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\title{An analysis of diffraction at edges illuminated by 
  transition fields}
\author{R. Tiberio and R. G. Kouyoumjian}
\address{Department of Electrical Engineering, Ohio State 
  University}
\received{(Received December 30, 1987; revised August 13, 1984;
  accepted August 13, 1984.)}
\maketitle

\begin{abstract}
A solution is obtained for the high-frequency scattering by a pair of
parallel wedges where the edge of one wedge is illuminated by the shadow
boundary field of the other. A generalized spectral extension of the uniform
GTD is used to determine the contribution from the doubly diffracted ray.
\end{abstract}

\section{INTRODUCTION}
The multiple-edge interaction problem is of considerable practical
importance. Periodic structures involved the diffraction from interacting
edges, and the same problem may occur in the propagation over mountain
ridges. 

\section{FORMULATION}
The ray geometry for the diffraction from a pair of staggered wedges with
parallel edges, illuminated at oblique incidence from a source located a O,
is depicted in Figure 1.

Let the field incident at P be
\begin{equation}
	\mbox{f}(\mbox{s}_{\mbox{o}}) = 
	\mbox{exp}( -\mbox{jks}_{\mbox{o}})1/\mbox{ks}_{\mbox{o}}
\end{equation}

\subsection{Doubly diffracted field}
Field incident on the second edge. Let us refer to the configuration in
\marginboxed{Fig. 1}
Figure 1, and for the sake of simplicity consider a cylindrical wave
illumination. The extension of the procedure to follow to other types of
illumination is straightforward. In order to evaluate the field
$\mbox{u}^{\mbox{d}}(\mbox{Q}_{\mbox{2}})$ at $\mbox{Q}_{\mbox{2}}$,
diffracted by $\mbox{Q}_{\mbox{1}}$, let us use the following high-frequency
integral representation of the diffracted field.\cite{KP}

\subsubsection{Diffraction of inhomogeneous waves} The preceding
interpretation of the diffracted field can be conveniently used to determine
the doubly diffracted field when the second edge is in the transition region
of the first edge.

\begin{acknowledgments}
The work reported in this paper was supported in part by Associate Joint
Services Electronics Program contract N00014-78-C-0049 between the Office of
Naval Research and the Ohio State University Research Foundation.
\end{acknowledgments}

\newpage
\begin{thebibliography}{}

\bibitem[Jones, 1973]{Jones}
Jones, D. S., Double knife-edge diffraction and ray theory, {\em Q. J. Mech.
Appl. Math.}, 26(1), 1--18, 1973.

\bibitem[Kouyoumjian, 1975]{Kou}
Kouyoumjian, R. G., The geometrical theory of diffraction and its
application, in {\em Numerical and Asymptotic Techniques in
Electromagnetics}, edited by R. Mittra, pp. 165--215, Springer-Verlag, New
York, 1975.

\bibitem[Pathak and Kouyoumjian, 1970]{KP}
Pathak, P. H. and R. G. Kouyoumjian, The dyadic diffraction coefficient for a
perfectly conducting wedge, {\em Sci. Rep. 5}, Dep. of Electr. Eng., Ohio
State Univ., Columbus, 1970.

\end{thebibliography}

\begin{addresses}
R. G. Kouyoumjian and R. Tiberio, Department of Electrical Engineering,
Ohio State University, Columbus, OH 43210.
\end{addresses}

\copyrightnotice{Copyright 1987 by the American Geophysical Union.}

\papernumber{4S1297. \\ 0048--6604/87/004S--1297\$08.00}

\runningheads{3}{TIBERIO AND KOUYOUMJIAN: ANALYSIS OF DIFFRACTION AT EDGES}

\printcaption{Fig.~1. The geometry of a two-dimensional ogive.}

\begin{table}
  \begin{center}
  TABLE 1. Comparison of $\beta$/k for $\mbox{TE}_{\mbox{y}}$ and
	$\mbox{HE}_{\mbox{y}}$ 
	Modes in the Case of a = D and d = D/2
  \end{center}
  \begin{tabular}{lcccc}	\hline
    Mechanism 	&        &  & Surface
      &             \\ 	
    Name $\!\!\!\!\!$       & \hspace*{-10pt} Sector $\!\!\!\!$
    & \hspace*{-10pt} Diffraction $\!\!\!\!\!\!\!\!\!$  & Ray \hspace*{-10pt}
      & \hspace*{-10pt} Fig.      \\    \hline
    Specular	        & Both		  & None	 & No
      & -          \\
    Front Edge	        & Both		  & Single	 & No
      & -          \\
    Rear Edge	        & Wide		  & Single	 & No
      & -          \\
    CCW-R-CW            & Low		  & Single       & Yes
      & -          \\
    CW-T-CW-D           & Low             & Double       & Yes
      &            \\
      \hline
  \end{tabular} 
\end{table}

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