Reciprocity relations and completeness of far-field pattern vectors for obstacle scattering of acoustic wave in a stratified medium

Under a generalized Sommerfeld radiation condition, we proved the uniqueness and existence of the direct obstacle scattering problem of time-harmonic acoustic waves in a stratified medium [8]. In this paper, we study the asymptotic behaviour of the scattered waves and prove three reciprocity relations among the free-wave far-field patterns and the guided-wave far-field pattern vectors corresponding to incident distorted plane waves and normal mode waves. Then we prove conditions under which a set of far-field patterns is complete in a Hilbert space based on the reciprocity relation. These properties are important in investigating the inverse obstacle scattering problems.     

The Far-Field Equations in Linear Elasticity for Disconnected Rigid Bodies and Cavities

In this paper the far-field equations in linear elasticity for scattering from disjoint rigid bodies and cavities are considered. The direct scattering problem is formulated in differential and integral form. The boundary integral equations are constructed using a combination of single- and double-layer potentials. Using a Fredholm type theory it is proved that these boundary integral equations are uniquely solvable. Assuming that the incident field is produced by a superposition of plane incident waves in all directions of propagation and polarization it is established that the scattered field is also expressed as the superposition of the corresponding scattered fields. A pair of integral equations of the first kind which hold independently of the boundary conditions are constructed for the far-field region. The properties of the Herglotz functions are used to derive solvability conditions for the far-field equations. It is also proved that the far-field operators, in terms of which we can express the far-field equations, are injective and have dense range. An analytical example for spheres illuminates the theoretical results.     

On the far-field operator for electromagnetic scattering in chiral media

We consider time-harmonic electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium where a perfect conductor has been immersed. Assuming that the incident electric field is a superposition of plane incident electric waves, the corresponding scattered field and the far-field pattern are expressed as the superposition of the scattered fields and the far-field patterns respectively. It is also proved that the sets of far-field patterns are complete if and only if there does not exist an eigenfunction to the interior perfect conductor problem that vanishes on the boundary of the scatterer which is an electric Herglotz field. The Left-Circularly Polarized and the Right-Circularly Polarized far-field operators are defined and studied and using them the electric far-field operator is defined too. The properties of the above operators and Herglotz functions are related to the solution of the interior perfect conductor boundary value problem.     

海洋波导的可穿透障碍物散射场远场分布的若干性质

研究了海洋波导中可穿透目标时谐声波散射传播远场分布的性质,构造了透射问题解的集合,使得所构造解的集合在边界上的限制在某个Hilbert空间中是稠密的,确定了传播远场分布的集合在某个Hilbert空间中是完备的．这些性质对研究海洋波导中的逆透射问题有重要的应用．     

3D elastic scattering theorems for point-generated dyadic fields

The problem of scattering of elastic spherical waves by a rigid body, cavity or a penetrable obstacle in 3D linear elasticity is considered. For two point sources, dyadic far-field pattern generators are defined, which are used for the formulation of a general scattering theorem. The main reciprocity principle and mixed scattering relations are also established. We provide the necessary energy considerations, presenting relative energy functionals and expressions for the differential and the scattering cross section due to point-source dyadic incidence. Finally, an application of the general scattering theorem for appropriate locations of the point source leads to an optical theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

Note on a Uniqueness Theorem of Schiffer

AMS (MOS): 35R30, 76Q05

The uniqueness of the inverse scattering problem is examined. It is shown that the boundary conditions must be suitably restricted to secure uniqueness. For one class the far-field pattern for one incident field is sufficient for uniqueness and, for a wider class, information from a finite number of distinct incident waves suffices. In either case, the far-field also dictates the boundary condition on the scatterer.     

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.     

ANALYSIS OF TWO-DIMENSIONAL ELECTROMAGNETIC SCATTERING BY A PERFECTLY CONDUCTING OBSTACLE IN A HOMOGENEOUS CHIRAL ENVIRONMENT

The time-harmonic electromagnetic plane waves incident on a perfectly conducting obstacle in a homogeneous chiral environment are considered.A two-dimensional direct scat- tering model is established and the existence and uniqueness of solutions to the problem are discussed by an integral equation approach.The inverse scattering problem to find the shape of scatterer with the given far-field data is formulated.Result on the uniqueness of the inverse problem is proved.     

The propagation problem and far-field patterns in a stratified finite-depth ocean

In this paper we investigate the direct problem associated with the scattering of ‘plane waves’ from an object submerged in an ocean of finite depth. An integral representation for the Dirichlet problem is found, from which a formula for the far-field pattern evolves. A density theorem is established concerning the set of all far-field patterns. This theorem is essential for the reconstruction of the submerged object, the ‘inverse’ problem [2], [4], [5].     

Fragments of the theory of nanotransistors: self-switching of a plane transverse hypersonic wave in nonlinearly elastic nanocomposite materials

We present fragments of the theory of nanotransistors concerning general information about transistors, models of nanocomposite materials, effects of nonlinear interaction of waves, and a theoretical analysis of the interaction of cubically nonlinear elastic plane harmonic waves in materials whose nonlinear properties are described by the Murnaghan potential. Using the method of slowly varying amplitudes, we investigate the interaction of two harmonic vertical transverse plane waves. Shortened and evolution equations, and Manley-Rowe relations, are obtained. We analyze analytically and numerically the mechanism of repumping of the energy of a strong pumping wave that propagates at a frequency of ω to a weak signal wave propagating at a frequency of 3ω . The described mechanism of switching of hypersonic waves in a nonlinearly elastic nanomaterial is similar to the mechanism of switching observed in optical and other transistors.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

A phase decomposition approach and the Riemann problem for a model of two-phase flows

We present a phase decomposition approach to deal with the generalized Rankine–Hugoniot relations and then the Riemann problem for a model of two-phase flows. By investigating separately the jump relations for equations in conservative form in the solid phase, we show that the volume fractions can change only across contact discontinuities. Then, we prove that the generalized Rankine–Hugoniot relations are reduced to the usual form. It turns out that shock waves and rarefaction waves remain on one phase only, and the contact waves serve as a bridge between the two phases. By decomposing Riemann solutions into each phase, we show that Riemann solutions can be constructed for large initial data. Furthermore, the Riemann problem admits a unique solution for an appropriate choice of initial data.     

Analytical Solutions of the Internal Gravity Wave Equation for a Semi-Infinite Stratified Layer of Variable Buoyancy

Computational Mathematics and Mathematical Physics - The problem of constructing asymptotics describing far-field internal gravity waves generated by an oscillating point source of perturbations...     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

Transient Nonlinear Disturbances and Shelves in a Stratified Fluid Layer

We revisit the classical problem of internal wave propagation in a stratified fluid layer bounded by rigid walls and point out a mechanism by which unsteady locally confined disturbances generate far-field shelves. Carrying the standard expansion procedure to fourth order in the wave amplitude reveals that weakly nonlinear long waves of a certain mode shed, in general, lower- and higher-mode shelves, which propagate upstream and downstream with the corresponding long-wave speeds. This phenomenon is brought about by the combined effect of nonlinear interactions and the presence of transience in the main disturbance. While the shelves accompanying small-amplitude waves are relatively weak, numerical solutions of the full Euler equations indicate that shelves induced by unsteady disturbances of finite amplitude close to breaking can be quite significant.     

Wave Properties of Double One-Dimensional Periodic Sheet Grating

We show that the double one-dimensional periodic sheet gratings always have waveguide properties for acoustic waves. In general, there are two types of pass bands: i.e., the connected sets of frequencies for which there exist harmonic acoustic traveling waves propagating in the direction of periodicity and localized in the neighborhood of the grating. Using numerical-analytical methods, we describe the dispersion relations for these waves, pass bands, and their dependence on the geometric parameters of the problem. The phenomenon is discovered of bifurcation of waveguide frequencies with respect to the parameter of the distance between the gratings that decreases from infinity. Some estimates are obtained for the parameters of frequency splitting or fusion in dependence on the distance between the simple blade gratings forming the double grating. We show that near a double sheet grating there always exist standing waves (in-phase oscillations in the neighboring fundamental cells of the group of translations) localized near the grating. By numerical-analytical methods, the dependences of the standing wave frequencies on the geometric parameters of the grating are determined. The mechanics is described of traveling and standing waves localized in the neighborhood of the double gratings.     

Optimal Shape Design of Blazed Diffraction Gratings

The problem of designing a periodic interface between two materials in such a way that time-harmonic waves diffracted from the interface have a specified far-field pattern is studied. A minimization problem for the interface is formulated, and it is shown that solutions of constrained bounded variation exist. The differentiability of the cost functional is then studied, with no restrictions on the smoothness of the interface. Some computational issues are discussed, and finally the results of some numerical experiments are presented. Accepted 3 February 1998     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.     

Inverse scattering from an open arc

A Newton method is presented for the approximate solution of the inverse problem to determine the shape of a sound-soft or perfectly conducting arc from a knowledge of the far-field pattern for the scattering of time-harmonic plane waves. Fréchet differentiability with respect to the boundary is shown for the far-field operator, which for a fixed incident wave maps the boundary arc onto the far-field pattern of the scattered wave. For the sake of completeness, the first part of the paper gives a short outline on the corresponding direct problem via an integral equation method including the numerical solution.