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1.The nonlinear interaction of a small monoenergetic beam and a plasma has also been considered by V. D. Shapiro, Zh. Eksp. Teor. Fiz. 44, 613 (1963)
1.[V. D. Shapiro, Sov. Phys. JETP 17, 416 (1963)]. However, he used quasilinear theory and thereby missed many of the effects we will discuss.
2.W. E. Drummond and J. H. Malmberg, Bull. Am. Phys. Soc. 13, 1542 (1968);
2.W. E. Drummond, J. H. Malmberg, T. M. O’Neil, and J. R. Thompson, Phys. Fluids 13, 2422 (1970).
3.The single wave model has been partially verified in beam plasma experiments and computer simulations: J. R. Apel, Phys. Fluids 12, 291 (1969);
3.R. L. Morse and C. W. Nielson, Phys. Fluids 12, 2418 (1969); , Phys. Fluids
3.and T. P. Armstrong and D. Montgomery, Phys. Fluids 12, 2094 (1969)., Phys. Fluids
4.A recent example is the computer simulation by R. L. Morse and C. W. Nielson, Phys. Fluids 12, 2418 (1969).
5.R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969), p. 37;
5.and J. H. Malmberg and C. B. Wharton, Phys. Rev. Letters 19, 775 (1967).
6.J. H. Malmberg and C. B. Wharton, Phys. Fluids 12, 2600 (1969).
7.The coefficients are time dependent but the equation is linear.
8.L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison‐Wesley, Reading, Mass. 1966), p. 102;
8.R. Z. Sagdeev and A. A. Galeev, Nonlinear Plasma Theory (Benjamin, New York, 1969), p. 24.
9.W. M. Manheimer, Bull. Am. Phys. Soc. 14, 1041 (1969).
10.J. R. Apel, Phys. Rev. Letters 19, 744 (1967);
10.J. R. Apel, Phys. Fluids 12, 640 (1969).
11.J. Dawson, W. Kruer and R. Sudan, Bull. Am. Phys. Soc. 13, 1504 (1968);
11.M. V. Goldman and H. L. Berk, Bull. Am. Phys. Soc. 14, 1034 (1968)., Bull. Am. Phys. Soc.
12.C. B. Wharton, J. H. Malmberg, and T. M. O’Neil, Phys. Fluids 11, 1761 (1968).
13.This description is similar to the theory of A. Nordsieck, Proc. IRE 41, 630 (1953). However, for the traveling wave amplifier, the background dielectric represents a rigid circuit structure rather than a plasma, and the single wave grows in space rather than time. Also, the equations for the traveling wave amplifier were only integrated to the beginning of nonlinear saturation, which is the natural interest in an amplifier; whereas, our turbulence interests really only begin at the point of nonlinear saturation.
14.T. M. O’Neil and J. H. Malmberg, Phys. Fluids 11, 1754 (1968).
15.Actually, at the point where cannot be zero. It has a small imaginary part where ῡ is the plasma thermal velocity. Nevertheless, we can neglect this term in the following Taylor expansion provided it is much smaller than
16.In other words, we compare the autocorrelation time to the nonlinear saturation time.
17.Direct integration schemes may introduce systematic errors in the acceleration, because they do not account for changes in the orbit during a time step, whereas the predictor‐corrector scheme produces only random walk errors.
18.Recall that we are working in a coordinate system moving with the initial beam velocity.
19.During an e‐folding, the beam electrons move relative to the wave by about one‐tenth of a wavelength.

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