\batchmode
\documentclass[12pt]{article}
\RequirePackage{ifthen}


\usepackage{wsh}



\setlength{\textwidth}{\paperwidth} 

\addtolength{\textwidth}{-1.6in} 

\addtolength{\oddsidemargin}{-0.8in} 

\setlength{\evensidemargin}{\oddsidemargin} 




\usepackage{xcolor}

\usepackage[latin1]{inputenc}



\makeatletter

\makeatletter
\count@=\the\catcode`\_ \catcode`\_=8 
\newenvironment{tex2html_wrap}{}{}%
\catcode`\<=12\catcode`\_=\count@
\newcommand{\providedcommand}[1]{\expandafter\providecommand\csname #1\endcsname}%
\newcommand{\renewedcommand}[1]{\expandafter\providecommand\csname #1\endcsname{}%
  \expandafter\renewcommand\csname #1\endcsname}%
\newcommand{\newedenvironment}[1]{\newenvironment{#1}{}{}\renewenvironment{#1}}%
\let\newedcommand\renewedcommand
\let\renewedenvironment\newedenvironment
\makeatother
\let\mathon=$
\let\mathoff=$
\ifx\AtBeginDocument\undefined \newcommand{\AtBeginDocument}[1]{}\fi
\newbox\sizebox
\setlength{\hoffset}{0pt}\setlength{\voffset}{0pt}
\addtolength{\textheight}{\footskip}\setlength{\footskip}{0pt}
\addtolength{\textheight}{\topmargin}\setlength{\topmargin}{0pt}
\addtolength{\textheight}{\headheight}\setlength{\headheight}{0pt}
\addtolength{\textheight}{\headsep}\setlength{\headsep}{0pt}
\setlength{\textwidth}{349pt}
\newwrite\lthtmlwrite
\makeatletter
\let\realnormalsize=\normalsize
\global\topskip=2sp
\def\preveqno{}\let\real@float=\@float \let\realend@float=\end@float
\def\@float{\let\@savefreelist\@freelist\real@float}
\def\liih@math{\ifmmode$\else\bad@math\fi}
\def\end@float{\realend@float\global\let\@freelist\@savefreelist}
\let\real@dbflt=\@dbflt \let\end@dblfloat=\end@float
\let\@largefloatcheck=\relax
\let\if@boxedmulticols=\iftrue
\def\@dbflt{\let\@savefreelist\@freelist\real@dbflt}
\def\adjustnormalsize{\def\normalsize{\mathsurround=0pt \realnormalsize
 \parindent=0pt\abovedisplayskip=0pt\belowdisplayskip=0pt}%
 \def\phantompar{\csname par\endcsname}\normalsize}%
\def\lthtmltypeout#1{{\let\protect\string \immediate\write\lthtmlwrite{#1}}}%
\usepackage[tightpage,active]{preview}
\newbox\lthtmlPageBox
\newdimen\lthtmlCropMarkHeight
\newdimen\lthtmlCropMarkDepth
\long\def\lthtmlTightVBox#1#2{%
    \setbox\lthtmlPageBox\vbox{\hbox{\catcode`\_=8 #2}}%
    \lthtmlCropMarkHeight=\ht\lthtmlPageBox \advance \lthtmlCropMarkHeight 6pt
    \lthtmlCropMarkDepth=\dp\lthtmlPageBox
    \lthtmltypeout{^^J:#1:lthtmlCropMarkHeight:=\the\lthtmlCropMarkHeight}%
    \lthtmltypeout{^^J:#1:lthtmlCropMarkDepth:=\the\lthtmlCropMarkDepth:1ex:=\the \dimexpr 1ex}%
    \begin{preview}\copy\lthtmlPageBox\end{preview}}%
\long\def\lthtmlTightFBox#1#2{%
    \adjustnormalsize\setbox\lthtmlPageBox=\vbox\bgroup %
    \let\ifinner=\iffalse \let\)\liih@math %
    {\catcode`\_=8 #2}%
    \@next\next\@currlist{}{\def\next{\voidb@x}}%
    \expandafter\box\next\egroup %
    \lthtmlCropMarkHeight=\ht\lthtmlPageBox \advance \lthtmlCropMarkHeight 6pt
    \lthtmlCropMarkDepth=\dp\lthtmlPageBox
    \lthtmltypeout{^^J:#1:lthtmlCropMarkHeight:=\the\lthtmlCropMarkHeight}%
    \lthtmltypeout{^^J:#1:lthtmlCropMarkDepth:=\the\lthtmlCropMarkDepth:1ex:=\the \dimexpr 1ex}%
    \begin{preview}\copy\lthtmlPageBox\end{preview}}%
    \long\def\lthtmlinlinemathA#1#2\lthtmlindisplaymathZ{\lthtmlTightVBox{#1}{#2}}
    \def\lthtmlinlineA#1#2\lthtmlinlineZ{\lthtmlTightVBox{#1}{#2}}
    \long\def\lthtmldisplayA#1#2\lthtmldisplayZ{\lthtmlTightVBox{#1}{#2}}
    \long\def\lthtmlinlinemathA#1#2\lthtmlindisplaymathZ{\lthtmlTightVBox{#1}{#2}}
    \def\lthtmlinlineA#1#2\lthtmlinlineZ{\lthtmlTightVBox{#1}{#2}}
    \long\def\lthtmldisplayA#1#2\lthtmldisplayZ{\lthtmlTightVBox{#1}{#2}}
    \long\def\lthtmldisplayB#1#2\lthtmldisplayZ{\\edef\preveqno{(\theequation)}%
        \lthtmlTightVBox{#1}{\let\@eqnnum\relax#2}}
    \long\def\lthtmlfigureA#1#2\lthtmlfigureZ{\let\@savefreelist\@freelist
        \lthtmlTightFBox{#1}{#2}\global\let\@freelist\@savefreelist}
    \long\def\lthtmlpictureA#1#2\lthtmlpictureZ{\let\@savefreelist\@freelist
        \lthtmlTightVBox{#1}{#2}\global\let\@freelist\@savefreelist}
\def\lthtmlcheckvsize{\ifdim\ht\sizebox<\vsize 
  \ifdim\wd\sizebox<\hsize\expandafter\hfill\fi \expandafter\vfill
  \else\expandafter\vss\fi}%
\providecommand{\selectlanguage}[1]{}%
\makeatletter \tracingstats = 1 


\begin{document}
\pagestyle{empty}\thispagestyle{empty}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength hsize=\the\hsize}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength vsize=\the\vsize}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength hoffset=\the\hoffset}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength voffset=\the\voffset}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength topmargin=\the\topmargin}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength topskip=\the\topskip}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength headheight=\the\headheight}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength headsep=\the\headsep}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength parskip=\the\parskip}\lthtmltypeout{}%
\lthtmltypeout{latex2htmlLength oddsidemargin=\the\oddsidemargin}\lthtmltypeout{}%
\makeatletter
\if@twoside\lthtmltypeout{latex2htmlLength evensidemargin=\the\evensidemargin}%
\else\lthtmltypeout{latex2htmlLength evensidemargin=\the\oddsidemargin}\fi%
\lthtmltypeout{}%
\makeatother
\setcounter{page}{1}
\onecolumn

% !!! IMAGES START HERE !!!

\stepcounter{section}
\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1417}%
${{\svector x}_s}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1419}%
${{\svector x}_r}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1421}%
$0 \leq t \leq T$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1423}%
${c({\svector x})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1425}%
${\rho({\svector x})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1427}%
${P_s {({\svector x}, t)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1939}%
$\displaystyle
\left [ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{{\partial ^2}\over{\partial t^2}}- {{\svector \nabla}\cdot}\left ( {1 \over {{\rho({\svector x})}}}{\svector \nabla}\right ) 
\right ] {P_s {({\svector x}, t)}}= {{f_s}(t)}\delta ({\svector x}-{{\svector x}_s}) \delta(t) ,
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1941}%
$\displaystyle
{{f_s}(t)}= {{\svector \nabla}\cdot}[ {\svector{f}(t)}/ \rho({{\svector x}_s})],
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1429}%
${\svector{f}(t)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1944}%
$\displaystyle
\left. {P_s {({\svector x}, t)}}\right|_{z \leq 0} = 0 ,~ 
\left. {P_s {({\svector x}, t)}}\right|_{t \leq 0} = 0 ,~ \mbox{and}
\left. {{\partial ^2}\over{\partial t^2}}{P_s {({\svector x}, t)}}\right|_{t \leq 0} = 0 ,
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1431}%
$z = x_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1433}%
${\svector x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1435}%
$P$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1439}%
$\partial \Omega$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1441}%
${V_{max}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1443}%
$T$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1953}%
$\displaystyle \mbox{ the surface }
\partial \Omega$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1954}%
$\textstyle =$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1955}%
$\displaystyle \{ {\svector x}~|~ z = 0 ~~ \mbox{or} ~ \parallel {\svector x}\parallel
= T {V_{max}}\},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1956}%
$\displaystyle \mbox{ for the region }
\Omega$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1958}%
$\displaystyle \{ {\svector x}~|~ z \geq 0 ~~ \mbox{and} ~
\parallel {\svector x}\parallel \leq T {V_{max}}\}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1962}%
$\displaystyle
\left. {P_s {({\svector x}, t)}}\right|_{\mathbf{x} \in \partial \Omega} = 0.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1965}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\left \{ \left [ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{{\partial ^2}\over{\partial t^2}}-
{{\svector \nabla}\cdot}\left ( {1 \over {{\rho({\svector x})}}}{\svector \nabla}\right ) \right ] ~ {P_s {({\svector x}, t)}}
- {{f_s}(t)}\delta ({\svector x}- {{\svector x}_s}) \delta(t) \right \}
{q {({\svector x}, t)}}{~d {\svector x}\, d t}= 0,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1449}%
$H^2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1451}%
${q {({\svector x}, t)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1453}%
$q$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1455}%
${D_{rs}(t)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1465}%
$\rho$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1467}%
$c$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1469}%
${{f_s}(t)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1979}%
$\displaystyle J [ \rho, c, {f_s}]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1981}%
$\displaystyle {1 \over 2} {\sum_{r,s}}{\int_0^T}
\left [ {D_{rs}(t)}- {1 \over \rho ({{\svector x}_r})}
{\partial P ({{\svector x}_r}, t ) \over \partial z}\right ]^2 {~w(t)}dt$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1983}%
$\displaystyle {1 \over 2} {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}
\left [ {D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ]^2
{\delta ({\svector x}- {{\svector x}_r})}{~d {\svector x}\, w(t) \, d t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1471}%
${~w(t)}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1477}%
$s$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1481}%
$w(t)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1995}%
$\displaystyle
{{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}L_1 [ {{\svector m}{({\svector x}, t)}}] {P_s {({\svector x}, t)}}{~d {\svector x}\, d t}= 0, ~ \forall {q {({\svector x}, t)}}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay1997}%
$\displaystyle
\mbox{where } {{\svector m}{({\svector x}, t)}}= [ {c({\svector x})}, {\rho({\svector x})}, {f_s}].
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1491}%
${{\svector m}{({\svector x}, t)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1493}%
$L_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1501}%
$1/c^2 \rho$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1503}%
$1/\rho$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1505}%
${f_s}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1509}%
${\svector m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2010}%
$\displaystyle
J [ {{\svector m}{({\svector x}, t)}}] = {1 \over 2} {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}
\left \{ {D_{rs}(t)}- L_2 [{{\svector m}{({\svector x}, t)}}, t ] {P_s {({\svector x}, t)}}\right \}^2 {\delta ({\svector x}- {{\svector x}_r})}{~w(t)}{~d {\svector x}\, d t}.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1511}%
$\delta {\svector m}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1513}%
${\delta P_s}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2015}%
$\displaystyle
{{\int_0^T}\! {\int_{\Omega}}}q L_1 {\delta P_s}{~d {\svector x}\, d t}= 
- {{\int_0^T}\! {\int_{\Omega}}}q {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}( L_1 P ) \cdot \delta {\svector m}{~d {\svector x}\, d t}, 
~ \forall q.
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2017}%
$\displaystyle {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}G({\svector m}) \cdot \delta {\svector m}
= \left. {d \over d \epsilon} G ( {\svector m}+ \epsilon \delta {\svector m}) 
\right|_{\epsilon = 0}, \mbox{ where } G({\svector m}) = L_1 P ,
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1523}%
${{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1527}%
$a$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1529}%
$b$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1531}%
$L^*$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1533}%
$L$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1535}%
$<a | Lb>_a = <L^* a| b>_b$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2032}%
$\displaystyle < q_1 | q_2 >_q$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2034}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}q_1 {({\svector x}, t)}q_2 {({\svector x}, t)}{~d {\svector x}\, d t},$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2035}%
$\displaystyle < P_1 | P_2 >_P$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2037}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}P_1 {({\svector x}, t)}P_2 {({\svector x}, t)}{~d {\svector x}\, d t}, \mbox{ and}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2038}%
$\displaystyle < m_1 | m_2 >_m$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2040}%
$\displaystyle {\int_{\Omega}}m_1 ({\svector x}) m_2 ({\svector x}) ~ d {\svector x}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2042}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{\delta P_s}L_1^* q {~d {\svector x}\, d t}=
- {\int_{\Omega}}\delta {\svector m}{\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)]^* q ~ dt {~d {\svector x}}, ~ \forall q .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1543}%
$L_1^*$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1549}%
$J$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1553}%
$L_2$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2052}%
$\displaystyle - \delta J$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2054}%
$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P_s) L_2 \delta P_s ~w~ {~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2055}%
$\textstyle +$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2056}%
$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P_s) {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_2 P_s) \cdot \delta {\svector m}~w~ {~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2058}%
$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}\delta P_s L_2^* [ {\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P_s) ~w] {~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2060}%
$\displaystyle {\sum_{r,s}}{\int_{\Omega}}\delta {\svector m}\cdot {\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_2 P_s) ]^*
[ {\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P_s) ~w] dt {~d {\svector x}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1559}%
$\delta P$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2065}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\delta P L_1^* q {~d {\svector x}\, d t}= {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}\delta P L_2^*
[ {\delta ({\svector x}- {{\svector x}_r})}({D_{rs}(t)}- L_2P) ~w ] {~d {\svector x}\, d t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2069}%
$\displaystyle L_1^* q = {\sum_{r,s}}L_2^* [ {\delta ({\svector x}- {{\svector x}_r})}({D_{rs}(t)}- L_2 P) w] .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2074}%
$\displaystyle \delta J$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2076}%
$\displaystyle {\int_{\Omega}}\delta {\svector m}{\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)]^* q ~ dt ~ d {\svector x}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2077}%
$\textstyle -$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2078}%
$\displaystyle {\sum_{r,s}}{\int_{\Omega}}\delta {\svector m}{\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_2 P) ]^*
[ {\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P) w ] ~ dt ~ d {\svector x}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1571}%
${\svector m}({\svector x})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2082}%
$\displaystyle {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}J [{\svector m}({\svector x})] =  \left. {\delta J \over \delta {\svector m}} \right |_{{\svector m}({\svector x})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2084}%
$\displaystyle {\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_1 P)]^* q ~dt$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2086}%
$\displaystyle {\sum_{r,s}}{\int_0^T}[ {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}(L_2 P) ]^* [ {\delta ({\svector x}- {{\svector x}_r})}( {D_{rs}(t)}- L_2 P) w ] ~dt .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1577}%
$\{{{\svector x}_r}\}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2091}%
$\displaystyle \min_\alpha J [ {\svector m}_n + \alpha {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}J ({\svector m}_n) ], \mbox{ and }
{\svector m}_{n+1} = {\svector m}_n + \alpha {{\svector \nabla}_{\mbox{\scriptsize\boldmath m}}}J({\svector m}_n) .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1581}%
${\svector m}_{n+1}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

\stepcounter{section}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2095}%
$\displaystyle - \delta J [ \rho, c, {f_s}]$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2097}%
$\displaystyle - {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- {{\svector x}_r})}
\left [ {D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ] {1 \over {{\rho({\svector x})}}}{{\partial}\over{\partial z}}{\delta {P_s {({\svector x}, t)}}}{~d {\svector x}\, w(t) \, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2099}%
$\displaystyle {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta ({\svector x}- {{\svector x}_r})}
\left [ {D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ] {{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}{\delta {\rho({\svector x})}\over {\rho({\svector x})}^2} {~d {\svector x}\, w(t) \, d t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1583}%
$z$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2102}%
$\displaystyle - \delta J [ \rho, c, {f_s}] = {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}{\delta {P_s {({\svector x}, t)}}}{{\partial}\over{\partial z}}
\left \{ {\delta ({\svector x}- {{\svector x}_r})}{1 \over {{\rho({\svector x})}}}\left [ {D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ] \right \} {~d {\svector x}\, w(t) \, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2103}%
$\displaystyle + {\sum_{r,s}}{{\int_0^T}\! {\int_{\Omega}}}\delta {\rho({\svector x})}
\left \{ {\delta ({\svector x}- {{\svector x}_r})}{1 \over {{\rho({\svector x})}}^2}\left [ {D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ] {{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}
\right \} {~d {\svector x}\, w(t) \, d t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1585}%
$\delta {P_s {({\svector x}, t)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2107}%
$\textstyle \hfil$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2108}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\left [ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{{\partial ^2}\over{\partial t^2}}-
{{\svector \nabla}\cdot}\left ( {1 \over {{\rho({\svector x})}}}{\svector \nabla}\right ) \right ] ~ {\delta {P_s {({\svector x}, t)}}}{~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2110}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}\left [ {1 \over {{c({\svector x})}^3 {\rho({\svector x})}}}{{\partial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}\right ] \delta {c({\svector x})}{~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2112}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}\left \{ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}^2 }}{{\partial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}
- {{\svector \nabla}\cdot}\left [ {1 \over {{\rho({\svector x})}}^2}{\svector \nabla}{P_s {({\svector x}, t)}}\right ]
\right \} \delta {\rho({\svector x})}{~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2114}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}{q {({\svector x}, t)}}\delta ({\svector x}) \delta {{f_s}(t)}{~d {\svector x}\, d t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2116}%
$\displaystyle {\int_{\Omega}}a {{\svector \nabla}\cdot}{\svector b}{~d {\svector x}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2118}%
$\displaystyle {\int_{\Omega}}{{\svector \nabla}\cdot}(a {\svector b}) {~d {\svector x}}- {\int_{\Omega}}{\svector b}\cdot {\svector \nabla}a {~d {\svector x}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2120}%
$\displaystyle {\int_{\partial \Omega}}{\svector {\hat n} }\cdot {\svector b}\, a \, d \sigma - {\int_{\Omega}}{\svector b}\cdot {\svector \nabla}a {~d {\svector x}}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1591}%
${\svector b}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1593}%
${\svector {\hat n} }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1597}%
$\Omega$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2127}%
$\displaystyle \left. {\int_{\Omega}}{q {({\svector x}, t)}}{1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{{\partial}\over{\partial t}}{\delta {P_s {({\svector x}, t)}}}{~d {\svector x}}\right |_{t=T}
-
\left. {\int_{\Omega}}{\partial {q {({\svector x}, t)}}\over \partial t} {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}
{\delta {P_s {({\svector x}, t)}}}{~d {\svector x}}\right |_{t=T}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2128}%
$\displaystyle +{{\int_0^T}\! {\int_{\Omega}}}{\delta {P_s {({\svector x}, t)}}}{1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{\partial^2 {q {({\svector x}, t)}}\over \partial t^2} {~d {\svector x}\, d t}
-
{\int_0^T}\! {\int_{\partial \Omega}}{q {({\svector x}, t)}}{1 \over {{\rho({\svector x})}}}
{\partial \over \partial n} {\delta {P_s {({\svector x}, t)}}}\, d \sigma \, dt$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2129}%
$\displaystyle - {{\int_0^T}\! {\int_{\Omega}}}{\delta {P_s {({\svector x}, t)}}}{{\svector \nabla}\cdot}\left [ {1 \over {{\rho({\svector x})}}}{\svector \nabla}{q {({\svector x}, t)}}\right ] {~d {\svector x}\, d t}
=$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2130}%
$\displaystyle {{\int_0^T}\! {\int_{\Omega}}}\delta {c({\svector x})}\left [ {1 \over {{c({\svector x})}^3 {\rho({\svector x})}}}{{\partial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}\right ] {q {({\svector x}, t)}}{~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2131}%
$\displaystyle - {{\int_0^T}\! {\int_{\Omega}}}\delta {\rho({\svector x})}\left [ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}^2 }}{{\partial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}\right.
-
\left. {1 \over {{\rho({\svector x})}}^2}{\svector \nabla}{P_s {({\svector x}, t)}}\cdot {\svector \nabla}\right ] {q {({\svector x}, t)}}{~d {\svector x}\, d t}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2132}%
$\displaystyle + {\int_0^T}\! {\int_{\partial \Omega}}\delta {\rho({\svector x})}
\left [ {1 \over {{\rho({\svector x})}}^2}{\partial {P_s {({\svector x}, t)}}\over \partial n} \right ] {q {({\svector x}, t)}}
\, d \sigma \, dt
- {{\int_0^T}\! {\int_{\Omega}}}\delta {{f_s}(t)}\delta ({\svector x}) {q {({\svector x}, t)}}{~d {\svector x}\, d t}.$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1599}%
$t=0$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1605}%
$\partial P / \partial n = {\svector {\hat n} }\cdot {\svector \nabla}P$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2141}%
$\displaystyle \left \{ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}}}{{\partial ^2}\over{\partial t^2}}- {{\svector \nabla}\cdot}\left [ {1 \over {{\rho({\svector x})}}}{\svector \nabla}\right ] \right \} {q {({\svector x}, t)}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2142}%
$\displaystyle = - {\sum_{r,s}}{{\partial}\over{\partial z}}\left \{ {\delta ({\svector x}- {{\svector x}_r})}{ {~w(t)}\over {\rho({\svector x})}} \left [
{D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ] \right \}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1615}%
$t=T$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2146}%
$\displaystyle \left. {q {({\svector x}, t)}}\right |_{t=T} =
\left. {{\partial}\over{\partial t}}{q {({\svector x}, t)}}\right |_{t=T} =
\left. {q {({\svector x}, t)}}\right |_{\partial \Omega} = 0 .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1623}%
$\delta J$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2154}%
$\displaystyle {\svector \nabla}_c J({\svector x})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2156}%
$\displaystyle {\delta J \over \delta {c({\svector x})}} =
{\int_0^T}{1 \over {{c({\svector x})}^3 {\rho({\svector x})}}}{{\partial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}{q {({\svector x}, t)}}dt .$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2157}%
$\displaystyle {\svector \nabla}_{\rho} J({\svector x})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2159}%
$\displaystyle {\delta J \over \delta {\rho({\svector x})}} =
{\int_0^T}\left [ {1 \over {{c({\svector x})}^2 {\rho({\svector x})}^2 }}{{\partial^2 {P_s {({\svector x}, t)}}}\over{\partial t^2}}- {1 \over {{\rho({\svector x})}}^2}{\svector \nabla}{P_s {({\svector x}, t)}}\cdot {\svector \nabla}\right ] {q {({\svector x}, t)}}dt$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2160}%
$\textstyle \,$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2161}%
$\displaystyle - {\sum_{r,s}}{\int_0^T}{{\delta ({\svector x}- {{\svector x}_r})}\over {\rho({\svector x})}^2} \left [ {D_{rs}(t)}- {1 \over {{\rho({\svector x})}}}{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}\right ]
{{\partial {P_s {({\svector x}, t)}}}\over{\partial z}}{~w(t)}dt$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2162}%
$\displaystyle {\svector \nabla}_s J(t)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2164}%
$\displaystyle {\delta J \over \delta {{f_s}(t)}} =
\left. {q {({\svector x}, t)}}\right |_{\mathbf{x} = \mathbf{0}}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1633}%
$\frac{1}{c^3 \rho}\frac{\partial^2P}{\partial t^2}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1639}%
${\svector \nabla}_{\rho} J_1$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}

{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline1645}%
${{f_s}(t)}= \delta(t)$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}


\end{document}
