Inverse scattering problem for nonstationary Dirac-type systems on the plane

In this paper the forward and inverse scattering problems for the nonstationary Dirac-type systems on the plane are considered. The scattering data for the inverse scattering problem (ISP) is defined and a unique restoration of the potential from the scattering data is proved.     

Inverse problem for wave propagation in a perturbed layered half-space

This paper is concerned with the inverse medium scattering problem in a perturbed, layered, half-space, which is a problem related to the seismologial investigation of inclusions inside the earth’s crust. A wave penetrable object is located in a layer where the refraction index is different from the other part of the half-space. Wave propagation in such a layered half-space is different from that in a homogeneous half-space. In a layered half-space, a scattered wave consists of a free wave and a guided wave. In many cases, only the free-wave far-field or only the guided-wave far-field can be measured.We establish mathematical formulas for relations between the object, the incident wave and the scattered wave. In the ideal condition where exact data are given, we prove the uniqueness of the inverse problem. A numerical example is presented for the reconstruction of a penetrable object from simulated noise data.     

Inverse scattering problem for a wave equation with absorption

We prove a uniqueness theorem for the inverse scattering problem for a wave equation with absorption and develop an algorithm for the solution of this problem on the basis of a given scattering operator. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1580–1584, November, 2007.     

Love waves in layered anisotropic media

A modified transfer-matrix method is proposed to describe Love waves in multilayered anisotropic (monoclinic) media. Dispersion relations for media consisting of one and two anisotropic elastic layers in contact with an anisotropic half-space are obtained in closed form. The conditions for Love waves to exist are analysed. Waves with horizontal transverse polarization of the non-canonical type are investigated.     

Asymptotic expansion of the coefficients of radiating plane waves in the scattering problem on a smooth periodic boundary

High-frequency asymptotic expansion of the coefficients of radiating plane waves in the problem of grazing scattering of a plane wave on a smooth periodic boundary is derived. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 71–86, 1990. Translated by V. V. Zalipaev.     

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.     

Inverse scattering in one-dimensional nonconservative media

The inverse scattering problem arising in wave propagation in one-dimensional non-conservative media is analyzed. This is done in the frequency domain by considering the Schrödinger equation with the potentialikP(x)+Q(x), wherek ² is the energy andP(x) andQ(x) are real integrable functions. Using a pair of uncoupled Marchenko integral equations,P(x) andQ(x) are recovered from an appropriate set of scattering data including bound-state information. Some illustrative examples are provided.Dedicated to M.G. Kreîn, one of the founding fathers of inverse scattering theory.     

Inverse scattering problem for the Schroedinger equation on the axis

Necessary and sufficient conditions on the reflection coefficient are obtained for the Schroedinger equation on the whole axis with a potential growing on one side by utilization of the inverse problem formalism.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 677–684, December, 1968.The author is grateful to L. D. Faddeev for formulating the problem, and for constant attention, and to V. S. Buslaev, whose remarks have been utilized.     

Exact solution of the problem of plane bending of a layered ring

An exact solution is given for the problem of a closed cylindrical transversely isotropic ring subjected to bending in its own plane under an arbitrary external load; it is assumed that a fiber straight before deformation and orthogonal to the neutral surface remains straight and undeformed but may no longer be orthogonal to the neutral surface. A variant of the theory developed by Korolev is employed.Moscow Institute of Electronic Machine Building. Translated from Mekhanika Polimerov, No. 1, pp. 168–171, January–February, 1973.     

Inverse scattering problem for nonstationary Dirac-type systems on the half-plane

In this paper was considered the scattering problem for the nonstationary Dirac-type systems of n (n?2) equations on the half-plane when the system has n₁ (1?n₁?n−1) incident and n₂ (n₂=n−n₁) scattered waves. In case n₁ is divisible by n₂, we formulate the inverse scattering problem for a nonstationary Dirac-type system when considering m () scattering problems on the half-plane with the same incident waves but different boundary conditions. Moreover, the scattering operator for the nonstationary Dirac-type system on half-plane was defined and unique restoration of the potential with respect to the scattering operator was proved.     

Approximate solution of the problem of scattering of surface water waves by a partially immersed rigid plane vertical barrier

The problem of scattering of two dimensional surface water waves by a partially immersed rigid plane vertical barrier in deep water is re-examined. The associated mixed boundary value problem is shown to give rise to an integral equation of the first kind. Two direct approximate methods of solution are developed and utilized to determine approximate solutions of the integral equation involved. The all important physical quantity, called the Reflection Coefficient, is evaluated numerically, by the use of the approximate solution of the integral equation. The numerical results, obtained in the present work, are found to be in an excellent agreement with the known results, obtained earlier by Ursell (1947), by the use of the closed form analytical solution of the integral equation, giving rise to rather complicated expressions involving Bessel functions.