2.5.6 Linear stability analysis of original PDE

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

$\begin{displaymath} \rho \! \left( x, t \right) = A \! \left( t \right) e^{- jk x} \end{displaymath}$

(130)

with $\bgroup\color{DEFcolor}$j^2=-1$\egroup$ we get

$\begin{displaymath} \frac{\mathrm{d} A}{\mathrm{d} t} + v_{} \left( - jk \right... ...ce \Rightarrow \enspace A = A_0 e^{ jv_{} k t} \enspace , \end{displaymath}$

(131)

$\begin{displaymath} \rho = A_0 e^{j\left( \omega t - k x \right)} \end{displaymath}$

(132)

with

$\begin{displaymath} \mbox{abs} \! \left( A \right) = \mbox{abs} \! \left( A_0 \right) = \mbox{const} \enspace , \end{displaymath}$

(133)

$\begin{displaymath} \omega = v_{} k \enspace . \end{displaymath}$

(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.