**3. Problems of periodic oscillations of Biot's medium**

Construction of motion equation solutions by periodic oscillations is very important for practice since existing power sources of disturbances are often periodic in time and therefore can be decomposed into a finite or infinite Fourier series in the form:

$$\begin{aligned} F'(\mathbf{x}, t) &= \sum\_{n} F'\_n(\mathbf{x}) e^{-i\alpha\_n t}, \\ F^f(\mathbf{x}, t) &= \sum\_{n} F^f\_n(\mathbf{x}) e^{-i\alpha\_n t} \end{aligned} \tag{4}$$

where periods of oscillation of each harmonic *Tn* ¼ 2*π=ω<sup>n</sup>* are multiple to the general period of oscillation *T*. Therefore, it is enough to consider the case of stationary oscillations, when the acting forces are periodic on time with an oscillation frequency *ω*:

$$\begin{aligned} F^\circ(\varkappa, t) &= F^\circ(\varkappa) e^{-i\alpha t}, \\ F^f(\varkappa, t) &= F^f(\varkappa) e^{-i\alpha t} \end{aligned} \tag{5}$$

The solution of Eq. (1) can be represented in the similar form:

$$u\_{\mathfrak{s}}(\mathfrak{x},t) = u\_{\mathfrak{s}}(\mathfrak{x})e^{-i\alpha t}, \quad u\_f(\mathfrak{x}) = u\_f(\mathfrak{x})e^{-i\alpha t} \tag{6}$$

and satisfy the radiation condition of type of Sommerfeld radiation conditions [10].

*Generalized and Fundamental Solutions of Motion Equations of Two-Component Biot's Medium*

*<sup>m</sup>*ð Þ *x*, *ω* , we use Fourier transformation, which for regular

*<sup>i</sup>*ð Þ *<sup>ξ</sup>*,*<sup>x</sup> dx*<sup>1</sup> … *dxN*

�*i*ð Þ *<sup>ξ</sup>*,*<sup>x</sup> dξ*1*:* … *dξ<sup>N</sup>*

\$ �*iξ <sup>j</sup>* (12)

*<sup>j</sup>* <sup>þ</sup> *<sup>ρ</sup>*12*ω*<sup>2</sup>

*<sup>j</sup>*þ*<sup>N</sup>* ¼ 0

*Uk*

*<sup>j</sup>*þ*<sup>N</sup>* <sup>þ</sup> *<sup>δ</sup><sup>k</sup>*

*<sup>j</sup>* ¼ 0

(13)

(14)

*φ*ð Þ *x e*

ð

*φ ξ*ð Þ*e*

*RN*

Matrix of such fundamental equations is named *Green tensor* of Eq. (8).

*RN*

Let us apply Fourier transformation to Eq. (10) and use property of Fourier

Then we get the system of 2 *N* linear algebraic equations for Fourier components

*<sup>j</sup>* <sup>þ</sup> *<sup>ρ</sup>*22*ω*<sup>2</sup>

By using gradient divergence method, this system has been solved by us. For this

<sup>2</sup>k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> ,

**Theorem 1.** *Components of Fourier transform of fundamental solutions have the form:*

*for j* ¼ 1, *N*, *k* ¼ 1, *N*,

*j*

� � �*iξ<sup>k</sup>* ð Þ *<sup>γ</sup>*1*f*<sup>21</sup> <sup>þ</sup> *<sup>γ</sup>*2*f*<sup>22</sup> <sup>þ</sup> *<sup>γ</sup>*<sup>3</sup> <sup>½</sup> *<sup>f</sup>*23��

*<sup>f</sup>*<sup>23</sup> � <sup>1</sup> *α*2

*ρ*11*δ<sup>k</sup>*

*<sup>j</sup>*þ*<sup>N</sup>* <sup>þ</sup> *<sup>ρ</sup>*12*δ<sup>k</sup>*

� �*f*03;

*j*

� � �*iξ<sup>k</sup>* ð Þ *<sup>β</sup>*1*f*<sup>21</sup> <sup>þ</sup> *<sup>β</sup>*2*f*<sup>22</sup> <sup>þ</sup> *<sup>β</sup>*<sup>3</sup> <sup>½</sup> *<sup>f</sup>*23�þ

*<sup>j</sup>*þ*<sup>N</sup>* � *<sup>ρ</sup>*22*δ<sup>k</sup>*

� �*f*03,

*<sup>j</sup>*þ*<sup>N</sup>* <sup>þ</sup> *<sup>ρ</sup>*11*ω*<sup>2</sup>

*Uk*

�*i<sup>ω</sup>* , *<sup>j</sup>* <sup>¼</sup> 1, 2;

*Uk*

*<sup>j</sup>*þ*<sup>N</sup>* <sup>þ</sup> *<sup>δ</sup><sup>k</sup>*

*∂ ∂x <sup>j</sup>*

*<sup>j</sup>* � *<sup>Q</sup><sup>ξ</sup> <sup>j</sup><sup>ξ</sup> jU<sup>k</sup>*

*Uk*

1 ð Þ <sup>2</sup>*<sup>π</sup> <sup>N</sup>*

**5. Fourier transform of fundamental solutions**

*<sup>F</sup>*½ �¼ *<sup>φ</sup>*ð Þ *<sup>x</sup> φ ξ*ð Þ¼ <sup>ð</sup>

½ �¼ *φ ξ*ð Þ *φ*ð Þ¼ *x*

where *ξ* ¼ *ξ*<sup>1</sup> ð Þ , … , *ξ<sup>N</sup>* are Fourier variables.

*<sup>j</sup>* � *μ ξ*k k<sup>2</sup>

*<sup>j</sup>* � *<sup>R</sup><sup>ξ</sup> <sup>j</sup>ξlU<sup>k</sup>*

*j* ¼ 1, … , *N*, *k* ¼ *N* þ 1, … , 2*N*

the next basic function were introduced:

*Uk*

*Uk*

**27**

*<sup>j</sup>* ¼ �*iξ <sup>j</sup>*

þ 1 *α*2

*<sup>j</sup>*þ*<sup>N</sup>* ¼ �*i<sup>ξ</sup> <sup>j</sup>*

� *μ α*2 *δk <sup>j</sup>*þ*<sup>N</sup>*k k*<sup>ξ</sup>* <sup>2</sup>

*Uk*

*<sup>j</sup>*þ*<sup>N</sup>* <sup>þ</sup> *<sup>ρ</sup>*12*ω*<sup>2</sup>

*<sup>f</sup>*0*<sup>k</sup>*ð Þ¼ *<sup>ξ</sup>*,*<sup>ω</sup>* <sup>1</sup> *сk*

and the next theorem has been proved [11, 12].

*ρ*12*δ<sup>k</sup>*

*fjk*ð Þ¼ *<sup>ξ</sup>*,*<sup>ω</sup> <sup>f</sup>*ð Þ *<sup>j</sup>*�<sup>1</sup> *<sup>k</sup>*ð Þ *<sup>ξ</sup>*,*<sup>ω</sup>*

To construct *U <sup>j</sup>*

functions has the form:

*F*�<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.92064*

transform of derivatives [10]:

of this tensor:

�ð Þ *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup> <sup>ξ</sup> <sup>j</sup><sup>ξ</sup> jU<sup>k</sup>*

�*Q<sup>ξ</sup> <sup>j</sup><sup>ξ</sup> jU<sup>k</sup>*

where complex amplitudes of displacements *us*ð Þ *x* , *u <sup>f</sup>*ð Þ *x* must be determined. If the solution has been known for any frequency ω, then we get similar decomposition for displacements of a medium:

$$\begin{aligned} u\_t(\mathbf{x}, t) &= \sum\_n u\_{sn}(\mathbf{x}) e^{-i\alpha\_n t}, \\ u\_f(\mathbf{x}, t) &= \sum\_n u\_{fn}(\mathbf{x}) e^{-i\alpha\_n t} \end{aligned} \tag{7}$$

which give us the solution of problem for forces (4).

We get equations for complex amplitudes by stationary oscillations, substituting (6) into the system (1):

$$\begin{aligned} (\lambda + \mu) \text{grad} \, \text{div} \, u\_t + \mu \Delta u\_t + Q \, \text{grad} \, \text{div} \, u\_f + \rho\_{11} a^2 u\_t + \rho\_{12} a^2 u\_f + F(\mathbf{x}) &= \mathbf{0} \\ Q \, \text{grad} \, \text{div} \, u\_t + R \mathbf{grad} \, \text{div} \, u\_f + \rho\_{12} a^2 u\_t + \rho\_{22} a^2 u\_f + F^f(\mathbf{x}) &= \mathbf{0} \end{aligned} \tag{8}$$

To construct the solutions of this system for different forces, we define Green tensor of it.
