Inviscid fluid flow
$\newcommand{\prtl}[2]{\frac{\partial #1}{\partial #2}}$
\begin{align} &\mbox{Mass: } & \prtl{\rho}{t} + u\prtl{\rho}{x} +\rho\prtl{u}{x}= 0 &\rightarrow \frac{D\rho}{Dt} &=-\rho\prtl{u}{x}\\ &\mbox{Mom: } & \rho\prtl{u}{t} + \rho u\prtl{u}{x} + \prtl{P}{x} = 0 &\rightarrow \frac{Du}{Dt} &=-\frac{1}{\rho}\prtl{P}{x}\\ &\mbox{Energy: } & \rho\prtl{e}{t} + \rho u\prtl{e}{x} + P\prtl{u}{x} = 0 &\rightarrow \frac{D e}{Dt} &=-\frac{P}{\rho}\prtl{u}{x} \end{align}Convert energy equation to a pressure equation. $P=\rho RT$, so
$$\frac{DP}{Dt} = RT\frac{D\rho}{Dt} + \rho R\frac{DT}{Dt} = RT\frac{D\rho}{Dt} + \frac{\rho R}{c_v}\frac{De}{Dt}$$Now insert mass and energy equations.
\begin{align} \frac{DP}{Dt} &= -\rho RT\prtl{u}{x} - \frac{R}{c_v}P\prtl{u}{x} \\ &= -P\prtl{u}{x} - P(\gamma-1)\prtl{u}{x} \\ &= -P\gamma\prtl{u}{x}, \end{align}where $R/c_v=\gamma-1$ is used, where $\gamma = c_p/c_v$.
So, Now our equations are
\begin{align} &\mbox{Mass: } & \prtl{\rho}{t} + u\prtl{\rho}{x} + \rho\prtl{u}{x} = 0 \\ &\mbox{Mom: } & \prtl{u}{t} + u\prtl{u}{x} + \frac{1}{\rho}\prtl{P}{x} = 0 \\ &\mbox{Energy: } & \prtl{P}{t} + \gamma P\prtl{u}{x} + u\prtl{P}{x} = 0 \end{align}We can write this in vector notation as $$\prtl{\mathbf{U}}{t} + \mathbf{A}\prtl{\mathbf{U}}{x},$$ where $$\mathbf{U} = \begin{pmatrix} \rho \\ u \\ P \end{pmatrix},$$ $$\mathbf{A} = \begin{pmatrix} u & \rho & 0 \\ 0 & u & 1/ \rho \\ 0 & \gamma P & u \end{pmatrix}$$
We can decouple the system by diagonalizing it: $Ax = \lambda x\rightarrow AV = V\Lambda\rightarrow A=V\Lambda V^{-1},$ where $\Lambda$ is a diagonal matrix of eigenvalues of $A$, and the columns of matrix $V$ are the corresponding eigenvectors.
$$\prtl{\mathbf{U}}{t} + \mathbf{A}\prtl{\mathbf{U}}{x} \rightarrow\mathbf{V}^{-1}\prtl{\mathbf{U}}{t} + \Lambda\mathbf{V}^{-1}\prtl{\mathbf{U}}{x} \rightarrow \prtl{\hat{\mathbf{U}}}{t}+\Lambda\prtl{\hat{\mathbf{U}}}{x}=0.$$(The last step assumes constant $V^{-1}$, which isn't true, but the point is to give a conceptual illustration.) This is the decoupled system, and each equation has the form of the 1-D wave equation, where the eigenvalues are wave speeds.
The eigenvalues are found analytically from $\det(\mathbf{A}-\lambda_iI)=0$. This gives
$$\det\begin{bmatrix} u-\lambda & \rho & 0 \\ 0 & u-\lambda & 1/\rho \\ 0 & \gamma P & u-\lambda \end{bmatrix} = 0.$$$$\rightarrow (u-\lambda)[(u-\lambda)(u-\lambda)-c^2]=0.$$Hence,
$$\begin{align} \lambda_1 &=u, \\ \lambda_2 &= u+c, \\ \lambda_3 &= u-c.\end{align}$$Here, $c$ is the sound speed ($c^2=\gamma P/\rho$). The Euler equation wave speeds are the advection velocity, and the advection velocity plus and minus the sound speed. Often, $c>>u$, and we have a stiff system.