Scattering of nonstationary waves by a one-dimensional obstacle

Problems for the three-dimensional wave equation with a boundary condition on an axis are considered.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 62, pp. 48–51, 1976.     

The scattering of sound waves by an obstacle

In this paper we study the scattering of acoustic waves by an obstacle ??. We establish the following relation between the scattering kernel S(s, θ, ω) and the support function h_?? of the obstacle: The right endpoint of the support of S(s, θ, ω) as function of s is h_??(θ-ω); h_?? is defined by For Dirichlet boundary condition the result is proved in full generality, for Neumann condition only for backscattering, i.e., for θ = -ω. Since the convex hull of ?? can be recovered from knowledge of h_??, the above result may be useful in reconstructing ?? from scattering data.     

On the one-dimensional scattering of plane waves on an arbitrary potential

Theoretical and Mathematical Physics -     

Reconstruction of a penetrable obstacle by complex spherical waves

We consider an inverse acoustic scattering problem for identifying a non-convex penetrable obstacle in three dimensions in a homogeneous medium. We apply the complex geometrical optics solutions with logarithmic phase, which is called complex spherical waves, to reconstruction problem. The reconstruction schemes will be demonstrated in the last section.     

On the scattering of waves by a disk

Zusammenfassung Es wird gezeigt, dass das Problem, die Beugung einer ebenen Welle an einer kreisförmigen Öffnung oder Scheibe zu bestimmen, auf die Lösung von regulären Fredholmschen Integralgleichungen zweiter Art zurückgeführt werden kann. Die Lösungen dieser Integralgleichungen liefern uns für die Scheibe im Falle der Neumannschen Bedingung die radiale Variation der Unstetigkeiten der Wellenfunktion und im Dirichletschen Falle die Unstetigkeiten ihrer normalen Ableitung. Ist das Produkt von Öffnungsradius und Wellenzahl klein, so können die Integralgleichungen gelöst werden. Für die Ableitung der Integralgleichungen verwenden wir einerseits die Poissonsche Darstellung für die Wellenfunktion und andererseits die Fortsetzung der Helmholtzschen Darstellung in die komplexe Ebene.     

Resonance-free region in scattering by a strictly convex obstacle

We prove the existence of a resonance-free region in scattering by a strictly convex obstacle \(\mathcal{O}\) with the Robin boundary condition \(\partial_{\nu}u+\gamma u|_{\partial\mathcal{O}}=0\) . More precisely, we show that the scattering resonances lie below a cubic curve ?ζ=?S|ζ|^1/3+C. The constant S is the same as in the case of the Neumann boundary condition γ=0. This generalizes earlier results on cubic pole-free regions obtained for the Dirichlet boundary condition.     

Electromagnetic scattering by a homogeneous chiral obstacle in a chiral environment

Time-harmonic electromagnetic waves are scattered by a homogeneouschiral obstacle embedded in a chiral environment. The correspondingtransmission problem is reduced, via Bohren's decomposition,to an integral equation over the interface between the obstacleand the surrounding medium. This integral equation is shownto be uniquely solvable except for a discrete set of electromagneticparameters of the obstacle.     

Inverse scattering by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium

This paper is concerned with the inverse problem of scattering of time-harmonic acoustic waves by an inhomogeneous penetrable obstacle in a piecewise homogeneous medium. The well-posedness of the direct problem is first established by using the integral equation method. We then proceed to establish two tools that play important roles for the inverse problem: one is a mixed reciprocity relation and the other is a priori estimates of the solution on some part of the interfaces between the layered media. For the inverse problem, we prove in this paper that both the penetrable interfaces and the possible inside inhomogeneity can be uniquely determined from a knowledge of the far field pattern for incident plane waves.     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

On the scattering of spherical electromagnetic waves by a layered sphere

A spherical electromagnetic wave is scattered by a layered sphere.The exact expressions of the scattered and interior fields areobtained by solving the corresponding boundary-value problem,by means of a combination of Sommerfeld's and T-matrix methods.A recursive algorithm with respect to the number of layers isextracted for the computation of the fields in every layer.The far-field pattern and the scattering cross-sections aredetermined in terms of the physical and geometrical characteristicsof the scatterer. As the point-source tends to infinity, theknown results for plane wave incidence are recovered. Numericalresults are presented for several cases and various parametersof the layered spherical scatterer.     

Traveling waves in a one-dimensional heterogeneous medium

We consider solutions of a scalar reaction–diffusion equation of the ignition type with a random, stationary and ergodic reaction rate. We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit. We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media.