Propagation of wave pulses in the regimes of single-particle and collective cherenkov effects during convective and abso
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MA OSCILLATIONS AND WAVES
Propagation of Wave Pulses in the Regimes of Single-Particle and Collective Cherenkov Effects during Convective and Absolute Instabilities M. V. Kuzelev Moscow State University, Vorob’evy gory, Moscow, 119899 Russia Received June 1, 2005
Abstract—Asymptotic solutions to the problem of the time evolution of delta-shaped wave pulses excited during resonant instabilities of electron beams in slowing-down electromagnetic media are found and investigated analytically. Convective and absolute instabilities developing under the conditions of collective and single-particle Cherenkov effects are considered. The results obtained apply to an arbitrary linear nonequilibrium dispersive medium that can be described by a set of first- or second-order differential transport equations. PACS numbers: 52.40.–w DOI: 10.1134/S1063780X06070051
1. The mathematical solution of the problem of the propagation of an arbitrarily shaped perturbation in a linear dispersive medium characterized by the eigenfrequencies ωm(k) is reduced to calculating the integrals of the form [1] J m ( t, z ) =
∫
p m ( k ) exp ( –iω m ( k )t + ikz ) dk.
(1)
Here, k is the wavenumber; t is the time; z is the spatial coordinate; m = 1, 2, …, N, with N being the number of branches of the dispersion curves of natural waves; and the functions pm(k) are determined by the structure of the initial perturbation in the medium. In the general case, the integration region in expressions (1) is the entire real axis, k ∈ (–∞, +∞). Integrals (1) are difficult to calculate because of the complex nature of the functions ωm(k) = ω 'm (k) + iω ''m (k) and their nonlinearity (dnωm/dkn ≠ 0; n ≥ 2).
frequencies ωm(k), asymptotic estimates are obtained mainly by the saddle point method [3]. For real frequencies ωm(k), the stationary-phase method [4] is used. In the present paper, a study is made of the propagation of wave pulses in a system that is important for applications and is characterized by a complicated dispersion law—specifically, a nonequilibrium system in which an electron beam undergoes a resonant Cherenkov interaction with electromagnetic waves in a certain slowing-down medium (such as plasma, dielectric, or a periodic material structure). In terms of the theory of electron beam instabilities, we consider two problems described by the following sets of linear equations with differential transport operators [5]: ∂ 2 ∂ ⎛ ---- + U -----⎞ A b – ia A w = f ( t, z ), ⎝ ∂t ⎠ ∂z
The integrals in question can be calculated approximately by two methods. The first method is based on the quasi-harmonic approximation, which is restricted to the expansion [2] dω m ( k 0 ) -. ω m ( k ) = ω m ( k 0 ) + ( k – k 0 ) ------------------dk
∂ ∂ ⎛ ---- + V -----⎞ A w + i A b = 0, ⎝ ∂t ∂z⎠ ∂ 2 3 ∂ ⎛ ---- + U -----⎞ A b + b A w = f ( t, z ), ⎝ ∂t ∂z⎠
(2)
Here, k0 is the point where the function pm(k) (or the imaginary part of the eigenfrequency, ω m'' (k)) has a maximum. This approximation is valid only for sufficiently short time scales t. For an infinite inte
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