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By Makhmudov O. I.
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Additional info for A Cauchy Problem for the System of Elasticity Equations
126) over the ensemble realizations of the random set of inhomogeneities. Because the ﬁeld E∗ (x) is the same for all the inclusions, we obtain E(x) = E0 (x) + p Λ0 (k∗ ) = 1 1 lim Ω→∞ p Ω G(x − x ) · ε1 · Λ0 (k∗ ) · E∗ (x )dx , λ(x, k∗ )dx = Ω 1 v0 λ(x, k∗ )dx. 128) v0 Here Λ0 (k∗ ) is a constant (with respect to x) tensor because λ(x, k∗ ) is a realization of a stationary random function, Ω is a region that occupies all 3D-space in the limit Ω → ∞, p is the volume concentration of inclusions.
51) in the long-wave region. 21) of the one-particle problem if we keep only the principal terms of the kernel G∗ (x) and of the ﬁeld E∗ (x) on the right-hand side of this equation. 10) G∗ (x) = k∗2 g∗ (x)1 + ⊗ g∗ (x), g∗ (x) = e−ik∗ |x| , 4πε∗ |x| k∗2 = ω 2 ε∗ . 62) Let us expand g∗ (x) in a series with respect to the wave number k∗ and keep the ﬁrst four terms of this series g∗ (x) = 1 1 e−ik∗ r 1 i ≈ − ik∗ − k∗2 r + k∗3 r2 , r = |x|. 62) and keeping only the terms of order not higher than k∗3 in the equation for G∗ (x), we obtain G∗ (x) ≈ Gs∗ (x) + ik∗3 Gω ∗, 1 1 1 Gs∗ (x) = Gω ∇⊗∇ , 1.
This conclusion does not depend on the volume concentrations of inclusions, and therefore, version II does not give the correct long wave asymptotics of the attenuation coeﬃcients even for small concentrations of inclusions. Let us consider version III of the EMM. As is shown in , (Chapter 8), the forward amplitude F(k∗ , n0 ) of the wave ﬁeld scattered by a coated inclusion is presented in the form F(k∗ , n0 ) = − i 2k∗ ∞ (2n + 1) Yn(7) (k∗ ) + Yn(8) (k∗ ) U∗ . 118) n=1 According to the condition of self-consistency III2 the eﬀective wave number k∗ should be chosen in order to decrease the vector F(k∗ , m).
A Cauchy Problem for the System of Elasticity Equations by Makhmudov O. I.