Perturbation of Boundary Conditions

Abstract

The perturbation method of solving boundary value problems with irregular boundary conditions has been re-examined and generalized. Problems are considered for which the boundary of the region is changed and the boundary conditions are unaltered, and also the inverse case where the boundary is fixed and the boundary conditions are changed. It is pointed out that the integrations involved in obtaining the perturbed eigenvalues will be in general very difficult, but that this difficulty can be avoided in two important cases. The perturbed eigenfunctions, however, can always be simply obtained. An application of the method was made to a problem in room acoustics which requires the calculation of second-order terms in the perturbation, and the results obtained for the pressure distribution and frequency were found to check well with experiment. A simple formula is found for the problem of the absorbing wall. The method is useful in calculating electromagnetic resonators of the "rhumbatron" type, and may also be useful in calculating electronic wave functions for metals. In Appendix III the method is applied to the scattering problem, and the results are related to Huygens' principle.