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\begin{document}
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% !!! IMAGES START HERE !!!

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$\displaystyle v({\svector x})^{-2} \ddot{p} ( {\svector x}, t )
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{\newpage\clearpage
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$\displaystyle =$%
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$\displaystyle w ( {\svector x}, t );$%
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$\displaystyle {\rm or } ~~ 4  \pi^2 f^2 s({\svector x})^2 p ( {\svector x}, f )+ \nabla^2p ( {\svector x}, f )$%
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$ p$%
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$ {\svector x}$%
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$ v$%
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$ s$%
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$ w$%
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$ f$%
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{\newpage\clearpage
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$ - i 2 \pi$%
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$ G$%
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{\newpage\clearpage
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$\displaystyle 4  \pi^2 f^2 s({\svector x})^2 G ({\svector x}, {\svector x}' ,f)$%
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{\newpage\clearpage
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$\displaystyle +$%
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{\newpage\clearpage
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$\displaystyle \nabla^2G ({\svector x}, {\svector x}' ,f)=
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{\newpage\clearpage
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$\displaystyle {\rm so ~ that } ~~ p ( {\svector x}, f )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay730}%
$\displaystyle \iiint G ({\svector x}, {\svector x}' ,f)w ( {\svector x}' ,f ) d {\svector x}' .$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay733}%
$\displaystyle 4  \pi^2 f^2 s({\svector x})^2 \Delta G ({\svector x}, {\svector x}' ,f)+ \nabla^2\Delta G ({\svector x}, {\svector x}' ,f)$%
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{\newpage\clearpage
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$\displaystyle -8  \pi^2 f^2 s({\svector x}) G ({\svector x}, {\svector x}' ,f)\Delta s({\svector x}),$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay740}%
$\displaystyle \Delta G ({\svector x}, {\svector x}'' ,f)=
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$\displaystyle \frac{\partial G ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )} =8  \pi^2 f^2 s( {\svector x}' ) G ({\svector x}, {\svector x}' ,f)G ( {\svector x}' , {\svector x}'' ,f).$%
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{\newpage\clearpage
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$\displaystyle \frac{\partial p ( {\svector x}, f )}{\partial s( {\svector x}' )}$%
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{\newpage\clearpage
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$\displaystyle \iiint
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay754}%
$\displaystyle \iiint 8  \pi^2 f^2 s( {\svector x}' ) G ({\svector x}, {\svector x}' ,f)G ( {\svector x}' , {\svector x}'' ,f)w( {\svector x}'' ,f) d {\svector x}'' .$%
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{\newpage\clearpage
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$\displaystyle \theta ({\svector x}, {\svector x}'' ,f)$%
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$\displaystyle \equiv$%
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$\displaystyle \log G ({\svector x}, {\svector x}'' ,f), { ~ \rm and }$%
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{\newpage\clearpage
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$\displaystyle \frac{\partial \theta ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )} G ({\svector x}, {\svector x}'' ,f)$%
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{\newpage\clearpage
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$\displaystyle \frac{\partial G ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )} ,$%
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{\newpage\clearpage
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$\displaystyle \iiint
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{\newpage\clearpage
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$\displaystyle {\rm where ~~}
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay781}%
$\displaystyle 8  \pi^2 f^2 s( {\svector x}' ) \frac{G ({\svector x}, {\svector x}' ,f)G ( {\svector x}' , {\svector x}'' ,f)}{G ({\svector x}, {\svector x}'' ,f)} .$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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$ {\svector x}' $%
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{\newpage\clearpage
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$ {\svector x}'' $%
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{\newpage\clearpage
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$ w( {\svector x}'' ,f)$%
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{\newpage\clearpage
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$\displaystyle \phi ( {\svector x}, f )$%
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{\newpage\clearpage
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$\displaystyle \frac{\partial \phi ( {\svector x}, f )}{\partial s( {\svector x}' )}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay805}%
$\displaystyle \frac
{\iiint 8  \pi^2 f^2 s( {\svector x}' ) G ({\svector x}, {\svector x}' ,f)G ( {\svector x}' , {\svector x}'' ,f)w( {\svector x}'' ,f) d {\svector x}'' }
{\iiint G ({\svector x}, {\svector x}''' ,f)w( {\svector x}''' ,f) d {\svector x}''' } .$%
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{\newpage\clearpage
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$ p ( {\svector x}, f )$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay811}%
$\displaystyle G ({\svector x}, {\svector x}' ,t)$%
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{\newpage\clearpage
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{\newpage\clearpage
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$ u$%
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$ \tau$%
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{\newpage\clearpage
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$\displaystyle p ( {\svector x}, t )= \iiint u ({\svector x}, {\svector x}' )w [  {\svector x}' , t - \tau ({\svector x}, {\svector x}' )] d {\svector x}' .$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
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$\displaystyle \left | {\svector \nabla}\tau ({\svector x}, {\svector x}' )\right |^2 = s ({\svector x})^2 ,$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay840}%
$\displaystyle {\rm ~ and ~ }
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$ {\svector \nabla}$%
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$ \nabla^2$%
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{\newpage\clearpage
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$\displaystyle \left | {\svector \nabla}v ({\svector x}) \right| \ll 2 \pi \left| f \right| .$%
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{\newpage\clearpage
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$\displaystyle \frac{\partial \theta ({\svector x}, {\svector x}'' ,f)}{\partial s( {\svector x}' )}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay860}%
$\displaystyle 8  \pi^2 f^2 s ( {\svector x}' )
\frac{ u ({\svector x}, {\svector x}' )u ( {\svector x}' , {\svector x}'' )}{u ({\svector x}, {\svector x}'' )} \exp \{ -i 2 \pi f
[ \tau ({\svector x}, {\svector x}' )+ \tau ( {\svector x}' , {\svector x}'' )- \tau ({\svector x}, {\svector x}'' )] \} ,$%
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{\newpage\clearpage
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$\displaystyle \frac{\partial \theta ({\svector x}, {\svector x}'' ,t)}{\partial s( {\svector x}' )}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay866}%
$\displaystyle -2 s ( {\svector x}' )
\frac{ u ({\svector x}, {\svector x}' )u ( {\svector x}' , {\svector x}'' )}{u ({\svector x}, {\svector x}'' )}
\ddot{\delta} [ t - \tau ({\svector x}, {\svector x}' )- \tau ( {\svector x}' , {\svector x}'' )+ \tau ({\svector x}, {\svector x}'' )] , {\rm and ~}$%
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$\displaystyle \frac{\partial G ({\svector x}, {\svector x}'' ,t)}{\partial s( {\svector x}' )}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay872}%
$\displaystyle -2 s ( {\svector x}' )
u ({\svector x}, {\svector x}' )u ( {\svector x}' , {\svector x}'' )\ddot{\delta} [t - \tau ({\svector x}, {\svector x}' )- \tau ( {\svector x}' , {\svector x}'' )+ \tau ({\svector x}, {\svector x}'' )] .$%
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$\displaystyle \frac{\partial p ( {\svector x}, t )}{\partial s( {\svector x}' )}$%
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{\newpage\clearpage
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$\displaystyle \int\iiint
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\lthtmlindisplaymathZ
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{\newpage\clearpage
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$\displaystyle \iiint -2 s (  {\svector x}' ) u ({\svector x}, {\svector x}' )u ( {\svector x}' , {\svector x}'' )
\ddot{w} [ {\svector x}'' , t - \tau ({\svector x}, {\svector x}' )- \tau ( {\svector x}' , {\svector x}'' )+ \tau ({\svector x}, {\svector x}'' )] d  {\svector x}'' .$%
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$\displaystyle w ( {\svector x}, f )= w(f) \delta ( {\svector x}- {\svector x}_0), {\rm ~~ then ~}
p ( {\svector x}, f )= G ({\svector x},{\svector x}_0,f) w(f) .$%
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$\displaystyle \int
\frac{\partial \theta ( {\svector x}, {\svector x}_0, t - t')}{\partial s( {\svector x}' )} p ({\svector x},t') dt'$%
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$\displaystyle -2 s( {\svector x}' ) \frac{ u ({\svector x}, {\svector x}' )u  (  {\svector x}' , {\svector x}_0) }{u  ( {\svector x}, {\svector x}_0) }
\ddot{p} [ {\svector x}, t - \tau ({\svector x}, {\svector x}' )- \tau  (  {\svector x}' , {\svector x}_0) + \tau  ( {\svector x}, {\svector x}_0) ]$%
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$\displaystyle -2 s( {\svector x}' ) u ({\svector x}, {\svector x}' )u  (  {\svector x}' , {\svector x}_0) \ddot{w} [ t - \tau ({\svector x}, {\svector x}' )- \tau  (  {\svector x}' , {\svector x}_0) + \tau  ( {\svector x}, {\svector x}_0) ] .$%
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$ \ddot{w}(t)$%
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$ u ({\svector x}, {\svector x}' )$%
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