Non-Stationary Electromagnetics, 2nd Edition

Here, F = θ(x − ut)E₀(t, x) is a free term of the integral equation and a constitutive relation for cold plasma is given by

$P=\frac{{\omega }_e^2}{4\pi }{\displaystyle \underset{0}{\overset{t}{\int }}\left(t-t^{\prime }\right)\theta \left(x-u{t}^{\prime }\right)E\left(t^{\prime },x^{\prime }\right)}d{t}^{\prime }.$

(6.101)

The solution to Eq. (6.100) is found by the resolvent, which satisfies Eq. (3.7)

$\langle p\left|\widehat{R}\right|p^{\prime }\rangle =\langle p\left|\widehat{K}\right|p^{\prime }\rangle +{\displaystyle \int d{p}_1}\langle p\left|\widehat{K}\right|p_1\rangle \langle p_1\left|\widehat{R}\right|p^{\prime }\rangle .$

(6.102)

The kernel of this equation $\langle p\left|\widehat{K}\right|p^{\prime }\rangle =\langle p\left|\frac{2\pi }{c}\frac{{\partial }^2}{\partial t^2}{\text{Γ}}_k\left(t-t^{\prime },x-x^{\prime }\right)\text{χ}\widehat{P}\right|p^{\prime }\rangle$, where $\widehat{P}$ is the operator determined by (6.101), depends on the direction of the boundary velocity. If u > 0 then

$\langle p\left|\widehat{K}\right|p^{\prime }\rangle ={\omega }_e^2\left\{\frac{p^2}{p^2+c^2{k}^2+{\omega }_k^2}\frac{i}{k-k^{\prime }-i0}+\frac{c}{2{\phi }_k}\frac{{\left(q-\beta {\phi }_k\right)}^2}{\left({\phi }_k-\beta p-ick\right)\left({\phi }_k-\beta q-ic{k}^{\prime }{\gamma }^{-2}\right)}\right\}\frac{1}{{\left(q-iu{k}^{\prime }\right)}^2\left(q-p^{\prime }\right)},$

(6.103)

where q = p + iuk, $\gamma =1/\sqrt{1-{\beta }^2},{\phi }_k=\sqrt{q^2+{\omega }_k^2{\gamma }^{-2}}$, and that branch of the square root is taken which has the positive real part, Reφ_k > 0.

If ...