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\includeonly{chap1,chap2,chap3,chap4,chap5,bib}


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{
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\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
\stepcounter{chapter}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\ddot{a}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2688: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{engraving.ps}

\label{engraving}
\end{figure}
}

\stepcounter{chapter}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{waveeq.ps}

\label{waveeq}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\nabla \cdot \left( {1 \over \rho } \nabla P \right)
      - {1 \over \rho c^2(z)} P_{tt}
    = -s( t ) {\delta ( z - z_s ) \delta ( r ) \over 2 \pi r} , \label{eq:wave}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P( r, z, t )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2692: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$s( t )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2700: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2704: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}P( r, 0, t ) & = & 0 ,  \nonumber \\ 
   P_z( r, D, t ) & = & 0 .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P( r, z, t ) \;\;\; \mbox{outgoing as} \; r \rightarrow \infty .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}s( t ) = e^{-i \omega t} ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$e^{-i \omega t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2706: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}P( r, z, t ) = p( r, z ) e^{-i \omega t} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{1 \over r} {\partial \over \partial r}
   \left( r {\partial p \over \partial r} \right) +
   \rho( z ) {\partial \over \partial z}
   \left( {1 \over \rho( z )} {\partial p \over \partial z} \right)
    + { \omega ^2 \over c^2( z ) } p
    = {-\delta ( z - z_s ) \delta ( r ) \over 2 \pi r} . \label{Helm}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p( r, z ) = Z( z ) R( r )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2708: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ {1 \over r} {\partial \over \partial r}
   \left( r {\partial R \over \partial r} \right) \right] +
   \left[ \rho(z) {\partial \over \partial z}
   \left( {1 \over \rho(z)} {\partial Z \over \partial z} \right)
    + { \omega ^2 \over c^2(z) } \right] = 0 . \label{Sep}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2714: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\rho (z) {d \over dz} \left( {1\over \rho(z) }{dZ( z ) \over dz}
   \right)
    + \left(
     { \omega ^2 \over c^2(z) } - k^2
     \right) Z(z) & = & 0,  \nonumber \\ 
   Z(0) & = & 0, \nonumber \\ 
   {dZ \over dz} (D) & = & 0. \label{NM}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2716: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2718: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m( z )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2720: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2722: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2724: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2726: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2728: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$[0,D]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2734: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2736: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_1^2 > k_2^2 > \cdots $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2738: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega / c_{min}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2740: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{min}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2742: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int_0^D { Z_m(z) Z_l(z) \over \rho(z)} \, dz = 0 ,
   \qquad \mbox{for $m \neq n$.}           \label{ortho}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int_0^D {Z_m^2(z) \over \rho(z)} \, dz = 1 . \label{norm}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = \sum _{m=1}^\infty R_m(r) Z_m(z).
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}& & \sum _{m=1} ^\infty
      \left\{
         {\rho(z) \over r} {\partial \over \partial r}
         \left(
           {r \over \rho(z)} {\partial R_m(r) \over \partial r}
         \right)
         Z_m(z) \right. \nonumber \\ 
	 & & + R_m(r)
         \left. \left[
            \rho (z) {d \over dz} \left( {1\over \rho(z) }{dZ(z) \over dz}
         \right)
            + { \omega ^2 \over c^2(z) }Z_m(z)
         \right]
      \right\}
    = {-\delta (r) \delta (z-z_s) \over 2 \pi r} . \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\sum _{m=1}^\infty \left[ {1 \over r} {\partial \over \partial r}
   \left( r {\partial R_m(r) \over \partial r} \right) Z_m(z) +
   k_m^2 R_m(r) Z_m(z) \right] =
       {-\delta (r) \delta (z-z_s) \over 2 \pi r} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int_0^D (\cdot) {Z_l( z ) \over \rho( z )} \, dz ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{1 \over r} {\partial \over \partial r}
   \left( r {\partial R_l(r) \over \partial r} \right) +
   k_l^2 R_l(r) = {-\delta (r) Z_l(z_s) \over 2 \pi r \rho(z_s)} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}R_l( r )= {i \over 4 \rho( z_s ) } Z_l( z_s ) H_0^{(1,2)}( k_l r ).
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$H_0^{(1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2748: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$H_0^{(2)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2750: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r
\rightarrow \infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2752: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$e^{-i \omega t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2754: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2756: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
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}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = {i \over 4 \rho(z_s)}
      \sum _{m=1}^\infty Z_m(z_s) Z_m(z) H_0^{(1)}(k_m r),
   \label{Mod_sum}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) \approx {i \over  \rho(z_s) \sqrt{8 \pi r} } e^{-i \pi / 4}
         \sum _{m=1}^\infty Z_m(z_s)Z_m(z) 
         { e ^{i k_m r} \over \sqrt { k_m } }. \label{risum} 
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}TL( r, z ) = -20 \log \left| { p(r,z) \over p^0(r=1) } \right|,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p^0( r ) = { e^{ i k_0 r } \over 4 \pi r }
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}TL( r, z ) \approx -20 \log
   \left| {1 \over \rho(z_s) }
      \sqrt{{2 \pi \over r} }
         \sum _{m=1}^\infty Z_m(z_s) Z_m(z) 
         { e ^{i k_m r} \over \sqrt { k_m } }
   \right|.  \label{TL}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}TL^{Inc}( r, z ) \approx -20 \log
   {1 \over \rho( z_s ) }
      \sqrt{ {2 \pi \over r} }
      \sqrt{ \sum _{m=1}^\infty
      \left|
	 { Z_m(z_s) Z_m(z) \over \sqrt { k_m } }
      \right| }.  \label{TLinc}
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{iso_schem.ps}

\label{iso_schem}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2762: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2764: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z_m(z) = A \sin \gamma z + B \cos \gamma z ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma = \sqrt{ {\omega^2 \over c^2 } - k^2 }.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}A \gamma \cos \gamma D =  0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\gamma D = \left( m+{1 \over 2} \right) \pi, \qquad m=1, 2, \ldots ,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}k_m = \sqrt{ {\omega^2 \over c^2 } - \left[ \left( m+{1 \over 2} \right)
   { \pi \over D} \right]^2}, \qquad m=1, 2, \ldots .
   \label{disper}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Z_m(z) = \sqrt{{2 \over D}} \sin \gamma_m z , \label{iso_mod}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2776: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2778: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2780: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2782: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nu_p(\omega) = \omega / k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2784: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nu_g(\omega) = d\omega / dk$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2786: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2790: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2792: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$e^{-i\omega t}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2794: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2796: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2804: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m = i a$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2806: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\omega /c$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2810: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{iso_loc.ps}

\label{iso_loc}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c=1500 \zw m/s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2812: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D=100 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2814: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f = 100 \zw Hz$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2816: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = {i \over 2 D} \sum _{m=1}^\infty
   \sin(\gamma_m z_s) \sin(\gamma_m z) H_0^{(1)}(k_m r) \label{isosum}.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-10 \log I$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2822: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}I(r,z) = | p( r, z ) |^2 =
   \left| { 1 \over D }
      \sqrt{{8 \pi \over r} }
         \sum _{m=1}^\infty \sin(\gamma_m z_s) \sin(\gamma_m z)
         { e ^{ i k_m r} \over \sqrt { k_m } }
   \right|^2.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z_s = 25 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2826: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z = 50 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2828: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}I(r,z) & = &                   {8 \pi \over r D^2}
   \left|
               \sum _{m=1}^\infty A_m e^{i k_m r} \right|^2 \nonumber \\ 
          & = &
                 {8 \pi \over r D^2}
	       \sum _{n \neq m} \sum _{m}
         A_m^2 + A_n^2 + 2 A_m A_n \cos ( \Delta k_{mn} r )
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta k_{mn} = k_m - k_n ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}A_m = { \sin( \gamma_m z_s ) \sin( \gamma_m z ) \over k_m } .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$e^{ i k r }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2830: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\cos[ ( k_1 - k_2 ) r
]$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2832: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
   \epsfxsize = 80 mm
   \leavevmode\epsfbox{iso_tl1.ps}

   \epsfxsize = 80 mm
   \leavevmode\epsfbox{iso_tl2.ps}

   \epsfxsize = 80 mm
   \leavevmode\epsfbox{iso_tl3.ps}
   
\end{figure}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{pek_schem.ps}

\label{pek_schem}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}Z_{HS}(z) = B e^{-\gamma_b z} + C e^{\gamma_b z} ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma_b  = \sqrt{ k^2 - {\omega^2 \over c_{b}^2 } }.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{b}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2844: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma_b$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2846: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z(D)  & = & B e^{-\gamma_b D} , \\ 
   { Z'(D) \over \rho } & = & -B { \gamma_b e^{-\gamma_b D} \over \rho_b } .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2850: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_b$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2852: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2854: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\rho Z(D) \over Z'(D)} = -{\rho_b \over \gamma_b(k^2)}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{ d^2 Z(z) \over dz^2}
    + \left(
     { \omega ^2 \over c^2 } - k^2
     \right) Z(z) & = & 0,  \nonumber \\ 
   Z(0) & = & 0, \nonumber \\ 
   f( k^2 ) Z( D ) + { g( k^2 ) \over \rho } Z'( D ) & = & 0 .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}f(k^2) = 1 , \qquad
   g(k^2) = \rho_b \bigg/ \sqrt{ k^2 - {\omega^2 \over c_{b}^2 } }.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2856: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}p(r,z) & = & { 1 \over 2 \pi }
                 \int _0^\infty G(z,z_r;k) J_0( k r ) k dk \nonumber \\ 
          & = & { 1 \over 4 \pi }
                 \int _{-\infty} ^{\infty}
	   G(z,z_s;k) H_0^{(1)}( k r ) k dk \label{transform}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$G(z,z_s;k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2860: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\rho (z) \left[ {1\over \rho(z) }G'(z) \right] '
   + \left( { \omega ^2 \over c^2(z) } - k^2 \right) G(z)
   & = & \delta (z-z_s), \nonumber \\ 
   f^T(k^2) G(0) + { g^T(k^2) \over \rho(0) } G'(0) & = & 0, \nonumber \\ 
   f^B(k^2) G(D) + { g^B(k^2) \over \rho(D) } G'(D) & = & 0 .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^{T,B}, g^{T,B}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2864: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\cal L}(k_m)G =  \delta (z-z_s), \qquad {\cal B}_1 G={\cal B}_2 G =0.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}G(z,z_s;k) =  { -p_1(z_< ;k) p_2(z_> ;k) \over W(z_s;k)}
   \label{Grep}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z_<=\min (z,z_s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2866: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$z_>=\max (z,z_s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2868: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W( z; k )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2870: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}W(z;k) = p_1(z;k) p_2'(z;k) - p_1'(z;k) p_2(z;k) ,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_1, p_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2872: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\cal L}(k)p_1 & = & 0, \qquad {\cal B}_1 p_1=0 , \\ 
   {\cal L}(k)p_2 & = & 0, \qquad {\cal B}_2 p_2=0 . \label{IVPs}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2874: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_{EJP}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2876: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int _{-\infty} ^{\infty} + \int _{C_\infty}
    + \int _{C_{EJP}} = 2 \pi i \sum _{m=1} ^M \mbox{res}(k_m)
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mbox{res}(k_m)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2878: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{ejp_loc.ps}

\label{ejp_loc}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$C_\infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2882: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = {i \over 2 } \sum _{m=1} ^M
   { p_1( z_< ; k_m ) p_2( z_> ; k_m ) \over
      \left.
         {\partial W(z_s;k) / \partial k}
      \right|_{k=k_m} }
      H_0^{(1)}(k_m r) k_m - \int _{C_{EJP}},
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2884: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Re\{ k_1 \} > Re \{ k_2 \} > \cdots $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2888: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W( z_s; k_m ) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2890: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W(k_m)=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2892: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{1,2}(z; k_m)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2894: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z)=p_{1}(z; k_m)=p_{2}(z; k_m)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2896: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\cal L}(k_m) Z_m =  0, \qquad
   {\cal B}_1 Z_m = {\cal B}_2 Z_m  = 0.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2898: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2900: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2902: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2904: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = {i \over 2} \sum _{m=1} ^M
   {   Z_m(z_s) Z_m(z) \over
      \left.
         {\partial W(z_s;k) / \partial k}
      \right|_{k=k_m} }
      H_0^{(1)}(k_m r) k_m - \int _{C_{EJP}},
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial W/\partial k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2906: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{1,2}(z;k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2908: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial W/\partial
k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2910: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left. { \partial W / \partial k \over \rho (z_s) } \right|_{k_m}
   = 2k_m \int _0^D
   { Z_m^2 (z) \over \rho (z) } dz -
   \left. {d (f/g)^T \over d k} \right|_{k_m}
      Z_m^2 (0) +
   \left. {d (f/g)^B \over d k} \right|_{k_m}
      Z_m^2 (D) .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2912: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left. \partial W(z_s;k) / \partial k \right|_{k=k_m} = 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2914: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = 
      {i \over 4\rho(z_s)} \sum _{m=1}^M Z_m(z_s) Z_m(z)
      H_0^{(1)}(k_m r) - \int _{C_{EJP}},
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int _0^D { Z_m^2 (z) \over \rho (z)} dz -
   \left.
      {1 \over 2 k_m} {d (f/g)^T \over d k}
   \right|_{k_m} Z_m^2 (0) +
   \left.
      {1 \over 2 k_m} {d (f/g)^B \over d k}
   \right|_{k_m} Z_m^2 (D) = 1.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{pek_loc.ps}

\label{pek_loc}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}Z(z) = A \sin \gamma z ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma = \sqrt{ { \omega^2 \over c^2 } - k^2 }.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}-\tan \gamma D =  {\gamma \over \gamma_b} .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_m(\omega )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2916: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{pek_mod.ps}

\label{pek_mod}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}p = \left(
         e^{i\gamma_m z} + e^{-i\gamma_m z}
       \right) e^{ i k_m r} .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 2918
$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2918: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 2920
$\tan \theta = k_m/\gamma_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2920: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k = \omega
/c_b$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2922: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial W/\partial k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2930: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\rho (z) \left[ {1\over \rho(z) }G'(z) \right] '
   + \left( { \omega ^2 \over c^2(z) } - k^2 \right) G(z)
   & = & \delta (z-z_s), \nonumber \\ 
   f^T(k^2) G(0) + { g^T(k^2) \over \rho(0) } G'(0) & = & 0, \nonumber \\ 
   f^B(k^2) G(D) + { g^B(k^2) \over \rho(D) } G'(D) & = & 0 .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\cal L}(k_m)g =  \delta (z-z_s), \qquad {\cal B}_1 g={\cal B}_2 g =0.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}W(z;k) = p_1(z;k)p_2'(z;k) - p_1'(z;k)p_2(z;k) ,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_1, p_2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2934: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\cal L}(k)p_1 & = & 0, \qquad {\cal B}_1p_1=0 , \\ 
   {\cal L}(k)p_2 & = & 0, \qquad {\cal B}_2p_2=0 .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2936: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\cal L}(k_m)Z_m =  0, \qquad {\cal B}_1 Z_m ={\cal B}_2 Z_m =0.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{equation}p_2{\cal L}(k_m)Z_m - Z_m {\cal L}(k)p_2 = 0,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ p_2 \left( {Z_m' \over \rho} \right) '
   -Z_m \left( {p_2'\over \rho} \right) ' \right]
      +(k^2-k_m^2) {Z_m p_2 \over \rho} = 0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left( p_2{Z_m'\over \rho} - Z_m{p_2'\over \rho} \right) '
      +(k^2-k_m^2) {Z_m p_2 \over \rho} = 0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left. \left[ {p_2 Z_m' - Z_m p_2' \over \rho} \right] \right|_0^D
      +(k^2-k_m^2) \int _0^D {Z_m p_2\over \rho}dz  =  0 \label{eq:boo}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${W(z) / \rho(z) }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2944: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left( {W \over \rho }\right) ' = 
   \left( {p_1 p_2' - p_1'p_2 \over \rho} \right) '
   = {p_1{\cal L}p_2 - p_2{\cal L}p_1 \over \rho}  = 0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{W(z_s) \over \rho(z_s)} = {W(0) \over \rho (0)} =
      {p_1(0)p_2'(0) - p_1'(0)p_2(0) \over \rho(0)}.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_2'(0)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2946: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p_2'(0) = 
      { p_1'(0)p_2(0) + W(z_s) \rho(0)/\rho (z_s) \over p_1(0)} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left.
      { p_2 Z_m' - Z_m p_2' \over \rho}
   \right|_0 = 
   { p_2(0) Z_m'(0) - p_1'(0) p_2(0) Z_m(0)/p_1(0) \over \rho (0) }
   - {W(z_s) \over \rho (z_s)} . \label{eq:a1}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z_m'(0)   & = & -{ \rho(0) f^T(k_m) \over g^T(k_m) } Z_m(0) \nonumber \\ 
   p_1'(0)   & = & -{ \rho(0) f^T(k)   \over g^T(k)   } p_1(0) ,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\left. {p_2 Z_m' - Z_m p_2' \over \rho} \right|_0 = 
   -\left[
      { f^T(k)   \over g^T(k)   }
    - { f^T(k_m) \over g^T(k_m) }
   \right]
   p_2(0) Z_m(0) - {W(z_s) \over \rho
   (z_s)}. \label{eq:topval}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\hspace{-15mm}
   \left. {p_2 Z_m' - Z_m p_2' \over \rho} \right|_D  =
   -\left[
      { f^B(k)   \over g^B(k)   }
    - { f^B(k_m) \over g^B(k_m) }
    \right] p_2(D)Z_m (D) . \label{eq:botval}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z_m'(D)   & = & -{ \rho(D) f^B(k_m) \over g^B(k_m) } Z_m(D) \nonumber \\ 
   p_1'(D)   & = & -{ \rho(D) f^B(k)   \over g^B(k)   } p_1(D) .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}-\left[
      { f^B(k)   \over g^B(k)   }
    - { f^B(k_m) \over g^B(k_m) }
    \right] p_2(D) Z_m(D)
  & + &
  \left[
      { f^T(k)   \over g^T(k)   }
    - { f^T(k_m) \over g^T(k_m) }
  \right] p_2(0) Z_m(0) \nonumber \\ 
  + {W(z_s;k) - W(z_s;k_m) \over \rho (z_s)}
  & + & (k^2-k_m^2) \int _0^D p_2{Z_m \over \rho}dz = 0
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W(z_s;k_m)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2952: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$W(z;k_m)=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2954: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_{1,2}(z;k)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2958: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2960: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_2(D;k)=Z_m(D)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2962: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k-k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2964: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k
\rightarrow k_m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2966: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left. { \partial W / \partial k \over \rho (z_s) } \right|_{k_m}
   = 2k_m \int _0^D
   { \left[Z_m (z)\right]^2 \over \rho (z) } dz -
   \left. {d (f/g)^T \over d k} \right|_{k_m}
      \left[Z_m (0)\right]^2 +
   \left. {d (f/g)^B \over d k} \right|_{k_m}
      \left[Z_m (D)\right]^2 .
\end{equation}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}c(z) = 1500.0
   \left[
      1.0 + \epsilon
      \left(
         \tilde{z} - 1 + e^{ -\tilde{z} }
      \right)
   \right] .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2974: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\epsilon = 0.00737
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{z}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2976: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\tilde{z} = { 2 ( z - 1300 ) \over 1300 } .
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$5000 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2978: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1600 \zw m/s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2980: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1 \zw g/cm^3$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2982: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$50 \zw Hz$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2984: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{munk_schem.ps}

\label{munk_schem}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{munk_mod.ps}

\label{munk_mod}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}Z(z) \approx
     A { e^{  i\int _0^z \gamma (s) \, ds} \over \sqrt{ \gamma(z)} }
   + B { e^{ -i\int _0^z \gamma (s) \, ds} \over \sqrt{ \gamma(z)} },
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\gamma^2( z ) =   { \omega ^2 \over c^2(z) } - k^2_m . \end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2990: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2992: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma^2(z)=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2994: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1/\gamma(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2996: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline2998: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3000: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$100 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3002: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$90^\circ$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3004: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{munk_tl.ps}

\label{munk_tl}
\end{figure}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf r}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3006: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left( r_1, r_2, r_3, r_4 \right)
   = \left( {u \over ik}, w,
            {\tau _{zx} \over ik}, \tau _{zz} \right)
                                          \label{rdef}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau _{zx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3012: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau _{zz}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3014: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau _{zx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3018: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf r}
' = {\bf E}
 {\bf r}

\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf E}
(z,k) = \left[ \begin{array}{cccc}
   0 & -1 & 1/(\rho c^2_s) & 0 \\ 
   k^2\eta (z) & 0 & 0 &  1/(\rho c^2_p) \\ 
   k^2 \zeta (z) - \rho \omega ^2 & 0 & 0 & - \eta (z) \\ 
   0 & - \rho \omega ^2 & k^2 & 0 \\ 
   \end{array} \right] \label{elastic}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\eta (z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3022: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\zeta (z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3024: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\eta (z) = { c^2_p-2c^2_s \over c^2_p }, \qquad
   \zeta (z) = { \rho \left[ c^4_p - (c^2_p - 2 c^2_s)^2 \right]
                     \over c^2_p }
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_p, c_s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3026: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(k_j, {\bf r}
_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3032: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(-k_j, {\bf r}
_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3034: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf r}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3036: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf r}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3038: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3040: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau _{zz}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3042: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3044: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau _{zx}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3046: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$3)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3048: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\omega ^2r_2(z) & = & p'(z) \nonumber \\ 
   r_3(z) & = & 0 \nonumber \\ 
   r_4(z) & = & -p(z) .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c} y_1 \\  y_2 \\  y_3 \\  y_4 \\  y_5
      \\ y_6 \end{array}\right]
   = \left[ \begin{array}{c}
        r_1s_2 - r_2s_1 \\ 
        r_3s_4 - r_4s_3 \\ 
        r_1s_3 - r_3s_1 \\ 
        r_2s_3 - r_3s_2 \\ 
        r_1s_4 - r_4s_1 \\ 
        r_2s_4 - r_4s_2 \\ 
   \end{array} \right] , \label{Y}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf r}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3050: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf s}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3052: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf y}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3054: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_i, s_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3056: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3058: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf y}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3060: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf y}
' = {\bf W}
 {\bf y}

\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf W}
 = \left[ \begin{array}{ccccc}
   0  &  0  &  0  &  1/(\rho c_s^2)  & -1/(\rho c_p^2) \\ 
   0  &  0  &  0  &  -\omega^2\rho   & -(k^2t(z)-\omega^2\rho) \\ 
   0  &  0  &  0  &  1               & \eta (z) \\ 
   k^2\zeta(z)-\omega^2\rho  & 1/(\rho c_p^2)  & -k^2\eta(z)& 0 & 0 \\ 
   \omega^2\rho  &  -1/(\rho c_s^2) &  -2k^2 & 0 & 0 \\ 
   \end{array} \right] \label{RDM}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y_6$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3062: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y_6 = -k^2y_5$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3064: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}f^B(k^2) Z_j(D) + g^B(k^2) {dZ_j(D) \over dz} = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}f^B(k^2) & = & \omega^2 y_4 , \nonumber \\ 
   g^B(k^2) & = & y_2            \label{elbc}
\end{eqnarray}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p( z )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3068: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f p( z ) + g p'( z ) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3070: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p( z ) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3072: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p'( z ) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3074: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf r}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3076: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf y}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3078: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}p(0) = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}r_3(0)=r_4(0)=0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\bf r}
 & = & ( 1, 0, 0, 0 ) \nonumber \\ 
   {\bf s}
 & = & ( 0, 1, 0, 0 )
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf y}
 = ( 0, 1, 0, 0, 0 ) .
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}p'(D) = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}r_1(D)=r_2(D)=0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\bf r}
 & = & ( 0, 0, 1, 0 ) \nonumber \\ 
   {\bf s}
 & = & ( 0, 0, 0, 1 ) .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf y}
 = ( 1, 0, 0, 0, 0 ) .
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{pb}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3080: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_b$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3082: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z_{HS}(z) = A e^{-\gamma_b z} + B e^{\gamma_b z}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma_b  = \sqrt{k^2 - { \omega^2 \over c_{pb}^2 } },
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z(D)  & = & A e^{-\gamma_b D} \\ 
   Z'(D) & = & -A \gamma_b e^{-\gamma_b D}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}Z(D) + { \rho_b \over \gamma_b } Z'(D) = 0.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Z(D) - { \rho_t \over \gamma_t } Z'(D) = 0.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho \to 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3086: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho \to \infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3088: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\phi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3090: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\psi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3092: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\phi(z) = A e^{-\gamma _pz}, \qquad   \psi(z) = B e^{-\gamma _sz}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma _p = \sqrt{k^2 - {\omega^2 \over c_{pb}^2} }, \qquad
   \gamma _s = \sqrt{k^2 - {\omega^2 \over c_{sb}^2} }.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}u = \phi_x - \psi_z , \qquad w = \phi_z + \psi_x .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c} u \\  w \end{array} \right]
   = \left[ \begin{array}{cc}
   ik        &   -\gamma_s ik \\ 
   -\gamma_p &   -k^2 \end{array} \right] 
   \left[ \begin{array}{c} A \\  B \end{array} \right]
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}( r_1, r_2 ) = \left( { u \over i k }, w \right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}r_3 & = & \mu (r_1'+r_2)   \\ 
   r_4 & = & (\lambda + 2\mu ) r_2' - k^2 \lambda r_1
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c} r_1 \\  r_2 \\  r_3 \\  r_4 \end{array} \right]
   = \left[ \begin{array}{cc}
      1                   &        \gamma_s  \\ 
      -\gamma_p           &        -k^2      \\ 
      -2\mu\gamma_p       &        -\mu(\gamma_s^2+k^2)  \\ 
      \mu(\gamma_s^2+k^2) &  2\mu\gamma_s k^2  \end{array} \right]
   \left[ \begin{array}{c} A \\  B \end{array} \right]
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c} y_1 \\  y_2 \\  y_3 \\  y_4 \\ y_5 \end{array} \right]
   = \left[ \begin{array}{c}
      (\gamma_s \gamma_p-k^2)/\mu  \\ 
      \left( ( \gamma_s^2 + k^2 )^2 - 4\gamma_s\gamma_pk^2 \right) \mu \\ 
      2\gamma_s\gamma_p-\gamma_s^2-k^2  \\ 
      \gamma_p(k^2-\gamma_s^2)  \\ 
      \gamma_s(\gamma_s^2-k^2)
      \end{array} \right]
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y_2 = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3098: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3100
$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3100: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3102
$R(\theta )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3102: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3104
$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3104: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p(z) = e^ {i\gamma z} + R e^ {-i\gamma z}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma = \sqrt{ {\omega ^2 \over c^2(0)} - k^2 }
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p'(z) = i\gamma \left( e^ {i\gamma z} + R e^ {-i\gamma z} \right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left. {p \over p_z } \right|_{z=0} = { 1 \over i\gamma} {1+R \over 1-R}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p - {1 \over i \gamma} {1 + R \over 1 - R } p' = 0 . \label{ZR}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p(z) = e^ {-i\gamma z} + R e^ {i\gamma z}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p + {1 \over i \gamma}{1 + R \over 1 - R} p' = 0 . \label{ZRR}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3108
$\theta$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3108: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3110: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1525
\tan \theta = { \gamma \over k } .
\end{equation}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1527
\alpha^{Thorpe} =
   { 40 f^2 \over 4100 + f^2 } + { 0.1 f^2 \over 1 + f^2 }
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3114
$\alpha^{Thorpe}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3114: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(z) = c_r(z) + i c_i(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3116: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3118
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3118: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3120
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3120: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1529
e^{-\alpha r} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}e^{-i \omega /c} = e^{-i \omega {c_r - ic_i \over c_r^2 + c_i^2 }}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1533
c_i = {\alpha |c|^2 \over \omega} .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3122
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3122: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3124
$\alpha ^{(m)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3124: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3126
$\alpha ^{(m)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3126: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1535
\alpha ^{(m)}  = -10 \log _{10} { I(x + 1 ) \over I(x) }
      = -20 \log _{10} { e^{-\alpha(x + 1 )} \over e^{-\alpha(x)} }
      = 20 \alpha \log _{10} e
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1537
\alpha \approx {\alpha ^{(m)} \over 8.69} .
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3128
$\alpha ^{(w)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3128: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3130
$\alpha ^{(m)} = \alpha ^{(w)} / \lambda$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3130: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1539
\alpha \approx {\alpha ^{(w)} \over 8.69 \lambda } .
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3132
$\alpha ^{(f)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3132: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3134
$\alpha ^{(m)} = (f/1000)\alpha ^{(f)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3134: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1541
\alpha \approx { \alpha ^{(f)} \over 8.69 } { f \over 1000 } .
\end{equation}
}

\stepcounter{subsubsection}
{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1543
\alpha \equiv {\pi \over \lambda Q} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{table}% latex2html id marker 645


\begin{center}
\begin{displaymath}
\begin{array}{|c||c|c|c|c|c|} \hline
            &\multicolumn{5}{c|}{$desired units$}                                               \\  \hline
$user units$& \alpha           & \alpha^{(m)}   & \alpha^{(w)} & \alpha^{(f)}  &  Q^{-1}        \\  \hline
\alpha      &        1         &      8.69      & 8.69\lambda  &   8686/f      & \lambda / 3.14 \\  \hline
\alpha^{(m)}&     0.115        &        1       & \lambda/1000 &   1000/f      & 0.037\lambda   \\  \hline
\alpha^{(w)}& 0.115 / \lambda  &  1000/\lambda  &     1        &   1000/c      &   0.037        \\  \hline
\alpha^{(f)}&     f/8686       &     f/1000     &   c/1000     &     1         &   c/27288      \\  \hline
Q^{-1}      &  3.14 / \lambda  &  0.115/\lambda &    27.29     &   27288/c     &      1         \\  \hline
\end{array}
\end{displaymath}
\end{center}
\end{table}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3142
$R(\theta)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3142: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1545
R(\theta ) = 1 - 2 \gamma^2 \sigma ^2 ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3144: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3146: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma^2 = { \omega^2 \over c^2 } - k^2 .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma \sigma = 1 / \sqrt{ 2 }$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3148: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma \sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3150: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3152: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c}
      p_2\\ 
      p_2'/\rho
   \end{array}\right] = \sigma^2 A
   \left[ \begin{array}{r}
      p_1\\ 
      p_1'/\rho
   \end{array}\right]
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}a_{11} & = & 0.5 ( \gamma_1^2 - \gamma_2^2 ) -
     { \rho_2 \gamma_1^2 - \rho_1 \gamma_2^2 ) ( \gamma_1 + \gamma_2
     ) \over \Delta } \nonumber \\ 
   a_{12} & = & { i ( \rho_2 - \rho_1 )^2 \gamma_1 \gamma_2 \over
      \Delta } \nonumber \\ 
   a_{21} & = & {-i ( \rho_2 \gamma_1^2 - \rho_1 \gamma_2^2 )^2 \over
      \rho_1 \rho_2 \Delta } \nonumber \\ 
   a_{22} & = & 0.5 ( \gamma_1^2 - \gamma_2^2 ) +
      {( \rho_2 - \rho_1 ) \gamma_1 \gamma_2 ( \gamma_1 + \gamma_2 )
      \over \Delta }
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}\Delta = \rho_1 \gamma_2 + \rho_2 \gamma_1
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma^2 = { \omega^2 \over c^2 } - k^2 .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3156: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3158: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\gamma \sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3160: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{eqnarray}\rho (z) \left[ {1\over \rho(z) }Z_m'(z) \right]'
            + \left( K^2(z) - k_m^2 \right) Z_m(z) & = & 0,  \nonumber \\ 
   Z_m(0)  & = & 0,  \nonumber \\ 
   Z_m'(D) & = & 0,  \nonumber \\  
   \int _0^D {Z_m^2 (z) \over \rho (z)} dz  & = & 1 ,     \label{mat}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^2(z) =\omega ^2 / c^2(z) $}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3164: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}K^2(z) = K^2_0(z) + \epsilon K^2_1(z) + \; \cdots
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^2_0(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3166: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^2(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3168: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z(z) = Z_0(z)  + \epsilon Z_1(z) + \; \cdots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}k^2 = k^2_0(z) + \epsilon k^2_1  + \; \cdots
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}O(1): \qquad
   \rho (z) \left[ {1\over \rho(z) }Z_0'(z) \right]'
           + \left( K^2_0(z) - k^2_0 \right) Z_0(z) & = & 0, \nonumber \\ 
   Z_0(0)  & = & 0,  \nonumber \\ 
   Z_0'(D) & = & 0,  \nonumber \\  
   \int _0^D { Z_0^2 (z) \over \rho (z)} dz   & = & 1. \label{O0}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}O(\epsilon): \qquad
   \rho (z) \left[ {1\over \rho(z) }Z_1'(z) \right]'
      + \left( K^2_0(z) - k^2_0 \right) Z_1(z) & = &
             -\left( K^2_1(z) - k^2_1 \right) Z_0(z), \nonumber \\ 
    Z_1(0)  & = & 0, \nonumber \\ 
    Z_1'(D) & = & 0, \nonumber \\  
   \int _0^D { Z_1^2 (z) \over \rho (z)} dz & = & 1 . \label{O1}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_0(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3170: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int _0^D \left( K^2_1(z) - k^2_1 \right)
      { Z_0^2(z) \over \rho(z) } dz = 0,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}k^2_1 =  { -\int_0^D  K^2_1(z) Z_0^2(z) / \rho(z) dz
         \over \int_0^D           Z_0^2(z) / \rho(z) dz } . \label{moss}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K_1^2(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3172: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(z) = c_r(z) + i c_i(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3174: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^2(z) =
K^2_r(z) + i K^2_i(z) = \omega / c(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3176: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K^2_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3178: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_r$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3180: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3182: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon K^2_1 = i K^2_i(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3184: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\epsilon
k^2_1 = i k^2_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3186: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}i k^2_i & = &  \epsilon k^2_1 \nonumber \\ 
           & = & { -\int_0^D  \epsilon K^2_1(z) Z^2(z) / \rho(z) dz
        \over       \int_0^D                    Z^2(z) / \rho(z) dz } .
                                 \nonumber \\ 
           & = & { -\int_0^D         i K^2_i(z) Z^2(z) / \rho(z) dz
              \over \int_0^D                    Z^2(z) / \rho(z) dz } .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}k^2_i = { -\int_0^D           K^2_i(z) Z^2(z) / \rho(z) dz
        \over \int_0^D                    Z^2(z) / \rho(z) dz } .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}k_i^{(scat)} = { \rho \sigma^2 \gamma_m \over 2 k_m }
   \left( { dZ_m(z) \over dz } \right)_{z=0}^2 ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3188: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\gamma_m = \sqrt{ {\omega^2 \over c^2(0)} - k_m^2 } .
\end{equation}
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}p^j(r,z) = \sum _{m=1}^M \left[
      a_m^j \widehat{H}1_m^j(r) + b_m^j \widehat{H}2_m^j(r) \right]
      Z_m^j(z) ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\widehat{H}1,2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3194: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\widehat{H}1_m^j(r) & = & { H_0^{(1)}(k_m^j r) \over
      H_0^{(1)}(k_m^j r_{j-1}) } , \\ 
   \widehat{H}2_m^j(r) & = & { H_0^{(2)}(k_m^j r) \over
      H_0^{(2)}(k_m^j r_{j-1}) } ,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r_{j-1} = r_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3196: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\widehat{H}1_m^j(r) & \sim & H1_m^j(r) =
      \sqrt{ { r_{j-1} \over r } } e^{ i k_m^j (r - r_{j-1}) } , \\ 
   \widehat{H}2_m^j(r) & \sim & H2_m^j(r) =
      \sqrt{ { r_{j-1} \over r } } e^{-i k_m^j (r - r_{j-1}) } .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{cm.ps}

\label{cm}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}\sum _{m=1}^M \left( a_m^{j+1} + b_m^{j+1} \right)
      Z_m^{j+1}(z) = 
   \sum _{m=1}^M \left[ a_m^j H1_m^j(r_j) + b_m^j H2_m^j(r_j) \right]
      Z_m^j(z) . \label{thesum}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${a_m,b_m}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3206: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int ( \cdot ) {Z_l^{j+1}(z) \over \rho_{j+1}(z) } \, dz ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$l=1, \ldots , M$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3212: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int { Z_m^{j+1}(z) Z_l^{j+1}(z)
      \over \rho_{j+1}(z) } \, dz = \delta _{lm} ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}a_l^{j+1} + b_l^{j+1} = 
      \sum _{m=1}^M
      \left[
         a_m^j H1_m^j(r_j) + b_m^j H2_m^j(r_j)
      \right]
      \tilde{c}_{lm}, \qquad l = 1, \ldots , M ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\tilde{c}_{lm} = \int { Z_l^{j+1}(z) Z_m^j(z) \over \rho_{j+1}(z)
   } \, dz .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf a}
^{j+1} + {\bf b}
^{j+1} = {\bf \tilde{C}}
^j
      \left( {\bf H}
_1^j {\bf a}
^j + {\bf H}
_2^j {\bf b}
^j \right),
      \label{pmat}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf H}
_1^j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3214: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf H}
_2^j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3216: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$H1_m^j(r_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3218: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$H2_m^j(r_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3220: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf \tilde{C}}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3222: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{c}_{lm}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3224: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf a}
, {\bf b}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3226: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a_l, b_l$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3228: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{ 1 \over \rho_j } {\partial p^j(r,z) \over \partial r} \approx
      { 1 \over \rho_j } \sum _{m=1}^M k_m^j
      \left[ a_m^j H1_m^j(r) - b_m^j H2_m^j(r) \right]
      Z_m^j(z) .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int ( \cdot ) Z_l^{j+1}(z) \, dz ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}a_l^{j+1} - b_l^{j+1} = 
   \sum _{m=1}^M \left[ a_m^j H1_m^j(r_j) - b_m^j H2_m^j(r_j) \right]
      \hat{c}_{lm}, \qquad l = 1, \ldots , M ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\hat{c}_{lm} =  { k_m^j \over k_l^{j+1} }
   \int { Z_l^{j+1}(z) Z_m^{j}(z) \over \rho_j(z) } \, dz .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat{c}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3230: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{c}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3232: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf a}
^{j+1} - {\bf b}
^{j+1} =
      {\bf \hat{C}}
^j
      \left( {\bf H}
_1^j {\bf a}
^j - {\bf H}
_2^j {\bf b}
^j \right).
      \label{umat}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf a}
^{j+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3234: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf b}
^{j+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3236: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c} {\bf a}
^{j+1} \\  {\bf b}
^{j+1} \end{array} \right]
   = \left[ \begin{array}{cc}
   {\bf R}
_1^j   &   {\bf R}
_2^j \\ 
   {\bf R}
_3^j   &   {\bf R}
_4^j \end{array} \right] 
   \left[ \begin{array}{c} {\bf a}
^{j}   \\  {\bf b}
^{j}   \end{array} \right] ,
   \label{int_mat}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\bf R}
_1^j & = & {1 \over 2} \left( {\bf \tilde{C}}
^j +
      {\bf \hat{C}}
^j \right) {\bf H}
_1^j \nonumber \\ 
   {\bf R}
_2^j & = & {1 \over 2} \left( {\bf \tilde{C}}
^j -
      {\bf \hat{C}}
^j \right) {\bf H}
_2^j \nonumber \\ 
   {\bf R}
_3^j & = & {1 \over 2} \left( {\bf \tilde{C}}
^j -
      {\bf \hat{C}}
^j \right) {\bf H}
_1^j \nonumber \\ 
   {\bf R}
_4^j & = & {1 \over 2} \left( {\bf \tilde{C}}
^j +
      {\bf \hat{C}}
^j \right) {\bf H}
_2^j .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$r \rightarrow \infty$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3240: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$b_m^{N} = 0,$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3242: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$m=1, \ldots , M$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3244: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}a_m^1 = {i \over 4 \rho(z_s) } Z_m (z_s) H_0^{(1)} ( k_m^1 r_1 )
   + b_m^1 { H_0^{(1)} ( k_m^1 r_1 ) \over  H_0^{(2)} ( k_m^1 r_1 ) },
   \qquad m = 1, \ldots , M .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{cccccccc}

{\bf I}
    &-{\bf D}
   &   0     &         &         &         &         &   \\ 
{\bf R}
_1^1&{\bf R}
_2^1&  {\bf I}
 &     0   &         &         &         &   \\ 
{\bf R}
_3^1&{\bf R}
_4^1&   0     & {\bf I}
  &         &         &         &   \\ 
          & \ddots   & \ddots  & \ddots  & \ddots  &         &         &   \\ 
   &   &         &{\bf R}
_1^{N-2}&{\bf R}
_2^{N-2}&  {\bf I}
    &     0   &   \\ 
   &   &         &{\bf R}
_3^{N-2}&{\bf R}
_4^{N-2}&   0     &    {\bf I}
  &   \\ 
   &   &         &         &         &{\bf R}
_1^{N-1}&{\bf R}
_2^{N-1}&{\bf I}
\\ 
   &   &         &         &         &{\bf R}
_3^{N-1}&{\bf R}
_4^{N-1}&    0 \\ 
   \end{array} \right]
   \left[ \begin{array}{c}
      {\bf a}
^1 \\ 
      {\bf b}
^1 \\    
      {\bf a}
^2 \\ 
      {\bf b}
^2 \\    
      \vdots    \\    
      {\bf a}
^{N-1} \\ 
      {\bf b}
^{N-1} \\ 
      {\bf a}
^N     \\ 
   \end{array} \right] =
   \left[ \begin{array}{c}
      {\bf s}
 \\ 
      {\bf 0}
 \\ 
      {\bf 0}
 \\ 
      {\bf 0}
 \\ 
      \vdots  \\    
      {\bf 0}
 \\ 
      {\bf 0}
 \\ 
      {\bf 0}
 \\ 
   \end{array} \right]
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf D}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3248: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}d_{ii} = { H_0^{(1)} ( k_i^1 r_1 ) \over H_0^{(2)} ( k_i^1 r_1 )
   } ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf s}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3250: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}s_m = {i \over 4 \rho(z_s) } Z_m (z_s) H_0^{(1)} ( k_m^1 r_1 ) .
\end{equation}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\left[ \begin{array}{c} {\bf a}
^{j+1} \\  {\bf b}
^{j+1} \end{array} \right]
   = \left[ \begin{array}{cc}
   {\bf R}
_1   &   {\bf R}
_2 \\ 
   {\bf R}
_3   &   {\bf R}
_4 \end{array} \right] 
   \left[ \begin{array}{c} {\bf a}
^{j}   \\  {\bf b}
^{j}   \end{array} \right] .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf b}
^{j+1} = {\bf 0}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3254: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf b}
^j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3256: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf b}
^j = -{\bf R}
_4^{-1} {\bf R}
_3 {\bf a}
^j .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf a}
^{j+1}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3258: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf a}
^{j+1} = \left( {\bf R}
_1 -
      {\bf R}
_2 {\bf R}
_4^{-1} {\bf R}
_3 \right) {\bf a}
^j ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf a}
^{j+1} = {\bf R}
_1 {\bf a}
^j .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf R}
_1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3260: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}{\rho \over r}{\partial \over \partial r}
   \left( { r \over \rho }{\partial p \over \partial r} \right)
   + \rho {\partial \over \partial z}
   \left( { 1 \over \rho }{\partial p \over \partial z} \right)
   + {\omega ^2 \over c^2(r,z)}p
   = {-\delta (z-z_s) \delta (r) \over 2 \pi r} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) = \sum_m R_m(r) Z_m(z,r) , \label{ansatz}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(r,z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3262: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\rho (r,z) \left[ {1\over \rho(r,z) }Z_m'(z,r) \right] '
   + \left( { \omega ^2 \over c^2(r,z) } - k_m^2(r) \right) Z_m(z,r)
    = 0 . \label{loc_modes}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(r,z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3268: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\sum_m { \rho \over r }{\partial \over \partial r}
   \left( { r \over \rho }{\partial (R_m Z_m) \over \partial r} \right)
   -\sum_m k_m^2(r) R_m Z_m
   = {-\delta (z-z_s) \delta (r) \over 2 \pi r} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\sum_m \left[
   { \rho \over r}{\partial \over \partial r}
   \left( { r \over \rho }{\partial R_m \over \partial r} \right) Z_m
   + 2{\partial R_m \over \partial r}
    {\partial Z_m \over \partial r}
   + {\rho \over r}{\partial \over \partial r}
   \left( { r \over \rho }{\partial Z_m \over \partial r} \right) R_m
   \right] \nonumber \\ 
   -\sum_m k_m^2(r) R_m Z_m
   = {-\delta (z-z_s) \delta (r) \over 2 \pi r} .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3272: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int (\cdot) { Z_l(z,r) \over \rho } dz,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{ 1 \over r}{\partial \over \partial r}
   \left( r {\partial R_l \over \partial r} \right)
   + \sum_m 2 B_{lm} {\partial R_m \over \partial r}
   + \sum_m A_{lm} R_m
   + k_l^2(r) R_l
   = {-Z_l(z_s) \delta (r) \over 2 \pi r} . \label{cm_cts}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}A_{lm} & = &  \int {1 \over r}{\partial \over \partial r}
    \left( r {\partial Z_m \over \partial r} \right)
     {Z_l \over \rho} dz ,  \nonumber \\ 
   B_{lm} & = &  \int {\partial Z_m \over \partial r}
      {Z_l \over \rho} dz .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B_{lm}=-B_{ml}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3276: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int { Z_m(z) Z_l(z)
      \over \rho(z) } \, dz = \delta _{lm} ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int {\partial Z_m(z) \over \partial r} {Z_l(z)
      \over \rho(z) } \, dz +
   \int {  Z_m(z) \over \rho(z) }
       {\partial Z_l(z) \over \partial r} \, dz
 = 0 .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$B_{lm}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3278: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{lm}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3280: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$A_{nn}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3282: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{ 1 \over r} {\partial \over \partial r}
   \left( r { \partial R_l \over \partial r } \right)
   + k_l^2(r) R_l
   = {-Z_l( z_s ) \delta (r) \over 2 \pi r} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}R_l( r ) \approx
         A { e^{ i\int _0^r k_l (s) \, ds} \over \sqrt{k_l(r)} } .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}A =  {i \over \sqrt{8 \pi r} } e^{-i \pi / 4} Z_l(z_s) .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}p(r,z) \approx {i \over \sqrt{8 \pi r} } e^{-i \pi / 4}
         \sum _{m=1}^\infty Z_m(z_s)Z_m(z,r) 
         { e^{ i\int _0^r k_m (s) \, ds} \over \sqrt{k_m(r)} } .
 \label{adiab}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(z,r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3286: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1000 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3288: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{eddy_ssp.ps}

\label{eddy_ssp}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$50 \zw Hz$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3290: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$300 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3292: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{eddy_tl.ps}

\label{eddy_tl}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$100 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3294: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$150 \zw km$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3298: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$90 \zw km$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3300: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$400 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3306: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$400 \zw m$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3308: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$50 \zw Hz$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3310: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{lant_tri.ps}

\label{lant_tri}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{lant_tl.ps}

\label{lant_tl}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{equation}\rho \nabla \cdot \left( { 1 \over \rho } \nabla p \right)
   + {\omega ^2 \over c^2(x,y,z)}p
   = {-\delta(x) \delta(y) \delta (z-z_s)} , \label{Helm3D}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\rho {\partial \over \partial x}
   \left( { 1 \over \rho }{\partial p \over \partial x} \right)
   + \rho {\partial \over \partial y}
   \left( { 1 \over \rho }{\partial p \over \partial y} \right)
   + \rho {\partial \over \partial z}
   \left( { 1 \over \rho }{\partial p \over \partial z} \right) & &
   \nonumber \\ 
   + {\omega ^2 \over c^2( x, y, z )}p
   & = & {-\delta(x) \delta(y) \delta (z-z_s)} . \nonumber
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}p(x,y,z) = \sum_m \phi_m( x, y ) Z_m(x,y,z) , \label{ansatz2}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_m(x,y,z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3312: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}\int (\cdot) { Z_l( x, y, z ) \over \rho } dz,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\partial^2 \phi_l \over \partial x^2}
   & + & {\partial^2 \phi_l \over \partial y^2}
   + k_l^2( x, y ) \phi_l
   + \sum_m A_{lm} \phi_m  \nonumber \\ 
   & + & \sum_m 2 B_{lm} {\partial \phi_m \over \partial x}
     + \sum_m 2 C_{lm} {\partial \phi_m \over \partial y}
   = {-\delta(x) \delta(y) \delta (z-z_s)} ,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}A_{lm} & = &  \int \left(
     {\partial^2  \over \partial x^2}
   + {\partial^2  \over \partial y^2} \right) Z_m
      {Z_l \over \rho} dz , \nonumber \\ 
   B_{lm} & = & -B_{ml} =  \int 
     {\partial Z_m \over \partial x} {Z_l \over \rho} dz , \nonumber \\ 
   C_{lm} & = & -C_{ml} = \int 
     {\partial Z_m \over \partial y} {Z_l \over \rho} dz .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{\partial^2 \phi_l \over \partial x^2}
   + {\partial^2 \phi_l \over \partial y^2}
   + k_l^2( x, y ) \phi_l
   = {-Z_l(z_s) \delta(x) \delta(y)} .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_l(x,y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3326: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{1 \over r} \left( r \phi_l' \right)' +
      k_l( r ) \phi_l = \phi_l( z_s; 0 ) { \delta( r ) \over r } .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_l (r)= Z_l( z_s; 0 )
      { e^{ -i \int_0^r k_l( \hat{ r } ) d \hat{r} } \over \sqrt{ k_l(r)r} }.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$L(s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3328: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K(s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3330: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{dx \over ds } & = & c_l \xi(s), \nonumber \\ 
   {d \xi \over ds } & = & -{1 \over c_l^2} {dc_l \over dx}, \nonumber \\ 
   {dy \over ds } & = & c_l \eta(s), \nonumber \\ 
   {d \eta \over ds } & = & -{1 \over c_l^2} {dc_l \over dy},
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_l(x,y)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3334: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(s,n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3338: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}g_l( s, n ) = A \sqrt{ c( s ) \over q( s ) }
   \exp
      \left\{ -i \omega
         \left[
           \tau ( s ) + { p( s ) \over 2 q( s ) } n^2
         \right]
      \right\}  \label{beam}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p(s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3342: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q(s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3344: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{dq \over ds} & = & c( s ) p( s ),\\ 
   {dp \over ds} & = & -{c_{nn} ( s ) \over c^2( s ) } q( s ).
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c_{nn}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3348: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau ( s )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3350: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{d \tau \over ds} =  {1 \over c( s )} .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}L(s) & = & \sqrt{ -2 \omega \mbox{Im}
                   \left\{ { p(s) \over q(s) } \right\} }, \\ 
   K(s) & = &            -c(s) \mbox{Re}
                   \left\{ { p(s) \over q(s) } \right\}.
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$L(s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3354: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1/e$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3356: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$K(s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3358: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left( x(s), y(s) \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3360: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left( \xi (s), \eta (s) \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3362: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left(  p(s), q(s) \right)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3364: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tau (s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3366: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}% latex2html id marker 1830
x(     0 ) & = & x_s , \nonumber \\ 
   y(     0 ) & = & y_s , \nonumber \\ 
   \xi  ( 0 ) & = & { \cos \alpha \over c( 0 ) } ,  \nonumber \\ 
   \eta ( 0 ) & = & { \sin \alpha \over c( 0 ) } ,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(x_s,y_s)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3368: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3370
$\alpha$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3370: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1832
u ( x, y, z ) = \int_0^{2 \pi} A( \alpha) g_l( s, n ) d\alpha ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_l(s,n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3376: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(x_r,y_r)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3380: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(s,n)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3382: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1834
u ( x, y ) = {A( \alpha _0 ) \over \sqrt{ 4 \pi } }
                e^ {-i k_l \sqrt{ x^2 + y^2 } } .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}u ( x, y ) = Z_l( z_s )
                e^ {-i k_l \sqrt{ x^2 + y^2 } } .
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{% latex2html id marker 3384
$A(\alpha)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3384: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_l(z_s)/ \sqrt{4 \pi}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3386: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}p( x, y, z ) = \sum _{ n = 1 }^m \phi_l( x, y ) Z_l( z ) ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}% latex2html id marker 1840
\phi_l ( x, y, z ) = \int g_l( s, n ) d\alpha ,
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}g_l( s, n ) = A \sqrt{ c( s ) \over q( s ) }
   \exp
      \left\{ -i \omega
         \left[
           \tau ( s ) + { p( s ) \over 2 q( s ) } n^2
         \right]
      \right\} .
\end{equation}
}

\stepcounter{chapter}
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}z_j= j h, \qquad j = 0, \ldots , N
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h=D/N$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3396: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_j = Z(z_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3398: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{fin_dif.ps}

\label{fin_dif}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{equation}Z''(z)
   + \left( { \omega ^2 \over c^2(z) } - k^2 \right) Z(z) =  0,
   \label{foo}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z_{j+1} & = & Z_{j} + Z'_{j}h + Z''_{j}{h^2 \over 2!} +
   Z_{j}^{(3)}{h^3 \over 3!} + \cdots
   \label{forward}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}Z'_{j} = {Z_{j+1}-Z_{j} \over h } - Z''_{j}{h \over 2} + \cdots.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$O(h)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3406: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z'_{j} \approx {Z_{j+1}-Z_{j} \over h }.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Z''(z) = -\left( { \omega ^2 \over c^2(z) } - k^2 \right) Z(z).
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$O(h^2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3408: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z'_{j} \approx {Z_{j+1}-Z_{j} \over h } -
   \left( { \omega ^2 \over c^2(z_j) } - k^2 \right) Z_{j}
   {h \over 2}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Z_{j-1} = Z_{j} - Z'_{j}h + Z''_{j}{h^2 \over 2!} +
      Z_{j}^{(3)}{h^3 \over 3!} + \cdots
   \label{backward}
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$O(h)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3410: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z'_{j} \approx {Z_{j}-Z_{j-1} \over h } ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$O(h^2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3412: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z'_{j} \approx {Z_{j}-Z_{j-1} \over h } +
   \left( { \omega ^2 \over c^2(z_j) } - k^2 \right) Z_{j}
   {h \over 2}.
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Z''_{j} = { Z_{j-1} - 2Z_{j} + Z_{j+1} \over h^2 } + O(h^2) .
 \label{centered}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z''(z)
        + \left( { \omega ^2 \over c^2(z) } - k^2 \right) Z(z) & = &  0,
      \nonumber \\ 
   f^{T}(k^2) Z(0) + {g^{T}(k^2) \over \rho} Z'(0) & = & 0,      \nonumber \\ 
   f^{B}(k^2) Z(D) + {g^{B}(k^2) \over \rho} Z'(D) & = & 0,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}Z_{j-1} +
   \left[ -2 + h^2\left( {\omega^2 \over c^2(z_j)}
   - k^2 \right) \right] Z_{j}
   + Z_{j+1} & = & 0 , \qquad j = 1, \ldots, N-1
      \nonumber \\ 
   { f^T \over g^T } Z_{0} +  
   { 1 \over \rho } \left[ {Z_{1}-Z_{0}   \over h } -
   \left( { \omega ^2 \over c^2(0) } - k^2 \right)
   Z_{0} {h \over 2} \right] & = & 0 , \label{top}
      \nonumber \\ 
   { f^B \over g^B } Z_{N} +
   { 1 \over \rho } \left[ {Z_{N}-Z_{N-1} \over h } +
   \left( { \omega ^2 \over c^2(D) } - k^2 \right)
   Z_{N} {h \over 2} \right] & = & 0 . \label{bot}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{equation}{1 \over h\rho} Z_{j-1} +
   { -2 + h^2\left[ \omega^2 / c^2(z_j) - k^2 \right] \over h\rho} Z_{j}
   - {1 \over h\rho} Z_{j+1} = 0 . \label{mid}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}{\bf A}
(k^2){\bf Z}
 = 0.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\bf Z}
$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3414: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_0,
Z_1, \ldots Z_N$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3416: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}A = \left[ \begin{array}{cccccccc}
   d_0 & e_1 &         &         &         &         &    \\ 
   e_1 & d_1 & e_2     &         &         &         &    \\ 
       & e_2 & d_2     & e_3     &         &         &    \\ 
       &     & \ddots  & \ddots  & \ddots  &         &    \\ 
       &     &         & e_{N-2} & d_{N-2} & e_{N-1} &    \\ 
       &     &         &         & e_{N-1} & d_{N-1} & e_N\\ 
       &     &         &         &         & e_N     & d_N\\ 
   \end{array} \right]
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3420: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$e_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3422: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}d_0 & = & {-2 + h^2 \left[ \omega^2 /c^2(z_0) - k^2 \right] \over 2h\rho }
   + { f^T(k^2) \over g^T(k^2) } , \nonumber \\ 
   d_j & = & {-2 + h^2 \left[ \omega^2 /c^2(z_j) - k^2 \right] \over  h\rho } ,
    \qquad  j = 1, \ldots, N-1 \nonumber \\ 
   d_N & = & {-2 + h^2 \left[ \omega^2 /c^2(z_N) - k^2 \right] \over 2h\rho }
   - { f^B(k^2) \over g^B(k^2) } ,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{displaymath}e_j= {1 \over h\rho } ,\qquad j = 1, \ldots, N.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$1/(h\rho)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3424: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f/g$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3426: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3428: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^{T,B}, g^{T,B}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3430: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}\centering
\epsfxsize = 140 mm
\leavevmode\epsfbox{fin_difint.ps}

\label{fin_difint}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_w$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3434: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3436: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}Z_{j-1} +
   \left[ -2 + h_w^2\left(
   {\omega^2 \over c^2(z_j)} - k^2 \right) \right] Z_{j}
   + Z_{j+1} = 0 , \qquad j=1, \ldots, N-1 \label{water}
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}Z_{j-1} +
   \left[ -2 + h_s^2\left(
   {\omega^2 \over c^2(z_j)} - k^2 \right) \right] Z_{j}
   + Z_{j+1} = 0 , \qquad j=N+1, \ldots 
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_N$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3438: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}{Z'(D) \over \rho_w} = {Z'(D) \over \rho_s} ,
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_w$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3440: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3442: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}& & \left[
     { Z_{N} - Z_{N-1} \over h_w } +
        \left(
          { \omega ^2 \over c^2(D^-) } - k^2
        \right)
     Z_{N} {h_w \over 2}
   \right]
   \bigg/ \rho_w \nonumber \\ 
   & = &
   \left[
     { Z_{N+1} - Z_{N} \over h_s } -
        \left(
          { \omega ^2 \over c^2(D^+) } - k^2
        \right)
     Z_{N} {h_s \over 2}
   \right] \bigg/ \rho_s ,
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(D^{\pm})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3444: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D^-$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3448: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$D^+$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3452: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}{Z_{N-1}    \over h_w \rho_w } & + &
   {-Z_{N} + \left[ \omega ^2 / c^2(D^-) - k^2 \right]
   Z_{N} h_w^2 / 2 \over h_w\rho_w}  \nonumber \\ 
   & + & 
   {-Z_{N} + \left[ \omega ^2 / c^2(D^+) - k^2 \right]
   Z_{N} h_s^2 / 2 \over h_s\rho_s}
   + {Z_{N+1} \over h\rho_s } = 0 .
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_w=h_s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3454: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\rho_w=\rho_s$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3456: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$c(z)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3458: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}N_m = \int _0^D {Z_m^2 (z) \over \rho (z)} dz -
   \left.
      {1 \over 2k_m} {d (f/g)^T \over d k}
   \right|_{k_m} Z_m^2 (0) +
   \left.
      {1 \over 2k_m} {d (f/g)^B \over d k}
   \right|_{k_m} Z_m^2 (D) .
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\int _0^D {Z_m^2 (z) \over \rho (z)} dz
   \approx {D \over N} \left( {1 \over 2} \phi_0
      + \phi_1 + \phi_2 + \cdots + \phi_{N-1} + {1 \over 2}\phi_N \right)
\end{equation}
}

{\newpage
\clearpage
\samepage \begin{equation}\phi_j = { Z_j^2 \over \rho(z_j) }.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d(f/g)^{T,B} / dk$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3460: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{equation}\Delta (k^2) = 0
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\Delta (k^2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3462: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{equation}A( k_j^2 ) Z^{(l+1)} = Z^{(l)}.
\end{equation}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3466: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$k_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3470: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z^{(1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3472: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z^{(l+1)}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3474: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${S_j}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline3482: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}S_1 & = & 1 , \\ 
   S_j & = & d_j S_{j-1} - e_j^2 S_{j-2}.
\end{eqnarray}
}

{\newpage
\clearpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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{\newpage
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\samepage \begin{figure}\centering
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\leavevmode\epsfbox{double_tl.ps}

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{\newpage
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{\newpage
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{\newpage
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\leavevmode\epsfbox{flused_tl.ps}

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{\newpage
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\samepage \begin{figure}\centering
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\leavevmode\epsfbox{atten_tl.ps}

\label{atten_tl}
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{\newpage
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{\newpage
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\samepage \begin{figure}\centering
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\leavevmode\epsfbox{normal_tl.ps}

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{\newpage
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\samepage \begin{figure}\centering
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\leavevmode\epsfbox{ice_tl.ps}

\label{ice_tl}
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\stepcounter{chapter}

\end{document}
