5. D. R. Rhodes, IEEE Trans., AP-19, No. 4, 485 (1971). 6. R. S. Hansen, TIIER, 69, No. 2, 35 (1981). 7. Ya. N. Fel'd, Radiotekh. Elektron., 27, No. ii, 2094 (1982). 8. I.M. Polishchuk, Radiotekh. Elektron., 27, No. 5, 885 (1982). 9. V. A. Pavlyuk, T. A. Sigova, M. A. Martynov, and V. P. Kismereshkin, Radiotekh. Elektron., 29, No. 5, 990 (1984). i0. L. D. Bakhrakh and S. D. Kremenetskii, Synthesis of Radiating Systems [in Russian], Soy. Radio, Moscow (1974). Ii. E. I. Krupitskii and T. N. Sergeenko, Radiotekh. Elektron., 15, No. 2, 252 (1970). 12. A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Ill-Posed Problems [in Russian], Nauka, Moscow (1979). HYBRID MODES OF OPEN WAVEGUIDE CAVITIES (NUMERICAL AND ANALYTICAL INVESTIGATION) I. E. Pochanina, V. P. Shestopalov, and N. P. Yashina UDC 621.372.831 The analytical and physical nature of the mode interaction phenomenon is investi- gated for a cavity open at both ends and bounded by different coaxial discon- tinuities of a circular waveguide. It is established that the generation of hy- brid modes of the cavity corresponds to the fact that the operator function of the problem has an isolated Morse critical point. An approximation of the spectral curves is formulated in the neighborhood of the Morse critical point. The data are compared with the results of a rigorous solution of the spectral problem. Hybrid modes of closed and open electrodynamic structures have been investigated pre- viously [1-6]. However, the analysis of this interesting physical phenomenon is reduced, for the most part, to the verification of its existence; attempts to explain the origin of mode interaction are of a specialized nature and do not have any rigorous mathematical founda- tion. Melezhik et al. [7] have successfully investigated the analytical nature of hybrid modes by combining the machinery of the theory of functions of several complex variables and catastrophe theory. This approach enables one, without finding the complete set of values of the spectral parameter of the problem, to determine the dimensions of the electrodynamic system and the frequencies around which mode interaction can be observed, to d~scribe its canonical forms, and to plot the corresponding coupling diagrams [7]. The investigated phenomenon refers to the existence of a set of values of nonspectral geometrical parameters of the structure such that a small variation of these parameters in- duces abrupt variations in the values of the eigenfrequencies, a significant increase or de- crease in the diffraction losses, a variation of the field configurations, etc. Unstable equilibrium is one analog of such a state of the system. Catastrophe theory [8] is concerned with the variation of an equilibrium state as a function of the values of control (nonspec- tral) parameters of the structure. According to this theory, an unstable equilibrium state of a system is described mathematically by the presence of an isolated Morse critical point of the function describing the state of the system. This function can be represented locally by a quadratic form in a neighborhood of the Morse point [8]. Such a representation well approximates the characteristic behavior of the spectral curves for the interacting oscilla- tory modes. The results obtained in [7] offer new possibilities for investigating the spec- tral properties of various electrodynamic objects, and they can be used to predict the onset of mode interaction and to approximate the behavior of the spectral curves. It would cer- tainly be interesting to apply the approach described in [7] to the solution of specific phys- ical problems, to further investigate the analytical nature of the mode interaction phenomenon Institute of Radiophysics and Electronics, Academy of Sciences of the Ukrainian SSR, Kharkov. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 32, No. 8, pp. 1000-1008, August, 1989. Original article submitted December 15, 1987. 744 0033-8443/89/3208-0744 $12.50 9 1990 Plenum Publishing Corporation