On the Inverse Elastic Scattering by Interfaces Using One Type of Scattered Waves

Abstract

We deal with the problem of the linearized and isotropic elastic inverse scattering by interfaces. We prove that the scattered p-parts or s-parts of the far field pattern, corresponding to all the incident plane waves of pressure or shear types, uniquely determine the obstacle geometry for both the penetrable and impenetrable obstacles. In addition, we state a reconstruction procedure. In the analysis, we assume only the Lipschitz regularity of the interfaces and, for the penetrable case, the Lamé coefficients to be measurable and bounded, inside the obstacles, with the usual jumps across these interfaces.

Appendix

1.1 A.1 Derivatives of the Helmholtz Fundamental Solution

We have, for $x,y\in\mathbb{R}^{3}$ with x≠y

$$G_p(x,y) = \frac{e^{i\kappa_p|x-y|}}{4\pi|x-y|}. $$

1st Order Partial Derivatives

The first partial derivatives of G _p can be written as:

$$\frac{\partial G_p(x,y)}{\partial x_l} = \frac{1}{4\pi}e^{i\kappa _p|x-y|} \biggl[ \frac{i\kappa_p(x_l-y_l)}{|x-y|^2}-\frac {(x_l-y_l)}{|x-y|^3} \biggr], $$

for all l=1,2,3.

2nd Order Partial Derivatives

For all l=1,2,3

$$\begin{aligned} \frac{\partial^2G_p}{\partial x_{l}^{2}} =& \frac{1}{4\pi}e^{i\kappa_p|x-y|} \biggl[(i \kappa_p)^2\frac {(x_l-y_l)^2}{|x-y|^3}-3i\kappa_p \frac{(x_l-y_l)^2}{|x-y|^4} \\ &{}+\frac{i\kappa_p}{|x-y|^2}-\frac{1}{|x-y|^3}+3\frac{(x_l-y_l)^2}{|x-y|^5} \biggr]. \end{aligned}$$

For l≠m with l,m=1,2,3, we have

$$\begin{aligned} \frac{\partial^2 G_p}{\partial x_m\partial x_l} =& \frac{1}{4\pi}e^{i\kappa_p|x-y|}(x_l-y_l) (x_m-y_m) \\ &{}\times \biggl[(i\kappa_p)^2\frac{1}{|x-y|^3}-3i \kappa_p\frac{1}{|x-y|^4}+3\frac {1}{|x-y|^5} \biggr]. \end{aligned}$$

3rd Order Partial Derivatives

For all l=1,2,3

$$\begin{aligned} \frac{\partial^3G_p}{\partial x_{l}^{3}} =& \frac{1}{4\pi}e^{i\kappa_p|x-y|}(x_l-y_l) \biggl[3(i\kappa_p)^2\frac {1}{|x-y|^3}-9(i \kappa_p)\frac{1}{|x-y|^4} \\ &{}+(i\kappa_p)^3\frac{(x_l-y_l)^2}{|x-y|^4}+3\frac{1}{|x-y|^5} \\ &{} -6(i\kappa_p)^2\frac{(x_l-y_l)^2}{|x-y|^5}+15i \kappa_p\frac {(x_l-y_l)^2}{|x-y|^6}-15\frac{(x_l-y_l)^2}{|x-y|^7} \biggr]. \end{aligned}$$

(A.1)

For l≠m with l,m=1,2,3 we have

$$\begin{aligned} \frac{\partial^3G_p}{\partial x_l\partial x_{m}^{2}} =&\frac{\partial^3G_p}{\partial x_m\partial x_l\partial x_m}=\frac {\partial^3G_p}{\partial x_m\partial x_m\partial x_l} \\ =& \frac{1}{4\pi}e^{i\kappa_p|x-y|}(x_l-y_l) \biggl[(i\kappa_p)^2\frac {1}{|x-y|^3}-3i \kappa_p\frac{1}{|x-y|^4} \\ &{}+(i\kappa_p)^3\frac{(x_m-y_m)^2}{|x-y|^4}+3\frac{1}{|x-y|^5} -6(i\kappa _p)^2\frac{(x_m-y_m)^2}{|x-y|^5} \\ &{}+15i\kappa_p\frac{(x_m-y_m)^2}{|x-y|^6}-15\frac {(x_m-y_m)^2}{|x-y|^7} \biggr]. \end{aligned}$$

(A.2)

At last for k≠l≠m with k,l,m=1,2,3 we have

$$\begin{aligned} \frac{\partial^3G_p}{\partial x_k\partial x_l\partial x_m} =& \frac{1}{4\pi}e^{i\kappa_p|x-y|}(x_k-y_k) (x_l-y_l) (x_m-y_m) \\ &{}\times \biggl[(i\kappa_p)^3\frac{1}{|x-y|^4}-6(i \kappa_p)^2\frac {1}{|x-y|^5}-9i\kappa_p \frac{1}{|x-y|^6}-15\frac{1}{|x-y|^7} \biggr]. \end{aligned}$$

(A.3)

1.2 A.2 Derivatives of the Elastic Fundamental Tensor

In (2.3), the fundamental tensor of the elastic model is given. Now the ij-th element of the fundamental tensor Φ(x,y) can be viewed as:

$$\begin{aligned} \varPhi_{ij}(x,y) = & \frac{1}{4\pi}\sum_{n=0}^{\infty} \frac{i^n}{(n+2) n!} \biggl(\frac {n+1}{{\mu_0}^{\frac{n+2}{2}}}+ \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \kappa^n\delta_{ij}|x-y|^{n-1} \\ &{} - \frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n(n-1)}{(n+2) n!} \biggl(\frac{1}{{\mu_0}^{\frac{n+2}{2}}} - \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n|x-y|^{n-3}(x_i-y_i) (x_j-y_j) \biggr\} , \end{aligned}$$

where $x,y \in\mathbb{R}^{3}$ with x≠y, see for more details ([15], Chap. 2).

For l≠i,j

$$\begin{aligned} \frac{\partial\varPhi_{ij}}{\partial x_l} = & \frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n}{(n+2)n!} \biggl(\frac {n+1}{{\mu_0}^{\frac{n+2}{2}}}+ \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n\delta_{ij}(n-1) (x_l-y_l)|x-y|^{n-3} \biggr\} \\ &{} -\frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n(n-1)}{(n+2) n!} \biggl(\frac{1}{{\mu_0}^{\frac{n+2}{2}}} - \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n(n-3) (x_l-y_l) (x_i-y_i) (x_j-y_j)|x-y|^{n-5} \biggr\} . \end{aligned}$$

(A.4)

Similarly, for l=i

$$\begin{aligned} \frac{\partial\varPhi_{ij}}{\partial x_i} = & \frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n}{(n+2)n!} \biggl(\frac {n+1}{{\mu_0}^{\frac{n+2}{2}}}+ \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n\delta_{ij}(n-1) (x_l-y_l)|x-y|^{n-3} \biggr\} \\ &{} -\frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n(n-1)}{(n+2) n!} \biggl(\frac{1}{{\mu_0}^{\frac{n+2}{2}}} - \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n \bigl[(n-3) (x_l-y_l) (x_i-y_i) (x_j-y_j)|x-y|^{n-5} \biggr\} \\ &{}+(x_j-y_j)|x-y|^{n-3} \bigr], \end{aligned}$$

(A.5)

and for l=j we obtain

$$\begin{aligned} \frac{\partial\varPhi_{ij}}{\partial x_j} =&\frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n}{(n+2)n!} \biggl(\frac {n+1}{{\mu_0}^{\frac{n+2}{2}}}+ \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n\delta_{ij}(n-1) (x_l-y_l)|x-y|^{n-3} \biggr\} \\ &{} -\frac{1}{4\pi}\sum_{n=0}^{\infty} \biggl\{ \frac{i^n(n-1)}{(n+2) n!} \biggl(\frac{1}{{\mu_0}^{\frac{n+2}{2}}} - \frac{1}{(\lambda_0+2\mu_0)^{\frac {n+2}{2}}} \biggr) \\ &{}\times\kappa^n \bigl[(n-3) (x_l-y_l) (x_i-y_i) (x_j-y_j)|x-y|^{n-5} \biggr\} \\ &{}+(x_i-y_i)|x-y|^{n-3} \bigr]. \end{aligned}$$

(A.6)