1. 3. ([12], Matsumura[13], Kikuchi[10] ) [12], [13], [10] ( [12], [13], [10]

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1 ) [2] ) ) Newton[4] Colton-Kress[2] ) ) OK) [5] [] ) [2] Matsumura[3] Kikuchi[] ) [2] [3] [] 2 ) 3 2 P P )+ P + ) V + + P H + ) [2] [3] [] P V P ) ) V H ) P V ) ) ) 2 C) 25473)

2 2 3 Dermenian-Guillot[3] M.Kawashita-W.Kawashita-oga [7] [8] M. Kawashita-oga[9] [6] ) ) 2. 3 [3] ) x = x x 2 x 3 ) = x x 3 ) R 2 R + = ux) = t u x) u 2 x) u 3 x)) C3 ρ > µ > λ R 3λ + 2µ > H = L 2 C 3 ρ dx) L L u = ρ λ + µ ) u + µ u} DL ) = u H C 3 ); ρ λ + µ ) u + µ u} H u λ u)δ 3 + µ + u ) 3 = in the trace sense = 2 3) x 3 x x3 = = t / x / x 2 / x 3 ) = 2 / x / x / x 2 3 L L k = k k 2 k 3 ) = k k 3 ) R 2 R + = p = p p 2 ) R 2 k = k ω = k ω ω 3 ) = k ω ω 2 ω 3 ) p = p ν = p ν ν 2 ) ξ l k) = c 2 c 2 l c 2 P = λ + 2µ c 2 = µ ρ ρ ) /2 ) c 2 /2 k 2 k 2 γ l ω) = ω 2 for l = P ) /2 ) /2 ξ l k) = k 2 c2 c 2 k 2 γ l ω) = ω 2 c2 l c 2 for l = P l ) /2 γ R = c2 R c 2 for = P c 2 R < c2 R < µ ρ } E V = k ; k 3 > c 2 P c 2 ) /2 k c 2 l E V = k ; < k 3 < c 2 P c 2 ˆk = k k 3 ) k P = k ξ P k)) = k ω γ P ω)) k ξ P k)) = k ω γ P ω)) k = k ω E V ) k P = k iξ P k)) = k ω iγ P ω)) k = k ω E V ) }. ) /2 k }

3 3 3 L ) k Ψ P x k) = 2π) 3 2 ρ 2 α ±) P ω) = c2 P c 2 a ) e iˆk x a ) P ω) + eik P x a 2) P ω) eik x a 3) P ω) } 2 ω 2 ) 2 ± 4 ω 2 ω 3 γ P ω) β P ω) = 4 ω c2 P c 2 2 ω 2 )ω 3 P ω) = t ω ω 2 ω 3 ) P ω) = β P ω) ω γ P ω) ω a 2) a 3) P 2) k Ψ V x k) = 2π) 3 2 ρ 2 t α +) P ω) ω) = α ) P ω) α +) P ω) t ω ω 2 ω 3 ). β P ω) = 4 ω 2 ω 2 )ω 3 α ±) P ω) = a ) a 2) V ω ) 2γ P ω) ω ω e iˆk x a ) V ω) + eik x a 2) V ω) + eik P x a 3) V ω) } c2 P c 2 2 ω 2 ) 2 ± 4 ω 2 ω 3 γ P ω) k = k ω E V ) c2 P c 2 2 ω 2 ) 2 ± 4i ω 2 ω 3 γ P ω) k = k ω E V ) V ω) = t ω ω 3 ω 2ω 3 ω ω α ) P ω) = ω) ω ω 3 ω a 3) V ω) = 3) k 4) p R 2 Ψ R x p) = ) a ) R ν) = 2 c2 R c 2 t α +) P ω) β P ω) α +) ) ω ω ) 2ω 3 ω ω t ω ω 2 γ P ω)) P ω) k = k ω E V ) β P ω) ω α +) ω 2 iγ P ω) P ω)) k = k ω E V ) Ψ H x k) = e iˆk x + e ik x )a 2π) 3 2 ρ H ω) 2 a H ω) = t ω 2 ω ω ) ω. C 2π p 2 e ip x e p γ RP x 3 a ) R ν) + e p γ Rx 3 a 2) R ν) } t iν iν 2 γ RP ) a 2) R ν) = 2 γ RP t iν γ R iν 2 γ R )

4 4 C 4π 2 Ψ R x p) 2 ρ dx 3 =. Ψ P x k) : incident P wave+refleced wave +reflected P wave Ψ V x k) : incident wave+refleced wave +reflected P wave Ψ H x k) : incident wave+refleced wave Ψ R x p) : Rayleigh wave F = P V H) F R u C R3 + C 3 ) ): F uk) = F R up) = Ψ x k) ux)ρ dx Ψ R x p) ux)ρ dx. F = P V H) H L 2 ) F R H L 2 R 2 ). F [3]. F H F u = F P u F V u F H u F R u) for u H Ĥ = =PVH L 2 ) L 2 R 2 ) unitary u DL ) F L u = c 2 P k 2 F P u c 2 k 2 F V u c 2 k 2 F H u c 2 R p 2 F R u). 3. z = λ + iε C ε L R z) R z) = L z) ) u H suppu compact u L R z)u = F c 2 k 2 z) F u + FRc 2 R p 2 z) F R u 3.) 3.2) = G x y; z) = =PVH =PVH G R x y; z) = c = G x y; z)uy)dy + G R x y; z)uy)dy c P = P ) = V H) c c 2 k 2 z ψ x k) t ψ y k)dk R 2 c 2 R p 2 z ψ Rx p) t ψ R y p)dp = P V H)

5 3 5 ) 2) ) G x y; λ ± i) = lim ϵ ± G x y; z) = P V H R). λ y ) x G x y λ ± i) x 3.) 3.2) 2) < a < b a < λ < b a b Φ l s) C R) l = 2) Φ s) = a/2 < s < 3b/2) s < a/3 2b < s) Φ 2 s) = Φ s). G x y; z) = G l) x y; z) = P V H R) l=2 G l) x y; z) = Φ l c 2 k 2 ) R 3 c 2 + k 2 z ψ x k) t ψ y k)dk G l) R x y; z) = Φ l c 2 R p ) c 2 R p 2 z ψ Rx p) t ψ R y p)dp R 2 = P V H) l = 3.3) 3.4) G ) x y; z) = G ) R x y; z) = µ 2 Φ c 2 µ2 ) c 2 µ2 z J x y µ)dµ = P V H) τφ c 2 R τ 2 ) c 2 R τ 2 z J Rx y τ)dτ J x y µ) = 2 + ψ x µω) t ψ y µω)dω J R x y τ) = ψ R x τν) t ψ R y τν)dν = P V H) J x y µ) = P V H) J R x y τ) ) x = x x 2 x 3 ) = x x 3 ) = x ϕ = r ϕ ϕ 3 ) = rϕ ϕ 2 ϕ 3 ) ˇϕ = ϕ ϕ 3 ) ϕ l = ϕ γ l ϕ)) x = x ϕ = r ϕ.

6 6 I P x y µ) =e iµr a ) P ˇϕ) t Ψ P y µ ˇϕ) e iµr a 3) χ ϕ < c e iµ c P cp c r c P ϕ 3 c γ P ω) a2) P P ϕ) t Ψ P y µϕ) ) cp ϕ P c t Ψ P y c c P µϕ P I V x y µ) =e iµr a ) V ˇϕ) t Ψ V y µ ˇϕ) e iµr a 2) V ϕ) t Ψ V y µϕ) ) e iµ c cp r c γ P ω) a 3) c V ϕ P c P ϕ 3 c t Ψ V y c ) µϕ P P c P I H x y µ) =e iµr a H ˇϕ) t Ψ H y µ ˇϕ) e iµr a H ϕ) t Ψ H y µϕ) I R x y τ) =e πi 4 e iτ r e τ γ RP x 3 a ) R ϕ) + e τ γ Rx 3 a 2) R ϕ) } t Ψ R y τ ϕ) + e πi 4 e iτ r e τ γ RP x 3 a ) R ϕ) + e τ γ Rx 3 a 2) R ϕ) } t Ψ R y τ ϕ) a i ) il=23 a i ) il=23 = max i=23 a i 2. M a b < a < b M > ) ) 3) ) y < M a/3 < c 2 P µ2 < 2b y µ J i P x y µ) 2πρ ) 2 µr I P x y µ) = or ) r ). 2) = V H y < M a/3 < c 2 µ2 < 2b y µ J i x y µ) 2πρ ) 2 µr I x y µ) = O r log r ) r x 3 > r r r ) or ) < x3 < r r r ). ) 3) y < M a/3 < c 2 R τ 2 < 2b y τ J C Rx y τ) I 2π) 3 2 τ r) R x y τ) 2 = O r 3 2 ) e a 3 c R γ RP x 3 + e ) a 3 c R γ Rx 3 r ) 3. 2) V i ) = 3 ) 3 2) 3 3) 3 ) 3 2) ) i J V x y µ) 2πρ ) 2 µr I V x y µ) = or ) r ) H i ) = 3 ) 3 2) 3 3) 3 ) 3 2) 2 [4] [2] Copson[] Lewis[5] i

7 ) 3.4) ϵ ) lim ϵ ± G) x y; z) = p.v. µ 2 Φ c 2 µ2 ) c 2 µ2 λ J x y µ)dµ ± πij x y c λ 2 ) = P V H) lim ϵ ± G) R x y; z) = p.v. τφ c 2 R τ 2 ) c 2 τ 2 λ J Rx y τ)dτ ± πij R x y c R λ 2 ) G ) x y; λ ± i) = P V H R) x y; z) = P V H R) G 2) ϵ = ) x ) 4. M a b < a < b M > ) ) 4) ) λ > y < M a < λ < b y λ G 2) P x y; λ) = or ) r ). 2) y < M y 3 a < λ < b y λ G 2) V x y; λ) = or ) r ). 3) y < M a < λ < b y λ G 2) H x y; λ) = or ) r ). 4) y < M a < λ < b y λ 5. 2) ) y 3 G 2) R x y; λ) = Or 2 ) G 2) V x y; λ) = Φ 2 c 2 k 2 ) E V c 2 k 2 λ ψ V x k) t ψ V y k)dk+ r ). E V Φ 2 c 2 k 2 ) c 2 k 2 λ ψ V x k) t ψ V y k)dk 2 2) ux) C 2 ) suppux) E V Φ 2 c 2 k 2 ) c 2 k 2 λ ψ V x k) t ψ V y k)dkuy)dy = o x ) References [] E.T.Copson Asymptotic Expansions. Cambridge Tracts in Mathematics 55 Cambridge Univ. Press965. [2] D.Colton and R.Kress Inverse Acoustic and Electromagnetic cattering Theory. Appl.Math.ci.Vol.5 pringer-verlag997. [3] Y. Dermenian and J.C. Guillot cattering of elastic waves in a perturbed isotropic half space with a free boundary. The limiting absorption principle Math. Meth. in the Appl. ci. 988) [4]. 976 [5] [6] M.Kadowaki On a framework of scattering for dissipative systems. Osaka J. Math.4 23) no

8 8 [7] M.Kawashita W.Kawashita and H.oga Relation between scattering theories of the Wilcox and Lax- Phillips types and a concrete construction of the translation representation. Comm. Partial Differentail Equations 28 23) no [8] cattering theory for the elastic wave equation in perturbed half-spaces. Tran. Amer. Math. oc ) no [9] M. Kawashita and H. oga ingular support of the scattering kernel for the Rayleigh wave in perturbed half-spaces. Method Appl. Anal. 7 2) no. 47 [] K. Kikuchi Asymptotic behavior at infinity of the Green function of a class of systems including wave propagation in crystals. Osaka J. Math ) no [] 2. [2] ) [3] M. Matsumura Asymptotic behavior at infinity for Green s functions of first order systems with characteristics of nonuniform multiplicity. Publ. Res. Inst. Math. ci ) no [4] R.G.Newton Inverse chrödinger cattering in Three Dimensions. Text and Monographs in Physics pringer-verlag989. [5] R. Lewis Asymptotic theory of transients Electromagnetic wave theory URI ymposium Proceedings Delft Pergamon Press New York 967.

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