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):  using the small angle approx, the component of the membrane movement +  some multiple of the membrane movement (alpha)? This seems kind of weak.
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?