Abstract
When we investigate the stability of a bounded distributed system or resonator, the problem is made more difficult than the corresponding lumped system by the fact that the spectrum of the complex fundamental frequencies is countable. By going through all the possible spatial disturbances, i.e., all the wave numbers k n acceptable given the boundary conditions, we can by determining all the roots of the characteristic equation D(ω, k n ) = 0, completely determine the stability. Obviously there will be difficulties, but these are technical.
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References
K. Rohlfs, Lectures on Density Wave Theory (Berlin, 1977) [Russian translation], Mir, Moscow (1980).
M.V. Vol’kenstein, General Physics [in Russian], Nauka, Moscow (1978), Ch. 8, Sec.8.4.
A.M. Zhabotinskii, Concentration Oscillations [in Russian], Nauka, Moscow (1974), pp.164–167.
H. Haken, Synergetics [Russian translation], Mir, Moscow (1980), pp.342–358.
G. Nicolis and I. Prigogine, Self-Organisation in Nonequilibrium Systems [Russian translation], Mir, Moscow (1979).
V.N. Shevchik, G.N. Shvedov, and A.V. Soboleva, Wave and Oscillation Effects in Electron Beams at High Frequencies [in Russian], Izd. Saratovskogo Univ., Saratov (1962), Ch.1.
V.N. Shevchik and D.I. Trubetskov, Analyticall Calculation Methods in Microwave Electronics [in Russian], Sovetskii Radio, Moscow (1970).
J.R. Pierce, Travelling-Wave Tube [Russian translation], Sovetskii Radio, Moscow (1952).
J.H. Piddington, “Growing electromagnetic waves. Growing electric waves of space charge and a twin-beam tube” in: Topics on Radio Location [Russian translation], Izd. Ino. Lit., Moscow (1956), No.6(36), pp.53–66. (from Phys. Rev., 101, No.1, 9, Jan.1, 1956).
V.N. Shevchik and D. I. Trubetskov, Electron Tube with Reverse Wave [in Russian], Izd. Saratovskogo Univ., Saratov (1975).
Yu.G. Al’tshuler and A.S. Taratenko, Low-Power Tube with Reverse Wave [in Russian], Sovetskii Radio, Moscow (1963), Table 1.1, p. 35.
V.Ya. Arsenin, Methods of Mathematical Physics and Special Functions [in Russian], Nauka, Moscow (1974).
R. Courant, Partial Differential Equations; R. Courant, D. Hilbert, Methods of Mathematical Physics [Vol.2. [Russian translation], Mir, Moscow (1964), Ch.5, Sec 1.
V.M. Lopukhin, Exciting Electromagnetic Oscillations and Waves by Electron Beams [in Russian], Gostekhizdat, Moscow (1953), Ch.8.
P.A. Sturrock, “Kinematics of growing waves” in: Microwave Oscillations in Plasmas [Russian translation], Izd. Ino. Lit., Moscow (1961), pp.72–112.
E.M. Lifshits and L.P. Pitaevskii, Physical Kinetics [in Russian], Nauka, Moscow (1979), Vol.10, Ch. 6.
R.J. Briggs, Electron Stream Interaction with Plasmas, MIT Press, Cambridge, Mass. (1964).
R.J. Briggs, “Twin-Beam Instability” in Advances in Plasma Physics [Russian translation], Mir, Moscow (1974), pp.132–171.
L.D. Landau and E.M. Lifshits, Mechanics of Continuous Media [in Russian], Gostekhizdat, Moscow (1954), Sec.30.
A.M. Fedorchenko and N.Ya. Kotsarenko, Absolute and Convective Instabilities in Plasmas and Solids [in Russian], Nauka, Moscow (1981).
A.I. Akhiezer and R.V. Polovin, “Criteria of Wave Growth,” UFN, 104, 185 (1971).
Yu.K. Pozhela, Plasma and Current Instabilities in Semiconductors [in Russian], Nauka, Moscow (1977), Ch.3.
A.S. Monin and A.M. Yaglom, Statistical Fluid Mechanics [in Russian], Nauka, Moscow (1967).
L.A. Vainshtein, “Pulse propagation,” UFN, 118, 338–367 (1971).
J.R. Pierce, “An interesting wave amplifier,” IRE Trans., ED-7, 73–74 (1960).
R.L. Kyhl and H.R. Webster, “Breakup of hollow cylindrical electron beams,” IRE Trans., ED-3, 172–183 (1956).
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© 1989 Kluwer Academic Publishers
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Rabinovich, M.I., Trubetskov, D.I. (1989). Stability of Distributed Systems with Continuous Spectra. In: Oscillations and Waves. Mathematics and Its Applications (Soviet Series), vol 50. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-1033-1_7
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DOI: https://doi.org/10.1007/978-94-009-1033-1_7
Publisher Name: Springer, Dordrecht
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