**1. Introduction**

Various mathematical models of deformable solid mechanics are used to study the seismic processes of earth's crust. The processes of wave propagation are most studied in elastic media. But these models do not take into account many real properties of an ambient array. These are, for example, the presence of groundwater, which affects the magnitude and distribution of stresses. Models, which take into account the water saturation of earth's crust structures, presence of gas bubbles, etc., are multicomponent media. A variety of multicomponent media, complexity of processes associated with their deformation, lead to a large difference in methods of analysis and modeling used in solving such problems.

Porous medium saturated with liquid or gas, from the point of view of continuum mechanics, is essentially a two-phase continuous medium, one phase of which is particles of liquid (gas) and other solid particles are its elastic skeleton. There are various mathematical models of such media, developed by various authors. The most famous of them are the models of Biot, Nikolaevsky, and Horoshun [1–5]. However, the class of solved tasks to them is very limited and mainly associated with the construction and study of particular solutions of these equations based on methods of full and partial separation of variables and theory of special functions in the works of Rakhmatullin, Saatov, Filippov, Artykov [6, 7], Erzhanov, Ataliev, Alexeyeva, Shershnev [8, 9], etc. In this regard, it is important to develop effective methods of solution of boundary value problems for such media with the use of modern mathematical methods.

Here *σij*ð Þ *x*, *t* are a stress tensor in a skeleton, and *p x*ð Þ , *t* is a pressure in a fluid

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

<sup>1</sup> <sup>¼</sup> *<sup>α</sup>*<sup>1</sup> <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *α*1

<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>1</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *α*1

*α*<sup>1</sup> ¼ ð Þ *λ* þ 2*μ ρ*<sup>22</sup> þ *Rρ*<sup>11</sup> � 2*Qρ*12,

The first two speeds *с*1,*c*2ð Þ *с*<sup>1</sup> >*c*<sup>2</sup> describe the velocity of propagation of two types of *dilatational waves*. The second slower dilatation wave is called *repackaging wave*. A third velocity *c*<sup>3</sup> corresponds to *shear waves* and at *ρ*<sup>12</sup> ¼ 0 coincides with

We introduce also two velocities of propagation of dilatational waves in

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

> ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup> *n Fs <sup>n</sup>*ð Þ *x e* �*iωnt* ,

ð Þ¼ *<sup>x</sup>*, *<sup>t</sup>* <sup>X</sup> *n F f <sup>n</sup>* ð Þ *x e*

ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>F</sup><sup>s</sup>*

ð Þ¼ *<sup>x</sup>*, *<sup>t</sup> <sup>F</sup> <sup>f</sup>*

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 oscilla-

> ð Þ *x e* �*iωt* ,

> > ð Þ *x e*

ffiffiffiffiffiffiffiffiffiffiffiffiffi *λ* þ 2*μ ρ*11 s

,

*:*

, *с <sup>f</sup>* ¼

ffiffiffiffiffiffi *R ρ*22 s

�*iωnt* (4)

�*iω<sup>t</sup>* (5)

2*α*<sup>2</sup>

2*α*<sup>2</sup>

p

p

ffiffiffiffiffiffiffiffiffi *ρ*22*μ α*2

r

*<sup>α</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ρ</sup>*11*ρ*<sup>22</sup> � *<sup>ρ</sup>*<sup>12</sup> ð Þ<sup>2</sup>

*<sup>α</sup>*<sup>3</sup> <sup>¼</sup> ð Þ *<sup>λ</sup>* <sup>þ</sup> <sup>2</sup>*<sup>μ</sup> <sup>R</sup>* � *<sup>Q</sup>*<sup>2</sup>

velocity of shear wave propagation in an elastic skeleton (*c*<sup>3</sup> < *c*1).

corresponding elastic body and in an ideal compressible fluid:

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

*Fs*

*F f*

*Fs*

*F f*

*cs* ¼

<sup>2</sup> � 4*α*2*α*<sup>3</sup>

<sup>2</sup> � 4*α*2*α*<sup>3</sup>

,

,

*j*

ð Þ *x*, *t ej* and on a liquid

*<sup>∂</sup>xk* , *<sup>u</sup> <sup>j</sup>*,*<sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>ku <sup>j</sup>*,

(3)

component. External mass forces acting on a skeleton *<sup>F</sup><sup>s</sup>* <sup>¼</sup> *<sup>F</sup><sup>s</sup>*

*c* 2

*c* 2

*c* 2 3 ¼

where the next constants were introduced as:

Further we use the next notations for partial derivatives: *<sup>∂</sup><sup>k</sup>* <sup>¼</sup> *<sup>∂</sup>*

*<sup>j</sup>*ð Þ *x*, *t ej*.

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

There are three sound speeds in this medium:

component *<sup>F</sup> <sup>f</sup>* <sup>¼</sup> *<sup>F</sup> <sup>f</sup>*

in the form:

tion frequency *ω*:

**25**

*<sup>Δ</sup>* <sup>¼</sup> *<sup>∂</sup>k∂<sup>k</sup>* is Laplace operator.

Periodic on time processes are very widespread in practice. By this cause, here we consider also processes of wave propagation in Biot's medium, posed by the periodic forces of different types. Based on the Fourier transformation of generalized functions, we constructed fundamental solutions of oscillation equations of Biot's medium. It is Green tensor, which describes the process of propagation of harmonic waves at a fixed frequency in the space–time of dimension N = 1, 2, 3, under the action concentrated at the coordinates origin. By using this tensor, we construct generalized solutions of these equations for arbitrary sources of periodic disturbances, which can be described as both regular and singular distributions. They can be used to calculate the stress-strain state of a porous water-saturated medium by seismic wave propagation.
