diff --git a/Doc/UserManual/Scattering.tex b/Doc/UserManual/Scattering.tex index 47e9c1615c32cdf7328ca0c294df02d96a3aa859..a618ffba725071e877ff6b7b5700b522d5d62fdf 100644 --- a/Doc/UserManual/Scattering.tex +++ b/Doc/UserManual/Scattering.tex @@ -742,7 +742,10 @@ Similarly, we introduce the vectorial Green function with final polarization sta \v{G}^\alpha(\r,\r') \coloneqq \ue_\alpha^* \TG(\r,\r'). \end{equation} -Its far-field limit has the simple form +The main result of this entire chapter\footnote +{To our knowledge, \cref{EmyG} has never before been stated. +Informations about similar expressions anywhere in the literature would be highly welcome.} +is the following simple expression for its far-field: \Emph{ \begin{equation}\label{EmyG} \v{G}^\infty_\alpha(\r,\r') = \phi(r) \v\Psi_\alpha^*(\r'), @@ -758,8 +761,13 @@ and $\v\Psi_\alpha$ is a solution of the unperturbed distorted wave equation \end{equation} with the boundary condition \begin{equation} - \Psi_\alpha(\r) = \ue_\alpha\text{~~for }r\to\infty. + \v\Psi_\alpha(\r) = \ue_\alpha\e^{i\k_\sf\r} \end{equation} +for $r\to\infty$ and +with an outgoing wavevector +\nomenclature[2f000]{f}{Subscript ``final''}% +$\k_\sf\coloneqq K \r / r$. + We now outline a proof for~\cref{EmyG}. We first consider wave propagation in vacuum,