Interpretation of dx/dt and u in CFD

Question

I am currently studying CFD and I have a question about how I should interpret the term u and dx/dt. I would like to find that out in the textbook, but I could not.

When I solve 1D Euler Equation with Mass conservation

, then I get the result

.

In the result I have a problem about how to interpret these two terms u and dx/dt. As I understand, both terms were the flow velocity like u=dx/dt. The fact that the meaning of these terms diverges confuses me.

Thanks for your answer in advance and have a nice weekend.

Answer 1

I think U is the flow velocity field. it can be pointing in any direction, but $\frac{dx}{dt}$ is only the x component.

Like in vortex U is turning but $\frac{dx}{dt}$ is not.

Answer 2

Tentatively, this looks like it's analysing a sound wave, $\mathsf{d}x/\mathsf{d}t$ is the phase velocity of the wave in the lab frame, $\pm c$ is the phase velocity of the wave in the rest frame of the fluid, and $u$ sometimes represents the local, instantaneous flow velocity, and sometimes represents the average flow velocity over the fluctuations in the wave. But in that case, it's odd that OP didn't mention that the phenomenon they're trying to analyse is a sound wave.