay too quickly for any higher-frequency modes, and therefore don't excite the system very much and can't propagate far enough to interfere with themselves to make a standing wave. Which is to say: I wouldn't dare try anything asymptotic here :P

I guess I can add that we know that the system is classical (nothing exotic!), and the observed phenomena are almost surely mediated by coupling that is not global (and mediated by interactions that would support traveling waves, if not for the boundary). The coupling might not be strictly local. Maybe instead of a polynomial in the Laplacian, we have integral coupling in the form of a convoltion kernel. I had sort of hoped this might not matter, since a polynomial in the Laplacian is just a low-order approximation of any nice coupling kernel via a Taylor expansion at zero frequency.

wait, so you are fitting the dispersion relation on how many resonances?

I mean, yes, five. It's embarrassing. But, hear me out! If you saw five resonance peaks that perfectly matched the d'Alembert wave equation, you'd probably say "okay, whatever is actually happening here, probably a linear wave equation captures the coarse phenomenology, and all models should be consistent with this". So, it's not that crazy. I suppose the problem creeps in when that doesn't work. There aren't really enough data to constrain anything more sophisticated.

and when you say 1s,2p,3d, since you are in 2D, I guess that the first number is the number of the Bessel function zero, but you don't have orbital number, only the "magnetic" one. Ok, but if you only have five points, the fit better be perfect. Perhaps including a graphe of the fit would help

I'm not really looking for a "true" physics model, but the sort of "next leading order" flavour of the "does it fit a d'Alembert wave equation?" reasoning. The idea is not to get the actual model, but to get some of intuition that constrains the type of relationships you can expect in any model, if all you can do is measure the first 5 frequencies. My argument for "if you saw $a=1$, you'd probably be convinced that it has to be a model that looks like linear, classical waves, to first order, at low frequencies" makes me feel like we're not completely lost here. Just, very data-poor.

I'm using the terms in the general sense that one sometimes sees when discussing 2D circular membranes. So, 1s has no angular nodes, and one radial node at the putative boundary. 2s has one radial node, no angular nodes. 2p has one angular node and single radial node at the boundary. 3d has two angular nodes and one radial node at the boundary.

ok in chat.

It's slightly unclear to me whether these chat rooms are public/archived? If they are private I could probably say a bit more. But generally in my field it's considered poor form the collaborating data scientists spills the experimental details before publication.

Yes I understand what you are trying to do, and I have no problem with the objective. It's just the methodology that I am investigating, since you don't leave equations

I'll say perhaps slightly more than I am permitted to, and no more than that ( :

I think they ar public

it's just to avoid cluttering the comment section

Alright; well, my hope is to sort of whittle down candidate models that are nonlinear, but somewhat nice. I hope a good model will have homogeneous and isotropic local coupling. While it may be nonlinear, its "nonlinearity" should probably be limited to a since monotonic, smooth, pointwise nonlinearity. Think: the sigmoid from neural networks. If I have a decent measurement of some of the lower frequencies, I feel like this is enough to at least start excluding parts of the model search space.

Seeing the $a=3$ fit to the dispersion sort of ruined my hoped of phrasing this as "something vaguely linear that we all know and love". I might have to go back to "exhaustively search the parameter space for the putatative nonlinear model class" which... I can do, but anything to narrow this down will make my day better ( :

Well, you can look at the literature for the triharmonic equation, but I don't know about physical applications on the top of my head. Perhaps the physically relevant quantity is not your original field $u$, but rather its derivatives?

My mind is slightly blown by this insight; This conversation has been helpful for this alone, if nothing else! I'd considered the possibility that I'm measuring some awkward projection of a 2-4 dimensional state space, but hadn't considered asked which, if any, of these states could have the interpretation of a derivative.

Also, I did not know the term "triharmonic equation"

If you'll phrase this as an answer I'll accept it. "Anything that reduces to a triharmonic equation at low frequencies" sounds like the sort of answer I was looking for. Although it would be lovely to have a concrete example of a physical system that obeys this, for entertainment purposes.

researchgate.net/publication/…

Or maybe I should change the title to "what are some intuitive examples of physics that reduces to a triharmonic equation"

(done), changed it to "Concerete examples of physical systems that can be (approximately) modelled using a 2D triharmonic equation?"