A Barkhausen tube is not really a single component oscillator. It is an active device that has gain and a self resonance that can be physically tuned to a centre frequency in a specific range. By loading it with a tuned network it can be stabilised and power extracted at the wanted frequency.
Since the bandwidth of the loading at microwave frequencies is usually insufficient to pass the second harmonic, only the fundamental is available so it will have a sinewave output.
The relative voltages on the electrodes decide the gain and whether it will oscillate or not. Operating voltages also decide the DC current and the output power available.
If you are into historical microwave oscillators then I would recommend you get a copy of; Volume 7. KLYSTRONS AND MICROWAVE TRIODES, in “The MIT Radiation Laboratory Series”. Vol 7 is available as a 24 Megabyte pdf download from Jefferson Lab. https://www.jlab.org/ir/MITSeries/V7.PDF
Adjust V7 in the above link for volumes V3 through to V28, the series index.
Massachusetts Institute Of Technology, Radiation Laboratory Series.
1. RADAR SYSTEM ENGINEERING — Ridenour (Use https://www.jlab.org/ir/MITSeries/V1-1.pdf )
2. RADAR AIDS TO NAVIGATION — Hall (Use https://www.jlab.org/ir/MITSeries/V2.pdf )
3. RADAR BEACONS — Roberts
4. LORAN — Pierce, McKenzie and Woodward
5. PULSE GENERATORS — Glasoe and Lebacqz
6. MICROWAVE MAGNETRONS — Collins
7. KLYSTRONS AND MICROWAVE TRIODES — Hamilton, Knipp and Kuper
8. PRINCIPLES OF MICROWAVE CIRCUITS — Montgomery, Dicke and Purcell
9. MICROWAVE TRANSMISSION CIRCUITS — Ragan
10. WAVEGUIDE HANDBOOK — Marcuvitz
11. TECHNIQUE OF MICROWAVE MEASUREMENTS — Montgomery
12. MICROWAVE ANTENNA THEORY AND DESIGN — Silver
13. PROPAGATION OF SHORT RADIO WAVES — Kerr
14. MICROWAVE DUPLEXERS — Smullin and Montgomery
15. CRYSTAL RECTIFIERS — Torrey and Whitmer
16. MICROWAVE MIXERS — Pound
17. COMPONENTS HANDBOOK — Blackburn
18. VACUUM TUBE AMPLIFIERS — Valley and Wallman
19. WAVEFORMS — Chance, Hughes, MacNichol, Sayre and Williams
20. ELECTRONIC TIME MEASUREMENTS — Chance, Hulsizer, MacNichol and Williams
21. ELECTRONIC INSTRUMENTS — Greenwood, Holdam and MacRae
22. CATHODE RAY TUBE DISPLAYS — Soller, Starr and Valley
23. MICROWAVE RECEIVERS — Van Voorhis
24. THRESHOLD SIGNALS — Lawson and Uhlenbeck
25. THEORY OF SERVOMECHANISMS — James, Nichols and Phillips
26. RADAR SCANNERS AND RADOMES — Cady, Karelitz and Tutner
27. COMPUTING MECHANISMS AND LINKAGES — Svoboda
28. SERIES INDEX — Linford
You can usually find a printed copy by searching
http://www.bookfinder.com with the title and first author's name.
The Information Resources of Jefferson Lab is often worth a visit.
https://www.jlab.org/ir/