Wave Propagation 2011 Part 2 pot

A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena 19 Note that F R in Eq. (106) is the Rayleigh function given by (107) where (108) Lemma 2 For f i ∈ L 2 (R + ) and η ∈ C \ B (109) [Proof] First, fix i and j and define (110) Then, the following is obtained by means of the Schwarz inequality: (111) where (112) As a result, the following is obtained: (113) where (114) Equation (113) concludes the proof. □ Theorem 1 The operator with the domain D( ) is self-adjoint. [Proof] It is sufficient to prove that ∀f i ∈ L 2 (R + ), there exist ∈ D( ) satisfying Wave Propagation in Materials for Modern Applications 20 (115) (116) where p is a positive real number. This fact is based on the results of a previous study (Theorem 3.1, Berthier, 1982). For the construction of , define (117) where η is chosen such that η 2 = ip. Note that η ∈ C \ B. The following equation: (118) yields Eq. (115), where (R + ) is the Schwartz space. During the derivation of Eq. (118), the following equation: (119) is based on the following properties of g ij (x 3 , y 3 , ξ r , η) at x 3 = y 3 (120) In addition, the following is obtained: (121) The order of the integral and differential operators of the properties of function g ij are changed such that A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena 21 (122) for an arbitrary positive integer n. According to Eq. (121), we have (123) It has been shown that u j ∈ L 2 (R + ) from Lemma 2, so that ∈ D( ). The construction of ∈ D( ) is also possible. As a result, the following conclusion is obtained. □ 4.3 Generalized Fourier transform for an elastic wave field in a half space The operator has been found to be self-adjoint and non-negative, which yields the following spectral representation: (124) where E ij is the spectral family. The spectral family is connected with the resolvent by means of the Stone theorem (Wilcox, 1976): (125) for u i , v i ∈ L 2 (R + ). Note that R ij is the resolvent of the operator and R ij (ζ)u j is defined by (126) Let and . Then, the right-hand side of Eq. (125) for the integral becomes (127) where ζ = μ 0 η 2 and η R is defined by F R (ξ r , η R ) = 0. The path of integration in the complex η plane shown in Fig. 5 is used for the evaluation of the integral. In the following, the relationship between the right-hand side of Eq. (127) and the eigenfunctions is presented. Let v i (x 3 , ξ r , η) ∈ D( ) satisfy (128) Wave Propagation in Materials for Modern Applications 22 Fig. 5. Path of the integral. and define the scalar function W( η) such that (129) It is easy to derive the following properties of W( η) by means of the boundary conditions for v i : (130) Note that v i (x 3 , ξ r , η R ) becomes the eigenfunction (Rayleigh wave mode) satisfying the free boundary conditions. Otherwise, v i (x, ξ r , η), (η ≠ η R ) cannot satisfy the free boundary conditions. As a result, Eq. (130) is established. Integration by parts of Eq. (129) yields (131) where (132) The following lemma can then be obtained: Lemma 3 The residue of g ij at η = η R can be expressed in terms of the eigenfunction such that (133) where ξ = (ξ 1 , ξ 2 , η R ) and ψ im (x 3 , ξ) is the eigenfunction defined by (134) A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena 23 [Proof] Based on a previous study (Aki & Richards, 1980), the function g ij can be constructed by (135) where w ij is defined by (136) In addition, Δ j is defined such that g ij satisfies the free boundary condition: (137) The definition of W( η) shown in Eq. (129) implies that the expression is valid: (138) Equations (137) and (138) yield (139) Now, let η approach η R . Due to reciprocity, it is found that (140) Therefore, (141) The residue of the resolvent kernel is expressed as (142) For the case in which the eigenfunction is normalized as I 2 (η R ) = 1, we have W ′(η R ) = −2μη R , which concludes the proof. □ Next, the function g ij : (143) is investigated for the case in which s = Re( η) > ξ r . The function g ij for this case is constructed by (144) where v ik is the definition function of the improper eigenfunction (Touhei, 2002). The definition function v ik satisfies the following: Wave Propagation in Materials for Modern Applications 24 (145) The relationship between the improper eigenfunction and the definition function is given as (146) Next, let us define the following function: (147) Substitution of the explicit forms of the eigenfunction and definition function of Eq. (147) yields the following: (148) In addition, note that (149) which is obtained from the definition of w ik shown in Eq. (136). Based on Eqs. (148) and (149), the following lemma is obtained. Lemma 4 For the region of s > ξ r , the function g ij satisfies the following equation: (150) where ψ im (x 3 , ξ) is the improper eigenfunction. [Proof] The requirement of the free boundary condition for g ij yields the following expression of Δ kβ : (151) Incorporating the following reciprocity relation: (152) into Eq. (151) yields (153) Therefore, the following is obtained: (154) Thus, Eqs. (146), (148), and (149) conclude the proof. □ A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena 25 Next, let us again consider Eq. (127). Equation (127) holds for an arbitrary v i ∈ L 2 (R + ), so that the following equation can be obtained by incorporating the results of Lemmas 3 and 4: (155) where u j (ξ r , · ) ∈ L 2 (R + ), ξ = (ξ 1 , ξ 2 , ξ 3 ) ∈ and (156) As mentioned earlier, and , so that (157) Therefore, Eqs. (125) and (155) yield (158) Let b in Eq. (158) approach infinity. Then, the following eigenfunction expansion form of u i is obtained: (159) Note that the eigenfunction expansion form shown in Eq. (159) is that for u i (ξ r , ·) having the compact support. This result can be extended to all u i (ξ r , · ) ∈ L 2 (R + ) by a limiting procedure, namely, (160) where the convergence is in L 2 (R + ). The transform of the function in L 2 (R + ) obtained here can be summarized as follows: Wave Propagation in Materials for Modern Applications 26 (161) At this point, the transformation of the elastic wave field in a half space can be presented. Let us define the subset of the wavenumber space as follows: (162) The following theorem is obtained based on Eqs. (85), (95), and (158): Theorem 2 There exists a map satisfying the free boundary condition of the elastic half space of the wave field from L 2 ( ) to L 2 (σ p ) ⊕ L 2 (σ c ) defined by (163) the inverse of which is (164) Here, and are expressed as follows: (165) where (166) Here, is referred to as the generalized Fourier transform of u j , and is referred to as the generalized inverse Fourier transform of . Based on the literature (Reed and Simon, 1975), the domain of the operators and could be extended from L 2 to the space of tempered distributions ′. 4.4 Method for the volume integral equation We have obtained the transform for elastic waves in a 3-D half space, which is to be applied to the volume integral equation. Preliminary to showing the application of the transform to A Volume Integral Equation Method for the Direct/Inverse Problem in Elastic Wave Scattering Phenomena 27 the volume integral equation, we have to construct the Green’s function for the elastic half space based on the proposed transform. The definition of the Green’s function for the half space is expressed as (167) The application of the generalized Fourier transform to Eq. (167) yields (168) where is the generalized Fourier transform of the Green’s function. Therefore, as a result of Eq. (168), the Green’s function for a half space can be represented as (169) Next, let the function w i (x) be given in the following form: (170) The formal calculation reveals that (171) where denotes (172) Note that 1(· ) in Eq. (172) is defined such that (173) At this point, the application of the generalized Fourier transform to the volume integral equation becomes possible and is achieved as follows: Wave Propagation in Materials for Modern Applications 28 (174) where is the generalized Fourier transform of v i and is the incident wave field due to the point source expressed by (175) The volume integral equation for the elastic wave equation in the wavenumber domain in a half space has the same structure as that in a full space. Therefore, almost the same numerical scheme based on the Krylov subspace iteration technique is available. Note that the difference in the numerical scheme between that for the elastic full space and that for the half space lies in the discretization of the wavenumber space. The discretization of the wavenumber space for elastic half space is as follows: (176) where Δ ξ j , (j = 1, 2, 3) are the intervals of the grids in the wavenumber space, (177) and N 1 , N 2 , and N 3 compose the set of integers defined by (178) where (N 1 ,N 2 ,N 3 ) defines the number of grids in the wavenumber space. Note that Eq. (176) corresponds to the decomposition of the Rayleigh and body waves. 4.5 Numerical example For the numerical analysis of an elastic half space, the Lam´e constants of the background structure is set such that λ 0 = 4 GPa, μ 0 = 2 GPa and the mass density is set at ρ =2 g/cm 3 . Therefore, the background velocity of the P and S waves are 2 km/s and 1 km/s, respectively. and that for the Rayleigh wave velocity is 0.93 km/s. In addition, the analyzed frequency is f = 1 Hz. First, let us investigate the accuracy of the generalized Fourier transform by composing the Green’s function. For the calculation of the generalized Fourier transform, N 1 = N 2 = N 3 = 256, Δx 1 = Δx 2 = 0.25 km, and Δx 3 =0.125 km are chosen to define D x and D ξ . The parameter ε for the Green’s function is set at 0.6. Figures 6(a) and 6(b) show the Green’s function calculated by the generalized Fourier transform and the Hankel transform. The distributions of the absolute displacements are shown is these figures. For the calculation of the Green’s function, the point source is set at a [...]... to (Gennarelli & Riccio, 20 09a), the elements of R and T are: R11 = R = R 12 + R 23 e j2k 2d cos 2 1 + R 12 R 23 e j2k 2d cos 2 R 22 = R ⊥ = R 12 + R 23 e j2k 2d cos 2 ⊥ ⊥ 1 + R 12 R 23 e j2k 2d cos 2 ⊥ ⊥ R 12 = R 21 = 0 T11 = T = T 12 T 23 e jk 2d cos 2 1+ R 12 R 23 e j2k 2d cos 2 T 22 = T⊥ = T 12 = T21 = 0 (57) (58) 12 23 T⊥ T⊥ e jk 2d cos 2 1 + R 12 R 23 e j2k 2d cos 2 ⊥ ⊥ (59) (60) ... ⎛ − ( R 21 + T21 ) cos θi M5 = ⎜ ⎜ 1+R − T 11 11 ⎝ (20 ) ( 1 − R 22 − T 22 ) cos θi ⎞ ⎟ (21 ) − ( R 12 + T 12 ) cos θi ⎞ ⎟ − ( 1 + R 22 − T 22 ) ⎟ ⎠ (22 ) ⎟ ⎠ R 12 − T 12 ⎛ ( 1 − R 11 − T11 ) cos θi M6 = ⎜ ⎜ − ( R 21 + T21 ) ⎝ 2. 3 Uniform asymptotic evaluation and diffracted field It is now necessary to perform the evaluation of the following integral: ∞ jk Is = − 0 4π ∫ ∞ e jk 0 x' sin β 'cos φ ' ∫ 2 e − jk... according to (Ferrara et al., 20 07b), the elements of R and T can be so expressed: R 11 = R = 2 R ⎡1 − ( Pda Patt ) Pa ⎤ ⎢ ⎥ ⎣ ⎦ 2 1 − R 2 ( Pda Patt ) Pa R 22 2 R ⊥ ⎡1 − ( Pda Patt ) Pa ⎤ ⎢ ⎥ ⎣ ⎦ = R⊥ = 2 2 1 − R ⊥ ( Pda Patt ) Pa R 12 = R 21 = 0 T11 = T = (1 − R2 ) PdaPatt Pt 2 1 − R 2 ( Pda Patt ) Pa T 22 = T⊥ = (47) (48) (1 − R2⊥ ) PdaPatt Pt 2 1 − R 2 ( Pda Patt ) Pa ⊥ T 12 = T21 = 0 (49) (50) in which... Moreover, jk 0 ( x' sin β 'cos φ '− z' cos β ') jk 0 ( x' sin β 'cos φ '− z' cos β ') (2) (3) 36 Wave Propagation in Materials for Modern Applications 2 2 2 − jk ρ−ρ ' + ( z − z') − jk ( x − x') + y 2 + ( z − z') − jk r − r' e 0 e 0 e 0 G(r, r') = = = 4 π r − r' 4 π x − x' 2 + y 2 + z − z' 2 ( ) ( ) 4 π ρ − ρ ' 2 + ( z − z' )2 2 (4) is the three-dimensional Green function To evaluate the edge diffraction confined... explicit form of the asymptotic evaluation of I ( Ω ) is: − jk ρ sin β '+ z cos β ') e 0( ⎛ ⎛ φ ± φ' ⎞ ⎞ Ft ⎜ 2k 0ρ sin β 'cos2 ⎜ ⎟⎟ = 2 2 πk 0ρ sin β ' sin β ' ( cos φ + cos φ ') ⎝ ⎝ 2 ⎠⎠ e − jπ I (Ω) = e − jπ 4 4 e − jk 0 s 2 2 πk 0 s ⎛ ⎛ φ ± φ' ⎞ ⎞ Ft ⎜ 2k 0 s sin 2 β 'cos2 ⎜ ⎟⎟ ⎝ 2 ⎠⎠ sin 2 β ' ( cos φ + cos φ ') ⎝ 1 (44) where the identities ρ = s sin β ' and z = s cos β ' are used on the diffraction... a Set of Diffraction Problems 37 ⎛ Er ⎞ ⎛ Ei ⎞ ⎛ R ⎜ ⎟ = R ⎜ ⎟ = ⎜ 11 ⎜ Er ⎟ ⎜ Ei ⎟ ⎝ R 21 ⎝ ⊥⎠ ⎝ ⊥⎠ i R 12 ⎞ ⎛ E ⎞ ⎜ ⎟ ⎟ R 22 ⎠ ⎜ Ei ⎟ ⎝ ⊥⎠ ( 12) ⎛ Et ⎞ ⎛ Ei ⎞ ⎛ T ⎜ ⎟=T⎜ ⎟ = ⎜ 11 ⎜ Et ⎟ ⎜ Ei ⎟ ⎝ T21 ⎝ ⊥⎠ ⎝ ⊥⎠ i T 12 ⎞ ⎛ E ⎞ ⎜ ⎟ ⎟ T 22 ⎠ ⎜ Ei ⎟ ⎝ ⊥⎠ (13) ^r e || ^i e || ^ e ^i s qi ^ n ^r s ^ e ^t e || ^ e ^t s Fig 2 Ray-fixed coordinate systems for the GO field By using the above results, the scattered... z') 2 2 ρ − ρ ' + ( z − z') −∞ 0 e 2 dz' dx' (23 ) By making the substitution z'− z = ρ − ρ ' sinh ζ and using one of the integral (2) representations of the zeroth order Hankel function of the second kind H0 , it results (see Appendix D in (Senior & Volakis, 1995) as reference): ∞ ∫ 2 e − jk 0 z' cos β ' e − jk 0 ρ−ρ ' + ( z − z') 2 2 ρ − ρ ' + ( z − z') −∞ dz' = − jπ e − jk 0z cos β ' H 0 (2) 2 (... Ω −1 2 - order term) since Ω 1 , it results: I (Ω) ⎡ π Gp ( 0 ) 2 Ft jΩ τp ±⎢ ⎢ Ω −τp ⎣ ( ( ) ⎤ )⎥⎥ (41) ⎦ in which Gp ( 0 ) ( −τp ) = G (0) = − jΩ sin β ' ⎡ ⎤ 1 1 Ae 2 Ωf α dα g ( αs ) e ( s ) e jπ 4 ⎥ = ⎢± 2 πj dτ τ=0 2 πj cos φ + cos φ ' ⎣ sin β ' ⎦ ( 42) and ∞ Ft ( η) = 2j η e jη ∫ 2 e − jξ dξ (43) η is the UTD transition function (Kouyoumjian & Pathak, 1974) By substituting ( 32) , (39) and ( 42) in... ⋅ z ⎠ ⎝ − sin φ (17) where 38 Wave Propagation in Materials for Modern Applications ⎛ 1 − sin 2 'cos2 φ − sin β 'cos β 'cos φ ⎞ ⎜ ⎟ 2 M 2 = ⎜ − sin β 'sin φ cos φ − sin β 'cos β 'sin φ ⎟ ⎜ − sin β 'cos β 'cos φ ⎟ sin 2 ' ⎜ ⎟ ⎝ ⎠ (18) 0 − sin β 'sin φ ⎞ ⎛ ⎜ ⎟ M 3 = ⎜ − cos β ' sin β 'cos φ ⎟ ⎜ sin β 'sin φ ⎟ 0 ⎝ ⎠ (19) ˆ ˆ − sin β 'cos φ ' ⎞ x⋅t⎞ ⎛ − cos β ' 1 ⎟= ⎜ ⎟ 2 2 ˆ ˆ − sin β 'cos φ ' cos β... = εr cos θi − εr − sin 2 θi εr cos θi + εr − sin 2 θi R⊥ = cos θi − εr − sin 2 θi cos θi + εr − sin 2 θi Pda = e − jβeq d cos θt Patt = e −αeq d (51) ( 52) (53) (54) 44 Wave Propagation in Materials for Modern Applications i Pa = e j2k 0d sin θ Pt = e ( jk 0d cos θi −θt tan θt ) cos θt (55) (56) where βeq and αeq are the equivalent phase and attenuation factors relevant to the propagation through the . 22 5 11 11 12 12 R T cos 1 R T cos M 1R T R T (21 ) ()() ()( ) ⎛⎞ − −θ−+θ = ⎜⎟ ⎜⎟ −+ −+− ⎝⎠ ii 11 11 12 12 6 21 21 22 22 1R T cos R T cos M RT 1RT (22 ) 2. 3 Uniform asymptotic evaluation and diffracted. φ ⎝⎠ ⎝⎠ 22 4 ˆ ˆˆ ˆ cos ' sin 'cos ' xe xt 1 M ˆ ˆ ˆˆ sin 'cos ' cos ' ze zt 1sin'sin' (20 ) ()( ) ⎛⎞ − +θ−−θ = ⎜⎟ ⎜⎟ +− − ⎝⎠ ii 21 21 22 22 5 11 11 12 12 R. ⎜⎟ == ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠   ri i 11 12 ri i 21 22 EE E RR R RR EE E ( 12) ⊥ ⊥⊥ ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ == ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠   ti i 11 12 ti i 21 22 EE E TT T TT EE E (13) e ^ || i s ^ r e ^ e ^ n ^ q i e ^ r || s ^ i s ^ t e ^ e ^ || t