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 Quote by Coto Not sure, but if he is deriving a boundary condition it is possible that the physics of the situation plays a role in determining how he derived it. For this reason it would help if we knew which paper you were talking about, or could provide more insight into the problem.
Sure thing. We're dealing with a membrane defined by a domain D and boundary B. No other constraints are made until this point. On part of the boundary the membrane may be free and somewhere else it could be forced at this external displacement (f).

Could it be that the author is saying that the boundary motion (f) is equal to 2 terms (in general): [1] using the small angle approx, the component of the membrane movement + [2] some multiple of the membrane movement (alpha)? This seems kind of weak.

 Quote by Coto In order to expand the function u(x), it makes sense to use some orthonormal functions, which the Sturm-Liouville equation ensures. Substituting in the expansion into the original PDE for u(x) and exploiting the orthogonality of the eigenfunctions, one finds a set of simpler PDEs that you can hope to solve and obtain solutions for the $$\phi_m$$'s.
I'm not so sure what you mean. Where are these Sturm-Liouville equations taken from and how are they relevant to the problem? Perhaps a simpler example (somewhere online) would help explain this?