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

To construct the non-stationary Green tensor, at first we also construct the originals of the basic functions in an initial space-time:

$$\Phi\_{0m}(\mathbf{x},t) = F^{-1}\left[\boldsymbol{f}\_{0m}(\xi,\boldsymbol{\alpha})\right] = F^{-1}\left[\left(\boldsymbol{c}\_m^2||\xi||^2 - \boldsymbol{\alpha}^2\right)^{-1}\right]$$

They are originals of the classic wave equation:

$$
\left(\frac{\partial^2}{\partial t^2} - c\_m^2 \Delta\right) \Phi\_{0m} = \delta(t)\delta(\mathbf{x}) \tag{18}
$$

Depending on the dimension of a space, solutions of this wave equation that satisfy the radiation conditions have the following form [10]:

$$\Phi\_{0m}(\varkappa, t) = \frac{1}{4\pi c\_m^2 r} \delta \left( t - \frac{r}{c\_m} \right), \quad N = 3; \tag{19}$$

$$\Phi\_{0m}(\mathbf{x},t) = \frac{1}{2\pi c\_m} \frac{H(ct-r)}{\sqrt{c\_m^2 t^2 - r^2}}, \quad N = 2; \tag{20}$$

$$\Phi\_{0m}(\mathbf{x},t) = \frac{1}{2}H(c\_mt - |\mathbf{x}|), \quad N = 1. \tag{21}$$

Here *H t*ð Þ is the Heaviside function, and singular function *δ*ð Þ *t* � *r=cm* is the *simple layer on the sound cone r* ¼ *cmt*, *r* ¼ k k*x* .

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

$$H(t)\delta(\mathbf{x}) \leftrightarrow \frac{1}{-i(a+i\mathbf{0})}$$

and the properties of Fourier transform of generalized functions convolution:

$$h = f \ast \mathfrak{g} \leftrightarrow \overline{h} = \overline{f} \times \overline{\mathfrak{g}}$$

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

**Lemma.** *The originals of the primitives of the basic functions satisfying the radiation conditions are representable in the following form:*

*Uk j*

*Uk*

where

*for N* ¼ 1

*for N* ¼ 2 *∂*2 *Ф*0*<sup>m</sup> ∂x <sup>j</sup>∂xk*

*for N* ¼ 3

*∂Ф*0*<sup>m</sup> ∂x <sup>j</sup>∂xk*

obtain from formulas for *U<sup>k</sup>*

which contain factor k k*<sup>ξ</sup>* <sup>2</sup>

dimension.

**30**

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

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

3

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

<sup>þ</sup> *<sup>μ</sup> <sup>α</sup>*2*ω*<sup>2</sup> *<sup>c</sup>*

*d*2 *Ф*0*<sup>m</sup> dx*<sup>2</sup> <sup>¼</sup> <sup>1</sup> 2*c*<sup>2</sup> *mkm*

¼ � *<sup>i</sup>* 4*c*<sup>2</sup> *m*

> <sup>¼</sup> <sup>1</sup> 4*πrc*<sup>2</sup> *m e*

*km* <sup>¼</sup> *<sup>ω</sup> cm*

�ΔΦ ¼ *c*

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

3

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

0*:*5*k*<sup>2</sup>

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

*Mathematical Theorems - Boundary Value Problems and Approximations*

þ 1 *α*2

*ρ*12*δ<sup>k</sup>*

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

*km*<sup>2</sup>

*ikr <sup>r</sup>*, *jr*, *<sup>k</sup> ikm* � <sup>1</sup>

. But using (16) we have:

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

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

*j*

*<sup>u</sup> <sup>j</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup><sup>k</sup>*

,*<sup>r</sup>* <sup>¼</sup> k k*<sup>x</sup>* , *<sup>r</sup>*, *<sup>j</sup>* <sup>¼</sup> *<sup>x</sup> <sup>j</sup>*

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

*<sup>j</sup>*þ*<sup>N</sup>* � <sup>1</sup> *α*2

ð Þ� sin *km*j j *<sup>x</sup>* <sup>2</sup>*kmδ*ð Þ *<sup>x</sup>* � �;

� �;

þ 1 *r*2

( ) � �

*r*

*<sup>j</sup>* in Theorem 1 the originals of all addends, besides that

*f*0*<sup>m</sup>* ¼ *c* �2 *<sup>m</sup>* <sup>þ</sup> *<sup>k</sup>*<sup>2</sup>

*<sup>m</sup>*ð Þ *<sup>H</sup>*0ð Þ� *kmr <sup>H</sup>*2ð Þ *kmr <sup>r</sup>*, *jr*, *<sup>k</sup>* <sup>þ</sup> *kH*<sup>1</sup>

*r* � �<sup>2</sup>

*Proof*. By using originals of basic functions, property (12) of derivatives, we can

*<sup>m</sup>*Φ0*<sup>m</sup>* \$ k k*<sup>ξ</sup>* <sup>2</sup>

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:

�*iωt*

ð Þ *x*,*ω* ∗ *Fk*ð Þ *x e*

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

*<sup>r</sup>* , *<sup>r</sup>*, *ij* <sup>¼</sup> <sup>1</sup>

!

� �

*j*

Φ<sup>03</sup>

*ρ*11*δ<sup>k</sup>*

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

<sup>1</sup>ð Þ *kmr r*, *jk*

*r*

*::*

*mf*0*<sup>m</sup>*

, *j*, *k* ¼ 1, 2*N* (17)

;

<sup>þ</sup> *<sup>r</sup>*, *jk ikm* � <sup>1</sup>

*<sup>δ</sup>ij* � *xix <sup>j</sup> r*2 � �

� �

*j*

Φ<sup>03</sup>

*for N = 3*

$$\begin{split} \Phi\_{1m}(\mathbf{x},t) &= \Phi\_{0m}(\mathbf{x},t) \ast H(t)\delta(\mathbf{x}) = \frac{H(c\_{m}t - r)}{4\pi c^{2}r}, \\ \Phi\_{2m}(\mathbf{x},t) &= \Phi\_{1m}(\mathbf{x},t) \ast H(t)\delta(\mathbf{x}) = \frac{(c\_{m}t - r)\_{+}}{4\pi c^{3}r}; \end{split} \tag{22}$$

for *N* = 2

for *N* = 3

*∂*2 **Ф**2*<sup>m</sup> ∂x <sup>j</sup>∂xk*

*∂*2 **Ф**2*<sup>m</sup> ∂x <sup>j</sup>∂xk*

> <sup>¼</sup> *<sup>t</sup>* 4*πc*<sup>2</sup>

tensor functional convolution:

Then also write:

*<sup>u</sup> <sup>j</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>U</sup><sup>k</sup>*

depending on the type of mass forces [10].

*<sup>u</sup>*1ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> H t*ð Þ <sup>ð</sup>*<sup>t</sup>*

*o dτ* ð

The convolution with singular part is equal to:

*<sup>u</sup>*2ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> H t*ð Þ <sup>ð</sup>*<sup>t</sup>*

To construct their integral presentation, use this rule:

<sup>¼</sup> *H t*ð Þ <sup>ð</sup>*<sup>t</sup>*

**33**

0 *dτ*

*α*ð Þ *x*, *t δ*ð Þ *cmt* � *r* ∗ *F x*ð Þ¼ , *t*

ð

k k¼ *y*�*x cmτ*

*RN*

*o*

*j*

functions and singular functions, which contain delta-function:

<sup>¼</sup> *H c*ð Þ *mt* � *<sup>r</sup>* 2*πc*<sup>3</sup> *mr*<sup>3</sup>

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

2*c*<sup>2</sup> *mt* <sup>2</sup> � *<sup>r</sup>*<sup>2</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *c*2

*mr*<sup>2</sup> *<sup>δ</sup>*ð Þ *cmt* � *<sup>r</sup> <sup>r</sup>*, *kr*, *<sup>j</sup>* � *tH c*ð Þ *mt* � *<sup>r</sup>*

*mt*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> <sup>p</sup> *<sup>r</sup>*, *kr*, *<sup>j</sup>* � *<sup>δ</sup>jk*

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

*r*, *<sup>j</sup>* ¼ *x <sup>j</sup>=r:*

Using the properties of Green tensor, we obtain generalized solutions of nonstationary Biot's equations under the action of arbitrary mass forces in the Biot's medium, which satisfy the radiation condition at infinity. They have the form of

**9. Generalized solutions of Biot's equations by non-stationary forces**

It's taken according to the rules of convolution of generalized functions

In order to get the classic solution, we must present formulas (28) in regular integral forms. For this, let us present matrix of Green tensor as sum of regular

*U x*ð Þ¼ , *t Ureg*ð Þþ *x*, *t Using* ð Þ*t δ*ð Þ *x :*

Here *u*1ð Þ *x*, *t* is representable by regular mass forces in the integral form:

*r*

� � � �, (27)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *c*2 *mt*<sup>2</sup> � *r*<sup>2</sup>

*δjk* � 3*r*, *kr*, *<sup>j</sup>*

q ! (26)

ð Þ *x*, *t* ∗ *Fk*ð Þ *x*, *t* , *j*, *k* ¼ 1, … , 2*N* (28)

*u x*ð Þ¼ , *t u*1ð Þþ *x*, *t u*2ð Þ *x*, *t* (29)

*F <sup>f</sup>*ð Þ *y*, *t* � *τ* !

*dτ*

*dy*

*Ureg* ð Þ� *<sup>x</sup>* � *<sup>y</sup>*, *<sup>τ</sup> Fs*ð Þ *<sup>y</sup>*, *<sup>t</sup>* � *<sup>τ</sup>*

*Using* ð Þ� *<sup>τ</sup> Fs*ð Þ *<sup>x</sup>*, *<sup>t</sup>* � *<sup>τ</sup>*

*<sup>α</sup>*ð Þ *<sup>x</sup>* � *<sup>y</sup>*, *<sup>τ</sup> F y*, *<sup>t</sup>* � k k *<sup>y</sup>* � *<sup>x</sup>*

In 3D space, there are convolutions with simple layers on sound cones (see (27)).

*F <sup>f</sup>*ð Þ *x*, *t* � *τ* !

*cm* � �*dS y* ð Þ

for N = 2

$$\begin{aligned} \Phi\_{1m}(\mathbf{x},t) &= \frac{1}{2\pi c^2} \ln\left(\frac{c\_m t + \sqrt{c\_m^2 t^2 - r^2}}{r}\right), \\\\ \Phi\_{2m}(\mathbf{x},t) &= \frac{1}{2\pi c^3} \left(c\_m t \ln\left(\frac{c\_m t + \sqrt{c\_m^2 t^2 - r^2}}{r}\right) - \sqrt{c\_m^2 t^2 - r^2}\right); \end{aligned} \tag{23}$$

for N = 1

$$\begin{aligned} \Phi\_{1m}(\mathbf{x}, t) &= 0, \mathbf{5}(c\_m t - r) H(c\_m t - r) \triangleq \mathbf{0}. \mathbf{5}(c\_m t - r)\_+, \\ \Phi\_{2m}(\mathbf{x}, t) &= \frac{1}{2c^2} (c\_m t - r)^2 H(c\_m t - r) \triangleq \frac{1}{2c^2} (c\_m t - r)\_+^2. \end{aligned} \tag{24}$$

Using these functions and the properties of the Fourier transform, we obtain the components of the Green tensor from the formulas of Theorem 1. We formulate the result in the next theorem.

**Theorem 3.** *The components of Green tensor of motion equations of two-component Biot's medium have the following forms:*

$$\begin{split} Forj=1,N,\qquad k=1,N,\\ U\_{j}^{k}(\mathbf{x},t) &= \sum\_{m=1}^{3} \rho\_{m} \frac{\partial^{2} \Phi\_{2m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k}} + \frac{1}{\alpha\_{2}} \left(\rho\_{12} \delta\_{j+N}^{k} - \rho\_{22} \delta\_{j}^{k}\right) \Phi\_{03}(\mathbf{x}),\\ U\_{j+N}^{k}(\mathbf{x},t) &= \sum\_{m=1}^{3} \gamma\_{m} \frac{\partial^{2} \Phi\_{2m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k}} + \frac{\mu}{\alpha\_{2} c\_{m}^{2}} \delta\_{j+N}^{k} (\Phi\_{0m} - t\_{+} \delta(\mathbf{x})) - \\ &\qquad - \frac{1}{\alpha\_{2}} \left(\rho\_{11} \delta\_{j+N}^{k} + \rho\_{12} \delta\_{j}^{k}\right) \Phi\_{03}(\mathbf{x}) \\ Forj=\overline{1,N}, \qquad k=N+1,2N,\end{split}$$

$$U\_{j}^{k}(\mathbf{x},t) = \sum\_{m=1}^{3} \eta\_{m} \frac{\partial^{2} \Phi\_{2m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k} - N} + \frac{1}{\alpha\_{2}} \left(\rho\_{12} \delta\_{j+N}^{k} - \rho\_{22} \delta\_{j}^{k}\right) \Phi\_{03}(\mathbf{x}),$$

$$\mathbf{J}\_{j} = \mathbf{J}\_{j}^{k}$$

$$U^{k}\_{j+N}(\mathbf{x},t) = \sum\_{m=1}^{3} \xi\_{m} \frac{\partial^{2} \Phi\_{2m}}{\partial \mathbf{x}\_{j} \partial \mathbf{x}\_{k-N}} + \frac{\mu}{\alpha\_{2} c\_{m}^{2}} \delta^{k}\_{j+N} (\Phi\_{0m} - t\_{+} \delta(\mathbf{x})) - \frac{\mu}{\alpha\_{2}}$$

$$-\frac{1}{\alpha\_{2}} \left(\rho\_{11} \delta^{k}\_{j+N} + \rho\_{12} \delta^{k}\_{j}\right) \Phi\_{03}(\mathbf{x})$$

Here for *N* = 1

$$\frac{d^2\Phi\_{2m}}{d\boldsymbol{\kappa}^2} = \frac{H(c\_m\boldsymbol{t} - |\boldsymbol{\kappa}|)}{c\_m^2} - \frac{2}{c\_m^2}(c\_m\boldsymbol{t} - |\boldsymbol{\kappa}|)\_+\delta(\boldsymbol{\kappa})\tag{25}$$

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

for *N* = 2

*for N = 3*

for N = 2

for N = 1

**Ф**1*m*ð Þ¼ *x*, *t*

**Ф**2*m*ð Þ¼ *x*, *t*

result in the next theorem.

1

1

**Ф**2*<sup>m</sup>*ð Þ¼ *x*, *t*

*Biot's medium have the following forms:*

ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup> 3

*<sup>j</sup>*þ*<sup>N</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup>

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

ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup> 3

*<sup>j</sup>*þ*<sup>N</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup>

*d*2 **Ф**2*<sup>m</sup>*

*m*¼1 *ηm ∂*2 **Ф**2*<sup>m</sup> <sup>∂</sup><sup>x</sup> <sup>j</sup>∂xk*�*<sup>N</sup>*

3

*m*¼1 *ςm ∂*2 **Ф**2*<sup>m</sup> <sup>∂</sup><sup>x</sup> <sup>j</sup>∂xk*�*<sup>N</sup>*

� 1 *α*2

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

3

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

� 1 *α*2

*For j* ¼ 1, *N*, *k* ¼ 1, *N*,

*Uk j*

*Uk*

*Uk j*

*Uk*

Here for *N* = 1

**32**

1

<sup>2</sup>*c*<sup>2</sup> ð Þ *cmt* � *<sup>r</sup>*

**Ф**1*m*ð Þ¼ *x*, *t* **Ф**0*m*ð Þ *x*, *t* ∗ *H t*ð Þ*δ*ð Þ¼ *x*

*Mathematical Theorems - Boundary Value Problems and Approximations*

**Ф**2*m*ð Þ¼ *x*, *t* **Ф**1*m*ð Þ *x*, *t* ∗ *H t*ð Þ*δ*ð Þ¼ *x*

<sup>2</sup>*πc*<sup>3</sup> *cmt* ln *cmt* <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*c*2 *mt*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> <sup>p</sup> *r* !

,

*H c*ð Þ *mt* � *<sup>r</sup>* <sup>≜</sup> <sup>1</sup>

q !

�

*c*2 *mt*<sup>2</sup> � *<sup>r</sup>*<sup>2</sup> <sup>p</sup> *r* !

**<sup>Ф</sup>**1*<sup>m</sup>*ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* 0, 5ð Þ *cmt* � *<sup>r</sup> H c*ð Þ *mt* � *<sup>r</sup>* <sup>≜</sup>0*:*<sup>5</sup> ð Þ *cmt* � *<sup>r</sup>* <sup>þ</sup>,

Using these functions and the properties of the Fourier transform, we obtain the components of the Green tensor from the formulas of Theorem 1. We formulate the

**Theorem 3.** *The components of Green tensor of motion equations of two-component*

<sup>þ</sup> *<sup>μ</sup> α*2*c*<sup>2</sup> *m δk*

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

� �

þ 1 *α*2 *j*

*ρ*12*δ<sup>k</sup>*

*j*

<sup>þ</sup> *<sup>μ</sup> α*2*c*<sup>2</sup> *m δk*

� 2 *c*2 *m*

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

� �

*ρ*12*δ<sup>k</sup>*

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

**Ф**03ð Þ *x*

� �

**Ф**03ð Þ *x*

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

*j*

*<sup>j</sup>*þ*<sup>N</sup>*ð**Ф**0*<sup>m</sup>* � *<sup>t</sup>*þ*δ*ð Þ *<sup>x</sup>* Þ�

ð Þ *cmt* � j j *<sup>x</sup>* <sup>þ</sup>*δ*ð Þ *<sup>x</sup>* (25)

**Ф**03ð Þ *x* ,

� �

*j*

*<sup>j</sup>*þ*<sup>N</sup>*ð**Ф**0*<sup>m</sup>* � *<sup>t</sup>*þ*δ*ð Þ *<sup>x</sup>* Þ�

**Ф**03ð Þ *x* ,

þ 1 *α*2

*ρ*11*δ<sup>k</sup>*

*ρ*11*δ<sup>k</sup>*

*dx*<sup>2</sup> <sup>¼</sup> *H c*ð Þ *mt* � j j *<sup>x</sup> c*2 *m*

2

<sup>2</sup>*πc*<sup>2</sup> ln *cmt* <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*H c*ð Þ *mt* � *r* <sup>4</sup>*πc*2*<sup>r</sup>* ,

ð Þ *cmt* � *<sup>r</sup>* <sup>þ</sup> 4*πc*3*r*

;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *c*2 *mt*<sup>2</sup> � *r*<sup>2</sup>

<sup>2</sup>*c*<sup>2</sup> ð Þ *cmt* � *<sup>r</sup>*

;

2 þ*:* (22)

(23)

(24)

$$\frac{\partial^2 \Phi\_{2m}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} = \frac{H(c\_m t - r)}{2\pi c\_m^3 r^3} \left( \frac{2c\_m^2 t^2 - r^2}{\sqrt{c\_m^2 t^2 - r^2}} r, \,\!r \,, \,\_j \quad - \delta\_{jk} \sqrt{c\_m^2 t^2 - r^2} \right) \tag{26}$$

for *N* = 3

$$\frac{\partial^2 \Phi\_{2m}}{\partial \mathbf{x}\_j \partial \mathbf{x}\_k} = \frac{t}{4\pi c\_m^2 r^2} \left( \delta(c\_m t - r) r\_k r\_{\cdot,j} - \frac{tH(c\_m t - r)}{r} \left( \delta\_{jk} - \mathfrak{F} r\_k r\_{\cdot,j} \right) \right), \tag{27}$$

$$r\_{\cdot,j} = \mathbf{x}\_j / r.$$
