# Wave equation for a static fluid that does not flow

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

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

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

$$$$\partial_t^2 \psi_1 = c^2 \nabla^2 \psi_1$$$$

What tricks or what mathematical tools do I need?

• What exactly are you trying to "solve"? If you set ${\bf v}_0=0$ your equation reduces immediately to the standard sound wave equation. – mike stone Oct 18 '18 at 12:52