Fixed Wave Number Research Articles

An Inverse Transmission Scattering Problem and the Enclosure Method

We consider an inverse scattering problem in two-dimensions for a penetrable polygonal obstacle having different density from the back ground medium, however, the speed of sound is constant in the whole space. Using a single set of the Cauchy data of the response for a single incident plane wave with a fixed wave number on a circle surrounding the obstacle, we give an extraction formula of the convex hull of the obstacle. An algorithm based on the formula is also described.

Noisy spectra, long correlations, and intermittency in wave turbulence.

We study the k-space fluctuations of the wave action about its mean spectrum in the turbulence of dispersive waves. We use a minimal model based on the random phase approximation (RPA) and derive evolution equations for the arbitrary-order one-point moments of the wave intensity in the wave-number space. The first equation in this series is the familiar kinetic equation for the mean wave-action spectrum, whereas the second and higher equations describe the fluctuations about this mean spectrum. The fluctuations exhibit a nontrivial dynamics if some long coordinate-space correlations are present in the system, as it is the case in typical numerical and laboratory experiments. Without such long-range correlations, the fluctuations are trivially fixed at their Gaussian values and cannot evolve even if the wave field itself is non-Gaussian in the coordinate space. Unlike the previous approaches based on smooth initial k-space cumulants, the RPA model works even for extreme cases where the k-space fluctuations are absent or very large and intermittent. We show that any initial non-Gaussianity at small amplitudes propagates without change toward the high amplitudes at each fixed wave number. At each fixed amplitude, however, the probability distribution function becomes Gaussian at large time.

Zero-energy solutions and vortices in Schrödinger equations

All two-dimensional Schr\"{o}dinger equations with symmetric potentials \break $(V_a(\rho)=-a^2g_a \rho ^{2(a-1)/2} {with} \rho=\sqrt{x^2+y^2} {and} a\not=0)$ is shown to have zero energy states contained in conjugate spaces of Gel'fand triplets. For the zero energy eigenvalue the equations for all $a$ are reduced to the same equation representing two-dimensional free motions in the constant potential $V_a=-g_a$ in terms of the conformal mappings of $\zeta_a=z^a$ with $z=x+iy$. Namely, the zero energy eigenstates are described by the plane waves with the fixed wave numbers $k_a=\sqrt{mg_a}/\hbar$ in the mapped spaces. All the zero energy states are infinitely degenerate as same as the case of the parabolic potential barrier (PPB) shown in ref. \cite{sk4}. Following hydrodynamical arguments, we see that such states describe stationary flows round the origin, which are represented by the complex velocity potentials $W=p_a z^a$, ($p_a$ being a complex number) and their linear combinations create almost arbitrary vortex patterns. Examples of the vortex patterns in constant potntials and PPB are presented.

Wrinkling of a compressed elastic film on a viscous layer

A compressively strained elastic film bonded to a viscous layer can form wrinkles. The present study provides a theoretical model for the wrinkling process. The elastic film is modeled with the nonlinear theory of a thin plate subject to in-plane and out-of-plane loads. The flow of the viscous layer is modeled with the theory of lubrication. The interface between the elastic film and the viscous layer is assumed to be perfect with no slipping or debonding. A set of partial differential equations evolves the deflection and the in-plane displacements as functions of time. A linear stability analysis identifies the critical wave number, below which the elastic film is unstable and the wrinkles can grow. For any fixed wave number less than the critical wave number, the wrinkles reach a kinetically constrained equilibrium configuration, in which the stress is partially relaxed in the elastic film and the viscous layer stops flowing. Numerical simulations reveal rich dynamics of the system with many unstable equilibrium configurations.

Ulysses observations of magnetohydrodynamic turbulence in corotating interaction regions

During its mission around the poles of the Sun, the Ulysses spacecraft encountered a number of well‐developed corotating interaction regions (CIRs) at midlatitudes. In this paper we discuss the character of magnetohydrodynamic (MHD) turbulence within defined regions of these CIRs. We examine magnetic field power spectra at a fixed wave number, and the normalized cross helicity, in order to describe solar wind fluctuations. We find that each CIR has a complicated turbulent signature. Magnetic field power spectra show evidence for the turbulent mixing of plasma at large distances from the Sun, and there are possible signs of instabilities arising at stream interfaces. Statistical analyses of our data set show that there does not appear to be any clear dependence of turbulence in the CIRs upon heliographic latitude but that there is a clear trend in magnetic field power and Alfvénicity with CIR structure. We also find that the variation of the solar cycle during Ulysses observations of these CIRs has a clear effect upon the turbulent character of the solar wind and thus provides further evidence that the source of the majority of turbulent fluctuations in the solar wind is the Sun.

A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations

Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as asmoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to the number of unknowns.

On the uniqueness of the shape of a penetrable, anisotropic obstacle

We consider the scattering of time-harmonic waves in an isotropic, homogeneous medium which contains a penetrable, anisotropic, inhomogeneous obstacle. After reviewing existence, uniqueness and regularity of a (variational) solution we prove that a knowledge of the scattered near fields for all incident plane waves at a fixed wave number uniquely determines the obstacle provided the matrix characterizing the anisotropic medium satisfies a suitable condition. To this end we examine the corresponding interior transmission problem and prove that it is a compact perturbation of an interior transmission problem which is well-posed.

Electromagnetic Scattering Problems in Chiral Media: A Review

ABSTRACT Time-harmonic electromagnetic waves are scattered by a homogeneous chiral obstacle. It is shown that the corresponding transmission problem has a unique solution. This transmission problem is reduced to a pair of uniquely solvable coupled integral equations over the interface between the obstacle and the surrounding medium. Some scattering relations are established, and the spectrum of the far-field operator is studied and related to that of the T-matrix. If the far-field pattern is known for all incident waves with a fixed wave number, uniqueness of the obstacle and its material parameters is established.

Uniqueness of the inverse scattering problem for a chiral obstacle

This paper deals with the inverse scattering problem to determine the shape of a chiral obstacle immersed in achiral free space from the knowledge of the scattered wave. A variational equation is established for the corresponding direct scattering problem. Then, under appropriate assumptions, it is shown that the chiral obstacle is uniquely determined by the far field patterns of all incident waves with a fixed wave number.

Inverse Obstacle Scattering Using Reduced Data

The classical result of Schiffer in acoustic scattering is that a knowledge of the far field pattern for all observation directions and all incident directions at a fixed wave number k uniquely determines the sound-soft scattering obstacle D. This is widely believed to be far greater information than is required for uniqueness, and more recent results have been obtained that considerably weaken the amount of data needed, provided certain restrictions are placed on the scatterer. Most of this work has concentrated on reducing the number of incident waves required for a unique determination. This paper takes another approach and seeks to determine sufficient information to recover the obstacle from measurements of the far field at isolated points. Our approach will be constructive and some numerical reconstructions will be presented.

An exactly solvable model for the large polaron

We present an exactly diagonalizable model Hamiltonian for the large polaron derived by analyzing the variational ansatz by Haga-Larsen (HL) for the Frohlich Hamiltonian. The lowest energy eigenvalue of the model Hamiltonian for fixed wave numbers reproduces the energy of the variational ansatz by Haga-Larsen and is, therefore, an upper bound with respect to the corresponding energy eigenvalue of the Frohlich Hamiltonian. This is valid for any momentum which is proven by extending the Haga-Larsen approach. Furthermore, since all integrations can be performed analytically, the model Hamiltonian is easily tractable. The energy eigenvalue spectrum of the model Hamiltonian is studied below and above the phonon-emission threshold. The quality of the model Hamiltonian is determined by the variational ansatz of Haga and Larsen. Incorporating an improved energy-momentum relation, a generalized model Hamiltonian is derived possessing a larger validity range with respect to the coupling strength. Furthermore, a second exactly diagonalizable model Hamiltonian based on improved Wigner-Brillouin perturbation theory due to Warmenbol, Peeters, and Devreese (WPD) is presented. It is briefly demonstrated that one is able to construct all mentioned model Hamiltonians also in the 2D polaron problem. In contrast to the 3D case, where the HL-type model Hamiltonian possesses the higher quality for any momentum, in the 2D case, it works well only for small momenta. For large momenta, only the WPD-type model Hamiltonian describes the energy-momentum relation correctly. We demonstrate the usefulness of the model Hamiltonian concept by exactly calculating the one-electron Green’s function for all mentioned model Hamiltonians and comment why significant advantages of the model Hamilton concept for the treating of low-dimensional systems (planar semiconducting quantum-well structures) can be expected.

The Effect of Thermal Fluctuation on Spectral Function for the Tomonaga-Luttinger Model

We examine the spectral function of the single electron Green function at finite temperatures for the Tomonaga-Luttinger model which consists of the mutual interaction with only the forward scattering. The spectral weight, which is calculated as a function of the frequency with the fixed wave number, shows that several peaks originating in the excitation spectra of charge and spin fluctuations vary into a single peak by the increase of temperature.

Azimuthal and depth-dependent transmission loss caused by shelf internal waves

The amplitude of an acoustic field propagating through a stratified continental shelf water column in the presence of internal wave packets is shown to be azimuthally and depth dependent when referenced to the propagation vector of the internal wave packet. Internal wave (IW) packets, which propagate across the continental shelf, are known to be regularly generated each tidal cycle at the shelf break edge in stratified water column conditions. They are dispersive in nature and change in physical structure as they propagate away from the generation site. Their wave directional spectrum is highly peaked. Consequently, the wave-number spectrum of the continental shelf sound velocity field is highly directional and will be shown to cause azimuthal dependency in the properties of the acoustic field as a function of range and bearing from the source. Preliminary results of numerical solutions to the parabolic equation approximation of the Helmholtz equation are presented to substantiate the above assertions, in particular for propagation through a uniform internal wave field of a fixed wave number and amplitude. [Work supported by the Office of Naval Research.]

The acoustical properties of fluid in a wall-driven elastic tube

The acoustics of a fluid in a long cylindrical tube is analyzed. The tube’s wall is thin, elastic, and externally driven by a phased array of ring-shaped piezoelectric ceramic transducers. The rings influence the radial motion of the tube’s wall. Applications of the theory to an active wall impedance tube and a toroidal tube are considered. In the toroidal tube, a digitally controlled electronic system drives the transducer array with a fixed wave number and a continuously variable frequency. If the tube wall were rigid, the fluid pressure reaches a resonant maximum when the wave number and frequency of the driven transducer array matches that allowed by the dispersion relation of sound in the homogeneous fluid. Experimentally determining the fluid’s homogeneous sound speed is more difficult, however, when the wall is not rigid. The idea of using the tube’s external airborne sound to noninvasively measure a fluid’s sound speed is introduced. The theory predicts a minimum in the external sound. At this minimum, the wall is effectively rigid and the transducer array’s phase velocity equals the homogeneous fluid’s phase velocity. Preliminary experimental efforts to make an elastic wall rigid are briefly described.

Uniqueness of the Inverse Conductive Scattering Problem for Time-Harmonic Electromagnetic Waves

The scattering of time-harmonic electromagnetic waves by an obstacle covered with a thin layer of high conductivity leads to conductive boundary conditions. This paper is related to the inverse obstacle problem to determine the shape of such obstacles from the knowledge of scattered waves. A variational equation for this scattering problem is established. Then, under suitable assumptions, it is shown that the obstacle is uniquely determined by the far field patterns of all incident waves with a fixed wave number. This information is also sufficient for uniqueness of the other parameters of the inverse problem.

Self-diffusion in dilute quasi-two-dimensional hard sphere suspensions: Evanescent wave light scattering and video microscopy studies.

We report measurements of the diffusion coefficient of a dilute quasi-two-dimensional hard sphere colloidal suspension using two independent experimental techniques: evanescent wave dynamic light scattering (EWDLS) and digital video microscopy (DVM). The system studied consists of 1 \ensuremath{\mu}m sterically stabilized, uncharged, poly(methylmethacrylate) spheres in a very thin cell (\ensuremath{\sim}3 \ensuremath{\mu}m). In principle, EWDSL and DVM yield identical information about the system, albeit by different analyses. When both methods are used to study the same system it is possible to directly compare measurements of preaveraged statistical dynamical quantities with their microscopic counterparts. Our EWDLS measurements yield the effective diffusion coefficient as a function of wave number and the mean square particle displacement, for fixed wave number, as a function of time. The DVM measurements generate particle trajectories; Fourier decomposition of the trajectories yields the dynamic scattering function, which is found to be in quantitative agreement with the same function measured by EWDLS. Analysis of the observed intermediate scattering function indicates that, as predicted by Cichocki and Felderhof [J. Phys. Condens. Matter 6, 7287 (1994)], in this quasi-two-dimensional system the time dependence of the evolution of the effective diffusion coefficient from its short time value to its long time value has the form (lnt)/t. To our knowledge, these results are the first experimental verification of the predicted temporal evolution of the diffusion coefficient for Brownian motion in a quasi-two-dimensional liquid. \textcopyright{} 1996 The American Physical Society.

Enhancement of the Raman spectrum of optically levitated microspheres by seeded nanoparticles

Optical levitation experiments were performed with diethyl sebacate and diethyl hexylsebacate particles with a diameter range of 20–30 μm. The particles were seeded with polystyrene–latex particles with a diameter of 90 nm. Elastically and inelastically scattered light was recorded simultaneously. Although no change in the Mie scattering could be observed, the nonresonant part of the Raman spectrum was raised by a factor of 2. The C–H stretching region between 2800 and 3100 cm−1 was investigated, and the time dependence of the Raman intensity was measured at a fixed wave number of Δν = 2920 cm−1.

Dynamic critical phenomena in aqueous protein solutions.

An aqueous protein solution is a binary mixture of large particles (protein molecules) and small particles (water molecules). In contrast, a binary liquid mixture is a mixture of two different kinds of particles that are approximately the same size. Thus, a comparison of the critical behavior in these two systems allows one to determine the effect on critical behavior of a relative difference of size in the two kinds of particles in a binary mixture. The correlation length j, osmotic compressibility kT , and coexistence curve of aqueous protein solutions in the vicinity of the critical point for liquid-liquid phase separation have been reported recently [1‐3]. These experimental results established that the static critical properties of these solutions are consistent with the behavior of members of the static universality class of three-dimensional systems with short range interactions and scalar order parameters. This universality class includes binary liquid mixtures. In contrast to the equilibrium properties, the dynamic properties of aqueous protein solutions in the vicinity of the critical point have not been investigated as thoroughly. In this Letter, we report our use of quasielastic light scattering (QLS) to investigate the time decay of spontaneous concentration fluctuations, as a function of both wave number q of the fluctuations and temperature T, along the critical isochore of aqueous solutions of the bovine eye lens protein gII-crystallin. The only previous study of critical dynamics in a protein solution of which we are aware is the work of Ishimoto and Tanaka who made QLS measurements at a single fixed wave number of the fluctuations using a 19 channel digital correlator on aqueous solutions of lysozyme [4] along the critical isochore for sT 2 TcdyTc $ 5 3 10 23 (q j, 0.1), where Tc is the critical temperature. The present study is a significant experimental advance over the work of Ishimoto and Tanaka in that (i) we obtain data for fluctuations at up to 12 different wave numbers, (ii) we obtain data using a 144 channel digital correlator, and (iii) we obtain data much closer to the critical point [ sT 2 TcdyTc $ 1.56 3 10 24 and q j# 3.18]. The more thorough investigation of critical dynamics presented here has revealed new and unexpected behavior. Specifically, we find that the wave number and temperature dependence of the average rate of decay of the concentration fluctuations are consistent with the theory for critical dynamics in binary liquid mixtures [5] only if we allow both the background viscosity and the background contribution to the decay rate to be unusually large. Furthermore, in contrast to the behavior seen in binary liquid mixtures we find that the concentration fluctuations exhibit very significant deviation from exponential decay with time. Of course, these protein solutions may be viewed as

Rayleigh-Taylor instability in elastic liquids

In this paper the problem of the stability of an incompressible elastic fluid at rest above a free surface is considered by carrying out a linear stability analysis. In the first case considered the Maxwell model is used to include elastic effects but inertial effects are neglected. A characteristic equation for the growth rate is found showing a dimensionless critical wave number. However, for each fixed wave number in the region of instability, this model is found to predict an unbounded growth rate at a certain value of the Weissenberg number. It is then shown how the introduction of inertial terms smooths out this singularity as does the change to a model which introduces a second elastic parameter, the linear Jeffreys model. Asymptotic expansions are developed to show how the singularity is approached in both cases.

Transmission of obliquely incident surface waves through a narrow gap

This note is concerned with the transmission of a train of surface water waves obliquely incident on a thin plane vertical barrier with a narrow gap. Within the framework of the linearized theory of water waves, the problem is reduced to the solution of an integral equation which is solved approximately. The transmission and reflection co–efficients are also obtained approximately and represented graphically against the different angles of incidence for fixed wave numbers.