**5. Fourier transform of fundamental solutions**

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

*Mathematical Theorems - Boundary Value Problems and Approximations*

*us*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup>

*<sup>u</sup> <sup>f</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup>

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

*Fs F f* � � *n*

*n*

We get equations for complex amplitudes by stationary oscillations, substituting

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

<sup>¼</sup> *<sup>δ</sup>* ½ �*j <sup>k</sup> ek δ* ½ �*j <sup>k</sup>*þ*<sup>N</sup>ek*

*k* ¼ 1, … , *N*, *j* ¼ 1, … , 2*N:*

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

*<sup>i</sup>* <sup>þ</sup> *QU<sup>k</sup>*þ*<sup>N</sup>*

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

!

*<sup>k</sup>* ¼ *δjk* is the Kronecker symbol, and *δ*ð Þ *x* is the singular delta-function.

*<sup>j</sup>*,*ji* � *<sup>ω</sup>*<sup>2</sup>

*<sup>i</sup>* <sup>þ</sup> *<sup>δ</sup>*ð Þ *<sup>x</sup> <sup>δ</sup><sup>k</sup>*

*U<sup>k</sup>*þ*<sup>N</sup>*

Since fundamental solutions are not unique, we'll construct such, which tend to

�*iωt*

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 decomposi-

*usn*ð Þ *x e*

*ufn*ð Þ *x e*

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

*δ*ð Þ *x e* �*iωt*

*<sup>m</sup>*ð Þ *x*,*ω* ð Þ *j*, *m* ¼ 1, … , 2*N* satisfy the next system

*<sup>i</sup>* <sup>þ</sup> *<sup>δ</sup>*ð Þ *<sup>x</sup> <sup>δ</sup><sup>k</sup>*

*<sup>j</sup>*þ*<sup>N</sup>* ¼ 0

*<sup>j</sup>* ¼ 0

(10)

*ρ*12*U<sup>k</sup>*þ*<sup>N</sup>*

*<sup>i</sup>*ð Þ! *x*,*ω* 0 at k k*x* ! ∞ (11)

, *u <sup>f</sup>*ð Þ¼ *x u <sup>f</sup>*ð Þ *x e*

�*iωnt* ,

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

*<sup>u</sup> <sup>f</sup>* <sup>þ</sup> *<sup>F</sup> <sup>f</sup>*

�*iω<sup>t</sup>* (6)

�*iωnt* (7)

*<sup>u</sup> <sup>f</sup>* <sup>þ</sup> *<sup>F</sup><sup>s</sup>*

ð Þ¼ *x* 0

ð Þ¼ *x* 0

, (9)

(8)

*us*ð Þ¼ *x*, *t us*ð Þ *x e*

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

ð Þ *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup>* grad div*us* <sup>þ</sup> *<sup>μ</sup>*Δ*us* <sup>þ</sup> *<sup>Q</sup>* grad div*<sup>u</sup> <sup>f</sup>* <sup>þ</sup> *<sup>ρ</sup>*11*ω*<sup>2</sup>

*<sup>Q</sup>* grad div *us* <sup>þ</sup> *<sup>R</sup>*grad div*<sup>u</sup> <sup>f</sup>* <sup>þ</sup> *<sup>ρ</sup>*12*ω*<sup>2</sup>

*F x*ð Þ¼ , *t*

tion for displacements of a medium:

(6) into the system (1):

Let us construct *U <sup>j</sup>*

system (1) for the forces in the form:

Their complex amplitudes *U <sup>j</sup>*

*<sup>j</sup>*,*ji* <sup>þ</sup> *<sup>μ</sup>U<sup>k</sup>*

*Uk*

*j* ¼ 1, … , 2*N*, *k* ¼ 1, … , 2*N:*

*<sup>i</sup>*,*jj* <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>

*<sup>i</sup>* , *tt* <sup>þ</sup> *RU<sup>k</sup>*þ*<sup>N</sup>*

*U j*

*ρ*11*U<sup>k</sup>*

tensor of it.

Here *δ <sup>j</sup>*

*k* ¼ *N* þ 1, … , 2*N*.

ð Þ *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup> <sup>U</sup><sup>k</sup>*

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

of equation:

*QU<sup>k</sup>*

zero at infinity:

**26**

To construct *U <sup>j</sup> <sup>m</sup>*ð Þ *x*, *ω* , we use Fourier transformation, which for regular functions has the form:

$$\begin{aligned} F[\varrho(\infty)] &= \overline{\varrho}(\xi) = \int\_{\mathbb{R}^N} \varrho(\infty) e^{i(\xi,\varkappa)} d\varkappa\_1 \dots d\varkappa\_N \\\ F^{-1}[\overline{\varrho}(\xi)] &= \varrho(\infty) = \frac{1}{(2\pi)^N} \int\_{\mathbb{R}^N} \overline{\varrho}(\xi) e^{-i(\xi,\varkappa)} d\xi\_1 \dots d\xi\_N \end{aligned}$$

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

Let us apply Fourier transformation to Eq. (10) and use property of Fourier transform of derivatives [10]:

$$\frac{\partial}{\partial \mathfrak{x}\_{j}} \leftrightarrow -i\mathfrak{x}\_{j} \tag{12}$$

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

$$\begin{aligned} & - (\lambda + \mu) \xi\_j \xi\_j \overline{U}\_j^k - \mu ||\xi||^2 \overline{U}\_j^k - Q \xi\_j \xi\_j \overline{U}\_{j+N}^k + \rho\_{11} a^2 \overline{U}\_j^k + \rho\_{12} a^2 \overline{U}\_{j+N}^k + \delta\_j^k = \mathbf{0} \\ & - Q \xi\_j \xi\_j \overline{U}\_j^k - R \xi\_j \xi\_l \overline{U}\_{j+N}^k + \rho\_{12} a^2 \overline{U}\_j^k + \rho\_{22} a^2 \overline{U}\_{j+N}^k + \delta\_{j+N}^k = \mathbf{0} \\ & j = 1, \dots, N, \quad k = N+1, \dots, 2N \end{aligned} \tag{13}$$

By using gradient divergence method, this system has been solved by us. For this the next basic function were introduced:

$$\begin{aligned} f\_{0k}(\xi, o) &= \frac{1}{c\_k^2 \left| \left| \xi \right| \right|^2 - o^2}, \\ f\_{jk}(\xi, o) &= \frac{f\_{(j-1)k}(\xi, o)}{-io}, \quad j = 1, 2; \end{aligned} \tag{14}$$

and the next theorem has been proved [11, 12]. **Theorem 1.** *Components of Fourier transform of fundamental solutions have the form:*

$$\begin{aligned} \text{for } j &= \overline{1}, N, \qquad k = \overline{1}, N, \\ \overline{U}\_j^k &= \left( -i\xi\_j \right) \left( -i\xi\_k \right) [\beta\_1 f\_{21} + \beta\_2 f\_{22} + \beta\_3 f\_{23}] + \\ &+ \frac{1}{\alpha\_2} \left( \rho\_{12} \delta\_{j+N}^k - \rho\_{22} \delta\_j^k \right) f\_{03}, \\ \overline{U}\_{j+N}^k &= \left( -i\xi\_j \right) \left( -i\xi\_k \right) [\gamma\_1 f\_{21} + \gamma\_2 f\_{22} + \gamma\_3 f\_{23}] - \\ &- \frac{\mu}{\alpha\_2} \delta\_{j+N}^k ||\xi||^2 f\_{23} - \frac{1}{\alpha\_2} \left( \rho\_{11} \delta\_{j+N}^k + \rho\_{12} \delta\_j^k \right) f\_{03}; \end{aligned}$$

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$\begin{aligned} for \quad j &= 1, \ldots, N \quad k = N+1, \ldots, 2N \\ \overline{U}\_j^k &= \left(-i\xi\_j\right) \left(-i\xi\_{k-N}\right) \left[\eta\_1 f\_{21} + \eta\_2 f\_{22} + \eta\_3 f\_{23}\right] + \\ &+ \frac{1}{\alpha\_2} \left(\rho\_{12}\delta\_{j+N}^k - \rho\_{22}\delta\_j^k\right) f\_{03} \\ \overline{U}\_{j+N}^k &= \left(-i\xi\_j\right) \left(-i\xi\_{k-N}\right) \left[\varepsilon\_1 f\_{21} + \varepsilon\_2 f\_{22} + \varepsilon\_3 f\_{23}\right] - \\ &- \frac{\mu}{\alpha\_2} \delta\_{j+N}^k ||\xi||^2 f\_{23} - \frac{1}{\alpha\_2} \left(\rho\_{11}\delta\_{j+N}^k + \rho\_{12}\delta\_j^k\right) f\_{03} \end{aligned}$$

Fundamental solutions of Helmholtz equation, which satisfy Sommerfeld con-

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

at *r* ! ∞

<sup>0</sup>*m*ð Þ� *<sup>r</sup> ikmФ*0*m*ð Þ¼ *<sup>r</sup> O r*�<sup>1</sup> � �, *<sup>N</sup>* <sup>¼</sup> 3,

<sup>0</sup>*m*ð Þ� *<sup>r</sup> ikmФ*0*m*ð Þ¼ *<sup>r</sup> O r*�1*=*<sup>2</sup> � �, *<sup>N</sup>* <sup>¼</sup> <sup>2</sup>*:*

*ikmr*

<sup>4</sup>*c*<sup>2</sup> *<sup>H</sup>*ð Þ<sup>1</sup>

*<sup>j</sup>* ð Þ *kmr* is the cylindrical Hankel function of the first kind:

*<sup>Ф</sup>*0*<sup>m</sup>* <sup>¼</sup> sin *km*j j *<sup>x</sup>* 2*kmc*<sup>2</sup> *m :*

The last property is true only for *N* = 2,3. In the case *N* = 1, all fundamental

Φ0*<sup>m</sup>* þ *c*

**Theorem 2.** *The components of Green tensor of Biot's equations at stationary oscillations with frequency ω, which satisfy the radiation conditions, have the form:*

> þ 1 *α*2

�2

*<sup>m</sup> δ*ð Þ¼ *x* 0,

*ρ*12*δ<sup>k</sup>*

<sup>3</sup>Φ0*<sup>m</sup>* � � � *<sup>ρ</sup>*11*δ<sup>k</sup>*

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

� �Φ03,

*j*

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

*α*2

*j*

Φ03;

, *km* <sup>¼</sup> *<sup>ω</sup>*

<sup>0</sup> ð Þ *kmr* ,

*cm* ;

) describe harmonic waves which move

are well known [10]. They are unique. Using them, we obtain:

*<sup>Ф</sup>*0*<sup>m</sup>* <sup>¼</sup> *<sup>i</sup>*

<sup>4</sup>*πrc*<sup>2</sup> *<sup>e</sup>*

*<sup>Ф</sup>*0*<sup>m</sup>* <sup>¼</sup> <sup>1</sup>

ditions of radiation:

for *N* ¼ 3

for *N* ¼ 2

where *H*ð Þ<sup>1</sup>

for *N* ¼ 1

solutions of Eq. (16):

do not decay at infinity.

*for j* ¼ 1, *N*, *k* ¼ 1, *N*,

ð Þ¼� *<sup>x</sup>*, *<sup>ω</sup> <sup>ω</sup>*�<sup>2</sup><sup>X</sup>

*<sup>j</sup>*þ*<sup>N</sup>*ð Þ¼� *<sup>x</sup>*, *<sup>ω</sup> <sup>ω</sup>*�<sup>2</sup><sup>X</sup>

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

þ *μδ<sup>k</sup> j*þ*N <sup>α</sup>*2*ω*<sup>2</sup> *<sup>c</sup>*

*Uk j*

*Uk*

**29**

*Ф*0

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

*Ф*0

These functions (subject to factor *e*�*iω<sup>t</sup>*

from the point *x* = 0 to infinity and decay at infinity.

*d*2 *dx*<sup>2</sup> <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *m*

3

*m*¼1 *βm ∂*2 Φ0*<sup>m</sup> ∂x <sup>j</sup>∂xk*

3

*m*¼1 *γm ∂*2 Φ0*<sup>m</sup> ∂x <sup>j</sup>∂xk* þ

> �2 <sup>3</sup> *<sup>δ</sup>*ð Þþ *<sup>x</sup> <sup>k</sup>*<sup>2</sup>

From Theorem 1, the next theorem follows.

!

*where the next constants have been introduced as:*

*<sup>D</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *α*2*υ*<sup>12</sup> , *υlm* ¼ *c* 2 *<sup>l</sup>* � *c* 2 *<sup>m</sup>*, *q*<sup>1</sup> ¼ *Qρ*<sup>12</sup> � ð Þ *λ* þ *μ ρ*12, *q*<sup>2</sup> ¼ *ρ*11*R* � *Qρ*12, *<sup>d</sup>*<sup>1</sup> <sup>¼</sup> ð Þ *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup> <sup>ρ</sup>*<sup>22</sup> � *<sup>Q</sup>ρ*12, *<sup>d</sup>*<sup>2</sup> <sup>¼</sup> *<sup>Q</sup>ρ*<sup>22</sup> � *<sup>R</sup>ρ*12, *<sup>d</sup>*3*<sup>j</sup>* <sup>¼</sup> *<sup>ρ</sup>*12*с*<sup>2</sup> *<sup>j</sup>* � *Q j* ð Þ ¼ 1, 2 *<sup>β</sup> <sup>j</sup>* ¼ �ð Þ<sup>1</sup> ð Þ *<sup>j</sup>*þ<sup>1</sup> *<sup>D</sup>*1*с*<sup>2</sup> *j α*2*υ*3*<sup>j</sup> d*1*bsj* þ *d*2*d*3*<sup>j</sup>* , *<sup>β</sup>*<sup>3</sup> ¼ � *<sup>с</sup>*<sup>2</sup> 3 *α*2*υ*31*υ*<sup>32</sup> ð Þ *d*1*bs*<sup>3</sup> þ *d*2*d*<sup>33</sup> ; *<sup>γ</sup> <sup>j</sup>* ¼ �ð Þ<sup>1</sup> *<sup>j</sup>*þ<sup>1</sup> *<sup>D</sup>*1*с*<sup>2</sup> *j α*2*υ*3*<sup>j</sup> q*1*bfj* þ *q*2*d*3*<sup>j</sup>* , *<sup>γ</sup>*<sup>3</sup> ¼ � *<sup>D</sup>*1*с*<sup>2</sup> <sup>3</sup>*υ*<sup>12</sup> *α*2*υ*31*υ*<sup>32</sup> *<sup>q</sup>*1*<sup>b</sup> <sup>f</sup>* <sup>3</sup> <sup>þ</sup> *<sup>q</sup>*2*d*<sup>33</sup> ; *<sup>η</sup> <sup>j</sup>* ¼ �ð Þ<sup>1</sup> *<sup>j</sup>*þ<sup>1</sup> *<sup>D</sup>*1*с*<sup>2</sup> *j α*2*υ*3*<sup>j</sup> <sup>d</sup> jd*3*<sup>j</sup>* <sup>þ</sup> *<sup>d</sup>*2*bjs* , *<sup>η</sup>*<sup>3</sup> ¼ � *<sup>с</sup>*<sup>2</sup> <sup>3</sup>*υ*<sup>12</sup> *α*2*υ*31*υ*<sup>32</sup> ð Þ *d*1*d*<sup>33</sup> þ *d*2*b*3*<sup>s</sup>* ; *<sup>ς</sup> <sup>j</sup>* ¼ �ð Þ<sup>1</sup> *<sup>j</sup>*þ<sup>1</sup> *<sup>D</sup>*1*с*<sup>2</sup> *j α*2*υ*3*<sup>j</sup> q*1*d*3*<sup>j</sup>* þ *q*2*b*ð Þ <sup>4</sup>�*<sup>j</sup> <sup>s</sup>* , *<sup>ς</sup>*<sup>3</sup> ¼ � *<sup>с</sup>*<sup>2</sup> <sup>3</sup>*υ*<sup>12</sup> *α*2*υ*31*υ*<sup>32</sup> *q*1*d*<sup>33</sup> þ *q*2*b*3*<sup>s</sup> bfj* ¼ *ρ*22*υfj*, *bsj* ¼ *ρ*11*υjs:*

This form is very convenient for constructing originals of Green tensor.
