Minnaert resonances for acoustic waves in bubbly media

Through the application of layer potential techniques and Gohberg–Sigal theory we derive an original formula for the Minnaert resonance frequencies of arbitrarily shaped bubbles. We also provide a mathematical justification for the monopole approximation of scattering of acoustic waves by bubbles at their Minnaert resonant frequency. Our results are complemented by several numerical examples which serve to validate our formula in two dimensions.     

Hydrodynamic action functional and absorption of electromagnetic waves in plasma

The path integration method is used to study the absorption of waves in plasma. The absorption is considered of longitudinal electromagnetic waves as a consequence of scattering on the transverse waves, which requires the introduction of an integration with respect to a supplementary field of the vector potential. The scattering on the transverse waves becomes the determining one for a one-component plasma in the long wavelength limit. The hydrodynamic action functional is constructed for a magnetized electron-ion plasma by the method of successive integration, first with respect to “fast” fields and, next, with respect to “slow” fields. The absorption of the longitudinal waves is computed with the aid of this functional. The domains of frequencies of the order of the plasma frequency and of the order of the cyclotron frequency are considered. In comparison with the case of a free plasma, here the Coulomb logarithm is varied.     

The effect of evanescent wave modes on scattering and near-trapping

The effect of evanescent modes on the scattering and near-trappingof small-amplitude waves over axisymmetric topography is investigated.A two-stage numerical implementation, which facilitates an examinationof the resonant frequencies associated with near-trapping, isdeveloped. This is achieved in the latter stage of the procedureby dealing with the progressing and decaying waves separately. Numerical results are given for a selection of bed shapes, andit is found that the evanescent waves can have a significanteffect on scattering. Numerical evidence is found that, forthe selection of bed profiles considered, no new resonant frequenciesare introduced by the inclusion of the decaying wave components,but that the inclusion of these decaying waves does improveprevious approximations to resonant frequencies.     

Reconstruction of a defect in an open waveguide

The paper is concerned with the reconstruction of a defect in the core of a two-dimensional open waveguide from the scattering data. Since only a finite numbers of modes can propagate without attenuation inside the core, the problem is similar to the one-dimensional inverse medium problem. In particular, the inverse problem suffers from a lack of uniqueness and is known to be severely ill-posed. To overcome these difficulties, we consider multi-frequency scattering data. The uniqueness of solution to the inverse problem is established from the far field scattering information over an interval of low frequencies.     

Scattering frequencies for the wave equation with a potential term

The existence of an infinite sequence of scattering frequencies for the equation □u + qu = 0 is established, where q is a real valued potential which may assume negative values. This result generalizes some of the results obtained by Lax and Phillips in Comm. Pure Appl. Math.22 (1969), 737–787.     

Scattering Frequencies and a Class of Perturbed Systems of Elastic Waves

We consider the system of elastic waves in three dimensions under the presence of an impurity of the medium which we represent by a real-valued function q(x) (or q(x,t)). The medium is assumed to be isotropic and occupies the whole space Ω = ℝ³. We study the location of the scattering frequencies associated with such phenomenon. We conclude that there is a large region on the complex plane which is free of scattering frequencies. In the remaining region they are discrete provided that q satisfies suitable assumptions concerning its behaviour at infinity.     

Scattering of elastic waves through a heterogeneous medium

In this paper we study the elastic wave diffraction in R ³ through a heterogeneous medium, with periodic structure, which occupies a bounded domain. We show that, as the period tends to zero, the solution tends, in some sense, to the solution corresponding to the diffraction by an obstacle made of the classical “homogenized medium”. An analogous result is also proved for the scattering frequencies and the associated scattering functions.     

An Explicit Example of a Local and a Nonlocal Potential Whose Hamiltonians Are Unitarily Equivalent and Whose Scattering Operators Are Identical

An explicit example is given for the one-dimensional Schrodinger equation in which two unitarily equivalent Hamiltonians, one with a local scattering potential and the other with a nonlocal scattering potential, have the same scattering operator and bound-state measure. The result has obvious implications for the inverse scattering problem. The unitary operator which maps one Hamiltonian to the other is of interest because it is expressed as the product of two operators, neither of which has an inverse.     

Uniqueness For An Inverse Boundary Value Problem For Dirac Operators

The uniqueness of both the inverse boundary value problem and inverse scattering problem for Dirac equation with a magnetic potential and an electrical potential are proved. Also, a relation between the Dirichlet to Dirichlet map for the inverse boundary value problem and the scattering amplitude for the inverse scattering problem is given     

Scattering of waves by thin periodic layers at high frequencies using the on-surface radiation condition method

This paper deals with the scattering problem of acoustic andelectromagnetic waves by thin periodic layers at high frequencies.The on-surface radiation condition method is employed to simplifythe numerical computations of reflected and transmitted waves.     

An ultra-weak method for acoustic fluid–solid interaction

We introduce the ultra-weak variational formulation (UWVF) for fluid–solid vibration problems. In particular, we consider the scattering of time-harmonic acoustic pressure waves from solid, elastic objects. The problem is modeled using a coupled system of the Helmholtz and Navier equations. The transmission conditions on the fluid–solid interface are represented in an impedance-type form after which we can employ the well known ultra-weak formulations for the Helmholtz and Navier equations. The UWVF approximation for both equations is computed using a superposition of propagating plane waves. A condition number based criterion is used to define the plane wave basis dimension for each element. As a model problem we investigate the scattering of sound from an infinite elastic cylinder immersed in a fluid. A comparison of the UWVF approximation with the analytical solution shows that the method provides a means for solving wave problems on relatively coarse meshes. However, particular care is needed when the method is used for problems at frequencies near the resonance frequencies of the fluid–solid system.     

Multipole Pseudopotential Method for Some Problems in Quantum Scattering

We construct a multipole pseudopotential that allows reconstructing the wave function in some problems in quantum scattering theory that are described by a nonlinear wave equation with a potential of compact support and a nonlocal boundary condition given in terms of the scattering amplitude. We establish that the structure of the wave function is completely defined by the scattering amplitude and is independent of the choice of the potential.     

Distribution of poles for scattering on the real line

The density of scattering poles is shown to be proportional to the length of the convex hull of the support of the potential. In the case of a potential with finite singularities at the endpoints of the support, asymptotic formulae for the poles are given, while in the C₀^∞ case, an example of a potential with infinitely many scattering poles on iR is constructed. The scattering amplitude of a compactly supported potential is also characterized.     

A uniqueness theorem in inverse scattering of elastic waves

The scattering of plane elastic waves in an isotropic inhomogeneousmedium is considered. The existence and uniqueness of the directproblem is stated, and a reciprocity principle for the far fieldsof the scattered waves formulated. Finally, it is proved thatthe knowledge of the far-field patterns for a bounded sequenceof different frequencies and certain sets of incoming planewaves uniquely determines the density of the medium.     

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.     

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

A direct and inverse scattering theory on the full line is developed for a class of first-order selfadjoint 2n×2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.Dedicated to Israel Gohberg on the Occasion of his 70^th Birthday     

Landscape of wave focusing and localization at low frequencies

High-contrast scattering problems are special among classical wave systems as they allow for strong wave focusing and localization at low frequencies. We use an asymptotic framework to develop a landscape theory for high-contrast systems that resonate in a subwavelength regime. Our from-first-principles asymptotic analysis yields a characterization in terms of the generalized capacitance matrix, giving a discrete approximation of the three-dimensional scattering problem. We develop landscape theory for the generalized capacitance matrix and use it to predict the positions of three-dimensional wave focusing and localization in random and non-periodic systems of subwavelength resonators.     

Two-dimensional nuclear Coulomb scattering of a slow quantum particle

We study two-dimensional scattering of a quantum particle by the superposition of a Coulomb potential and a central short-range potential. We analyze the low-energy asymptotic behavior of all radial wave functions, partial phases, and scattering cross sections of such a particle. We propose two approaches for evaluating the scattering length and the effective radius.     

Variational approach to scattering of plane elastic waves by diffraction gratings

The scattering of a time‐harmonic plane elastic wave by a two‐dimensional periodic structure is studied. The grating profile is given by a Lipschitz curve on which the displacement vanishes. Using a variational formulation in a bounded periodic cell involving a nonlocal boundary operator, existence of solutions in quasiperiodic Sobolev spaces is investigated by establishing the Fredholmness of the operator generated by the corresponding sesquilinear form. Moreover, by a Rellich identity, uniqueness is proved under the assumption that the grating profile is given by a Lipschitz graph. The direct scattering problem for transmission gratings is also investigated. In this case, uniqueness is proved except for a discrete set of frequencies. Copyright © 2010 John Wiley & Sons, Ltd.