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2.5.6 Linear stability analysis of original PDE

Lets see what happens to waves in the linear advection equation (63). For the ansatz

\rho \! \left( x, t \right) = A \! \left( t \right) e^{- jk x}
\end{displaymath} (130)
with \bgroup\color{DEFcolor}$j^2=-1$\egroup we get
\frac{\mathrm{d} A}{\mathrm{d} t} + v_{} \left( - jk \right...
...ce \Rightarrow \enspace
A = A_0   e^{ jv_{} k t}
\enspace ,
\end{displaymath} (131)
\rho = A_0   e^{j\left( \omega t - k x \right)}
\end{displaymath} (132)
\mbox{abs} \! \left( A \right) = \mbox{abs} \! \left( A_0 \right) = \mbox{const}
\enspace ,
\end{displaymath} (133)
\omega = v_{} k
\enspace .
\end{displaymath} (134)
Dispersion relation (134): no dispersion, all waves move with the same speed \bgroup\color{DEFcolor}$v_{}$\egroup.

Eq. (133): amplitude remains constant - without any diffusion.