The Solvability of the 3-D Elastic Wave Equations in ... - UKM

(1)

1063

The Solvability of the 3-D Elastic Wave Equations in Inhomogeneous Media

Purwadi A. Darwito^a, Gunawan Nugroho*^,a, Murry Raditya^b & Tavio^c

aDepartment of Engineering Physics, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Indonesia 60111

bDepartment of Instrumentation Engineering, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Indonesia 60111

cDepartment of Civil Engineering, Institut Teknologi Sepuluh Nopember, Kampus ITS Sukolilo, Indonesia 60111

*Corresponding author: [email protected]

Received 31 August 2021, Received in revised form 28 February 2022 Accepted 27 March 2022, Available online 30 November 2022

ABSTRACT

In this research, the three-dimensional elastic wave equations with variable coefficients (i.e. propagate through inhomogeneous media) are solved with the application of the Fourier transform in the spatial coordinates. The wave equation is coupled variable coefficients PDEs whose solutions may have significant in engineering applications. The method utilizes the second order ODE as the baseline for obtaining the complete solution. The solution of second order ODEs is expressed in one integration because the variable coefficients are broken down into several functions and resulted in first order reduction. Moreover, the coupled equations are performed by the order reduction of the higher order ODEs into the second order. The extended procedure for integral equation is implemented for the solutions from the transformed wave equations to generate the explicit expression. It is shown that the proposed method of integral evaluation is resulted in finding the roots of polynomials. Hence, it is concluded that the solvability of the elastic wave equations is ensured by the proposed method.

Keywords: Wave equation; Inhomogeneous media; Reduction of order; Integral evaluation; Reduction of polynomial order Jurnal Kejuruteraan 34(6) 2022: 1063-1075

https://doi.org/10.17576/jkukm-2022-34(6)-07

INTRODUCTION

The growing interests have been taking place in wave modeling fields. The transmittance and reflectance of sound wave with modulated speed is investigated by Mikhalevich

& Streltsov (2009). It is shown that the generated parameters are determined significantly by wave intensities and phase shift. The high-speed acoustic wave with dissipation in saturated sediment is also considered (Naugolnykh &

Esipov 2005). The consideration leads to the nonlinear evolution equation which the shock profile depends on the relaxation effects. The modified Kudraysov method with distinct integration schemes is utilized for ion sound wave (Seadawy et al. 2018). It is concluded that the approach is practically effective and can be exerted to several coupled PDEs. The similar method of reduction of the wave equation is found in Dzyuba & Romashko (2020). In this case, it is found that the speed of sound will have significant effect instead of pressure.

The geophysical geophysical problem of baroclinic wave packets is studied by Xie & Meng (2018). The one- and two-soliton solutions are obtained and the amplitude of the solitons become higher with the increasing parameters.

An exact solution for the geophysical water wave is investigated (Henry 2013). It is noted that the vorticity is not affected for the constant underlying current. Meanwhile, the losses of coal measurement related to elastic wave is

investigated. The low-frequency elastic wave with constant loss coefficient is considered and numerically solved to give the intrinsic absorption and scattering (Guo et al. 2020).

The large number of databanks are produced by the strong motion instrument motion in earthquake and seismology (Stamatovska 2012). They may become the validating instrument for the theoretical and mathematical solutions of elastic wave equation because better results will be obtained if the instrument position network is permanent. The Sine- Gordon expansion method is implemented for generating the exact solutions of the coupled Drinfeld-Sokolov-Wilson equation (Tarbozan et al. 2018). The method is also assisted by the perturbation iteration algorithm and it is concluded that the method is powerful and reliable. The study of Boussinesq equation is also performed to produce solitary wave solutions. It is concluded that the method is useful for extracting the exact solutions for shallow water problem and other nonlinear evolution equations (Hossain et al. 2018).

There are other related fields are studied which shows the importance of the wave equations and their solutions (Jleli et al. 2020; Wilk et al. 2017; Yu-Ting et al. 2013).

The considered problems depend on the medium and type of applications. The present study deals with the initial value problem of the three-dimensional elastic wave with variable coefficients. The considered problem is a system of linear PDE with possible spatially distributed forcing functions in time and x, y, z directions, which can be applied

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1064

to the geophysical problems and seismology (Kanth 2008;

Zhang et al. 1991; Pasternak & Dyskin 2008). Since the development of geological exploration for oil and gas including seismology need more description, indeed the study of the three-dimensional case is a challenging task.

In fact, the knowledge of the time and spatial distributions by both analytical and numerical method has attracted researchers in many fields (Guidotti et al. 2006).

In this work, the governing equation is simplified by application of Fourier transform in the spatial directions.

Starting from the x – displacement, the equation is solved analytically and is substituted sequentially in y – and z – displacements. Since the higher order ODE with variable

coefficients is produced, the method of order reduction is developed in this research. After second order equation is achieved, the method for solving second order ODE with variable coefficients is investigated and proposed. The evaluation of integral is also presented to compute the obtained solutions.

PROBLEM FORMULATION

Consider the initial value problem of 3-dimensional wave equation with variable coefficients as in the following (Yang, 2014),

Jurnal Kejuruteraan 34(6) 2022: xxx-xxx https://doi.org/10.17576/jkukm-2022-34(6)-07 Drinfeld-Sokolov-Wilson equation (Tarbozan et al.

2018). The method is also assisted by the perturbation iteration algorithm and it is concluded that the method is powerful and reliable. The study of Boussinesq equation is also performed to produce solitary wave solutions. It is concluded that the method is useful for extracting the exact solutions for shallow water problem and other nonlinear evolution equations (Hossain et al. 2018). There are other related fields are studied which shows the importance of the wave equations and their solutions (Jleli et al. 2020; Wilk et al. 2017; Yu-Ting et al.

2013).

The considered problems depend on the medium and type of applications. The present study deals with the initial value problem of the three- dimensional elastic wave with variable coefficients.

The considered problem is a system of linear PDE with possible spatially distributed forcing functions in time and directions, which can be applied to the geophysical problems and seismology (Kanth

2008; Zhang et al. 1991; Pasternak & Dyskin 2008).

Since the development of geological exploration for oil and gas including seismology need more description, indeed the study of the three- dimensional case is a challenging task. In fact, the knowledge of the time and spatial distributions by both analytical and numerical method has attracted researchers in many fields (Guidotti et al. 2006).

In this work, the governing equation is simplified by application of Fourier transform in the spatial directions. Starting from the displacement, the equation is solved analytically and is substituted sequentially in and displacements. Since the higher order ODE with variable coefficients is produced, the method of order reduction is developed in this research. After second order equation is achieved, the method for solving second order ODE with variable coefficients is investigated and proposed. The evaluation of integral is also presented to compute the obtained solutions.

PROBLEM FORMULATION

(1a)

(1b)

(1c)

(1d)

where is the forcing function in each and directions, is solid density, and are the displacements in and directions.

and are the P-wave and

shear wave velocities. The variant parameters and are elastic moduli of the solid which depend on the Young modulus and Poisson ratio. In this case the functions and are also smooth. In this case, equation (1) can be rewritten as,

(2a) , ,

x y z

x-

y- z-

( ) ( ) ( ) ( )

2 2 2 2

2 1

tt p x y z s y z s y x s x z

u c u v w c v w c u v c w u f

x x y z

¶ é ù ¶ é ù ¶ é ù ¶ é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 2

tt p x y z s x z s z y s y x

v c u v w c u w c v w c u v f

y y z x

¶ é ù ¶ é ù ¶ é ù ¶ é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 3

tt p x y z s x y s x z s z y

w c u v w c u v c w u c v w f

z z x y

¶ é ù ¶ é ù ¶ é ù ¶ é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

1 2

3 4

5 6

, , ,0 , , , , , ,0 , ,

t t t

u x y z g x y z u x y z g x y z v x y z g x y z v x y z g x y z w x y z g x y z w x y z g x y z

= =

( , , , )

f x y z ti

x y, z

r u v ,

w x y, z

p 2

c l µ

r

= + c_s µ

= r

µ l

fi g_i

( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

1

2 2

tt p xx s yy s zz p s xy p s xz px x sx y sx z sx x px sx y

sx x px y sx z

u c u c u c u c c v c c w c u c u c u c v c c v c w c w c w f

= + + + - + - + + + + + -

+ + - +

where f_i (x, y, z, t) is the forcing function in each x, y and z directions, ρ is solid density, u, v and w are the displacements in x, y and z directions.

2013).

PROBLEM FORMULATION Consider the initial value problem of 3-dimensional wave equation with variable coefficients as in the following (Yang, 2014),

(1a)

(1b)

(1c)

(1d)

(2a) , ,

x y z

x-

y- z-

( ) ( ) ( ) ( )

2 2 2 2

2 1

x x y z

¶é ù ¶é ù ¶é ù ¶é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 2

y y z x

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 3

z z x y

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

1 2

3 4

5 6

, , ,0 , , , , , ,0 , , , , ,0 , , , , , ,0 , ,

, , ,0 , , , , , ,0 , ,

t t t

= =

( , , ,)

f x y z ti

,

x y z r u v,

w x y, z

p 2

c l µ

r

= + cs µ

= r

µ l

fi gi

( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

1

2 2

u c u c u c u c c v c c w c u c u c u c v c c v

c w c w c w f

= + + + - + - + + + + + -

+ + - +

and

2013).

PROBLEM FORMULATION Consider the initial value problem of 3-dimensional wave equation with variable coefficients as in the following (Yang, 2014),

(1a)

(1b)

(1c)

(1d)

(2a) , ,

x y z

x-

y- z-

( ) ( ) ( ) ( )

2 2 2 2

2 1

x x y z

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 2

y y z x

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 3

z z x y

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

1 2

3 4

5 6

, , ,0 , , , , , ,0 , , , , ,0 , , , , , ,0 , ,

, , ,0 , , , , , ,0 , ,

t t t

= =

( , , ,)

f x y z ti

,

x y z r u v,

w x y, z

p 2

c l µ

r

= + cs µ

= r

µ l

fi gi

( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

1

2 2

u c u c u c u c c v c c w c u c u c u c v c c v

c w c w c w f

= + + + - + - + + + + + -

+ + - +

are the P-wave and μ shear wave velocities. The variant parameters λ and

are elastic moduli of the solid which depend on the Young modulus and Poisson ratio. In this case the functions f_i and g_i are also smooth. In this case, equation (1) can be rewritten as,

(1a)

(1b)

(1c)

(1d)

Jurnal Kejuruteraan 34(6) 2022: xxx-xxx https://doi.org/10.17576/jkukm-2022-34(6)-07

(2b)

(2c)

Considering the equation in direction, the Fourier transform, of the spatial coordinate is given by,

(3a) where the index represents the spatial coordinates, . The convolution of the first term in r.h.s. of (3a),

can be rearranged as,

which is only eligible when the range of . Applying the formulation to the rest of (3a) to produce,

(3b)

Performing the other transformation, will produce the following equation,

or

(4a) The same procedure is applied to the and directions and produce,

(4b) (4c) where,

and

( ) ( ) ( )

( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

2

2 2

tt s xx p yy s zz p s xy p s yz py sy x sx y sx x py y sz z

sz y py sy z

v c v c v c v c c u c c w c c u c u c v c v c v

c w c c w f

= + + + - + - + - + + + +

+ + - +

( ) ( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

3

2 2

tt s xx s yy p zz p s xz p s yz pz sz x sx z pz sz y sy z

sx x sy y pz z

w c w c w c w c c u c c v c c u c u c c v c v

c w c w c w f

= + + + - + - + - + + - +

+ + + +

x-

( ) ( )

² ³²

( )

^{i i}

i

i x i ik x i

H k = p ^-

ò

h x e^- dx

( )

( ) ( )

3 2 2 2 2 2 2

2 1 2 3

2 2 2 2 2

1 2 1 3 1 1

2 2 2 2

1 2 1 3 1 1 1 2

2 2 2

1 2 1 1 1 2

1 2 3 1

2 * * *

* * *

* * * *

2 * * *

2 *

tt p s s

p s p s p

s s s p

s s p

s

U c k U c k U c k U c c k k V c c k k W k c k U k c k U k c k U k c k V k c k V

k c k V k c k W k c k W k c k W F

p ^- = - - -

- - - - -

- - - -

+ - -

+ +

1,2,3 i j= =

(x y z, , )

( ) ( )

2 2 2 2

1 1

p* l p i i

c k U=

ò

c k -l l U l dl

( ) ( )

( ) ( ) ( )

2 2

1 1 1 2 2

i

j

p i i i i

l

p i j j j

l

c k l l U l dl k l c k l U l dl

- =

- -

ò ò

1

j 2 i

l = l

( )

( ) ( )

3 2 2 2 2 2 2

2 1 2 3

2 2 2 2

1 2 1 3

2 2 2 2 2 2

1 1 2 1 3 1 1

2 2 2 2

1 2 1 2 1

2 2

1 2 1 3 1

2 * * *

* *

* * * *

* 2 * *

* 2 *

tt p s s

p s p s

p s s s

p s s

p s

U k c U k c U k c U k k c c V k k c c W k c U k k c U k k c U k c V k k c V k k c V k c W

k k c W k k c W F

p ^- = - - -

- - - -

- + -

- + +

( ) ( )

2 ³²

( )

^{i i}

i

i x i im k i

I m = p ^-

ò

I k e^- dk

!

( ) ( )

( )

3

2 1 2 3 4 5

6 7 1 5 7 8

4 5 8 4 7 1

2 * * * *

* * * * *

* 2 * * * 2 *

Utt a U a U a U a a V a a W a U a U a U a V a V a V a W a W a W F

p ^- = - - - - -

- - - -

- + - - + +

( )

^!

(

^{! !}

)

(

! !

)

^! ^! ^!

! ! !

3 1 2 3 4 5

6 7 1 5 7 8 4

5 8 4 7 1

2

2 2

Utt a U a U a U a a V a a W a U a U a U a V a V a V a W a W a W F

p ^- = - - - - -

- - - -

+ - - + +

( )

2p ^-³Utt= -b U b V b W F1 + 2 + 3 + 1

y z-

( )

2p ^-³Vtt = -2b V4 +2b U5 +2b W F6 +^!2

( )

2p ^-³W^!_tt = -2b W7 +2b U8 +2b V F9 +^!3

2 2 2 2 2 2 2

1 1 2 2 3 3 4 1 2

2 2 2

5 1 2 6 1 3 7 1 3

2 2 2 2 2

8 1 9 2 10 2 3

2 2 2

11 2 3 12 3

, , , ,

, , ,

,

p s s p

s p s

s p p

s p

a k c a k c a k c a k k c a k k c a k k c a k k c a k c a k c a k k c a k k c a k c

= = = =

= = =

= =

! ! ! ! !

! ! ! !

! ! ! ! ! !

! ! ! !

1 1 2 3 5 7 2 5 4 8

3 7 4 6 8 4 8 9 3

5 5 4 6 11 10 7 8 2 12

8 7 6 9 11 10

2 , 3 2 ,

3 , ,

, , ,

,

b a a a a a b a a a

b a a a a b a a a b a a b a a b a a a b a a b a a

= + + + + = - -

= - - - = + +

= - = - = + +

= - = -

2013).

PROBLEM FORMULATION

(1a)

(1b)

(1c)

(1d)

(2a) x y z, ,

x- y- z-

( ) ( ) ( ) ( )

2 2 2 2

2 1

x x y z

¶ é ù ¶ é ù ¶ é ù ¶ é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 2

y y z x

¶ é ù ¶ é ù ¶ é ù ¶ é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

2 2 2 2

2 3

z z x y

¶ é ù ¶ é ù ¶ é ù ¶ é ù

=¶ ë + + û- ¶ ë + û+¶ ë + û+¶ ë + û+

( ) ( ) ( ) ( )

1 2

3 4

5 6

, , ,0 , , , , , ,0 , ,

t t t

= =

( , , , )

f x y z ti

x y, z

r u v ,

w x y, z

p 2

c l µ

r

= + c_s µ

= r

µ

l

fi g_i

( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

1

2 2

u c u c u c u c c v c c w c u c u c u c v c c v c w c w c w f

= + + + - + - + + + + + -

+ + - + ^(2a)

(2b)

(2c)

Considering the equation in x – direction, the Fourier transform,

Jurnal Kejuruteraan 34(6) 2022: xxx-xxx https://doi.org/10.17576/jkukm-2022-34(6)-07

(2b)

(2c)

Considering the equation in direction, the Fourier transform, of the spatial coordinate is given by,

(3a) where the index represents the spatial coordinates, . The convolution of the first term in r.h.s. of (3a),

can be rearranged as,

which is only eligible when the range of . Applying the formulation to the rest of (3a) to produce,

(3b)

Performing the other transformation, will produce the following equation,

or

(4a) The same procedure is applied to the and directions and produce,

(4b) (4c) where,

and

( ) ( ) ( )

( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

2

2 2

tt s xx p yy s zz p s xy p s yz py sy x sx y sx x py y sz z

sz y py sy z

v c v c v c v c c u c c w c c u c u c v c v c v

c w c c w f

= + + + - + - + - + + + +

+ + - +

( ) ( ) ( ) ( )

2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

3

2 2

tt s xx s yy p zz p s xz p s yz pz sz x sx z pz sz y sy z

sx x sy y pz z

w c w c w c w c c u c c v c c u c u c c v c v c w c w c w f

= + + + - + - + - + + - +

+ + + +

x-

( ) ( )2 ³² ( ) ^{i i}

i

i i ik x i

H k = p ^-

ò

xh x e^- dx

( )

( ) ( )

3 2 2 2 2 2 2

2 1 2 3

2 2 2 2 2

1 2 1 3 1 1

2 2 2 2

1 2 1 3 1 1 1 2

2 2 2

1 2 1 1 1 2

1 2 3 1

2 * * *

* * *

* * * *

2 * * *

2 *

tt p s s

p s p s p

s s s p

s s p

s

U c k U c k U c k U c c k k V c c k k W k c k U k c k U k c k U k c k V k c k V

k c k V k c k W k c k W k c k W F

p ^- = - - -

- - - - -

- - - -

+ - -

+ +

1,2,3 i j= = (x y z, , )

( ) ( )

2 2 2 2

1 1

p* l p i i

c k U=

ò

c k -l l U l dl

( ) ( )

( )

( ) ( )

2 2

1 1 12 2 i

j

p i i i i

l

p i j j j

l

c k l l U l dl k l c k l U l dl

- =

- -

ò ò

1

j 2 i

l = l

( )

( ) ( )

3 2 2 2 2 2 2

2 1 2 3

2 2 2 2

1 2 1 3

2 2 2 2 2 2

1 1 2 1 3 1 1

2 2 2 2

1 2 1 2 1

2 2

1 2 1 3 1

2 * * *

* *

* * * *

* 2 * *

* 2 *

tt p s s

p s p s

p s s s

p s s

p s

U k c U k c U k c U

k k c c V k k c c W

k c U k k c U k k c U k c V k k c V k k c V k c W k k c W k k c W F

p ^- = - - -

- - - -

- + -

- + +

( ) ( )2 ³² ( ) ^{i i}

i

i x i im k i

I m = p ^-

ò

I k e^- dk

!

( ) ( )

( )

3

2 1 2 3 4 5

6 7 1 5 7 8

4 5 8 4 7 1

2 * * * *

* * * * *

* 2 * * * 2 *

Utt a U a U a U a a V a a W a U a U a U a V a V a V a W a W a W F

p ^- = - - - - -

- - - - - -

- + - - + +

( ) ^!

(

^{! !}

)

(

! !

)

^! ^! ^!

! ! !

3 1 2 3 4 5

6 7 1 5 7 8 4

5 8 4 7 1

2

2 2

Utt a U a U a U a a V a a W a U a U a U a V a V a V a W a W a W F

p ^- = - - - - -

- - - - - - -

+ - - + +

( )2p ^-³Utt= -b U b V b W F₁ + ₂ + ₃ + ₁

y z-

( )2p^-³Vtt= -2b V4 +2b U5 +2b W F6 +^!2

( )2p^-³W^!_tt= -2b W₇ +2b U₈ +2b V F₉ +^!₃

2 2 2 2 2 2 2

1 1 2 2 3 3 4 1 2

2 2 2

5 1 2 6 1 3 7 1 3

2 2 2 2 2

8 1 9 2 10 2 3

2 2 2

11 2 3 12 3

, , , <