Propagation of Non-stationary Skew-Symmetric Waves from a

Spherical Cavity in a Porous-elastic Half-space

MAMURA MUSURMONOVA

Department of Applied Mathematics,

Karshi State University,

17, St. Kuchabog, Karshi,

UZBEKISTAN

Abstract: - Problems of propagation and diffraction of non-stationary waves in porous-elastic mediums are of

great theoretical and practical importance in such fields of science and technology as geophysics, seismic

exploration of minerals, seismic resistance of structures, and many others. The work considers the problem of

propagation of non-stationary skew-symmetric waves from a spherical cavity in a porous-elastic half-space

saturated with liquid. To solve the problem, the integral Laplace transform in dimensionless time and the

method of incomplete separation of variables were used. In the space of Laplace images in time, known and

unknown functions are expanded into Gegenbauer polynomials. The problem is reduced to solving an infinite

system of linear algebraic equations, the solution of which is sought in the form of an infinite exponential

series. Recurrence relations for the coefficients of the series and initial conditions for them are obtained, which

makes it possible to obtain a solution to the infinite system without using the reduction method. Recurrence

relations make it possible to determine the coefficients of a series in the form of rational functions, which

makes it possible to calculate their originals using the theory of residues. In image space, formulas are obtained

for the coefficients of the series of components of the displacement vector and stress tensor. Numerical

experiments were carried out, the results of which are presented in the form of graphs. The results obtained can

be used in geophysics, seismology, and design organizations during the construction of structures, as well as in

the design of underground reservoirs.

Key-Words: - propagation of shear wave, spherical cavity, Gegenbauer polynomials, Laplace transform,

residues, unsteady wave, porous-elastic medium, stress, displacement.

1 Introduction

The study of non-stationary wave processes in

continuous media is a complex and, at the same

time, relevant direction in the wave dynamics of

continuous media. The relevance of the problems of

continuum dynamics is due to the development of

various fields of technology, the creation of new

structures operating under dynamic loads, as well as

problems of geophysics, seismology, gas

exploration, oil exploration, mining industry, and

construction of civil and industrial structures.

Currently, there is a large number of scientific

works devoted to the study of wave propagation and

diffraction in continuous media. The existence of a

contact layer of soil during shear interaction of a

solid body with soil is demonstrated based on the

analysis of the numerical solution of a one-

dimensional non-stationary problem of the

interaction of a rigid strip with a nonlinearly

deformable soil medium, [1]. The appearance of a

peak value of shear stress and subsequent structural

destruction on the contact layer of soil is shown,

which is consistent with the experimental results.

Solutions to the problem of constant-velocity

expansion of a spherical cavity in a soil medium are

analyzed: the cavity expands from a point in the

half-space occupied by an elastic-plastic soil

medium, [2]. The previously obtained linearized

analytical solution of this problem is presented,

obtained under the assumption that the medium

behind the shock wave front is incompressible. As a

result of comparison with the results of the

numerical solution of the problem in the full

formulation, it is shown that the approximate

solution is a good approximation of the dependence

of the pressure at the boundary of the cavity on the

rate of its expansion. The propagation of non-

stationary transverse waves from a spherical

inclusion in an elastic half-space was studied in the

study [3]. Formulas for the components of the

displacement vector and stress tensor are obtained.

The stress-strain state of the medium in the vicinity

Received: April 13, 2024. Revised: October 19, 2024. Accepted: November 5, 2024. Published: December 17, 2024.

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of a spherical inclusion has been studied. An exact

solution to the problem of joint (coupled) seismic

vibrations of an underground pipeline and an

infinite elastic medium is given in the article [4].

Based on the established theorem on the separation

of boundary conditions for wave potentials on the

surface of a cylinder, a method is proposed that

significantly simplifies the solution of the external

problem for the medium. In [5], an exact analytical

solution to the problem of the dynamic expansion of

a spherical cavity in an elastic medium (soil) with

an arbitrary constant speed was obtained. The

solution found makes it possible to judge the impact

(or control the impact) of underground explosions

on objects in the “far” zone, at distances

significantly exceeding the size of the cavity. Article

[6] presents a unified mathematical approach to

describing the dynamic stress-strain state of

mechanical structures made of heterogeneous

materials with a double connected system of pore

channels filled with fluid. New dynamic equations

have been obtained that describe the vibrations of

poroelastic systems based on the developed model

of a continuous medium with additional degrees of

freedom in the form of different pressures of the

components that make up the liquid phase of the

material. In [7], the propagation of non-stationary

longitudinal waves from a spherical cavity

supported by a thin spherical shell in an elastic-

porous space saturated with liquid was studied. An

analytical solution to the problem in the image space

of the Laplace transform is obtained. Numerical

results are presented in the form of graphs. The

article [8] considered the problem of non-stationary

transverse oscillations of an elastic half-space with a

rigid ball. To solve the problem, the Laplace integral

transform and the method of incomplete separation

of variables were used. Formulas for the

components of the displacement vector and stress

tensor are obtained. The paper [9] studies the

interaction of spherical elastic SH waves of

harmonic type with a spherical layer when the

source is placed outside the layer. The exact

solution to this scattering problem is studied in

detail after establishing the generalized Debye

expansion. The problem of the diffraction of plane

waves by a system of two concentric spherical shells

surrounded by acoustic media was studied in the

article [10]. In this case, the solution is represented

in the form of a superposition of elementary waves.

The study [11] considered the problem of the

propagation of shear disturbances from a spherical

cavity in an infinite elastic medium. In this case, the

Fourier transform in time was used. Expressions are

obtained for the displacement and stress over time

caused by an axially symmetric shear stress applied

to the inner surface of a spherical cavity in an

infinite isotropic medium. In [12], the problems of

scattering and diffraction of a cylindrical transverse

shear wave in a viscoelastic isotropic medium by a

spherical inhomogeneity are solved analytically.

The waves are generated by harmonic longitudinal

vibrations of the cylinder walls. The spherical

inclusion is located in the radial center of the

cylinder and differs from the cylindrical material

only in its complex shear modulus. Numerical

examples are given to show the effect of changes in

inclusion rigidity on displacement fields. The paper

[13] addresses the problem of transient

elastodynamics analysis of a thick-walled, fluid-

filled spherical shell embedded in an elastic medium

with an analytical approach. Various constitutive

relations for the elastic medium, shell material, and

fill fluid are considered, as well as various

excitation sources (including S/P wave or

plane/spherical wave incident at different locations).

The statements and solutions of parabolic problems

modeling the physical phenomena in soils in the

case of discontinuous velocity on the boundaries at

the initial time are given in [14]. The notion of

generalized vorticity diffusion is introduced and the

cases of self-similarity existence are classified. In

the case of a physically linear medium, new self-

similar solutions are obtained which describe the

process of unsteady axially symmetric shear in

spherical coordinates. A semi-infinite circular

cylindrical cavity filled with an ideal compressible

liquid that contains a spherical body located near its

face is considered in [15]. A plain acoustic wave

propagates along the axis of the cavity. The problem

of determining of hydrodynamic characteristics of

the system depending on the frequency of the plane

wave and geometrical parameters is solved. The

developed method can detect the anomalous features

of the diffraction of a plane wave due to the

influence of the face wall. This study [16] considers

the propagation of harmonic plane waves in a

double-porosity solid saturated by a viscous fluid.

Two different porosities are supported with different

permeabilities to facilitate the wave-induced fluid

flow in this composite material. Relevant equations

of motion are solved to explain the propagation of

three longitudinal waves and one transverse wave in

the double-porosity dual-permeability medium. A

numerical example is considered to illustrate

dispersion in velocity and attenuation of the four

waves. The effect of wave-induced fluid flow is

analysed with changes in wave-inhomogeneity,

pore-fluid viscosity, and double-porosity structure.

In paper [17] slow motion of a porous spherical

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shell with radially varying permeability in a

spherical container at the instant it passes through

the center of the spherical container is discussed.

The exact solution of the problem is obtained. The

influence of the permeability parameter on the flow

has been discussed and exhibited graphically. A

statement of the problem of determination of the

acoustic radiation force acting on the rigid spherical

particle is formulated in the study [18]. The particle

is located in the fluid-filled thin flexible tube The

problem is solved by the use of the method of

separation of variables. The characteristics of the

acoustic radiation force are studied depending on

the primary wave frequency, the radius of the rigid

spherical particle, the radius of the compliant

cylindrical elastic tube, the properties of the filling

liquid, the thickness of the tube wall, and elastic

properties of the tube material.

This work is devoted to the study of problems

on the propagation of non-stationary skew-

symmetric waves from a spherical cavity in a

porous-elastic half-space saturated with liquid. The

purpose of the work is to develop an algorithm for

solving the problem and study non-stationary wave

processes during the propagation of non-stationary

skew-symmetric waves from a spherical cavity in a

porous-elastic saturated half-space.

2 Problem Formulation

Let the centre

of a spherical cavity of radius

(

Rh

) be located in a saturated porous-elastic half-

space

0z

at a depth

from the plane

0z

on the

axis

(the point

lies on the boundary of the

half-space) (Figure 1). We will consider two

coordinate systems: spherical

,,r

with the centre

at the point

and cylindrical

,,z

with the origin

at the point

At a moment

0

in time, an axisymmetric

specified tangential surface load

( , )q

is applied

to the surface of the cavity, which forms a rotational

motion of the medium around an axis passing

through the center of the cavity (Figure 1).

( , )

rrR q



   

. (1)

Taking into account the axial symmetry of the

problem, the motion of the medium relative to the

non-zero component



of the vector displacement

potential is described by the wave equation

sinr



     

. (2)

Here



is the Laplace operator in a spherical

coordinate system

,,r

Fig. 1: Geometry of the problem

The flat boundary of a half-space is either a rigid

wall:

u

, (3)

or free surface





. (4)

Initial conditions are homogeneous

 

   

(5)

and the problem closes with the requirement that the

solution is bounded at infinity

lim 0

r 

. (6)

In this case, the functions

u

U

r



and





are connected by the following relations:

urr



 









(7)







   

  







uu ctg



 



   

   







. (8)

The problem statement is given in the following

dimensionless quantities (the prime indicates a

dimensionless quantity)

























12 22

   





2H P Q R  

2P A N





( , )q



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where

u

and

U

are components of the

displacement vector of the skeleton and fluid,

respectively;





( , , , )r    

are stress tensor

components;

and

are the elastic constants of

the medium’s skeleton;

is pressure applied to the

liquid;

is the amount of adhesion between solid

and liquid components;



is coefficient of

dynamic coupling between solid and liquid

components;



is the effective mass of the fluid

during its relative motion;

is the speed of

propagation of transverse waves in the medium

(below, the prime in the designations of

dimensionless quantities is omitted).

3 Problem Solution

The initial-boundary value problem (1)-(6) is solved

using the integral Laplace transform in time



and

the method of incomplete separation of variables. In

the image space, we represent the potential



components

u

displacement vector and

r



stress

tensor in the form of infinite series of Gegenbauer

polynomials

 

1cos

C

, [19], and the

representation of the infinite series for the stress

tensor component

()lL





has the following form (

denotes the transformant of the Laplace transform,

is transformation parameter), [20]

(1)

( , ) (cos )

rsP



 



    



(2) 3 2

cos ( , ) (cos )

r s C



 



   



, (9)

(1) ( 1)( )

nn u

 

   

  

(2) LL

 

  



where

(cos )

P

are the Legendre polynomials [19].

In image space, taking into account the

boundedness condition (6) the solution of equation

(2) is written in the form:

L L L

    

, (10)

0 1 2 1

sin ( ) ( ) (cos )

n n n

A s K r s C









    



1 1 2 1 1 1

sin ( ) ( ) (cos )

n n n

B s K r s C









    



Here

()

and

()

are arbitrary functions;





is an additional spherical coordinate

system obtained by transferring along the axis

of the centre

of the original spherical system to a

point

symmetrical to the point

relative to the

plane

0z

Taking into account the connection between

coordinates



and



on flat boundary of the

half-space

00zz





00zz

    

, (11)

as well as properties of Gegenbauer polynomials

[19]:

3 2 3 2

( ) ( 1) ( )

C x C x



  

, (12)

from the boundary conditions (3) and (4), we obtain

the connection between the functions

()

and

()

( ) ( 1) ( )

L n L

B s A s  

. (13)

Here and below, the upper sign corresponds to a

rigid wall, and the lower sign corresponds to the free

boundary of the half-space.

Now, substituting (13) into (10) and using the

addition theorem [21] for Bessel functions of the

second kind

12()



, as well as expressing these

functions in terms of elementary ones [19], we

present the image of the potential in the form of the

following series:

   

sin , cos

r s C







     



, (14)

where

 

, ( ) ( )

()

L r s

n n n

r s R r s A s e





   





( ) ( ) ( ) hs

n np p

G r s S s A s e







 





, (15)

( 1 1) 0

(2 )

( 1) (2 1)

() 2 ( 1) (2 )

ppn np

np pn

R h s

S s b

n n h s







 



  





0 0 0

( ) ( ) ( )

n n n

G s R s e R s e

  

()

nnk

n nk

R s D s 





( )!

( )!2

nk k

Dnk





D

0k

kn

Here

( 1 1)np

b

are the Clebsch-Gordon

coefficients, [21].

Similarly to (14), we expand the images of

displacement

u

, stress

r



and a given function

into series using Gegenbauer polynomials, and

arrive at the following expressions and the boundary

condition regarding the coefficients of the series

Lnn

urr



 





Lnn









  

  







, (16)

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1()

r n n

rqs





. (17)

Next, taking into account (15) from formulas

(16) we arrive at the following expressions for the

coefficients

u

rn



( , ) ( ) ( )

()

L r s

n n n

u r s R r s A s e





   







3( ) ( )

J r s S s

, (18)

 

, ( ) ( )

L r s

r n n n

n n n

r s R r s A s e





   

   







4( ) ( )

r s S sJ 

, (19)

( ) ( ) ( )

nm nm nm

J s R s e R s e

  

3, 4m

3 1 0

( ) ( ) 2 ( )

n n n

R s R s R s

4 2 0

( ) ( ) ( )

n n n

R s R s R s

()

nnk

n nk

R s B s







()

nnk

n nk

R s C s







,1nk nk n k

B D kD 



,1nk nk n k

C B kB 



( ) ( ) ( ) hs

n np p

S s S s A s e







Substituting (19) into the boundary condition

(17), we obtain an infinite system of linear algebraic

equations for the functions

()

, which we write

in the form of a matrix equation:

2 (1)

( ) ( ) ( ) ( )s s y s s xM A F A

(2) 2

( ) ( ) ( )s s xy s yF A p

, (20)

2hs









Here

()sM

is an infinite diagonal matrix with

elements

()

;

()

lsF

are infinite matrices of

elements

()

(

1,2l

);

()sp

is an infinite

column vector with elements

()

;

()sA

is an

infinite unknown column vector with components

()

, and functions

()

and

()

have the form:

( ) ( )

M s R R s

( ) 2 ( ) / ( )

L n n L

p s s q s



  

(1) ( ) ( ) ( )

np n np

F s M s S s

(2) ( ) ( ) ( )

np n np

F s M s S s

We look for the solution to matrix equation (20)

in the form of an infinite exponential series:

( ) ( ) ij

s s x y







, (21)

Here

()

ij sa

are infinite column vectors with

elements

()

1,2,3,...n

Substituting series (21) into equation (20) and

equating the coefficients of the left and right sides

for the same degrees of variables

and

(the series

on the right side contains only one non-zero term),

we obtain a recurrent system of equations for

functions

()

and the corresponding initial

conditions for them:

1 1, 1 2 1, 1

( ) ( ) ( ) ( ) ( )

ij i j i j

s s s s s

   

a E a E a

1, 1ij

1( ) ( )

s S sE

()

( ) ( )

()

nnp

s S s



E

0 1 1,0

( ) ( ) ( )

s s s



a E a

1i

1( ) 0

isa

0i

0( ) 0

jsa

1j

()

() ()

asMs



, 1,2,...np

These relations make it possible to determine all

the required images without using the reduction of

an infinite system of equations. Analysis of

recurrence relations shows that images are rational

functions of the Laplace transform parameter, which

makes it possible to calculate their originals, and

therefore the originals of the displacement and stress

coefficients in the medium using the theory of

residues, [22].

The final formulas for the representations of the

coefficients of the series in Gegenbauer polynomials

of the sought functions follow from (18), (19) and

(21)



()

( , ) ( ) ( )

()

L n r

n n ij

u r s R r s a s y





   



( ) 1

( ) ( ) ( )

p i j

n np ij

J R s S s a s x x y







 





, (22)



()

( , ) ( ) ( )

L n r

r n n ij

n n n ij

r s R r s a s y



 

  

   



( ) 1

( ) ( ) ( )

p i j

n np ij

J r s S s a s x x y







 





3.1 Numerical Experiments

As an example, we consider the propagation of non-

stationary skew-symmetric waves from spherical

cavity in a half-space of sandstone saturated with

kerosene with parameters

0.4026 10 PaA

0.2493 10 PaN

0.0672 10 PaR

0.0295 10 PaQ

00.26

2600

skg m

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820 /

fkg m

12 1.9 kg m  

, which

corresponds to the following dimensionless

parameters

0.0088331

1

0.8772

0.392

. The center of the cavity is located at a

distance

1.8h

, the flat boundary of the half-space

is the free surface (4). The results of numerical

experiments are presented in the form of graphs of

changes in components

r







of the stress tensor

and

u

of displacement over dimensionless time t.

As the law of change of the given load

( , ) ( )q q H   

01q

, a constant function in time

was chosen, where h is the Heaviside function.

Numerical results were obtained taking into account

seven terms of the series of Gegenbauer

polynomials.

In Figure 1, Figure 2 and Figure 3 were

obtained, respectively, for coordinate values

1.2;1.4;1.6r

. In Figure 2, Figure 4 and Figure 5

were plotted at the coordinate value

4  

, and

the graphs presented in Figure 3 are constructed at

the coordinate value

34  

Consequently, curve 1 in Figure 2 demonstrate

the change in voltage coordinate

r



at a point in

the environment with coordinates

1.2r

4  

Fig. 2: Variation in time of the component

r



when

( , ) ( )q q H   

4  

Fig. 3: Variation in time of the component

r



when

( , ) ( )q q H   

34  

Fig. 4: Variation in time of the component





when

( , ) ( )q q H   

4  

Fig. 5: Variation in time of the component

u

when

( , ) ( )q q H   

4  

In the following example, in the form of a given

load on the surface of the cavity, function

( , ) ( )q e H



   

, exponentially decreases in time.

For the initial values of the dimensionless

parameters, numerical results were obtained for

changes in the components

r







of the stress

tensor, and

u

of the displacement over

dimensionless time



. In this case, the components

r







the stress tensors and

u

displacement

vector are demonstrated in Figure 6, Figure 7 and

Figure 8. In all of them, Figure 1, Figure 2 and

Figure 3 are plotted, respectively, at the above

coordinate values

. Graphs in Figure 6 and Figure

8 were obtained at

4  

, and the graphs

presented in Figure 7 built at

34  

Fig. 6: Variation in time of the component

r



when

( , ) ( )q e H



   

4  

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Fig. 7: Variation in time of the component





when

( , ) ( )q e H



   

4  

Fig. 8: Variation in time of the component

u

when

( , ) ( )q e H



   

4  

The graphs show that with the arrival of a wave,

a jump appears. As the distance increases, the jumps

1.2, 1.4, 1.6r

decrease.

Waves reflected from the flat boundary of the

half-space influence the stress-strain state of the

medium. For the moments time

5

, the stress-

strain state of the medium practically passes into a

stationary state.

4 Conclusion

An algorithm has been developed for solving the

problem of the propagation of non-stationary

transverse shear waves from a spherical cavity in a

porous-elastic half-space. The propagation of non-

stationary skew-symmetric waves in the vicinity of

a cavity under various given loads has been studied

in the form of a constant and exponentially

decreasing function.

Numerical results were obtained, which are

presented in the form of graphs, which show the

influence of the flat boundary of the half-space on

the stress-strain state of the medium. The results

obtained can be used in geophysics, seismology, and

design organizations during the construction of

structures, as well as in the design of underground

reservoirs.

The proposed approach to solving the problem

can be applied to similar problems without

fundamental changes, such as the problem with a

displacement specified at the boundary of a cavity,

the problem of the diffraction of plane waves by a

cavity, as well as the corresponding problems for a

porous-elastic half-space.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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(Attribution 4.0 International, CC BY 4.0)

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Creative Commons Attribution License 4.0

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