Here, *F* = *θ*(*x* − *ut*)*E*_{0}(*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 \text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\langle p\left|\frac{2\pi}{c}\frac{{\partial}^{2}}{\partial {t}^{2}}{\text{\Gamma}}_{k}\left(t-{t}^{\prime},x-{x}^{\prime}\right)\text{\chi}\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*, $\gamma \text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}1/\sqrt{1-{\beta}^{2}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\phi}_{k}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\sqrt{{q}^{2}+{\omega}_{k}^{2}{\gamma}^{-2}}$, and that branch of the square root is taken which has the positive real part, Re*φ*_{k} > 0.

If ...

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