SummaryA functionφ satisfies the equation
$$\phi _{xx} + \phi _{yy} + k^2 \phi = 0$$
in a regionR bounded by a closed curveC on which mixed boundary conditions are specified, namelyφ=0 on a partA of the boundary andδφ/δn=0 on a partB, whereC=A+B. It is required to find the values ofk for which equation (1) possesses solutions satisfying the mixed boundary conditions. Two variational principles are given for these eigenvalues.An example is worked out in detail for a case where the shape of the region is such that the variational expressions can be determined by separation of variables. The variational principles give upper and lower bounds for the lowest eigenvalue and these are close together in the case which is considered numerically. The corresponding eigenfunctions are also in close agreement.
[1]
W. W. Hansen,et al.
Disk‐Loaded Wave Guides
,
1949
.
[2]
E. Chu.
Upper and Lower Bounds of Eigenvalues for Composite‐Type Regions
,
1950
.
[3]
W. W. Hansen.
On the Resonant Frequency of Closed Concentric Lines
,
1939
.
[4]
A. Erdélyi,et al.
Higher Transcendental Functions
,
1954
.
[5]
F. Borgnis,et al.
Randwertprobleme der Mikrowellenphysik
,
1955
.
[6]
W. C. Hahn,et al.
A New Method for the Calculation of Cavity Resonators
,
1941
.
[7]
P. Morse,et al.
Methods of theoretical physics
,
1955
.
[8]
F. B. Hildebrand,et al.
Introduction To Numerical Analysis
,
1957
.