Consider a cold, uniform, and unmagnetized plasma, where ions are stationary and the electrons have velocity v_0, that is, the reference frame is moving with the ion stream. Let the electrostatic waves be of the form: \mathbf{E}_1 = \xi_1 \exp[i(kx - \omega t)] \mathbf{\hat{x}} Applying linearization techniques to the equation of motions for both species, to the equation of continuity, and
Poisson's equation, and introducing the spatial and temporal harmonic operators \partial_t \rightarrow -i\omega, \nabla \rightarrow ik we can get the following expression: 1 = \omega_{pe}^2 \left[\frac{m_e/m_i}{\omega^2} + \frac{1}{(\omega - kv_0)^2} \right], which represents the dispersion relation for longitudinal waves, and represents a quartic equation in \omega. The roots can be expressed in the form: \omega_j = \omega_j^R + i\gamma_j If the imaginary part (\operatorname{Im}(\omega_j)) is zero, then the solutions represent all the possible modes, and there is no temporal wave growth or damping at all: \mathbf{E} = \xi \exp[i(kx - \omega t)] \mathbf{\hat{x}} If \operatorname{Im}(\omega_j) \ne 0, that is, any of the roots are complex, they will occur in
complex conjugate pairs. Substituting in the expression for electrostatic waves leads to: \mathbf{E} = \xi \exp\left[i(kx - \omega_j^R t) + \gamma t\right] \mathbf{\hat{x}} Because of the second
exponential function at the right, the temporal dynamics of the wave amplitude depends strongly on the parameter \gamma; if \gamma , then the waves will be exponentially damped; on the other hand, if \gamma > 0, then the waves are unstable and will grow at an exponential rate. In both the hot-beam and cold-beam case, the instability grows until the beam particles are trapped in the
electric field of the wave. This is when the instability is said to
saturate. == Bibliography ==