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% \catcode`\"=\active  \let"=\"  \let\3=\ss
\title{MALYUZHINETS' PARABOLIC EQUATION IN SHORT WAVE DIFFRACTION PROBLEMS}
\author{Alexei V. Popov}{
IZMIRAN, Russian Academy of Sciences\\
Troitsk, Moscow Region, 142092, Russia\\
Email: popov@charley.izmiran.rssi.ru}
\coauthor{Ning Yan Zhu}{
Institut f\"{u}r Hochfrequenztechnik der Universit\"{a}t Stuttgart\\
Pfaffenwaldring 47, D-70550 Stuttgart, Germany\\
E-mail: zhu@ihf.uni-stuttgart.de}


Malyuzhinets' version of the parabolic wave equation which is written in ray
coordinates and approximates asymptotically the exact Helmholtz equation, 
combines the geometric optics with the principle of transversal
diffusion \cite{malyuzhinets:59}
\begin{eqnarray*}
& &ik\left(2 \displaystyle \frac{\partial u}{\partial \xi} + \frac{1}{h_s}
\frac{\partial h_s}{\partial \xi} u\right) +\frac{1}{h_s}
\frac{\partial}{\partial s}\left(\frac{1}{h_s} \frac{\partial u}{\partial s}\right) = 0,\\
& &u(0,s) = u_0(s),
\end{eqnarray*}
where $(\xi,s)$ are ray coordinates and $h_s, h_{\xi}$ the corresponding 
Lam\'e coefficients.

Via transformation of variables, it can be 
reduced to the original form of Leontovich-Fock parabolic equation with 
constant coefficients \cite{leontovich-fock}
\begin{eqnarray*}
& &2 i \displaystyle \frac{\partial W}{\partial t}  + \frac{\partial^2 W}{\partial \psi^2} = 0,\;\;\;
0<t<\frac{\gamma(L)}{kL}, 0<\psi< \gamma(L), \\
& &W(0,\psi) = \displaystyle u_0\left[\frac{L\psi}{\gamma(L)}\right]e^{ik\left[
\int_0^{\frac{L\psi}{\gamma(L)}}\gamma(s)ds-\frac{L}{2\gamma(L)}\psi^2\right]},
\end{eqnarray*}
with 
\[ t = \displaystyle \frac{\gamma(L) \xi}{kL\left[\xi + \frac{L}{\gamma(L)}\right]},\;\;\;
\psi = \displaystyle \gamma(L)\frac{s+\xi \gamma(s)}{L+\xi\gamma(L)}, \;\;\;
\gamma(s) = \int_0^s \frac{d s'}{\rho(s')} \]
and $L$ is the circumference of a \underline{closed} wave front.

This enables one to get a closed-form approximate solution of the short wave 
radiation problem 
\begin{eqnarray*}
u(\xi,s) &=& \displaystyle \frac{W(t,\psi)}{\sqrt{1+\frac{\xi}{L}\gamma(L)\;}}e^{ikP(\xi,s)},\\
P(\xi,s) &=& \displaystyle \frac{\left[s+\xi\gamma(s)\right]^2}{2\left[ \xi+\frac{L}{\gamma(L)}\right]}
-\int_0^s\gamma(s')\left[1+\frac{\xi}{\rho(s')}\right] ds',
\end{eqnarray*}
and provides an efficient tool for numerical calculations.

\begin{thebibliography}{99}

\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{leontovich-fock}
M.~A. Leontovich and V.~A. Fock, ``Solution of the problem of propagation of
  electromagnetic waves along the earth's surface by the method of parabolic
  equation,'' in {\em Electromagnetic Diffraction and Propagation Problems}
  (V.~A. Fock), ch.~11, Oxford: Pergamon Press, 1965.

\end{thebibliography}

\end{document}
