Non-Stationary Electromagnetics, 2nd Edition

The initial problem with initial moment t_n for the longitudinal component B is described by the equation below that follows from (7.61) and is similar to (7.55)

$B\left(t,r\right)=\tilde{F}\left(t,r\right)+{\displaystyle \underset{t_n}{\overset{\infty }{\int }}d{t}^{\prime }}{\displaystyle \underset{-\infty }{\overset{\infty }{\int }}d{z}^{\prime }}{\displaystyle \underset{-b/2}{\overset{b/2}{\int }}d{x}^{\prime }\widehat{K}(t-t^{\prime },|r-r^{\prime }|)B\left(t^{\prime },r^{\prime }\right)},$

(7.64)

where $\widehat{K}=G^{(2)}\left(t-t^{\prime },|r-r^{\prime }|\right)\widehat{V}\left(t^{\prime }\right),r=\left(x,z\right)$. The propagator in this equation is represented by the formula (7.62) and the medium operator determined by the expression (7.59) describes an arbitrary medium in the waveguide core. The free term in (7.64) is determined by the prehistory of the field before the moment t_n

$\begin{array}{l}\tilde{F}\left(t,r\right)=B^{(0)}\left(t,r\right)+{\displaystyle \underset{-\infty }{\overset{0}{\int }}d{t}^{\prime }}{\displaystyle \int d{r}^{\prime }\widehat{K}}\left(t-t^{\prime },|r-r^{\prime }|,t^{\prime }\right)B\left(t^{\prime },r^{\prime }\right)\\ +{\displaystyle \underset{0}{\overset{t_n}{\int }}d{t}^{\prime }}{\displaystyle \int d{r}^{\prime }\widehat{K}\left(t-t^{\prime },|r-r^{\prime }|,t^{\prime }\right)B\left(t^{\prime },r^{\prime }\right)}\end{array}$

(7.65)

and contains the preceding intervals before the zero moment as well as from the zero moment till ...