Normal modes of vibration of a plate vs a membrane

Question

I have been studying Chladni patterns but recently I have stumbled on some conceptual questions that I seem to not have an answer. At first I thought that the theory would be the same of a vibrating membrane, with very well defined solutions for given geometries like a square or a circular one like it is described in this document (found here) giving resonance at very well defined and discrete frequencies.. But then I saw the computational work done by kai5z in GitHub, where its script is able to generate a Chladni pattern for any frequency. At first I thought that it would have something to be with the fact that it considers a forced 2D oscillator but that doesn't make sense.

Further investigation showed me that Chladni patterns come from the eigenfunctions of a bi-harmonic operator and the vibration modes for a membrane come from the eigenfunctions of the Laplacian, but why? How are the problems different? What makes it imperative to have two theories for the two problems? Are the membrane solutions an approximation for the plate solutions? Are they somewhat related?

Answer 1

The difference depends on the physical structures: the tension of the membrane must be imposed by means of external forces whereas that of a plate naturally exists in its interior. In other words the relation between deformations and stress is different.

The small deformations of the structure, in the first case, are dynamically described in terms of a transversal D'Alembert equation and the spatial part of this equation of motion includes the spatial 2D Laplacian. In the second case, the model is more complicated and we have a similar equation of motion of the transversal deformations where however the Laplace operator is now replaced by (minus) its square. I am assuming that the membrane and the plate are uniform and isotropic, otherwise the equations become more complicated.

What the two structures share are the eigenfunctions of the relevant spatial operator: they only depends on the form of the structure (and on the boundary conditions). The eigenvalues are different (in fact the eigenvalues of $\Delta^2$ are the square of those of $-\Delta$). So the Chladni figures should be quite similar in some regime for equal shapes.