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A semi-analytical solution method for two-dimensional Helmholtz equation

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A semi-analytical solution method for two-dimensional Helmholtz equation
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  Applied Ocean Research 28 (2006) 193–207www.elsevier.com/locate/apor A semi-analytical solution method for two-dimensional Helmholtz equation Boning Li, Liang Cheng ∗ , Andrew J. Deeks, Ming Zhao School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia Received 18 November 2005; received in revised form 31 March 2006; accepted 15 June 2006Available online 25 January 2007 Abstract A semi-analytical solution method, the so-called Scaled Boundary Finite Element Method (SBFEM), is developed for the two-dimensionalHelmholtz equation. The new method is applicable to two-dimensional computational domains of any shape including unbounded domains. Theaccuracy and efficiency of this method are illustrated by numerical examples of wave diffraction around vertical cylinders and harbour oscillationproblems. The computational results are compared with those obtained using analytical methods, numerical methods and physical experiments. Itis found that the present method is completely free from the irregular frequency difficulty that the conventional Green’s Function Method (GFM)often encounters. It is also found that the present method does not suffer from computational stability problems at sharp corners, is able to resolvevelocity singularities analytically at such corners by choosing the structure surfaces as side-faces, and produces more accurate solutions thanconventional numerical methods with far less number of degrees of freedom. With these attractive attributes, the scaled boundary finite elementmethod is an excellent alternative to conventional numerical methods for solving the two-dimensional Helmholtz equation.c  2007 Published by Elsevier Ltd Keywords:  Scaled boundary finite element method; Helmholtz equation; Wave diffraction; Harbour oscillation; Irregular frequency; Singularity 1. Introduction Surface water wave diffraction around surface piercingstructures and harbour oscillations excited by waves havebeen the topics of many research projects in the past.These two classes of problems, in general, are governedby the two-dimensional Helmholtz equation associated withboundary conditions on the structure surface and a radiationcondition at infinity. Analytical solutions exist only fora few simple objects such as a circular cylinder [16].Most problems of engineering significance, however, needto be solved numerically. Conventional numerical methods,including the Finite Element Method (FEM)/Infinite ElementMethod (IEM) [2,3] and the Boundary Element Method (BEM)/Green Function Method (GFM) [1,4,11,21] have been employed for solving these problems.The FEM was applied to solve wave diffraction problemsin the early 1970s [3,30]. The FEM is well known for its flexibility in handling irregular boundary problems. This isparticularly attractive for wave diffraction around offshore ∗ Corresponding author. Tel.: +61 8 9380 3073; fax: +61 8 9380 1018.  E-mail address:  cheng@civil.uwa.edu.au (L. Cheng). structures, because most offshore structures are of irregularshape. However, one of the difficulties encountered in usingthe finite element method to solve wave diffraction problemsin an unbounded domain is the implementation of the radiationboundary condition at the infinity. In the FEM, a calculationdomain of a finite size is normally used to approximate theinfinite domain on which the wave diffraction problem isdefined. To satisfy the radiation boundary condition at the outerboundary of the truncated domain, this outer boundary has tobe far away from the object investigated. The further awaythe outer boundary is, the more the nodes needed inside thedomain to maintain the level of accuracy of the solution. Agreaternumberofnodesnormallyimplieshighercomputationalcost. This problem becomes more severe for three-dimensionalproblems. To avoid this difficulty, the so-called hybrid elementmethod has been used by many researchers with moderatesuccess [5]. The hybrid element method combines the finiteelement solution of the problem in a finite domain next to theobject with an analytical (or infinite element) solution at theouter boundary of the finite domain. It employs the advantagesof both the finite element method and the analytical method forthis particular kind of problems. Zienkiewicz et al. (1978) [30]provides a detailed review of such methods. 0141-1187/$ - see front matter c  2007 Published by Elsevier Ltddoi:10.1016/j.apor.2006.06.003  194  B. Li et al. / Applied Ocean Research 28 (2006) 193–207  TheBEM/GFMhavealsobeenwidelyusedforwavediffrac-tion problems. These methods make use of a singular solutionthat satisfies the free surface, seabed and radiation boundaryconditions.Thescatteredwavepotentialisrepresentedbyanin-tegral of the singular solution multiplied by a distribution func-tion of singularities on the surface of the object. The strength of the singularities is then determined by enforcing the boundarycondition on the surface of the object. The GFM is normallyvery efficient, because only the surface of the object needs tobe discretized, and is widely used for calculating wave forceson offshore structures. However, the GFM does suffer fromsome fundamental problems such as irregular frequencies dueto the use of the source distribution representation of the solu-tion [9,19], and some numerical difficulties such as modelling re-entrant structure geometries or structures with sharp cornersor small openings [20].The Scaled Boundary Finite Element Method (SBFEM) isa semi-analytical computational procedure for solving linearpartial differential equations, combining the advantages of the finite element and the boundary element methods withappealing features of its own. The method, based on the so-called multi-cell cloning technique, was srcinally establishedto model dynamic problems in an unbounded domain byWolf and Song [26], and has gained considerable success in modelling structural mechanics problems. The key advantageof the method is the derivation of fundamental equations basedon a scaled boundary coordinate transformation that leads toa system of linear second-order ordinary differential equationsin displacements with the radial coordinate as the independentvariable [22]. Recent developments in the solution of problems of elasto-statics and elasto-dynamics have demonstrated itsabilities and advantages for soil-structure interaction problemsin unbounded domains [27,29]. Two recent extensions of the srcinal scaled boundary finiteelement method [27] to fluid flow problems have demonstrated the promising potential of the method in solving fluid flowproblems. Deeks and Cheng [6] established a scaled boundary finite element solution to two-dimensional Laplace equationfor potential flow around obstacles. Their work demonstratedthe excellent ability and advantages of the technique to themodel velocity singularities at the corners of a square cylinder.Flow around multiple obstacles is solved using a substructuretechnique where the scaled boundary finite element solutionsfor bounded and unbounded domain problems are matched atthe domain interface. However, this solution is not applicableto computational domains with parallel side-faces because theconventional SBFEM requires existence of a scaling centre.Li et al. [14] proposed a modified SBFEM for problems withparallel side-faces. Li et al. [15] also applied this modifiedmethod to calculate wave diffraction by fixed structures andwaveradiationexcitedbyoscillatingstructuresinwateroffinitedepth.However, the previous studies [6,14,15] of the SBFEM are only applicable to problems governed by the Laplaceequation. The objective of the present paper is to extend theSBFEM to solving the two-dimensional Helmholtz equationthat governs many practical problems such as wave diffraction Fig. 1. Definition sketch of wave diffraction around obstacles. problems around vertical structures. In the following sections,commencing with the summary of the linearized wavediffraction formulae and the fundamental theory of thescaled boundary finite element method, the solution of two-dimensional Helmholtz equation for a bounded domain andan unbounded domain are derived. Then the substructuretechniqueisintroduced. Finally, theproposedmethod isappliedto several examples and the numerical results are comparedwith those obtained using other methods. 2. Formulation of the boundary-value problem The typical wave diffraction problem shown in Fig. 1 isconsidered here. Applying the assumption of linearized wavetheory, the total velocity potential  Φ  T   may be expressed as thesummation of the incident wave potential  Φ   I   and the scatteredwave potential  Φ  S  . Employing the method of the separation of variables, the velocity potential may be written as Φ  (  x ,  y ,  z , t  ) = φ(  x ,  y )  Z  (  z ) e − i ω t  (1)where  Φ   denotes any one of   Φ  T  ,  Φ   I   and  Φ  S  ,  Z  (  z )  isthe corresponding water depth function,  ω  is the angularfrequency,  t   is the time and i is the imaginary unit. Thevelocity potential  φ(  x ,  y )  is governed by the two-dimensionalHelmholtz equation ∇  2 φ + k  2 φ  = 0 in domain  Ω   (2)where  k   is the wave number. On the boundaries (wettedbody boundary  Γ  c  and infinity boundary Γ  ∞ ) of the domain Ω  , Neumann boundary condition and Sommerfeld radiationcondition are applied respectively: ∂φ T  ∂ n = 0 at the body surface  Γ  c  (3)lim r  →∞ √  r   ∂φ S  ∂ r  − i k  φ S   = 0 at infinity  Γ  ∞  (4)in which  n  designates the unit normal to the boundary and  r  is the radial coordinate. The boundary condition specified byEq. (3) will result in a homogeneous scaled boundary finiteelementequationthatisconvenienttosolve,whiletheboundary   B. Li et al. / Applied Ocean Research 28 (2006) 193–207   195Fig. 2. Substructuring configuration and scaled boundary coordinate definition. condition specified by Eq. (4) will result in satisfaction of theradiation condition at infinity.The scaled boundary finite element solution of wave diffrac-tion problems around multiple obstacles in an unbounded do-main can be constructed in a way similar to that illustrated inFig. 2(a). The entire computational domain is divided into anunbounded domain and a bounded domain with a common in-terface of  Γ  b . Thescaled boundary finite elementsolution of thetwo-dimensional Helmholtz equation is modified to seek solu-tions of Eq. (2) in the unbounded domain with relevant bound-ary conditions on  Γ  b  and  Γ  ∞  and in the bounded domain withappropriate boundary conditions on  Γ  b  and  Γ  c . The boundarycondition on Γ  b  can be eliminated by matching the unboundeddomain solution and bounded domain solution on Γ  b . 3. Scaled boundary finite element solutions 3.1. Scaled boundary finite element equation As illustrated in Fig. 2, the computational domain can bedivided into a number of subdomains, denoted as  Ω  i  ( i  = 1 , 2 , 3 ,...,  N   + 1 ) . A typical subdomain ( Ω  i  in Fig. 2(a)) hasa scaling centre (solid point in Fig. 2(b)) and is bounded by twoside-faces ( s 0  and  s 1  in Fig. 2(b)) or Γ  ci , Γ  bi , Γ  i  and Γ  i + 1 . Thescaled boundary finite element solution in the bounded domaincan be obtained by matching the solutions of Eq. (2) on internalboundaries Γ  i  ( i  = 1 to  N  ). The scaled boundary finite elementsolution in a typical subdomain  Ω  i  is presented here and thesolutions in the other subdomains can be obtained using thesame approach.Suppose  Γ  Ω  i  is comprised of   Γ  bi , Γ  i  and  Γ  i + 1 . The totalvelocity potential on Γ  Ω  i  satisfies ∂φ T  ∂ n =−¯ v n  on Γ  Ω  i  (5)where  ¯ v n  is the velocity component outward normal to theboundary Γ  Ω  i .ApplyingtheweightedresidualapproachtoEqs.(2) and (3) and employing Green’s identity result in   Ω  i ∇  T w ∇  φ T  d Ω   −   Ω  i w k  2 φ T  d Ω   −   Γ  Ω  i w ¯ v n d Γ   = 0 (6)where  w  is a weighting function.To apply the SBFEM to Eq. (6), the so-called scaledboundary coordinate system is introduced. A typical scaledboundary coordinate system [28] is shown in Fig. 2(b). The scaled radial coordinate  ξ   is 1 on  ξ   =  ξ  e  ( Γ  bi  ∪  Γ  i  ∪ Γ  i + 1 ) and zero at the scaling centre for a bounded domain.Consequently, each value of   ξ   defines a scaled version of thecurve  S  . Thecircumferentialcoordinate s  measuresthedistanceanticlockwise around a defining curve  S  . The bounded domain Ω  i  is defined as the region enclosed by 0  ≤  ξ   ≤  1 and s 0  ≤  s  ≤  s 1 . For the case of unbounded domain, the scaledradial coordinate  ξ   equals 1 at the boundary  Γ  bi  ∪  Γ  i  ∪  Γ  i + 1 and  ξ  e  approaches to infinity (see Ref. [6] for more details). The defining curve  S   can be discretized by shape functions  [  N  ( s ) ] in the classical finite element manner. Thus, an approximatesolution  φ h (ξ, s )  to Eq. (6) is sought in the form φ h (ξ, s ) =[  N  ( s ) ]{ a (ξ) }  (7)wherevector { a (ξ) } representsradialnodalfunctionsanalogousto nodal values in the standard finite element method. Ateach node  i  the function  a i (ξ)  designates the variation of thetotal potential in the radial direction. Since no discretization iscarried out in the radial direction, the scaled boundary finiteelement method keeps the radial solution  { a (ξ) }  analytical.Using the scaled boundary transformation detailed by Deeksand Cheng (2003) [6], the Laplace operator ∇   can be expressedas ∇ ={ b 1 ( s ) } ∂∂ξ  + 1 ξ  { b 2 ( s ) } ∂∂ s (8)where  { b 1 ( s ) }  and  { b 2 ( s ) }  are vectors that are only dependenton the definition of the curve  S  . The approximate velocityvector { v h (ξ, s ) } can be formulated as { v h (ξ, s ) }=[  B 1 ( s ) ]{ a (ξ) } , ξ   + 1 ξ  [  B 2 ( s ) ]{ a (ξ) }  (9)where, for convenience, [  B 1 ( s ) ]={ b 1 ( s ) }[  N  ( s ) ]  (10) [  B 2 ( s ) ]={ b 2 ( s ) }[  N  ( s ) ] , s  .  (11)Applying the Galerkin approach, the weighting function  w  canbe formulated by employing the same shape functions as theapproximation for the potential (Eq. (7)). w(ξ, s ) =[  N  ( s ) ]{ w(ξ) }={ w(ξ) } T [  N  ( s ) ] T (12)The following SBFEM equation can be obtained by substitutingEqs. (7), (8) and (12) into (6) and integrating all terms containing { w(ξ) } , ξ   by parts with respect to  ξ  : { w(ξ  e ) } T  [  E  0 ] ξ  e { a (ξ  e ) } , ξ   +[  E  1 ] T { a (ξ  e ) }−   s [  N  ( s ) ] T ¯ v n τ  ξ  d s  −{ w(ξ  i ) } T  [  E  0 ] ξ  i { a (ξ  i ) } , ξ   +[  E  1 ] T { a (ξ  i ) }+   s [  N  ( s ) ] T ¯ v n τ  ξ  d s  −    ξ  e ξ  i { w(ξ) } T  [  E  0 ] ξ  { a (ξ) } , ξξ   196  B. Li et al. / Applied Ocean Research 28 (2006) 193–207  + ( [  E  0 ]+[  E  1 ] T −[  E  1 ] ) { a (ξ) } , ξ   −[  E  2 ] 1 ξ  { a (ξ) }+  k  2 ξ  [  M  0 ]{ a (ξ) }  d ξ   = 0 (13)where [  E  0 ]=   s [  B 1 ( s ) ] T [  B 1 ( s ) ]|  J  | d s  (14) [  E  1 ]=   s [  B 2 ( s ) ] T [  B 1 ( s ) ]|  J  | d s  (15) [  E  2 ]=   s [  B 2 ( s ) ] T [  B 2 ( s ) ]|  J  | d s  (16) [  M  0 ]=   s [  N  ( s ) ] T [  N  ( s ) ]|  J  | d s .  (17)The following conditions hold if Eq. (13) is satisfied for all setsof weighting functions { w(ξ) } . [  E  0 ] ξ  i { a (ξ  i ) } , ξ   +[  E  1 ] T { a (ξ  i ) }=   s  N  T ( s )( −¯ v n )τ  ξ  d s  (18) [  E  0 ] ξ  e { a (ξ  e ) } , ξ   +[  E  1 ] T { a (ξ  e ) }=   s [  N  ( s ) ] T ¯ v n τ  ξ  d s  (19) [  E  0 ] ξ  2 { a (ξ) } , ξξ   + ( [  E  0 ]+[  E  1 ] T −[  E  1 ] )ξ  { a (ξ) } , ξ   −[  E  2 ]{ a (ξ) }+ k  2 ξ  2 [  M  0 ] a (ξ) = 0 .  (20)Eqs. (18) and (19) indicate the relationships between the nodal potential and the integrated nodal flow on the boundary withconstant radial coordinate  ξ  . For the case of bounded domainconsidered in Fig. 2(b),  ξ  e  represents the sum of  Γ  bi Γ  i  and Γ  i + 1 and  ξ  i  is actually the scaling centre with  ξ  i  =  0. For the caseof unbounded domain,  ξ  i  represents the sum of  Γ  bi Γ  i  and Γ  i + 1 and  ξ  e  tends to infinity. Eq. (20) is the scaled boundary finiteelement equation. It can be seen that Eq. (20) is a homogeneoussecond-order ordinary differential equation, due to the use of the total velocity potential as an unknown in the boundeddomain. 3.2. Bounded domain solution Song and Wolf  [23] presented the solution to Eq. (20) for elasto-dynamic problems in a bounded domain. However,this solution is based on a specific property of rigid bodymovements in solid mechanics that is not available for thewave diffraction problem considered in this study. Thereforethe application of this solution to the problem investigated hereis not straightforward. An improved method has been proposedin this study to obtain an analytical solution to Eq. (20).To solve Eq. (20), a transformation of Eq. (20) to a set of  first-order ordinary differential equations of order 2 n  can beperformed [23]. Considering the satisfaction of Eqs. (18) and (19), no generality is lost by assuming { q (ξ) }=[  E  0 ] ξ  { a (ξ) } , ξ   +[  E  1 ] T { a (ξ) } .  (21)Combining Eq. (20) with Eq. (21) and introducing the independent variables ¯ ξ   = k  ξ   (22)and {  X  ( ¯ ξ) }=  { a (ξ) }{ q (ξ) }  =[  X  ( ¯ ξ) ]{ c }  (23)result in ¯ ξ  [  X  ( ¯ ξ) ] , ¯ ξ   =−[  Z  ][  X  ( ¯ ξ) ]− ¯ ξ  2 [  M  ][  X  ( ¯ ξ) ]  (24)with the coefficient matrices [  Z  ]=   [  E  0 ] − 1 [  E  1 ] T −[  E  0 ] − 1 −[  E  2 ]+[  E  1 ][  E  0 ] − 1 [  E  1 ] T −[  E  1 ][  E  0 ] − 1   (25)and [  M  ]=   0 0 [  M  0 ]  0  .  (26)For the Hamiltonian matrix [  Z  ] , the eigenvalues consist of twogroups of values with opposite signs [¯ Λ ]=  [ λ  j ]  00  −[ λ  j ]  ,  j  = 1 , 2 ,..., n  (27)where Re (λ  j )  ≥  0. The corresponding eigenvalue problem canbe formulated as [  Z  ][ Φ  ]=[ Φ  ][¯ Λ ]  (28)where [ Φ  ] denotes the eigenvector matrix.Based on the physical meaning of Eqs. (27) and (28), zero eigenvalues appear in pairs in Eq. (27). A zero eigenvalue leadsto a constant potential in the entire fluid domain. This meansthat there is no flow in the fluid domain. Thus the eigenvectorsof zero eigenvalues are not linearly independent any more,leading to a singular matrix [ Φ  ] . The method proposed by Songand Wolf  [23] to solve Eq. (24) is not applicable here, because the eigenvectors of zero eigenvalue are linearly independent forthe solid mechanics problems [23] investigated. Therefore, to obtain the solution for wave problems, Jordan decompositionof the matrix [  Z  ] is performed, namely, [  Z  ][ T  ]=[ T  ][ Λ ]  (29)with [ Λ ]=  [ λ  j ]   0 10 0  −[ λ  j ]  ,  j  = 1 , 2 ,..., n − 1(30)in which the transform matrix  [ T  ]  is invertible. Althoughprogramming of Jordan decomposition for general matrix couldbe tedious and unstable, Jordan decomposition of the matrix [  Z  ] does not reduce the efficiency of the computer program toomuch when advantage is taken of the simple structural form of the matrix  [ Λ ]  and known eigenvalues. In fact, based on theeigenvector matrix  [ Φ  ] , the matrix  [  Z  ]  can be easily formedusing the characteristic of the Jordan chain.Following the procedures provided by Ref. [7], through a series of matrix transformations and the solution of a system of recursion equations, the analytical solution of Eq. (24) may be   B. Li et al. / Applied Ocean Research 28 (2006) 193–207   197 expressed as [  X  ( ¯ ξ) ]=[ T  ][  R ( ¯ ξ) ]¯ ξ  [ Λ ] ¯ ξ  [ U  ]  (31)where the matrix  [  R ( ¯ ξ) ]  is formulated as a power series in  ¯ ξ  with a leading identity matrix [  I  ][  R ( ¯ ξ) ]=[  R 0 ]+ ¯ ξ  2 [  R 1 ]+ ¯ ξ  4 [  R 2 ]+···+ ¯ ξ  2 m [  R m ]+··· (32)with [  R 0 ]=[  I  ]  (33)and the matrix  [ Λ ]  is an upper triangular matrix witheigenvalues being on the diagonal entries and  [ U  ]  is alsoan upper triangular matrix with zero diagonal entries. Forconvenience, the matrix function  ¯ ξ  [ U  ]  may be written as ¯ ξ  [ U  ]  =[ Y  ( ¯ ξ) ] .  (34)Partitioning all the matrices on the right hand side of Eq. (31)and constant vector { c } into block matrix with  n × n  dimensionsand block vector with  n  ×  1 dimensions respectively, thensubstituting Eqs. (31) and (34) into Eq. (23) yields {  X  ( ¯ ξ) }=  [ T  11 ] [ T  12 ][ T  21 ] [ T  22 ]  [  R 11 ( ¯ ξ) ] [  R 12 ( ¯ ξ) ][  R 21 ( ¯ ξ) ] [  R 22 ( ¯ ξ) ]  ×  ¯ ξ  [ λ ]  ¯ ξ  [ P ] 0  ¯ ξ  −[ λ ]  [ Y  11 ( ¯ ξ) ] [ Y  12 ( ¯ ξ) ][ Y  21 ( ¯ ξ) ] [ Y  22 ( ¯ ξ) ]  { c 1 }{ c 2 }  .  (35)To obtain a finite solution for  ¯ ξ   =  0,  { c 2 }  must be zero. Forbrevity, introducing  [ K  11 ( ¯ ξ) ] [ K  12 ( ¯ ξ) ][ K  21 ( ¯ ξ) ] [ K  22 ( ¯ ξ) ]  =  [ T  11 ] [ T  12 ][ T  21 ] [ T  22 ]  ×  [  R 11 ( ¯ ξ) ] [  R 12 ( ¯ ξ) ][  R 21 ( ¯ ξ) ] [  R 22 ( ¯ ξ) ]   (36)into Eq. (35) yields { a (ξ) }=[  A ( ¯ ξ) ]{ c 1 }  (37) { q (ξ) }=[ Q ( ¯ ξ) ]{ c 1 }  (38)with [  A ( ¯ ξ) ]=[ K  11 ( ¯ ξ) ]¯ ξ  [ λ ] [ Y  11 ( ¯ ξ) ]{ c 1 }  (39) [ Q ( ¯ ξ) ]=[ K  21 ( ¯ ξ) ]¯ ξ  [ λ ] [ Y  11 ( ¯ ξ) ]{ c 1 } .  (40)Eliminating the constant vector { c 1 } in Eqs. (37) and (38) leads to { q (ξ) }=[  H  b ( ¯ ξ) ]{ a (ξ) }  (41)with [  H  b ( ¯ ξ) ]=[ Q ( ¯ ξ) ][  A ( ¯ ξ) ] − 1 .  (42)Once the nodal potential vector  { a ( ¯ ξ) }  is obtained, theintegration constant vector { c 1 } can be determined as { c 1 }=[  A ( ¯ ξ) ] − 1 { a (ξ) } .  (43)With the known integration constant vector  { c 1 } , the entirepotential field can be found by substituting Eq. (37) into Eq. (7). The velocity field can be determined by substituting theintegration constant vector  { c 1 }  into Eq. (38) to determinethe nodal flow vector  { q (ξ) } , then employing Eq. (21) toderive the derivative of the nodal potential vector  { a ( ¯ ξ) }  withrespect to the radial coordinate  ξ   and substituting the derivativeinto Eq. (9). 3.3. Unbounded domain solution For an unbounded domain  ( 1  ≤  ξ   ≤ ∞ ) , Eq. (24) is moredifficult to solve because it has an irregular singular point at ξ   =∞ , and all eigenvalues of the matrix [  M  ] are zero. In solidmechanics problems, a high-frequency asymptotic expansion of the dynamic stiffness can be constructed using the radiationcondition [23]. The displacements obtained from the high- frequency asymptotic expansion method outlined do not satisfythe Sommerfeld radiation condition automatically in the sameway as the Hankel function does for the scalar case. Therefore,the solution procedure for an unbounded domain provided bySong and Wolf (1998) [23] cannot be applied directly to the problem at hand.In the work discussed here, an asymptotic expansionfor the scattered wave potential as  ξ   → ∞  is obtainedusing procedures suggested by Wasow [25]. The resulting solution is able to satisfy the Sommerfeld radiation conditionautomatically. This is one of the major contributions of thecurrent work. Due to its generality, the asymptotic expansionpresented here is also valid in the solid mechanics problems.Since the derivation of the procedure is rather long, only asummary is presented here. For full details the reader shouldrefer to Li et al. (2004) [13], or to the text by Wasow [25]. Eq. (24) can be re-written as ¯ ξ  − 1 [  X  ( ¯ ξ) ] , ¯ ξ   =[−[  M  ]− ¯ ξ  − 2 [  Z  ]][  X  ( ¯ ξ) ] .  (44)Since the eigenvalues of the square matrix  [  M  ]  are all zero, itis not possible to obtain an asymptotic expansion for  [  X  ( ¯ ξ) ] directly. Instead a transformation is introduced so that  [  X  ( ¯ ξ) ] is replaced by [ G ( ¯ ξ) ] , where [  X  ( ¯ ξ) ]=[¯ T  ][ P ( ¯ ξ) ][ G ( ¯ ξ) ]  (45)with [ ¯ P ( ¯ ξ) ]=[  I  ]+ ∞  m = 1 [ ¯ P m ]¯ ξ  − m .  (46)The coefficient matrices  [ ¯ P m ]  and  [ T  ]  are selected so that thetransformed differential equation (44) becomes ¯ ξ  − 1 [ ¯ G ( ¯ ξ) ] , ¯ ξ   =[¯  B ( ¯ ξ) ][ ¯ G ( ¯ ξ) ]  (47)where the matrix function [¯  B ( ¯ ξ) ] may be expressed as [¯  B ( ¯ ξ) ]= ∞  m = 0 [¯  B m ]¯ ξ  − m (48)and each of the coefficient matrices  [¯  B m ]  in this expansionare block diagonal. Insertion of Eq. (46) into Eq. (48) with
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