Editing Euler equations (fluid dynamics) (section)

===Waves in 1D inviscid, nonconductive thermodynamic fluid===
If one considers Euler equations for a thermodynamic fluid with the two further assumptions of one spatial dimension and free (no external field: ''g''&nbsp;=&nbsp;0):
<math display="block">\begin{align} 
  {\partial v \over \partial t} + u{\partial v \over \partial x} - v {\partial u \over \partial x} &= 0,\\[1.2ex]
  {\partial u \over \partial t} + u{\partial u \over \partial x} - e_{vv} v {\partial v \over \partial x} - e_{vs}v {\partial s \over \partial x} &= 0,\\[1.2ex]
  {\partial s \over \partial t} + u{\partial s \over \partial x} &= 0.
\end{align}</math>

If one defines the vector of variables:
<math display="block">\mathbf{y} = \begin{pmatrix}v \\ u \\ s\end{pmatrix},</math>
recalling that <math>v</math> is the specific volume, <math>u</math> the flow speed, <math>s</math> the specific entropy, the corresponding jacobian matrix is:
<math display="block">{\mathbf A}=\begin{pmatrix}u & -v & 0 \\ - e_{vv} v & u & - e_{vs} v \\ 0 & 0 & u \end{pmatrix}.</math>

At first one must find the eigenvalues of this matrix by solving the [[characteristic equation (calculus)|characteristic equation]]:
<math display="block">\det(\mathbf A(\mathbf y) - \lambda(\mathbf y) \mathbf I) = 0,</math>

that is explicitly:
<math display="block">\det\begin{bmatrix}u-\lambda & -v & 0 \\ - e_{vv} v & u-\lambda & - e_{vs} v \\ 0 & 0 & u-\lambda \end{bmatrix}=0.</math>

This [[determinant]] is very simple: the fastest computation starts on the last row, since it has the highest number of  zero elements.
<math display="block">(u-\lambda) \det \begin{bmatrix}u-\lambda & -v \\ - e_{vv} v & u -\lambda \end{bmatrix}=0.</math>

Now by computing the determinant 2×2:
<math display="block">(u - \lambda)\left((u - \lambda)^2 - e_{vv} v^2\right) = 0,</math>
by defining the parameter:
<math display="block">a(v,s) \equiv v \sqrt {e_{vv}},</math>
or equivalently in mechanical variables, as:
<math display="block">a(\rho,p) \equiv \sqrt {\partial p \over \partial \rho}.</math>

This parameter is always real according to the [[second law of thermodynamics]]. In fact the second law of thermodynamics can be expressed by several postulates. The most elementary of them in mathematical terms is the statement of convexity of the fundamental equation of state, i.e. the [[hessian matrix]] of the specific energy expressed as function of specific volume and specific entropy:
<math display="block"> \begin{pmatrix}e_{vv} & e_{vs} \\ e_{vs} & e_{ss} \end{pmatrix},</math>
is defined positive. This statement corresponds to the two conditions:
<math display="block">\left\{\begin{align} 
                   e_{vv} &> 0 \\[1.2ex]
  e_{vv}e_{ss} - e_{vs}^2 &> 0
\end{align}\right.</math>

The first condition is the one ensuring the parameter ''a'' is defined real.

The characteristic equation finally results:
<math display="block">(u - \lambda)\left((u - \lambda)^2 - a^2\right) = 0</math>

That has three real solutions:
<math display="block">\lambda_1(v,u,s) = u-a(v,s), \qquad \lambda_2(u)= u, \qquad \lambda_3(v,u,s) = u+a(v,s).</math>

Then the matrix has three real eigenvalues all distinguished: the 1D Euler equations are a ''strictly'' hyperbolic system.

At this point one should determine the three eigenvectors: each one is obtained by substituting one eigenvalue in the eigenvalue equation and then solving it. By substituting the first eigenvalue &lambda;<sub>1</sub> one obtains:
<math display="block">\begin{pmatrix}a  & -v & 0 \\ - e_{vv} v & a & - e_{vs} v \\ 0 & 0 & a \end{pmatrix} \begin{pmatrix}v_1\\ u_1  \\s_1 \end{pmatrix}=0.</math>

Basing on the third equation that simply has solution ''s''<sub>1</sub>=0, the system reduces to:
<math display="block">\begin{pmatrix}a  & -v \\-a^2 /v& a  \end{pmatrix} \begin{pmatrix}v_1\\ u_1 \end{pmatrix}=0</math>

The two equations are redundant as usual, then the eigenvector is defined with a multiplying constant. We choose as right eigenvector:
<math display="block"> \mathbf p_1=\begin{pmatrix}v\\ a  \\0\end{pmatrix}.</math>

The other two eigenvectors can be found with analogous procedure as:
<math display="block"> \mathbf p_2=\begin{pmatrix} e_{vs} \\ 0\\ - \left(\frac a v \right)^2 \end{pmatrix}, \qquad \mathbf p_3 = \begin{pmatrix}v\\ -a \\0\end{pmatrix}.</math>

Then the projection matrix can be built:
<math display="block"> \mathbf P (v,u,s)=( \mathbf{p}_1, \mathbf{p}_2, \mathbf{p}_3) =\begin{pmatrix} v & e_{vs} & v\\ a & 0 & -a \\ 0 &  - \left(\frac a v \right)^2 & 0 \end{pmatrix}.</math>

Finally it becomes apparent that the real parameter ''a'' previously defined is the speed of propagation of the information characteristic of the hyperbolic system made of Euler equations, i.e. it is the ''[[group velocity|wave speed]]''. It remains to be shown that the sound speed corresponds to the particular case of an [[Isentropic process|isentropic transformation]]:
<math display="block">a_s \equiv \sqrt {\left({\partial p \over \partial \rho} \right)_s}.</math>