**6. Stationary Green tensor construction: radiation conditions**

In this case let us construct the originals of the basic function but only over *ξ* by constant frequency:

$$\Phi\_{0m}(\mathfrak{x}, \alpha) = F\_{\xi}^{-1}\left[\left.f\_{0m}(\xi, \alpha)\right\vert\right]$$

which, in accordance with its definition (14), satisfies the equation:

$$\left(c\_m^2||\xi||^2 - \alpha^2\right)f\_{0m} = \mathbf{1} \tag{15}$$

Using property (12) for derivatives from here, we get Helmholtz equation for fundamental solution (accurate within a factor *c*�<sup>2</sup> *<sup>m</sup>* ):

$$\left(\Delta + k\_m^2\right)\Phi\_{0m} + c\_m^{-2}\delta(\varkappa) = \mathbf{0}, \quad k\_m = \frac{\alpha}{c\_m} \tag{16}$$

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

Fundamental solutions of Helmholtz equation, which satisfy Sommerfeld conditions of radiation:

$$\begin{aligned} \text{at } r &\to \infty \\\\ \Phi\_{0m}'(r) - ik\_m \Phi\_{0m}(r) &= O\left(r^{-1}\right), \qquad N = 3, \\\\ \Phi\_{0m}'(r) - ik\_m \Phi\_{0m}(r) &= O\left(r^{-1/2}\right), \quad N = 2. \end{aligned}$$

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

for *N* ¼ 3

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

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

ð Þ �*iξk*�*<sup>N</sup> η*<sup>1</sup> *f* <sup>21</sup> þ *η*<sup>2</sup> *f* <sup>22</sup> þ *η*<sup>3</sup> *f* <sup>23</sup>

*f*03

ð Þ �*iξk*�*<sup>N</sup> ς*1*f*<sup>21</sup> þ *ς*2*f*<sup>22</sup> þ *ς*<sup>3</sup> ½ *f*23��

*ρ*11*δ<sup>k</sup>*

*<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,

3 *α*2*υ*31*υ*<sup>32</sup>

<sup>3</sup>*υ*<sup>12</sup> *α*2*υ*31*υ*<sup>32</sup>

> <sup>3</sup>*υ*<sup>12</sup> *α*2*υ*31*υ*<sup>32</sup>

<sup>3</sup>*υ*<sup>12</sup> *α*2*υ*31*υ*<sup>32</sup>

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

*j*

*f*03

*<sup>j</sup>* � *Q j* ð Þ ¼ 1, 2

ð Þ *d*1*bs*<sup>3</sup> þ *d*2*d*<sup>33</sup> ;

*q*1*b <sup>f</sup>* <sup>3</sup> þ *q*2*d*<sup>33</sup> ;

ð Þ *d*1*d*<sup>33</sup> þ *d*2*b*3*<sup>s</sup>* ;

*f*0*<sup>m</sup>* ¼ 1 (15)

(16)

*cm*

*q*1*d*<sup>33</sup> þ *q*2*b*3*<sup>s</sup>* 

*j*

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

<sup>þ</sup>

*Uk*

*Uk*

, *υlm* ¼ *c*

*<sup>D</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup> *α*2*υ*<sup>12</sup>

*<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>

*<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>

*<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>

*<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>

constant frequency:

**28**

*bfj* ¼ *ρ*22*υfj*, *bsj* ¼ *ρ*11*υjs:*

*<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*N*k k*<sup>ξ</sup>* <sup>2</sup>

*where the next constants have been introduced as:*

2 *<sup>l</sup>* � *c* 2

*j α*2*υ*3*<sup>j</sup>*

*j α*2*υ*3*<sup>j</sup>*

*j α*2*υ*3*<sup>j</sup>*

*j α*2*υ*3*<sup>j</sup>*

*<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>β</sup>*<sup>3</sup> ¼ � *<sup>с</sup>*<sup>2</sup>

, *<sup>γ</sup>*<sup>3</sup> ¼ � *<sup>D</sup>*1*с*<sup>2</sup>

, *<sup>η</sup>*<sup>3</sup> ¼ � *<sup>с</sup>*<sup>2</sup>

, *<sup>ς</sup>*<sup>3</sup> ¼ � *<sup>с</sup>*<sup>2</sup>

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

In this case let us construct the originals of the basic function but only over *ξ* by

*<sup>ξ</sup> <sup>f</sup>* <sup>0</sup>*<sup>m</sup>*ð Þ *<sup>ξ</sup>*, *<sup>ω</sup>*

*<sup>m</sup>* ):

*<sup>m</sup> <sup>δ</sup>*ð Þ¼ *<sup>x</sup>* 0, *km* <sup>¼</sup> *<sup>ω</sup>*

**6. Stationary Green tensor construction: radiation conditions**

<sup>Φ</sup>0*<sup>m</sup>*ð Þ¼ *<sup>x</sup>*,*<sup>ω</sup> <sup>F</sup>*�<sup>1</sup>

*c* 2

fundamental solution (accurate within a factor *c*�<sup>2</sup>

*<sup>Δ</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup> *m* Φ0*<sup>m</sup>* <sup>þ</sup> *<sup>c</sup>*

which, in accordance with its definition (14), satisfies the equation:

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

Using property (12) for derivatives from here, we get Helmholtz equation for

�2

*d*1*bsj* þ *d*2*d*3*<sup>j</sup>*

*q*1*bfj* þ *q*2*d*3*<sup>j</sup>*

*d jd*3*<sup>j</sup>* þ *d*2*bjs*

*q*1*d*3*<sup>j</sup>* þ *q*2*b*ð Þ <sup>4</sup>�*<sup>j</sup> <sup>s</sup>*

*ρ*12*δ<sup>k</sup>*

*Mathematical Theorems - Boundary Value Problems and Approximations*

$$\Phi\_{0m} = \frac{1}{4\pi c^2} e^{ik\_m r}, \quad k\_m = \frac{\alpha}{c\_m};$$

for *N* ¼ 2

$$
\Phi\_{0m} = \frac{i}{4c^2} H\_0^{(1)}(k\_m r),
$$

where *H*ð Þ<sup>1</sup> *<sup>j</sup>* ð Þ *kmr* is the cylindrical Hankel function of the first kind:

for *N* ¼ 1

$$\phi\_{0m} = \frac{\sin k\_m |\boldsymbol{\chi}|}{2k\_m c\_m^2}.$$

These functions (subject to factor *e*�*iω<sup>t</sup>* ) describe harmonic waves which move from the point *x* = 0 to infinity and decay at infinity.

The last property is true only for *N* = 2,3. In the case *N* = 1, all fundamental solutions of Eq. (16):

$$
\left(\frac{d^2}{d\mathfrak{x}^2} + k\_m^2\right)\Phi\_{0m} + c\_m^{-2}\delta(\mathfrak{x}) = \mathbf{0},
$$

do not decay at infinity.

From Theorem 1, the next theorem follows.

**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:*

$$\begin{aligned} \text{for } j = \overline{1, N}, \qquad k = \overline{1, N}, \\ U\_j^k(\mathbf{x}, \boldsymbol{\alpha}) &= -\boldsymbol{\alpha}^{-2} \sum\_{m=1}^3 \beta\_m \frac{\partial^2 \Phi\_{0m}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} + \frac{1}{\alpha\_2} \left(\rho\_{12} \boldsymbol{\delta}\_{j+N}^k - \rho\_{22} \boldsymbol{\delta}\_j^k \right) \Phi\_{03}, \\\\ U\_{j+N}^k(\mathbf{x}, \boldsymbol{\alpha}) &= -\boldsymbol{\alpha}^{-2} \sum\_{m=1}^3 \gamma\_m \frac{\partial^2 \Phi\_{0m}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} + \\\\ &\quad + \frac{\mu \boldsymbol{\delta}\_{j+N}^k}{\alpha\_2 \boldsymbol{\alpha}^2} \left(\boldsymbol{\varepsilon}\_3^{-2} \boldsymbol{\delta}(\mathbf{x}) + k\_3^2 \boldsymbol{\Phi}\_{0m} \right) - \frac{\rho\_{11} \boldsymbol{\delta}\_{j+N}^k + \rho\_{12} \boldsymbol{\delta}\_j^k}{\alpha\_2} \boldsymbol{\Phi}\_{03}; \end{aligned}$$

$$\begin{split} U^{k}\_{j}(\mathbf{x},\boldsymbol{\alpha}) &= -\boldsymbol{\alpha}^{-2} \sum\_{m=1}^{3} \eta\_{m} \frac{\partial^{2} \Phi\_{0m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k}} + \frac{1}{\alpha\_{2}} \left(\rho\_{12} \boldsymbol{\delta}^{k}\_{j+N} - \rho\_{22} \boldsymbol{\delta}^{k}\_{j}\right) \boldsymbol{\Phi}\_{03} \\ U^{k}\_{j+N}(\mathbf{x},\boldsymbol{\alpha}) &= -\boldsymbol{\alpha}^{-2} \sum\_{m=1}^{3} \xi\_{m} \frac{\partial^{2} \Phi\_{0m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k}} + \\ &\quad + \frac{\mu}{\alpha\_{2} \boldsymbol{\alpha}^{2}} \left(\boldsymbol{\varepsilon}\_{3}^{-2} \boldsymbol{\delta}(\mathbf{x}) + k\_{3}^{2} \boldsymbol{\Phi}\_{0m}\right) \boldsymbol{\delta}^{k}\_{j+N} - \frac{1}{\alpha\_{2}} \left(\rho\_{11} \boldsymbol{\delta}^{k}\_{j+N} + \rho\_{12} \boldsymbol{\delta}^{k}\_{j}\right) \boldsymbol{\Phi}\_{03} \end{split}$$

The obtained solutions allow us to study the dynamics of porous water- and gassaturated media at the action of periodic sources of disturbances of a sufficiently arbitrary form. In particular, they are applicable in the case of actions of certain forces on surfaces, for example, cracks, in porous media that can be simulated by

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

There is another interesting feature of the Green tensor of the Biot's equations, which contains, as one of the terms, the delta-function that complicates the application of this tensor for solving boundary value problems based on analogues of Green formulas for elliptic systems of equations or the boundary element method. Here, when constructing the model, the viscosity of the liquid is not taken into account, which, apparently, leads to the presence of such terms, and it requires

simple and double layers on the crack surface.

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

improvement of this model taking into account a viscosity.

originals of the basic functions in an initial space-time:

They are originals of the classic wave equation:

*∂*2 *<sup>∂</sup>t*<sup>2</sup> � *<sup>c</sup>* 2 *<sup>m</sup>*Δ

satisfy the radiation conditions have the following form [10]:

**Φ**0*<sup>m</sup>*ð Þ¼ *x*, *t*

**Φ**0*<sup>m</sup>*ð Þ¼ *x*, *t*

*simple layer on the sound cone r* ¼ *cmt*, *r* ¼ k k*x* .

It is easy to show that the next lemma is true.

*conditions are representable in the following form:*

**31**

**Ф**0*<sup>m</sup>*ð Þ¼ *x*, *t*

**8. Green tensor of Biot's equations by non-stationary motion**

**<sup>Φ</sup>**0*<sup>m</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>F</sup>*�<sup>1</sup> *<sup>f</sup>* <sup>0</sup>*<sup>m</sup>*ð Þ *<sup>ξ</sup>*, *<sup>ω</sup>* � � <sup>¼</sup> *<sup>F</sup>*�<sup>1</sup> *<sup>c</sup>*

To construct the non-stationary Green tensor, at first we also construct the

Depending on the dimension of a space, solutions of this wave equation that

*<sup>δ</sup> <sup>t</sup>* � *<sup>r</sup> cm*

*H ct* ð Þ � *<sup>r</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *c*2

1 4*πc*<sup>2</sup> *mr*

1 2*πcm*

> 1 2

*H t*ð Þ*δ*ð Þ\$ *x*

Here *H t*ð Þ is the Heaviside function, and singular function *δ*ð Þ *t* � *r=cm* is the

Using regularization of the general function *ω*�1in the space of distribution [10]:

and the properties of Fourier transform of generalized functions convolution:

*h* ¼ *f* ∗ *g* \$ *h* ¼ *f* � *g*

**Lemma.** *The originals of the primitives of the basic functions satisfying the radiation*

1 �*i*ð Þ *ω* þ *i*0

2

� �**Φ**0*<sup>m</sup>* <sup>¼</sup> *<sup>δ</sup>*ð Þ*<sup>t</sup> <sup>δ</sup>*ð Þ *<sup>x</sup>* (18)

*<sup>m</sup>*k k*<sup>ξ</sup>* <sup>2</sup> � *<sup>ω</sup>*<sup>2</sup> � ��<sup>1</sup> � �

� �, *<sup>N</sup>* <sup>¼</sup> 3; (19)

*mt*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> <sup>p</sup> , *<sup>N</sup>* <sup>¼</sup> 2; (20)

*H c*ð Þ *mt* � j j *x* , *N* ¼ 1*:* (21)

where

*for N* ¼ 1

$$\frac{d^2\Phi\_{0m}}{d\boldsymbol{\chi}^2} = \frac{1}{2c\_m^2 k\_m} \left( k\_m \, ^2(\sin k\_m |\boldsymbol{\chi}|) - 2k\_m \delta(\boldsymbol{\chi}) \right);$$

*for N* ¼ 2

$$\frac{\partial^2 \Phi\_{0m}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} = -\frac{i}{4c\_m^2} \left( 0.5k\_m^2 (H\_0(k\_m r) - H\_2(k\_m r))r\_{,j}r\_{,k} + k H\_1^1(k\_m r)r\_{,jk} \right);$$

*for N* ¼ 3

$$\frac{\partial \Phi\_{0m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k}} = \frac{1}{4\pi r c\_{m}^{2}} e^{ikr} \left\{ r, \, \_{j} r, \_{k} \left( \left( ik\_{m} - \frac{1}{r} \right)^{2} + \frac{1}{r^{2}} \right) + r, \, \_{jk} \left( ik\_{m} - \frac{1}{r} \right) \right\};$$

$$k\_{m} = \frac{\alpha r}{c\_{m}}, r = ||\mathbf{x}||, \quad r, \, \_{j} = \frac{\mathbf{x}\_{j}}{r}, \quad r, \, \_{\vec{\eta}} = \frac{1}{r} \left( \delta\_{\vec{\eta}} - \frac{\boldsymbol{\varkappa}\_{i} \boldsymbol{\varkappa}\_{j}}{r^{2}} \right).$$

*Proof*. By using originals of basic functions, property (12) of derivatives, we can obtain from formulas for *U<sup>k</sup> <sup>j</sup>* in Theorem 1 the originals of all addends, besides that which contain factor k k*<sup>ξ</sup>* <sup>2</sup> . But using (16) we have:

$$-\Delta \Phi = \mathfrak{c}\_m^{-2} \delta(\mathfrak{x}) + k\_m^2 \Phi\_{0m} \quad \leftarrow \quad ||\xi||^2 f\_{0m} = \mathfrak{c}\_m^{-2} + k\_m^2 f\_{0m}$$

Then formulas of Theorem 2 follow from formulas of Theorem 1.

## **7. Generalized solutions by arbitrary periodic forces**

Under the action of arbitrary mass forces with frequency *ω* in Biot's medium, the solution for complex amplitudes has the form of a tensor functional convolution:

$$u\_j(\mathbf{x}, t) = U\_j^k(\mathbf{x}, a) \* F\_k(\mathbf{x}) e^{-i\alpha t}, j, k = \overline{1, 2N} \tag{17}$$

Note that mass forces may be different from the space of generalized vector function, singular and regular. Since Green tensor is singular and contains deltafunctions, this convolution is calculated on the rule of convolution in generalized function space. If a support of acting forces are bounded (contained in a ball of finite radius), then all convolutions exist. If supports are not bounded, then the existence conditions of convolutions in formula (17) requires some limitations on behavior of forces at infinity which depends on a type of mass forces and space dimension.

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

The obtained solutions allow us to study the dynamics of porous water- and gassaturated media at the action of periodic sources of disturbances of a sufficiently arbitrary form. In particular, they are applicable in the case of actions of certain forces on surfaces, for example, cracks, in porous media that can be simulated by simple and double layers on the crack surface.

There is another interesting feature of the Green tensor of the Biot's equations, which contains, as one of the terms, the delta-function that complicates the application of this tensor for solving boundary value problems based on analogues of Green formulas for elliptic systems of equations or the boundary element method. Here, when constructing the model, the viscosity of the liquid is not taken into account, which, apparently, leads to the presence of such terms, and it requires improvement of this model taking into account a viscosity.
