A sealed resonant cavity is shaped like a capsule: a circular cylindrical tube of…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Partial differential equations and boundary-value problems · Partial differential equations and boundary-value problems
Conditions of Validity, Ch Separation of variables as a chain of Sturm-Liouville problems
Problem
A sealed resonant cavity is shaped like a capsule: a circular cylindrical tube of radius a closed at each end by a hemispherical cap of the same radius a, so the whole boundary is smooth, with the cylindrical wall meeting each cap along a circle. Using the level-set criterion, predict whether the three-dimensional Helmholtz equation can be solved on this cavity by separation of variables in any single orthogonal chart among Cartesian, cylindrical and spherical coordinates, and justify the prediction.