Burton D. Fried

LONGITUDINAL ION OSCILLATIONS IN A HOT PLASMA

Linearized, longitudinal waves in a hot plasma include, besides the familiar electron plasma oscillations, in which the frequency ω is of order ωp = (4πne2/m)½, also ion plasma oscillations with ω ≈ ωp(m/M)½. The properties of the latter are explored using a Vlasov equation description of the plasma. For equal ion and electron temperatures, Te = Ti, there exists a discrete sequence of ion oscillations, but all are strongly damped, i.e., have ‐Im ω/Re ω ⪞ 0.5, and hence are not likely to be observable. The ratio Im ω/Re ω can be made to approach zero (facilitating detection of the waves) by either increasing Te/Ti or by producing a current flow in the plasma. In the latter case, Im ω can even be made positive (corresponding to growing waves), the current required for this being smaller the larger the value of Te/Ti. This growing wave is just the familiar two‐stream instability which is thus seen to be an unstable ion oscillation. It is also noteworthy that the ion oscillations, which for small k have the p...

Mechanism for Instability of Transverse Plasma Waves

Determination is made of the physical mechanism responsible for growing transverse waves resulting from plasma instabilities associated with anisotropic velocity distributions.

Stability Limits for Longitudinal Waves In Ion Beam‐Plasma Interaction

The two‐stream instability is examined for the case of an ion beam traversing a plasma. The dispersion equation for linearized, longitudinal waves in a plasma where collisions are negligible is used to find the restrictions on beam velocity, temperature, and density which will lead to growing waves.

Theory of double resonance parametric excitation in plasmas

Parametric instabilities in a plasma driven by a long wavelength electric field with two “pump” frequencies ω1 and ω2 which lie near the resonant frequency for Langmuir oscillations, their difference Δ = ω1 − ω2 being chosen close to a low frequency resonance, linear or nonlinear, at Ω − i Γ are studied. A general dispersion relation in terms of linear susceptibilities, χ, is derived by retaining, on a selective basis, terms of fourth order in the pump amplitudes. Illustrative calculations are carried out using resonant fluid approximations for the χ. The most interesting cases occur when Δ = Ω or Δ = 2Ω. A lowering of the net power threshold for instability is found in both cases, when the linear damping rate of the electronic wave is large compared with Ω. In addition, a coupling between the “decay” and “oscillating two‐stream” instabilities occurs when Δ = Ω, the threshold for exciting the latter with the ω2 pump being arbitrarily small when the ω1 pump amplitude is near the usual decay instability thr...

Optimum Rocket Trajectories

Some general considerations regarding maximum range missile trajectories are presented. It is shown that under certain simplifying assumptions maximum range is achieved if the thrust attitude angle is held constant during the entire powered flight. This result is exact for a constant gravitational force field if aerodynamic forces and altitude dependence of rocket thrust are neglected. For motion in the earths inverse square, central force field, constant thrust attitude is a good approximation to the optimum program, provided the distance covered during powered flight is small compared to the radius of the earth. Equations for the value of the optimum attitude angle are presented for both constant and Kepler force fields. The results provide a convenient means of estimating an upper limit for the range which could be achieved with a given missile.

Theory of Electron Driven Shock Waves

Previous calculations of the time required for ion heating in the discharge or driver section of an electrical shock tube yielded values much greater than that in which formation and acceleration of the first luminous front is observed to occur. The model of an electron driven shock presented here shows that the relation between shock velocity V and electron temperature Te, which has been established experimentally over a wide range of parameters, remains valid even though the conventional picture of a shock driven by hot ions must be abandoned. Thermal expansion of the hot electron gas accelerates the cold ions, resulting in a shock front or moving electrostatic double layer. Assuming conditions behind the shock to be coupled to those in the discharge region through a simple rarefaction wave, it is found that MV2/kTe is a universal function of W/MV2, where W is the effective ionization potential. This is shown to be in excellent agreement with a wide variety of experimental data.

PLASMA OSCILLATIONS IN AN EXTERNAL ELECTRIC FIELD

The fluctuation longitudinal electric field, Ek, is studied for a two component plasma immersed in a uniform, external electric field E0. Since the time dependence of the space‐averaged part of the distribution functions precludes the usual Laplace transform approach, computational techniques are developed to solve the integral equation for Ek directly in the time domain. Solutions are given for the case where E0 is large compared to the critical field for runaway and compared with the familiar constant‐drift solutions. Growth in the vicinity of resonance (vdrift ≈ ωp/k) is observed, followed by Landau damping at longer times.

Elementary Dynamics of the Nonadiabatic Transverse Pinch

The characteristics of the nonadiabatic transverse (Eθ,Bz) pinch are studied for configurations in which the discharge tube, current coils, and condenser form an integral structure, allowing very high voltage gradients (105 to 106 v/cm). The dynamics of the first pinch contraction are analyzed using the Garwin‐Rosenbluth model of an infinitely conducting, collisionless plasma, and expressions are derived for the parameter values corresponding to maximum energy transfer to the gas for given energy per particle. Theoretical curves of pinch radius, current, and plasma kinetic energy as functions of time for typical cases are shown, and the relative merits of this transverse pinch and the more common longitudinal (Ez,Bθ) pinch are discussed.