Reduction to the 2.5D case

Let us consider an environment with the following set of conditions:

\begin{displaymath}
\mbox{$\mathbf{C}$}=
\left[
\begin{array}{cc}
c_{nn} & 0 \\
0 & 0
\end{array} \right]
\end{displaymath}

(i.e., there is no dependence on the second component of $\mbox{$\mathbf{n}$}$, and indexing is being omitted by obvious reasons) and

\begin{displaymath}
\mbox{$\mathbf{Q}$}(s) =
\left[
\begin{array}{cc}
q(s) & 0 \\
0 & q_{\perp} (s)
\end{array} \right]
\end{displaymath}

so the matrix contains only diagonal elements, and we introduced the notation

\begin{displaymath}q(s) = q_{11}(s) \hskip5mm , \hskip5mm q_{\perp}(s) = q_{22}(s)  , \end{displaymath}

so det $\mbox{$\mathbf{Q}$}= q(s)q_{\perp}(s)$. It follows from the system given by Eq.(3.2) that

\begin{displaymath}p_{12}(s) = p_{21}(s) = 0  , \end{displaymath}

and

\begin{displaymath}
\begin{array}{ccl}
\displaystyle { \frac{d}{ds}} q = c_0 \; ...
..., &
\displaystyle { \frac{d}{ds}} p_{\perp} = 0 \
\end{array}\end{displaymath}

where $p = p_{11}$ and $p_{\perp} = p_{22}$. The solution of the last equation is trivial and corresponds to

\begin{displaymath}p_{\perp}(s) = p_{\perp}(0) = \makebox{constant}  . \end{displaymath}

It follows further that

\begin{displaymath}q_{\perp}(s) = p_{\perp}(0)I_c(s) + q_{\perp}(0)  , \end{displaymath}

where

\begin{displaymath}I_c(s) = \int c_0(s) ds  . \end{displaymath}

Recalling $\mbox{$\mathbf{n}$}$ as two-dimensional vector one gets that
\begin{displaymath}
\mbox{$\mathbf{M}$}\mbox{$\mathbf{n}$}\cdot\mbox{$\mathbf{n}$}= \frac {p(s)}{q(s)}n_1^2 + \frac {1}{I_c(s)}n_2^2  .
\end{displaymath} (3.6)

In the case of a point source it follows that

\begin{displaymath}q_{\perp}(s) = \frac {\cos\theta(0)}{c(0,0,0)} I_c(s) \sim s  , \end{displaymath}

which indicates that the parameter is proportional to the ray path.

Orlando Camargo Rodríguez 2012-06-21