these operations will be proved in the next sections. Taking into account our substitution (\[eqn13\]) we see that equations $(\ref{eqn7})\; - \;(\ref{eqn9})$ are in fact system of linear partial differential equations with constant coefficients.

Solution of this system will be presented by the following steps:

On the **first step** of our analysis we use Fourier transform $(\ref{A3})$ to solve equations $(\ref{eqn7})\; - \;(\ref{eqn14})$. We have got:

$$\\U_{k}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\;F[u_{k} ( x_{1}, x_{2}, x_{3}, t)]$$

$$\quad\quad\quad\quad\quad\quad -\gamma^{2}_{s}U_{k}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;\;=\;F{\mbox{\Large [ \normalsize}} \frac{\partial^{2}u_{k}( x_{1}, x_{2}, x_{3}, t)}{\partial x^{2}_{s}}{\mbox{\Large ] \normalsize}}\;\;\;\;\rm{[use \;(\ref{eqn10})]}$$

$$\\U_{k}^{0}( \gamma_{1} ,\gamma_{2} ,\gamma_{3})\;=\;F[u_{k}^{0} ( x_{1}, x_{2}, x_{3})]$$

$$\\P( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\;F[p\, ( x_{1}, x_{2}, x_{3}, t)]$$

$$\\\tilde{F}_{k}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t)\;=\;F[\tilde{f}_{k} ( x_{1}, x_{2}, x_{3}, t)]$$

$$\\  k,s\;=\;1,2,3$$

and then:

$$\label{eqn134}
\frac{d U_{1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{d t}  \;=\;-\nu
( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) U_{1}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\;i\gamma_{1} P( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\; \tilde{F}_{1}( \gamma_{1}, \gamma_{2}, \gamma_{3},t )$$

$$\label{eqn135}
\frac{d U_{2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{d t}  \;=\;-\nu
( \gamma_{1}^{2} +\gamma_{2}^{2} +\gamma_{3}^{2}) U_{2}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\;i\gamma_{2} P( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )\;+\; \tilde{F}_{2}( \gamma_{1}, \gamma_{2}, \gamma_{3},t )$$

$$\label{eqn136}
\frac{d U_{3}( \gamma_{1}, \gamma_{2}, \gamma_{3}, t )}{d t}  \;=\;-\nu
( \gamma_{1}^{2} +\gamma_{2}^{2} 