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\title{
NUMERICAL STUDY OF WAVE DIFFRACTION ON SCALAR AND
TENSOR IMPEDANCE WEDGES WITH THE METHOD OF PARABOLIC EQUATION}

\author{Ning Yan Zhu and Friedrich M. Landstorfer}
{Institut f\"{u}r Hochfrequenztechnik der Universit\"{a}t Stuttgart\\
Pfaffenwaldring 47, D-70550 Stuttgart, Germany\\
E-mail: zhu@ihf.uni-stuttgart.de}

An exact method to investigate wedge diffraction problems as pioneered
by Sommerfeld and further developed by Malyuzhinets and others,
consists principally in transforming the original partial differential Helmholtz 
equation into a difference equation for the corresponding spectral function 
and then solving the latter (e.~g. \cite{senior}). As an alternative, Malyuzhinets 
suggested to solve the parabolic approximation of the wave equation 
in a ray coordinate system \cite{malyuzhinets:59}. Plane wave and cylindrical
wave diffraction on perfectly conducting wedges have been studied by 
Malyuzhinets, Ufimtsev and Popov in this way and it is Popov who 
demonstrated also numerically the accuracy of the method of parabolic 
equation (PE) (\cite{malyuzhinets:59}, \cite{ufimtsev:65} and
\cite{popov:69}).

Based on these works PE has been applied to calculate space wave diffraction on
scalar impedance wedges illuminated by an cylindrical space wave. Using the
diffracted wave as the incident field successively, multiple diffraction has been 
studied as well. Numerical results of single, slope- and multiple-diffraction 
coefficients show a very good agreement with exact or UTD results where 
they are available \cite{zhu:95c}.

Recently tensor impedance wedges were tackled by using the Sommerfeld integral 
method. Due to the coupling of the resulting functional equations a uniform 
diffraction coefficient was obtained only for weak anisotropy 
\cite{ljalinov:94}. Although a parabolic approximation likewise leads to two
coupled equations, they can be solved efficiently for general physical tensor 
impedance boundary conditions and for a spatially fast varying incident cylindrical
wave, however. A comparison with the UAT of \cite{ljalinov:94} confirms the 
validity of PE in case of weak anisotropy and plane wave incidence 
\cite{zhu:96}.

It is exactly the coupling of spectral functions which limits an exact
solution of the diffraction of an obliquely incident plane wave on a scalar impedance
wedge to few special cases \cite{senior}. For a PE numerical solution no such a limit
exists. By neglecting the same higher order terms as in
the derivation of the parabolic equation, an approximate impedance boundary 
condition has been obtained. This in turn leads to an equivalent tensor 
impedance wedge for normal plane wave incidence like in \cite{zhu:96} and
consequently this problem can be solved in a similar way.

From the point of view of energy flow, an explanation of the accuracy of the
parabolic approximation is given for the case of a perfectly conducting half-plane.
It defines the statement of Malyuzhinets \cite{malyuzhinets:59} precisely and shows
clearly that edge diffracted rays originate from the shadow boundaries
of the incident and reflected waves.

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\bibitem{senior}
T.~B.~A. Senior and J.~L. Volakis, {\em Approximate boundary conditions in
  electromagnetics}, 
\newblock London: IEE, 1995.

\bibitem{malyuzhinets:59}
G.~D. Malyuzhinets, ``Developments in our concepts of diffraction phenomena,''
  {\em Soviet Physics: Usp.}, vol.~69(2), no.~5, pp.~749--758, 1959.

\bibitem{ufimtsev:65}
P.~Y. Ufimtsev, ``Transverse diffusion for diffraction by a wedge,'' {\em Radio
  Engineering and Electronic Physics}, vol.~10, no.~6, pp.~866--875, 1965.

\bibitem{popov:69}
A.~V. Popov, ``Numerical solution of the wedge diffraction problem by the
  transverse diffusion method,'' {\em Soviet Physics: Acoustics}, vol.~15,
  no.~2, pp.~226--233, 1969.

\bibitem{zhu:95c}
N.~Y. Zhu and F.~M. Landstorfer, ``Numerical determination of diffraction,
  slope- and multiple-diffraction coefficients of impedance wedges by the
  method of parabolic equation: Space waves,'' {\em IEEE Trans. Antennas
  Propagat.}, vol.~AP--43, no.~12, 1995.

\bibitem{ljalinov:94}
M.~A. Ljalinov, ``Diffraction by a wedge with anisotropic face impedances,''
  {\em Ann. T\'el\'ecommun.}, vol.~49, no.~11--12, pp.~667--672, 1994.

\bibitem{zhu:96}
N.~Y. Zhu and F.~M. Landstorfer, ``Numerical study of diffraction and
  slope-diffraction on anisotropic impedance wedges by the method of parabolic
  equation: Space waves,'' submitted for publication, Dec. 1995.

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