A thin plate occupies a circular sector, , (a "pizza slice" of opening angle )
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Special functions and orthogonal polynomials
Conditions of Validity, Ch Separation of variables as a chain of Sturm-Liouville problems
Problem
A thin plate occupies a circular sector, 0≤r≤a, 0≤ϕ≤α (a "pizza slice" of opening angle α), and is clamped along its entire boundary: the arc r=a and both straight edges ϕ=0, ϕ=α.
((a)) Using the level-set criterion, predict whether the two-dimensional Helmholtz equation can be solved on this region by separation of variables in plane polar coordinates.
((b)) If it can, state the ordinary differential equation and boundary conditions satisfied by Φ(ϕ), and say what boundary condition R(r) carries at r=a and at r=0.