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From the fluid dynamics, applying a linear approximation, I arrived at the following master sound-wave equation:

\begin{equation} \partial_t \left( - c_S^{-2} \rho_0\left( \partial_t \psi_1 + \mathbf{v}_0 \cdot \boldsymbol{\nabla} \psi_1\right) \right) + \boldsymbol{\nabla} \cdot \left( - c_S^{-2} \rho_0 \mathbf{v}_0 \left( \partial_t \psi_1 + \mathbf{v}_0 \cdot \boldsymbol{\nabla} \psi_1\right) + \rho_0 \boldsymbol{\nabla} \psi_1\right) = 0 \end{equation}

I have to assume a static background and it does not flow to reach the famous wave equation:

\begin{equation} \partial_t^2 \psi_1 = c^2 \nabla^2 \psi_1 \end{equation}

What tricks or what mathematical tools do I need?

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The standard wave equation only applies to stationary background flows. There are various ways to write the equation for non-zero background flows, and your general case is probably one of them. For some of these you can look at our old paper: arXiv:cond-mat/0106255 and the references it cites.

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  • $\begingroup$ I'm sorry but I can not solve it. The second term I think is easy, static and after the chain rule and using properties of nabla I get it. The first term I do not know how it is. Should we use the Bernouilli equation and the barotropic condition? How can I write math symbols here, at comments? $\endgroup$ – Álvaro Ferrández Oct 18 '18 at 10:13
  • $\begingroup$ What exactly are you trying to "solve"? If you set ${\bf v}_0=0$ your equation reduces immediately to the standard sound wave equation. $\endgroup$ – mike stone Oct 18 '18 at 12:52

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