Non-Stationary Electromagnetics, 2nd Edition

where the representation ${\widehat{X}}_1+p^2{w}_e{I}_{\perp }$ can be deduced from X in (4.65) after substituting < ϕ₁ for < φ₁. In this representation, we have

${\widehat{\text{X}}}_1=-v_1^2{w}_m\left(\begin{array}{cc}{s}_{\perp }^2& i\frac{{\varphi }_1}{v_1}{s}_{\perp }\\ -i\frac{{\varphi }_1}{v_1}{\tilde{s}}_{\perp }& {\widehat{\text{s}}}_{\perp }\end{array}\right)+2v{v}_1{w}_{em}\left(\begin{array}{cc}0& 0\\ 0& {\widehat{\text{s}}}_{\perp }\end{array}\right),$

(4.87)

where ${\widehat{s}}_{\perp }=\left(\begin{array}{cc}{s}_2^2& s_2{s}_3\\ s_2{s}_3& s_3^2\end{array}\right)$.

The first term in (4.86) is the residue in the pole p = iω,

$E_2^{\left(1\right)}\left(x\right)=\theta \left(t-x/v_1\right)\frac{v_1^2-v^2}{v^2}\frac{{\widehat{\text{X}}}^{\prime }}{2v_1^2{s}_1^{\prime }\left(s_1+s_1^{\prime }\right)}{E}_0{e}^{i\left(\omega t-s^{\prime }r\right)}.$

(4.88)

Here, ${\widehat{\text{X}}}^{\prime }$ is found from ${\widehat{X}}_1+p^2{w}_e{\widehat{\text{I}}}_{\perp }$ after substituting iω for p. The term ℑ₂ occurs owing to the other poles and branching points in (4.86).

Consider the transient field, which is determined by other singularities than the pole p = iω in the integrand of the integral:

${\Im }_2=\frac{v^2-v_1^2}{v^2}{\displaystyle \underset{-i\infty }{\overset{i\infty }{\int }}\frac{\text{dp}}{2\pi i}}\frac{{\text{e}}^{\text{pt-i }{s}_{\perp }{r}_{\perp }}}{2{\varphi }_1\left(p-i\omega \right)}\frac{\overline{X}}{{\varphi }_1+i{v}_1{s}_1}{e}^{-\frac{{\varphi }_1}{v_1}x}{E}_0{|}_{p\ne i\omega }.$

(4.89)

In addition to the branching points in Eq. (4.89) ...