1.5.4 Eigenvalues for the Euler equations

For the 1D Euler equations (40) we get the eigenvalues

$${ \renewedcommand{arraystretch}{1.2} \left( \! \begin{array}{l} v \\ v+c \\ v-c \end{array} \! \right) \enspace , }$$ (50)

corresponding to the flow with velocity $v$ and two acoustic waves travelling with sound speed $\pm c$ relative to the flow.