High frequency scattering by a classically invisible body (Q2903535)

The authors analyze a plane wave scattering problem by a polyhedron which is invisible to an observer viewing only the paths of geometrical optics rays. In fact, such an object \({\mathcal O} \subset {\mathbb{R}^3}\) with Lipschitz boundary \(\partial {\mathcal O}\), as proposed by \textit{A.~Aleksenko} and \textit{A.~Plakhov} [Nonlinearity 22, No.~6, 1247--1258 (2009; Zbl 1173.37035)], has the property that geometrical optical rays, coming from a particular direction and reflected twice from the boundary of \({\mathcal O}\), continue to propagate parallel to each other in the same way as if the obstacle was absent. Thus, the object has zero classical total scattering cross-section and appears invisible to an observer.NEWLINENEWLINEHigh frequency asymptotics are obtained for the scattering of a plane wave by the obstacle \({\mathcal O}\). In view of this, a constant phase shift \(\rho\) for all rays meeting \({\mathcal O}\) is considered. Therefore, in the absence of absorption, the obstacle \({\mathcal O}\) is almost invisible at a sequence of high frequencies \(k=k_n={(k_0+2\pi n)}/{\rho}\), \(n\to\infty\), where \(k_0\) is determined by the boundary condition, and the invisibility effect disappears for other frequencies.NEWLINENEWLINEWithin this context, for \(k > 0\), the authors consider the corresponding scattering problem, where the scattered field \(u\) fulfils the Helmholtz equation NEWLINE\[NEWLINE \Delta u +k^2 u =0 \qquad \text{in}\quad \Omega :={\mathbb R}^3 \backslash{\mathcal O}, NEWLINE\]NEWLINE subjected to the impedance boundary condition NEWLINE\[NEWLINE \left(\frac{\partial}{\partial n} + k \lambda \right)\left(u+e^{ik(r\cdot p_0)}\right) = 0 \qquad \text{on} \quad \partial {\mathcal O} NEWLINE\]NEWLINE (where \(n\) denotes the unit outer normal vector on the smooth parts of \(\partial{\mathcal O}\), \(p_0:=(0,0,1)\) and \(\lambda\) is a constant with \({\Im}m(\lambda) \geq 0\)), and satisfying a Sommerfeld radiation condition at infinity. The authors refer to [\textit{W.~McLean}, Strongly elliptic systems and boundary integral equations. Cambridge: Cambridge University Press (2000; Zbl 0948.35001)] for the existence and uniqueness of solution in \(H^1_{loc}(\Omega)\) which satisfies this problem, recalling that every consequent solution \(u\) has the following behavior at infinity NEWLINE\[NEWLINE u(r)=\frac{e^{ik|r|}}{|r|} u_{\infty}(\theta)+o \left (\frac{1}{|r|} \right ), \quad r \rightarrow \infty, \quad \theta=r/|r| \in S^2, NEWLINE\]NEWLINE where \(u_\infty(\theta)=u_\infty(\theta,k)\) represents the scattering amplitude. Having in mind the total cross-section, NEWLINE\[NEWLINE \sigma(k)=\|u_\infty\|^2_{L_2(S^2)}=\int_{S^2} |u_\infty(\theta)|^2 d\mu (\theta)\,, NEWLINE\]NEWLINE where \(d\mu\) is the surface element of the unit sphere, and the transport cross-section, NEWLINE\[NEWLINE \sigma_T(k)= \int_{S^2} (1-\theta \cdot p_0))|u_\infty(\theta)|^2 d\mu (\theta)\,, NEWLINE\]NEWLINE the authors conclude, as the main result for \({\Im}m(\lambda)>0\), that: (i) the transport cross-section vanishes as \(k\) goes to infinity, NEWLINE\[NEWLINE \lim_{k \rightarrow \infty} \sigma_T=0; NEWLINE\]NEWLINE (ii) the total cross-section has the following asymptotic behavior for large \(k\): NEWLINE\[NEWLINE \sigma(k)= \frac{1}{2} \left |A^2e^{ik\rho}-1 \right |^2 +o(1), \qquad k \rightarrow \infty, \quad A=\frac{i-2\lambda}{i+2\lambda}; NEWLINE\]NEWLINE (iii) \(|u_\infty (\theta)|^2 \sim \sigma(k) \delta(p_0)\), as \(k \rightarrow \infty\), in distributional sense, i.e., for any \(\varphi \in C(S^2)\), it holds NEWLINE\[NEWLINE \int_{S^2} \varphi(\theta) |u_\infty (\theta)|^2 dS_\theta = \frac{1}{2} {\left|A^2e^{ik\rho} -1 \right|}^2 \varphi(p_0) + o(1), \qquad k \rightarrow \infty.NEWLINE\]