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The eigenvalues of the flux Jacobian $f'(q)$ for the 1D Euler equations are: This relation is called the equation of state of the gas, as discussed further below. For gas dynamics, a different expression must be used to compute the pressure $p$ from the conserved quantities. This is very similar to the conservation of momentum equation in the shallow water equations, as discussed in Shallow_water.html, in which case $hu$ is the momentum density and $\frac 1 2 gh^2$ is the hydrostatic pressure. Thus the momentum conservation equation is These velocity components also lead to a net flux of momentum. These random velocities are what accounts for the pressure of the fluid, which we'll denote by $p$. Each molecule also has some additional random velocity component. The individual molecules in that region are not all moving with exactly velocity $u$ that's just their average. The density and velocity we are modeling are average values over some small region of space. To understand the second term in the momentum flux, we must realize that a fluid is made up of many tiny molecules. First, the momentum is transported in the same way that the density is this flux is given by the momentum density times the velocity: $\rho u^2$. The momentum density is given by the product of mass density and velocity, $\rho u$. For convenience, we review the ideas here. We discussed the conservation of momentum in a fluid already in Acoustics.html.