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The problems involved in the design of wide-tuning-range resonators suitable for use with reflex-klystron oscillators have been discussed elsewhere. When these resonators are tuned with noncontacting short circuits, parasitic resonances, caused by waves propagating circumferentially around the small gaps between the plunger and the inner and outer conductors, may occur. In this paper it is shown that for a typical coaxial-line resonator incorporating noncontacting S-type plungers: (1) These resonances may occur in the outer gap when the wavelength is somewhat less than a submultiple of the mean circumference of this gap. (2) Parasitic resonances occurring in the inner gap will be affected by the resonator tuning since these fields can propagate in the resonator. For the usual 3/4-wave resonator, two resonances may be expected; one at a wavelength somewhat greater than and the other at a wavelength somewhat less than the mean circumference of the inner gap. (3) Plunger eccentricity produces a strong coupling between the principal resonator mode and the "one-cycle" circumferential resonances, but only a slight coupling to the "multiple-cycle" circumferential resonances. Hence, all one-cycle resonances must be made to occur at a wavelength outside of the tuning range either by suitable choice of resonator dimensions or by a slotting technique to be described in another paper. (4) Plunger eccentricity splits the circumferential resonances into "direct-axis" and "quadrature-axis" types which have differing resonant wavelengths.