In Fig. 4, the contour plots of the space-time dependences of
electron concentration perturbations corresponding to the
\(x\)-branch (in this case, the dipole is oriented along the \(OX\)
axis) in the absence of electron drift (\(V_{0}=0\)) are exhibited. In
particular, in Fig. 4,\(a\), the spatial distribution of the concentration
at the time moment \({\tau_{x}={3\pi}/{2}}\) is shown. It has a
minimum and a maximum to the left and to the right, respectively,
from the coordinate origin, where the dipole is located. This
distribution arose owing to the corresponding arrangement of dipole
charges at that moment. The time-space illustration (Fig. 4,\(b\))
demonstrates that those maxima and minima alternate in time with a
period of \(\pi\). The spatial scale of perturbations has an order of
several \(h\).
The contour plots of the space-time dependences of the electron
concentration in the case where electrons drift along the \(OX\) axis
(\({V_{0}\approx 0.94}\)), are shown in Fig. 5. In the general case, if electrons
drift, the dipole becomes oriented in the plane \(y=0\) (see the next section).
One can see that the concentration minima and maxima, which are exhibited in
Fig. 4, are “blown” by the electron stream,
with the perturbation boundaries moving faster than its center. The
nonzero phase velocity of plasmon waves, which is
perpendicular to the electron drift, results in a smearing of perturbations
located far from the dipole.
\begin{equation}
\par
{n^{\prime\>(z)}(\mathbf{r},\tau_{z})=}\par
\nonumber \\
\end{equation}
A similar analysis of the time-space distributions of electrons can be
carried out for the \(z\)-branch of the frequency dispersion. For this case, instead
of Eq. (\ref{nrokRe}), we obtain
\begin{equation}
\par
{=e^{\frac{\omega_{z}^{\prime\prime}}{\omega_{z}^{\prime}}\tau_{z}}\biggl{[}(K_{z}^{\prime}J_{x}^{\prime}(\mathbf{r})-K_{z}^{\prime\prime}J_{z}^{\prime\prime}(\mathbf{r})+J_{z}^{\prime}(\mathbf{r}))\cos{\tau_{z}}+}\par
\nonumber \\
\end{equation} \begin{equation}
\label{nrokRe_z}{+(K_{z}^{\prime}J_{x}^{\prime\prime}(\mathbf{r})+K_{z}^{\prime\prime}J_{z}^{\prime}(\mathbf{r})+J_{z}^{\prime\prime}(\mathbf{r}))\sin{\tau_{z}}\biggr{]}}.\\
\end{equation}
The properties of this function are a little similar to those of the
function \({n^{\prime(x)}(\mathbf{r},\tau_{x})}\). In particular, this
function is also
even with respect to \(y\). However, in contrast to \({n^{\prime(x)}(\mathbf{r},\tau_{x})}\), \({n^{\prime(z)}(\mathbf{r},\tau_{z})}\) is
an even function of the coordinate \(x\) at \(V_{0}=0\).