Proceedings of the 3rd ICMEM
International Conference on Mechanical Engineering and Mechanics
October 2123, 2009, Beijing, P. R. China
The Transition Matrix Formalism for the Scattering of a Buried Pipeline in an Elastic-Half
Space
Wen-I Liao1, Jenn-Shin Hwang 2, Shiang-Jung Wang 3
1
2
Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan
Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan
3 National Center for Research on Earthquake Engineering, Taipei, Taiwan
Abstract: It is investigated in this paper on the transition matrix formulation for the analysis of
responses of an elastic half-space with a buried pipeline subjected to obliquely incident
waves. The basis functions are constructed using the moving P-, SV-, and SH-wave source
potentials to represent the scattered and refracted wave fields in a series form. The
associated T-matrix expression of the elastic inclusion is derived using Betti’s third identity.
Typical numerical results by obliquely incident plane waves are obtained for verification
purpose.
Keywords: Transition matrix, Betti’s identity, Scattering, plane wave
1 Introduction
The transition matrix (T-matrix) theory for the scattering of acoustic waves by a single scatterer in an infinite
medium was first proposed by Waterman [1, 2]. Initially, the theory was derived using Helmholtz integral formula. Pao [3,
4] has shown that the derivation can be greatly simplified if Green’s identity is applied to the acoustic wave problem as
well as Betti’s third identity is applied to the elastic wave problem. Since then, there have been numerous publications on
the applications and extensions to electromagnetic and elastic wave propagation problems. Detailed references for
electromagnetic and elastic wave propagation problems can be found in Varatharajula and Pao [5] and Varadan and
Varadan, [6]). Yeh and Pao [7] have recently proposed the T-matrix formulation for the scattering of acoustic waves by a
multiple layered inclusion in an infinite medium. Chai et al. [8] have successfully employed the cylindrical wave
functions as basis functions for SH-wave scattering in a two-dimensional alluvial valley to study its resonant phenomenon
according to the property of the T-matrix. However, it should be noted that only Neuman’s boundary condition can be
satisfied by the angular part of the cylindrical wave function.
As to research on dynamic responses of a buried layered inclusion (tunnel or pipeline) in an elastic semi-infinity
medium subjected to incident plane waves, Lee and Trifunac [9] have studied the scattering of two-dimensional anti-plane
problem using the method of series expansion. The cylindrical pipeline subjected to incident SH-waves was also analyzed
and discussed. El-Akily and Datta [10] have used the Flugge shell theory to simplify the behavior of the cylindrical
pipeline, and expand the scattering fields of the exterior region by circular cylindrical wave functions. Wong et al. [11]
combined the finite element method with the wave function expansion to analyze the responses of a buried pipeline with
non-circular cross-sections. Luco and de Barros [12] adopted the indirect boundary integral method to construct the
scattering field and combined it with the Donell shell theory to study the responses of a cylindrical pipeline subjected to
obliquely incident plane waves. Yeh et al. [13] have recently proposed a hybrid method for analyzing the dynamic
responses of shells buried in an elastic half-plane. More recently, Chen et al. [14, 15] used the boundary integral method
with degenerating kernel of fundamental solutions to transform the improper boundary integrals to a series of sums for the
study of the exterior radiation and scattering problems with circular boundaries.
In this paper, Yeh and Pao’s [7] concept is extended to analyze the scattering of elastic waves by a lined pipeline
subjected to obliquely incident waves in an elastic half-space. The basis functions and their regular parts are constructed
using moving P-, SV- and SH-wave source potentials, Betti’s third identity and orthogonality conditions for the elastic
·1·
half-space. The T-matrix relating the coefficient of the scattered waves to those of the free field is developed among the
basis functions. Although the theory presented herein is only for a buried pipeline in an elastic half-space, the extension
theory for the multiple layered inclusion is straightforward and easy to derive. Finally, numerical results for the stress and
displacement components of a buried pipeline within the elastic half-space with different obliquely incident waves are
calculated.
Free surface Sf
Incident wave
h
h
D1
D2
Inclusion
x
S2
Half-Space D
x
S1
Incident wave
Half-Space D
(b)
z
v
y
(a)
Figure 1 Configuration and coordinate system. (a) section view (b) plan view.
2 Statement of Problems and Basis Functions
Considering a layered elastic inclusion (pipeline) embedded in an elastic half-space D given in Fig. 1, the interface
between the elastic half-space and the inclusion surface is denoted as S1. The elastic inclusion is divided into two parts
denoted as D1 and D2 with an interface S2. For the half-space D, the density and Lame' constants are ρ , λ and μ ,
respectively, and those for the domain Dj (j=1,2) are denoted correspondingly as ρ j , λ j and μ j . The elastic half-space
is subjected to obliquely incident plane waves with harmonics to the time variable
t and to the space variable y. The
i (t  k y y )
vertical and horizontal incident angles of the incident waves are  v and  h , respectively. A factor e
is assumed
throughout this paper for the waves with specified circular frequency  and wave-number ky which is the y-component
wave-number of the incident waves. The wave-number ky for different types of incident waves are listed in the follows
Incident P-wave: k y  k p sin  sin  h
Incident S-wave: k y  k s sin  sin  h
Incident Rayleigh wave: k y  k R sin  sin  h
where k p   c p and ks   cs corresponding to the longitudinal and shear wave velocities c p  (  2 )  and
cs    , and k R   cR is with respect to the Rayleigh wave velocity cR .
As the incident wave impinges on the elastic inclusion, one part is transmitted into domains D1 and D2 , and the
other is reflected back into the half-space. Hence the wave field in the half-space comprises the free field u f and the
scattered field u s . Then the total displacement field for the half-space can be expressed as
u  u f  us
,
xD
(1)
The scattered field can be represented by suitable independent sets of basis functions satisfying the free-surface and
radiation conditions at infinity. The basis functions adopted in this study are singular solutions of the responses of a
half-space subjected to moving P-, SV- and SH-wave source potentials and their high-order terms, and are expressed in
integral forms. The detailed derivation of those basis functions is available in [16]. The scattered wave field of the exterior
region is expressed as
N
N
N
N
m 0
m 0
m 0
us   cm um   cmp ump   cm um   cmh uhm
 m 0
(2)
where u m (  =p, v ,h) is the basis functions for the elastic half-space, the superscript   p, v, and h denotes the wave
fields generated by moving P-, SV- and SH-wave source potentials, respectively. The subscript m ranges from 0 to N in
which N is the approximated order. Similarly, the displacement field of the free field can be expanded into a series of
regular basis function as
·2·
u f  Ym uˆ m
(3)
 ,m
where uˆ m is the regular part of the basis function u m . For domains D1 and D2 , the refracted wave fields can be
expressed as


u(1)   bm (1) um(1)   bm (2) uˆ m(1)
 m 0
 m 0

u(2)   f m(1) uˆ m(2)
 m 0
where uˆ m( j ) (j = 1, 2;

, x  D1
, x  D2
(4)
(5)
= p, v, h) is the regular part of the basis function um( j ) of domain Dj. The basis function u m
for the half-space should satisfy the traction-free conditions on the free surface, but the basis function um( j ) for the
elastic inclusion needs not satisfy the traction-free conditions. Hence, the basis functions for domains D1 and D2
include only the source term of the source potentials and have the following component form
ui( m( j)) 
1
2



Sm (k x / k *p ) H i ( j ) eikx dk x ;
x  D j , j  1,2 , i  x, y, z
(6)
where k p  k p2  k y2 . For   p (P-wave source potential), components of H ip ( j ) are given by
H xp 
H yp 
ik x

ik y

e
e
 z
 z
H zp   sgn  z  e
(7)
 z
For   v (SV-wave source potential), components of H iv ( j ) are
H xv  sgn  z  e
  z
H yv  0
H zv 
(8)
ik x   z
e

For   h (SH-wave source potential), components of H ih ( j ) are
 k x k y   z
e

k    z
H yh  s e

ik   sgn  z    z
H yh  y
e

H xh 
(9)
where ks  ks2  k y2 , and Sm (kx / k *p ) is properly selected as
i mTn ( k x / k *p ),
Sm ( k x / k p )   m
*
i Ql ( k x / k p ),
n  m / 2,
m  0, 2, 4,...
l  (m  1) / 2,
m  1,3,5,...
(10)
where Tn and Ql are Chebyshev polynomials of the first and second kinds, respectively. The basis functions um  j  (j =
1, 2;  = p, v, h) can be easily evaluated using the method presented in Section 4. By applying Betti's third identity of
elasticity
 t(u)  v  t(v)  u dS    [ σ(u)]  v  [ σ(v)]  u dV
S
V
(11)
In which S is the surface of volume V , t(u) and t( v) are tractions of displacement fields u and v , respectively.
·3·
The orthogonality conditions for domain D are

 tm  u n  t n  um  dS  0

S
 tˆm  uˆ n  tˆ n  uˆ m  dS  0

S

 tm  uˆ n  tˆ n  um  dS  Enm
S
(12)
(13)
(14)
where S can be any arbitrary closed bounded surface that outside the domain surface D1 . Similarly, the orthogonality
conditions for domain D1 are

S
 tm1  u n 1  t n 1  um(1)  dS  0


(15)
 tˆm1  uˆ n 1  tˆ n 1  uˆ m1  dS  0


(16)

S

S

 tm1  uˆ n 1  tˆ n 1  um1  dS  Enm
1


(17)
where S can be any arbitrary closed surface inside domain D1 as described by Figure 2.
Free Surface Sf
D1
D2
x
S2
S-
S1
S+
Half-Space D
z
Figure 2
Domains and relative surfaces of this problem.
3. Transition Matrix Formulism
If the elastic half-space is assumed to be continuous with the elastic inclusion, the continuity conditions on the
interface S1 are
t   t 1
u  u1
, x  S1
(18)

where u and u(1)
are the total displacements on the interface S, approaching respectively from the positive and

negative directions of the unit normal. t  ( ti   ij n j ) and t (1)
( ti(1)   ij(1) n j ) are the associated tractions of u and

, respectively. n j is the component of unit outer normal vector n of surface S1 . For interface S2 between
u(1)
domains D2 and D1 , in the case that domain D2 is an elastic medium and continuous with D1 , the boundary
conditions on surface S2 are
t 1 (x)  t 2 (x)
u1 (x)  u2 (x) , x  S2
（19）
In the case that domain D2 is a cavity, then the boundary conditions are
t 1 (x)  0
t 2 (x)  0 , x  S2
·4·
(20)
In this study, domain D2 is presumed to be a cavity. This is simply because to represent a buried lined pipeline or a
pipeline by assuming domain D2 as a cavity is more meaningful and has more applications than as an elastic medium.
If the volume in Eq. (11) is the domain bounded by surfaces S1 and S shown in Fig. 2, and let v  u n , then

S1
 t   u n  t n  u   dS    t  u n  t n  u  dS
S
(21)
Substituting Eqs. (2) and (3) into the right-hand side of Eq. (21), and applying the orthogonality condition of Eq. (12),
Eq.(21) can then be reduced to the following form

S1
 t   u n  t n  u   dS    t f  u n  t n  u f  dS  an
S
(22)
From the continuity conditions of Eq.(18), the coefficient vector a n can be expressed as






 dS     bm1 tm1   bm 2 tˆm1   un  t n    bm1 um1   bm2uˆ m1  dS
an    t (1)
 u n  t n  u(1)
S1
S1
 ,m
 ,m

, m

  ,m


  Pnm 1bm1   Qnm
1 bm 2 
(23)
Pnm(1)    tm(1)  un  t n  um(1)  dS
S1
(24)
 ,m
 ,m
where

 ˆ (1)  un  t n  uˆ m(1)  dS
Qnm
(1)    t m
S1
(25)
For the same domain bounded by surfaces S1 and S , letting v  uˆ n yields

S1




[t   uˆ n  tˆ n  u ] dS    t f   cm tm   uˆ n  tˆ n   u f   cm uˆ m  dS
S
 ,m
 ,m




(26)
Substituting the expansion form of Eq. (3) of free-field displacement into the right-hand side of Eq. (26) and using the
orthogonality condition, Eq.(26) can be reduced to the following form

S1
 





 t   uˆ n  tˆ n  u  dS   cm S  t m  uˆ n  tˆ n  um  dS   Enm cm
 ,m
 ,m
(27)

where the matrix Enm
is defined as

Enm
   tm  uˆ n  tˆ n  um  dS
S
(28)

In addition, from the continuity conditions of Eq.(18), the matrix Enm
is reduced to
 E  c  

nm
,m
m
S1






 t (1)
 dS     bm (1) tm(1)   bm (2) tˆm(1)   uˆ n  tˆ n    bm (1) um(1)   bm (2) uˆ m(1)  dS
 uˆ n  tˆ  u(1)
S1
 ,m
 ,m

  ,m

  ,m
 ,

  Pˆnm ,1 bm 1   Qˆ nm
1 bm  2 
 .m
(29)
 ,m
In which

  1 ˆ  ˆ   1 
Pˆ nm
1  S1  t m  u n  t n  u m  dS

 ˆ 1 ˆ  ˆ  ˆ  1 
Qˆ nm
1  S1  t m  u n  t n  u m  dS
Now, For volume V in Eq. (11) enclosed by surfaces S2 and S _ of Fig. 2, letting v  u n (1) deduces
·5·
(30)
(31)

S2
 t 1  un 1  t n 1  u1  ds



 1
 1
 1 
 1
 1 
  (  bm1 t m    bm 2 tˆ m  )  un (1)  t n      bm1 um    bm 2 uˆ m    dS
S_
 ,m
 ,m
  ,m
  ,m

(32)
Applying the orthogonality condition of Eq.(15), Eq.(32) can be reduced to

S2


 t 1  u n 1  t n 1  u1  dS   bm  2   tˆm1  u n 1  t n 1  uˆ m1  dS    Emn
1 bm  2 
S _ 



 ,m
 ,m
(33)

where matrix Emn
1 is defined as

  1 ˆ  1 ˆ  1  1 
Emn
1  S _  t m  un  t n  um  dS
(34)
and Eq.(33) can be rewritten as
 1
 1



 

  Emn
1 bm  2   S2  t  2   u n  t n  u  2   dS
 ,m
(35)
If domain D2 is a cavity, from the boundary conditions of Eq. (20), we can obtain the relation between
coefficients bm (2) and f m as





  1  
  1 ˆ   2  
  Emn
1 bm  2   S2   t n  u  2   dS   f m S2   t n  u m  dS   Qnm 2  f m
 ,m
 ,m
 ,m
(36)
where

  1 ˆ   2 
Qnm
 2   S2   t n  u m  dS
(37)
Now, for the domain enclosed by surfaces S _ and S2 , letting v  uˆ n 1 yields

S2

 t 1  uˆ n 1  tˆ n 1  uˆ 1  dS   bm 1  tm1  uˆ n 1  tˆ n (1)  um1  dS   bm1 Enm
1
S 



 ,m
 ,m
(38)
From the boundary conditions of Eq.(20), Eq.(38) becomes


ˆ  
ˆ  
ˆ ˆ 
bm 1 Enm
1    t n  u  2  dS   f m   t  u m dS   Qnm  2  f m
 1
 2
S2
 ,m
S2
(39)
 ,m
where

ˆ  1 ˆ  2
Qˆ nm
 2   t n  um dS
S2
(40)
In summary, the relations between the unknown coefficients cm , bm1 , bm 2 and f m in Eqs. (23), (29), (36) and
(39) can be expressed in matrix forms as the follows
 P1  b1  Q1  b2   a
(41)
ˆ ]b    E c
[Pˆ1 ]b1   [Q
1
2
(42)
  E1  b2   Q 2 
(43)

f 
 E1 b1  [Qˆ 2 ]  f 
Then the relation between the coefficient vector
f 
(44)
and vector a is given by
a  [T ]{ f }  {[ P1 ][E1 ]1[Qˆ 2 ]-[Qˆ 1 ][E1 ]T 1[Q2 ]}{ f }
where
·6·
(45)
ˆ ]-[Q
ˆ ][E ]T 1[Q ]
[T ]  [ P1 ][E1 ]1[Q
2
1
1
2
(46)
This is the T-matrix relating the coefficient of the scattered waves to that of the free field for the elastic half-space with
layered inclusions.
4 Numerical Examples
The problems of a cylindrical pipeline with r0 /ri  1.1 buried at different depths and subjected to incident plane
waves are investigated in the follows. r0 and ri are the outer and inner radius of the pipeline, respectively. For all cases
included in this study, the Poisson ratio for the pipeline and for the elastic half-space were assumed to be 0.2 and 1/3,
respectively. The ratio of density of the pipeline to the half-space was taken as 1/  1.5 . The ratio of shear modulus
was set to 1/  6 . The excitation frequency of incident wave was normalized as    1  cs to be consistent with
other studies. Basis functions of the half-space are calculated by the method of steepest descend [16]. The choice of
approximate order N of wave series are based on the convergence of calculated results. In all numerical examples of this
study, N=5 is enough to have a satisfactory results as compared with those results with more high order terms. For the
calculations of surface integrals in Equations (24), (31), (34), (37), and (40) are accomplished by Simpson’s rule with
integration points M =61.
To verify the validity and accuracy of the T-matrix formulation in this study, the plane strain cases (k y  0) were
first studied and the results obtained were compared with those obtained by Yeh et al. [17]. Figures 3 and 4 show the
calculated displacement amplitudes and hoop stresses along the inner surface of the cylindrical pipeline buried at different
depths respectively subjected to incident P-wave and SV-wave (   0.5 ). The results determined by the least square
method (Yeh et al., [17]) are denoted by circles in these figures, and it can be seen that the agreement is excellent. For the
responses of a buried pipeline subjected to obliquely incident waves, Figs. 5 and 6 show the displacements and stresses
along the inner surface of the cylindrical pipeline subjected to obliquely incident P-wave and SH-wave (   0.5 ,
θh  90 , θv  45 ) with buried depths of h/ri  2.0
and h/ri  5.0 . In these figures, the displacement
Ui | ui / A | ( i  x, y, z ) is normalized with respect to the amplitude A of incident wave, and the stress fields are
normalized with respect to the factor  cs . The responses in Figs. 5 and 6 should be symmetric with respect to the
vertical z-axis (   90o and 270o ) because the wave is propagating in the direction of the axis of the pipeline. The effect
of the embedment depth of the pipeline on the response can also be found in Figs. 5 and 6. Obviously, the embedment
depth has a significant effect on the response and, particularly, on the case of incident P-wave.
5 Conclusions
The T-matrix, which relates the unknown scattering coefficients to the free-field coefficients, can be derived directly
by the orthogonality conditions as well as the continuity conditions in the interface between the buried two-layered
scatterer and the surrounding half-space. The basis functions and their regular parts are constructed in a systematic
manner for the problems of obliquely incident waves. The results for wave scattering of a buried pipeline subjected to
incident plane waves were calculated and verified. Although only a limited number of numerical results are presented in
this paper, applications of the proposed method may be extended to problems of wave scattering in multiple layered
scatter, embedded cavities, surface scatter and other problems of oblique incidence. In addition to elastic waves, the
proposed T-matrix method could also be applicable to coupled field problems, such as thermoelasticity and poroelasticity.
Acknowledgement
The authors gratefully acknowledge the financial support granted by the National Science Council, R.O.C., the
facilities provided by the National Taipei University of Technology is also highly appreciated.
References
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pp. 1417-1429.
0, No. 3,
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(a)
(b)
(c)
4
4
30
2
2
15
0
0
0
90 180 270 360
0
90 180 270 360
 (degrees)
 (degrees)
 (degrees)
Figure 3 Dynamic responses for (a) horizontal displacements, (b) vertical displacements, and (c) hoop stresses along inner surface of a buried tunnel subjected
to incident P-wave with  v  30o .(solid line and dash line are results of h ri  1.5 and 5.0 , respectively. Circles are the results obtained by Yeh et al. [17])
0
90
180
270
360
0
(a)
(b)
(c)
4
4
20
2
2
10
0
0
0
90 180 270 360
0
90 180 270 360
 (degrees)
 (degrees)
 (degrees)
Figure 4 Dynamic responses for (a) horizontal displacements, (b) vertical displacements, and (c) hoop stresses along inner surface of a buried pipeline subjected
to incident SV-wave with  v  0o .(solid line and dash line are results of h ri  1.5 and 5.0 , respectively. Circles are the results obtained by Yeh et al. [17])
0
90
180
270
360
0
·8·
2
Ux
20
(a)
(d)

1
10
0
0
0
90
180
270
360
0
90

180
270
360

2
(e)
8
(b)
Uy
y 4
1
0
0
0
90
180
270
360
0
90

180
270
(f)
20
2
(c)
Uz
yy 10
1
0
0
0
90
180
270
360
0
90

Figure 5
180
270
360

Dynamic responses for (a)-(c) displacement amplitude, and (d)-(f) stress amplitudes along inner surface of a buried pipeline subjected to obliquely
incident P-wave with  v  45o and  h  90o . (solid line and dash line are results of h ri  1.5 and h ri  5.0 , respectively.)
2
Ux
10
(a)
(d)

1
0
5
0
0
90
180
270
360
0
90

180
270
20
(e)
(b)
Uy
y 10
0
0
0
0
90
180
270
360
0
90

180
270
360

1.0
8
(f)
(c)
Uz
360

1
yy 4
0.5
0.0
0
0
90
180
270
360

Figure 6
360

0
90
180
270
360

Dynamic responses for (a)-(c) displacement amplitude, and (d)-(f) stress amplitudes along inner surface of a buried pipeline subjected to obliquely
incident SH-wave with  v  45o and  h  90o . (solid line and dash line are results of h ri  1.5 and h ri  5.0 , respectively.)
·9·