\begin{table}%t1
\par
\caption {\label{tab:expansions}Sine expansions of the first 5 eigenmodes~($k$) when expanded linearly in the stratification parameter.}
%\centerline
 {\begin{tabular}{c|cccccc}
\hline\hline
\multicolumn{7}{c}{Displacement:}\\
\hline
$k j$ & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
1 & 1 & 0 & --.021221 & 0 & --.001011 & 0\\
2 & 0 & 1 & 0 & --.064672 & 0 & --.004042\\
3 & 0.190986 & 0 & 1 & 0 & --.113682 & 0\\
4 & 0 & 0.258690 & 0 & 1 & 0 & --.164621\\
5 & 0.025263 & 0 & 0.315784 & 0 & 1 & 0\\
\hline\hline
\end{tabular} }
%\centerline
 {\begin{tabular}{c|cccccc}
\multicolumn{7}{c}{Compression:}\\
\hline
$k j$ & 1 & 2 & 3 & 4 & 5 & 6\\
\hline
1 & 1 & 0 & --.190986 & 0 & --.025263 & 0\\
2 & 0 & 1 & 0 & --.258690
 & 0 & --.036378\\
3 & 0.021221 & 0 & 1 & 0 & --.315784 & 0\\
4 & 0 & 0.064672 & 0 & 1 & 0 & --.370397\\
5 & 0.001011 & 0 & 0.113682 & 0 & 1 & 0\\
\hline
\end{tabular}}
\par
\smallskip 
The off-diagonal numbers (i.e. $\left((RX)^{{\rm ex}j}_k\right)_1$ and $\left((RP)^{{\rm ex}j}_k\right)_1$) are the coefficients of $\alpha_1$ in the linear expansion.
\end{table}