Complex-variable, hodograph methods are used to solve the free-boundary problem of a plasma confined by its own skin currents and the field due to two parallel external conductors. In one extreme case the external field is a quadrupole field in the vicinity of the plasma; another extreme case describes the local perturbation of a large plasma by a single, nearby conductor.

By Whitham's method of averaged Lagrangian developed by Dysthe, two coupled nonlinear Schrödiriger equations are obtained for the evolution of the amplitude of two plane polarized transverse waves in a cold plasma, both propagating in the same direction. Relativistic corrections are included and amplitude-dependent frequency shifts are derived. It is found that two transverse waves are modulationally unstable. The relativistic terms play an important part in this instability.

A second-order theory for electrostatic instabilities driven by the free energy in the relative drifts of plasma components in an infinite homogeneous, Vlasov plasma is constructed. General expressions for the wave–particle exchange rates of momentum, energy and heat flux are derived. Under the boundary condition of constant current, new terms arise which significantly modify the conventional quasi-liner results. The theory is applied to the B = 0 ion-acoustic heat flux instability, and exchange rates are obtained which are suitable for inclusion in electron fluid models of thermal transport.

This paper (Hirose 1978) appears to contain an error. The space averaged beam kinetic energy density is, to second order where n and V are the space averaged beam density and velocity, ñ and V are the oscillating quantities. The last term is significant, as will be shown, but it has been omitted by Hirose.

The points raised by Schmidt (1979) are irrelevant and do not invalidate my paper (Hirose 1978). He has overlooked the beam deceleration and consequent obvious relationship, where V (t) is the beam drift velocity at arbitrary time, V(0) is the initial beam velocity, nb1, is the perturbed beam density and vb1 is the perturbed beam velocity. It is obvious that V(t) < V(0) due to beam deceleration caused by the instability as shown by Shapiro (1963) and also in the subject paper (Hirose 1978).