Closed-form eigenfrequencies in prolate spheroidal conducting cavity

In this paper, an efficient approach is proposed to analyze the interior boundary-value problem in a spheroidal cavity with perfectly conducting wall. Since the vector wave equations are not fully separable in spheroidal coordinates, it becomes necessary to double-check validity of the vector wave functions employed in analysis of the vector boundary problems. In this paper, a closed-form solution has been obtained for the eigenfrequencies f_ns0 based on TE and TM cases. From a series of numerical solutions for these eigenfrequencies, it is observed that the f_ns0 varies with the parameter ξ among the spheroidal coordinates (η, ξ, φ) in the form of f_ns0(ξ) =f_ns(0)[1+g⁽¹⁾/ξ²+g⁽²⁾/ξ⁴+g⁽³⁾/ξ⁶+···]. By means of the least squares fitting technique, the values of the coefficients, g⁽¹⁾, g⁽²⁾, g⁽³⁾, ..., are determined numerically. It provides analytical results and fast computations of the eigenfrequencies, and the results are valid if ξ is large (e.g., ξ≥100).