Complex-valued Wave Research Articles

The Limiting Absorption Principle and a Radiation Condition for the Scattering by a Periodic Layer

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These nonzero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply nonuniqueness of any solution to the scattering problem. In this paper, we consider a medium that is defined in the upper two-dimensional half-space by a penetrable and periodic contrast. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. Our method of proof seems to be new: By the Floquet--Bloch transform we first reduce the scattering problem to a finite-dimensional one that is set in the linear space spanned by all surface waves. In this space, we then compute explicitly which modes propagate along the periodic structure to the left or to the right. This finally yields a representation for our limiting absorption solution which leads to a proper extension of the well known upward propagating radiation condition. Finally, we prove the uniqueness of a solution under this radiation condition.

Interaction of Lamb waves with rivet hole cracks from multiple directions

This paper presents the interaction of Lamb waves with rivet hole cracks from multiple directions of incident using the finite element approach. Lamb waves undergo scattering and mode conversion after interacting with the damage. Shear horizontal waves appear in the scattered waves because of the mode conversion. Instead of analyzing the whole large structure, the local damage area is analyzed using finite element analyses and analytical formulation is used to analyze the whole structure. The scatter fields are described in terms of wave damage interaction coefficients that involve scattering and mode conversion of Lamb waves. Lamb wave mode (S0 and A0) hit the damage from multiple directions and corresponding wave damage interaction coefficients are obtained around the damage. Harmonic analysis has been performed over the fundamental frequency domain and “scatter cubes” of complex-valued wave damage interaction coefficients are formed. The scatter cube provides the information of relative amplitude and phase of scattered waves around the damage that can be used for designing the sensor installation. An application based on real time domain signal has been illustrated for the problem of multiple-rivet-hole cracks using the scatter cubes with the analytical framework.

Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding

This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwell’s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions.

Visco-elastic effects on wave dispersion in three-phase acoustic metamaterials

This paper studies the wave attenuation performance of dissipative solid acoustic metamaterials (AMMs) with local resonators possessing subwavelength band gaps. The metamaterial is composed of dense rubber-coated inclusions of a circular shape embedded periodically in a matrix medium. Visco-elastic material losses present in a matrix and/or resonator coating are introduced by either the Kelvin–Voigt or generalized Maxwell models. Numerical solutions are obtained in the frequency domain by means of k(ω)-approach combined with the finite element method. Spatially attenuating waves are described by real frequencies ω and complex-valued wave vectors k. Complete 3D band structure diagrams including complex-valued pass bands are evaluated for the undamped linear elastic and several visco-elastic AMM cases. The changes in the band diagrams due to the visco-elasticity are discussed in detail; the comparison between the two visco-elastic models representing artificial (Kelvin–Voigt model) and experimentally characterized (generalized Maxwell model) damping is performed. The interpretation of the results is facilitated by using attenuation and transmission spectra. Two mechanisms of the energy absorption, i.e. due to the resonance of the inclusions and dissipative effects in the materials, are discussed separately.It is found that the visco-elastic damping of the matrix material decreases the attenuation performance of AMMs within band gaps; however, if the matrix material is slightly damped, it can be modeled as linear elastic without the loss of accuracy given the resonator coating is dissipative. This study also demonstrates that visco-elastic losses properly introduced in the resonator coating improve the attenuation bandwidth of AMMs although the attenuation on the resonance peaks is reduced.

Lens-free coherent modulation imaging with collimated illumination

We propose a lens-free coherent modulation imaging (CMI) method for reconstructing a general complex-valued wave field from a single frame of a diffraction pattern. A numerical Fourier transform is introduced in the iterative reconstruction process to replace the lens or zone plate used in the current CMI technique to adopt the constraint on the Fourier components of the exit wave field of the sample. While the complexity of the experimental setup is remarkably reduced by replacing the zone plate and additional accessories with the numerical processing, the energy fluence loss induced by the undesired diffraction orders of the zone plate can be also avoided. The feasibility of the proposed technique is verified experimentally with visible light.

A wave equation interpolating between classical and quantum mechanics**We dedicate this paper to the memory of Richard Lewis Arnowitt—a pioneer of many-body theory, a path finder at the interface of gravity and quantum mechanics, and a true leader in non-relativistic and relativistic quantum field theory.

We derive a ‘master’ wave equation for a family of complex-valued waves whose phase dynamics is dictated by the Hamilton–Jacobi equation for the classical action . For a special choice of the dynamics of the amplitude R which eliminates all remnants of classical mechanics associated with our wave equation reduces to the Schrödinger equation. In this case the amplitude satisfies a Schrödinger equation analogous to that of a charged particle in an electromagnetic field where the roles of the scalar and the vector potentials are played by the classical energy and the momentum, respectively. In general this amplitude is complex and thereby creates in addition to the classical phase a quantum phase. Classical statistical mechanics, as described by a classical matter wave, follows from our wave equation when we choose the dynamics of the amplitude such that it remains real for all times. Our analysis shows that classical and quantum matter waves are distinguished by two different choices of the dynamics of their amplitudes rather than two values of Planck’s constant.

The first solution of a long standing problem: Reconstruction formula for a 3-d phaseless inverse scattering problem for the Schrödinger equation

Abstract A long standing problem is completely solved here for the first time. This problem was posed by K. Chadan and P. C. Sabatier in their classical book “Inverse Problems in Quantum Scattering Theory”, Springer, New York, 1977. The inverse scattering problem of the reconstruction of the unknown potential with a compact support in the three-dimensional Schrödinger equation is considered. Only the modulus of the scattering complex-valued wave field is known, whereas the phase is unknown. It is shown that the unknown potential can be reconstructed via the inverse Radon transform. This solution has potential applications in imaging of nanostructures.

Blow-up profile for the complex-valued semilinear wave equation

In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power nonlinearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real case. In particular, there is one more neutral-direction in our problem, which we control thanks to a modulation technique.

Hybridization of Off-Axis and In-line High-Resolution Electron Holography

Reconstruction of the complex-valued wave function that gives rise to the probability density of fast electrons in transmission electron microscopy (TEM) seems, at first glance, to be a solved problem. However, it is still challenging when considered more closely. In conventional TEM experiments, only the intensity (i.e., the square of the amplitude) of the wave function can be measured. Direct information about the phase of the wave, which carries information about electrostatic and magnetic fields that the electron has passed through, is lost during detection. In order to recover the phase information, it is necessary to interfere the electron wave function with a reference wave, in order to create an interference pattern. Denis Gabor introduced an approach that could be used to solve this problem 66 years ago [1]. In Gabor s original setup, which is the pioneering scheme for in-line holography, the wave that has been scattered by the specimen (the object wave) interferes with a reference wave propagated along the same axis. Using laser light, Leith and Upatnieks [2] showed that separation of the axes of propagation of the reference and object waves could be used to solve the twin-image problem. Mollenstedt later translated this idea back to electron microscopy, creating the field of off-axis electron holography [3,4].

Determination of the effective mechanical properties of inclusionary materials using bulk elastic waves

For many years, the study of material with locally-resonant inclusions has known a great interest. Due to their attractive macroscopic acoustic properties, which do not exist in natural materials, many applications governed by the contrast between the constituent properties can be achieved: enhanced absorption, flat perfect lens, wave trapping... Such structures containing inclusions are usually modelled as effective homogeneous materials with complex-valued and frequency-dependent effective wave number and mass density. In this context, we derived a method for the simultaneous determination of both parameters using an immersed sample. First, its reliability is demonstrated using noisy synthesised temporal-waveform. Finally, the characterisation method is applied to experimental waveforms measured on a standard glass panel.

Iterative projection approach for phase retrieval of semi-sparse wave field

Abstract In the paper, we consider the problem of two-dimensional (2D) phase retrieval, which recovers a 2D complex-valued wave field from magnitudes of both wave field and its Fourier transform. Due to the absence of the phase measurements, prior information on wave field is needed in order to recover phase, which is feasible when the phases of the wave field are sparse. In this paper, we improve the phase retrieval accuracy by incorporating phase sparse constraint of wave field. As a sequel to previous iterative projection approaches, iterative projection approaches with phase sparse constraint are realized based on ‘soft thresholding’. It has superior performances in terms of convergence, residual error, noise stability, and suitability in large-scale phase retrieval problems. Numerical experiments illustrate that the proposed approach outperforms existing iterative projection approaches.

Spaces of electromagnetic and mechanical constitutive parameters for dissipative media with either positive or negative index

Propagation of electromagnetic and acoustic plane waves in dissipative isotropic homogeneous media is described in terms of the Poynting vector and of the complex-valued wave vector. The negative sign of the refractive index, which is explained by the presence of backward bulk waves, is then directly related to the phase angle of the complex-valued wavenumber. Attention is focused on an alternative description dealing with the complex-valued dynamic material parameters: the relative permittivity ϵ and the relative permeability μ for the electromagnetic wave motion, and the bulk modulus κ and the mass density ρ for the acoustic wave motion. The 2D spaces of material parameters (ϵ,μ) and (κ,ρ) are found to be split into regions characterized by their abilities both to induce wave attenuation and to exhibit opposite directions between the energy flow and the direction of the plane wave propagation. Finally, the relevance of such representations is illustrated by superimposing experimentally retrieved and simulated constitutive parameters of media supporting both forward and backward wave motions.

Complex-valued travelling wave solutions to the Casimir equation for the Ito system via the first integral method

The Ito system was previously been shown to admit a reduction to a single nonlinear Casimir equation. In the present paper, we reduce this nonlinear partial differential equation into an ordinary differential equation governing a travelling wave solution. The ordinary differential equation takes the form of a second order nonlinear equation, and the form of this nonlinearity is a rational function. As such, the nonlinearity can become singular. This makes the problem interesting and somewhat challenging to solve for standard methods. Therefore, we make use of the first integral method in order to obtain exact solutions for this equation of second order and thus obtain exact solutions to the Casimir equation for the Ito system. The first integral method actually allows one to account for multiple solutions, and we demonstrate that there indeed can exist multiple complex-valued solutions for this Casimir equation. The results demonstrate the utility of the first integral method, suggesting that the approach can be used to solve many nonlinear differential equations that may admit multiple solutions.

Fast and Accurate Evaluation of Nonlocal Coulomb and Dipole-Dipole Interactions via the Nonuniform FFT

We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\mathbf{x})$ and a density function $\rho(\mathbf{x})=|\psi(\mathbf{x})|^2$ for some complex-valued wave function $\psi(\mathbf{x})$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\widehat{U}(\mathbf{k})$ has a singularity and/or $\widehat{\rho}(\mathbf{k})\ne0$ at the origin $\mathbf{k}={\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\mathbf{k}$ which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in $\hat{U}(\mathbf{k})$ at the origin is canceled so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function $(\rho\mathbf{x})$ is smooth and decays sufficiently quickly as $|\mathbf{x}| \rightarrow \infty$. More precisely, the calculation requires $O(N\log N)$ operations, where $N$ is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.

The Ginzburg--Landau Order Parameter near the Second Critical Field

In the Ginzburg--Landau theory of superconductivity, the density and location of the superconducting electrons are measured by a complex-valued wave function, the order parameter. In this paper, when the intensity of the applied magnetic field is close to the second critical field, and when the order parameter minimizes the Ginzburg--Landau functional defined over a two-dimensional domain, the leading order approximation of its $L^2$-norm in “small” squares is given as the Ginzburg--Landau parameter tends to infinity.

Optimal design of composite plates based on wave propagation characteristics

The present work is concerned with the design of composite plates that are optimized with respect to their wave propagation characteristics. For example, one may wish to design a composite plate such that the attenuation of a dominant wave is maximized. The optimization proposed here considers a composite plate with a specified number of layers. The material properties and thicknesses of the layers are considered as optimization parameters and these parameters are typically constrained in some way. For each choice of the optimization parameters, complex wave numbers and their associated wave shapes are calculated by using the semi-analytical finite element method developed by others. This method uses a finite element discretization in the thickness coordinate and a propagating wave solution in the lateral coordinates, resulting in a quadratic eigenvalue problem for the complex-valued wave number of each wave that the plate supports. One then computes a scalar cost function based on the complex wave numbers and their associated wave shapes, and proceeds by adjusting the optimization parameters to minimize the cost function. This presentation will describe the computational aspects of the optimization approach and will illustrate its potential by example. [Work supported by ONR under grant N000141210428.]

Cognition in Hilbert space

Use of quantum probability as a top-down model of cognition will be enhanced by consideration of the underlying complex-valued wave function, which allows a better account of interference effects and of the structure of learned and ad hoc question operators. Furthermore, the treatment of incompatible questions can be made more quantitative by analyzing them as non-commutative operators.

Analysis and optimization of constrained layer damping treatments using a semi-analytical finite element method

The present work investigates the physics of constrained layer damping treatments for plates and beams by a semi-analytical finite element method and presents applications of the method to the optimization of damping treatments. The method uses finite element discretizations in the thickness coordinate and propagating wave solutions in the remaining coordinates and therefore provides more generality and accuracy than existing analytical approximations. The resulting dispersion equation is solved for complex-valued wave numbers at each frequency of interest. By choosing sufficiently fine discretizations in the thickness coordinate, the method gives accurate estimates of wave numbers. The numerical implementation of the method is an efficient yet general tool for optimizing damping treatments with respect to material properties and dimensions. It explicitly allows for the possibility of analyzing structures with several layers where the material of each layer may be isotropic or orthotropic. Examples illustrate the numerical efficiency of the implementation and use this efficiency to provide optimizations of constrained layer damping treatments.

When holography meets coherent diffraction imaging

The phase problem is inherent to crystallographic, astronomical and optical imaging where only the intensity of the scattered signal is detected and the phase information is lost and must somehow be recovered to reconstruct the object's structure. Modern imaging techniques at the molecular scale rely on utilizing novel coherent light sources like X-ray free electron lasers for the ultimate goal of visualizing such objects as individual biomolecules rather than crystals. Here, unlike in the case of crystals where structures can be solved by model building and phase refinement, the phase distribution of the wave scattered by an individual molecule must directly be recovered. There are two well-known solutions to the phase problem: holography and coherent diffraction imaging (CDI). Both techniques have their pros and cons. In holography, the reconstruction of the scattered complex-valued object wave is directly provided by a well-defined reference wave that must cover the entire detector area which often is an experimental challenge. CDI provides the highest possible, only wavelength limited, resolution, but the phase recovery is an iterative process which requires some pre-defined information about the object and whose outcome is not always uniquely-defined. Moreover, the diffraction patterns must be recorded under oversampling conditions, a pre-requisite to be able to solve the phase problem. Here, we report how holography and CDI can be merged into one superior technique: holographic coherent diffraction imaging (HCDI). An inline hologram can be recorded by employing a modified CDI experimental scheme. We demonstrate that the amplitude of the Fourier transform of an inline hologram is related to the complex-valued visibility, thus providing information on both, the amplitude and the phase of the scattered wave in the plane of the diffraction pattern. With the phase information available, the condition of oversampling the diffraction patterns can be relaxed, and the phase problem can be solved in a fast and unambiguous manner. We demonstrate the reconstruction of various diffraction patterns of objects recorded with visible light as well as with low-energy electrons. Although we have demonstrated our HCDI method using laser light and low-energy electrons, it can also be applied to any other coherent radiation such as X-rays or high-energy electrons.

Wave propagation analysis of carbon nano-tube conveying fluid including slip boundary condition and strain/inertial gradient theory

This article addresses wave propagation in carbon nano-tube (CNT) conveying fluid. CNT structure is modeled by using size-dependent strain/inertia gradient theory of continuum mechanics, CNT wall-fluid flow interaction by slip boundary condition and Knudsen number (Kn). Complex-valued wave dispersion relations and corresponding characteristic equations are derived. Fluid viscosity, gyroscopic inertial force, flow velocity, wave number, wave frequency, and decaying ratio are among parameters that their variations are discussed and some remarkable results are drawn. It was observed Kn could impress complex wave frequencies at both lower and higher ranges of wave numbers, while small-size had impression at higher range.