**4. Green tensor of Biot's equations by stationary oscillations**

Let us construct *U <sup>j</sup> <sup>m</sup>*ð Þ *<sup>x</sup>*, *<sup>ω</sup> <sup>e</sup>*�*iω<sup>t</sup>* ð Þ *<sup>j</sup>*, *<sup>m</sup>* <sup>¼</sup> 1, … , 2*<sup>N</sup>* fundamental solutions of the system (1) for the forces in the form:

$$F(\mathbf{x}, t) = \begin{pmatrix} F' \\ F^f \end{pmatrix} = \begin{pmatrix} \delta\_k^{[j]} \mathbf{e}\_k \\ \delta\_{k+N}^{[j]} \mathbf{e}\_k \end{pmatrix} \delta(\mathbf{x}) e^{-i\alpha t}, \tag{9}$$
 
$$k = \mathbf{1}, \dots, N, j = \mathbf{1}, \dots, 2N.$$

Here *δ <sup>j</sup> <sup>k</sup>* ¼ *δjk* is the Kronecker symbol, and *δ*ð Þ *x* is the singular delta-function. They describe a motion of Biot's medium at an action of sources of stationary oscillations, concentrated in the point *x* = 0. The upper index of this tensor ( … [k]) fixes the current concentrated force and its direction. The lower index corresponds to component of movement of a skeleton and a fluid, respectively, *k* ¼ 1, … , *N* and *k* ¼ *N* þ 1, … , 2*N*.

Their complex amplitudes *U <sup>j</sup> <sup>m</sup>*ð Þ *x*,*ω* ð Þ *j*, *m* ¼ 1, … , 2*N* satisfy the next system of equation:

$$(\lambda + \mu)U\_{j, \vec{\mu}}^k + \mu U\_{i, \vec{\mu}}^k + a^2 \rho\_{11} U\_i^k + Q U\_{j, \vec{\mu}}^{k+N} - a^2 \rho\_{12} U\_i^{k+N} + \delta(\mathbf{x}) \delta\_j^k = \mathbf{0}$$

$$\mathbf{Q} U\_{j, \vec{\mu}}^k + \rho\_{12} a^2 U\_i^k, \mathbf{u}\_t + \mathbf{R} U\_{j, \vec{\mu}}^{k+N} + \rho\_{22} a^2 U\_i^{k+N} + \delta(\mathbf{x}) \delta\_{j+N}^k = \mathbf{0} \tag{10}$$

$$j = 1, \dots, 2N, \quad k = 1, \dots, 2N.$$

Since fundamental solutions are not unique, we'll construct such, which tend to zero at infinity:

$$U\_i^j(\mathfrak{x}, \mathfrak{o}) \to \mathbf{0} \text{ at } \|\mathfrak{x}\| \to \infty \tag{11}$$

*Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot's Medium DOI: http://dx.doi.org/10.5772/intechopen.92064*

and satisfy the radiation condition of type of Sommerfeld radiation conditions [10]. Matrix of such fundamental equations is named *Green tensor* of Eq. (8).
