Theoretical and numerical results concerning the nonlinear interaction of an initially cold beam with a single wave (ω ≈ k∥ vb) in either magnetized or field- free plasma are presented and discussed. By using the energy and momentum conservation laws, the equations describing the development of the wave amplitude and phase, both in the temporal and spatial problem, are derived and then solved numerically together with the equations of motion of the beam particles. The fundamental role of an absorption (or transformation) of the wave in its interaction with a beam modulated at a frequency belonging to the upper branch of the electron plasma oscillation is recognized in two model situations.

A closed solution for the increment in the logarithmic derivative &dgr;(Q) over the inner (‘resistivity’) region for resistive tearing theory in cylindrical geometry including ‘parallel’ viscosity is compared with an earlier numerical solution. The singular point Qcrit about which δ(Q) exhibits branch-point behaviour in the presence of compressibility is identified; modification of the viscous theory, to include (for example) finite Larmor radius effects, is suggested. In the incompressible limit, a familiar (inviscid) stability criterion is recovered, in the presence of ‘parallel’ viscosity.

The kinetic equation obtained in parts 1 and 2 is treated stochastically: the external field is stochastic, with an average and a fluctuating part. The turbulence of the system is described by the induced fluctuations in the plasma, and a general equation is derived for the average distribution function. As a particular case, the stochastic Vlasov equation is treated explicitly, and compared with the descriptions of Dupree, Weinstock and Benford & Thomson.

In close analogy with part 1, the dynamics of stochastic correlations is used to reformulate the preceding treatment of turbulent equations. The result is an exact and closed equation for an average function, involving only asymptotic stochastic correlations, and no initial fluctuation. This stochastic kinetic equation is valid for all initial conditions, and the corresponding function is a significant particular projection on an independent subspace R(t).

It is shown that the multiple water-bag model, with both analytic and numerical properties, is fairly well suited to the study of forced oscillations launched by grid systems, for different geometries with possible non-uniform charge distribution, and for any distance within the range 0–100 Debye lengths.

Freeston (1968) and Freeston & Malik (1971) compared measurements precisely with theoretical values of the electric field near a system of grids and antennas immersed in a plasma. Results provided a valuable diagnostic tool in space research; and radio-frequency probing was used successfully (Beghin & Debrie 1972; Chasseriaux, Debrie & Renard 1972).

The relationship between energy absorption and the continuous frequency spectrum of the linearized equations of ideal MHD is investigated. We limit ourselves to incompressible fluid perturbations for which the continuum stems from resonant surfaces where the oscillation frequency equals the local frequency of an Alfvén wave. Details of this absorption process are illustrated by obtaining the response of the diffuse sheet pinch to an external current source. Expressions are derived that give the rate at which energy is transferred to the plasma, and show the spatial distribution of the absorbed energy. The results indicate that the absorption is enhanced if the plasma profile has regions of large spatial gradients. This enhanced absorption is a consequence of the presence of surface waves.

The paper studies the confluence of three shock waves meeting at a point, in an ideally conducting perfect gas, for transverse shocks and shocks in an aligned MHD field. The method used involves the hodograph mapping technique. A presentation of the shock polars in the total pressure P*, flow deflexion δ(δ, P*/P*0) plane is thus given. The algebraic complexity of the problem prohibits discussion of an alternative mathematical approach.

Limiting currents in annular unneutralized relativistic electron beams are calculated and compared with experiment. The limiting factor for short beams is found to be the space-charge depression of the potential due to the unneutralized beam; in longer beams self-magnetic and electric fields impose a power limit. Typical beam impedances are found to be of the order of 15 Ω with a 350 keV beam. The role of shear in the axial velocity of the beam is discussed.

Physically acceptable solutions of the time-independent Vlasov–Poisson system are found by applying a method which is analogous to the soliton method in Korteweg–de Vries theory. Cnoidal wave solutions for small-amplitude Langmuir and ion acoustic waves are derived, in which the nonlinearity is determined by trapped electrons. The class of weakly distorted waves is shown to be modulational unstable.

It is common practice to define reflexion and transmission coefficients for waves propagating through isotropic and anisotropic inhomogeneous plasma in terms of coefficients of waves existing within homogeneous regions of the plasma below and above the inhomogeneity. This concept is shown to be very restricted, and corresponding coefficients are defined for solutions existing within the inhomogeneous regions of the plasma. The extended concept is shown to satisfy conditions usually associated with the more common definitions, notably energy conservation and reciprocity. The theory is developed using integral equations and matrix theory, and special precautions are taken so that the results extend from the inhomogeneous plasma into free-space domains.

It is found that, for a fully-ionized, homogeneous plasma in a high-frequency electric field, the fundamental term 1 in the kinetic-energy correction is defined by the intensity and frequency of the electric field and by the collision frequency v1. We obtain the Debye–Hükel energy correction in 2, too, as did Theimer & Wright, but this term is much smaller than 1, and its coefficient is no longer a constant.

In this paper we consider the stability of one-dimensional stationary waves set up by two counter-streaming beams of electrons in a background of stationary ions. The perturbations considered are long-wave in a direction perpendicular to the wave. The presence of a uniform magnetic field in the direction of the wave and the effect of a perpendicular pressure are taken into account. In the long-wavelength limit growth rates are diminished by the nonlinear wave. When the amplitude of this wave tends to its maximum value, the growth rates tend to zero. Thus the wave has a stabilizing effect for long-wave perturbations. Three- dimensional effects lead to additional instabilities which are also quenched by the nonlinear wave, but not as fast as the one-dimensional calculation indicates.

Demehenko & Hussein (1973) discussed some properties of nonlinear magneto-sonic waves in a collisionless plasma. The relevant equation describing the space dependence x of the magnetic field may be written in the form d2y/dx2+f(y) = 0, (1) where f(y) is a nonlinear function of y only.