Function Of Wavenumber Research Articles

Investigation of probe–solvent interactions: color effects in optical line widths

We have measured the optical hole width as a function of wave number across the inhomogeneous lines for two aromatic probe molecules, namely thionin and BODIPY. Upon tuning the laser to the red edge of the inhomogeneous band the hole width of thionin decreases, whereas the hole width of BODIPY increases. The color effect is correlated with the solvent shift in the former case and anti-correlated in the latter case. We suggest possible scenarios to explain these observations and back up our reasoning by site dependent Stark-effect experiments.

Studies of step stiffnesses and relaxation on Pt(1 1 1), Pd(1 1 1) and Mo(0 1 1)

We describe investigations of equilibrium step structure on Pt(1 1 1), Pd(1 1 1) and Mo(0 1 1), using low energy electron microscopy. From capillary wave analyses we obtained fluctuation amplitudes and relaxation times for Fourier components of the step edge displacement as functions of wave number q. The τ q −1 at ∼ T m/2 vary quite accurately with wavevector q as q 3 and show that surface diffusion is responsible for the relaxation at these lower temperatures for all clean single crystal surfaces studied. The temperature dependence of the step stiffnesses is very weak in the intervals 1190–1520 K for Pt, 990–1300 K for Pd and 1350–1680 K for Mo.

Fourier-domain low-coherence interferometry for light-scattering spectroscopy.

We present a novel method for obtaining depth-resolved spectra for determining scatterer size through elastic-scattering properties. Depth resolution is achieved with a white-light source in a Michelson interferometer with the mixed signal and reference fields dispersed by a spectrograph. The spectrum is Fourier transformed to yield the axial spatial cross correlation between the signal and reference fields with near 1-microm depth resolution. Spectral information is obtained by windowing to yield the scattering amplitude as a function of wave number. The technique is demonstrated by determination of the size of polystyrene microspheres in a subsurface layer with subwavelength accuracy. Application of the technique to probing the size of cell nuclei in living epithelial tissues is discussed.

Time-resolved small angle neutron scattering measurements of asphaltene nanoparticle aggregation kinetics in incompatible crude oil mixtures

We use time-resolved-small angle neutron scattering to study the kinetics of asphaltene nanoparticle aggregation in incompatible crude oil mixtures. We induce asphaltene aggregation by mixing asphaltene-rich Syrian crude oil (SACO) with a paraffinic British crude oil and observe the scattered neutron intensity, I, as a function of wave number, q, over times, t, ranging from twenty minutes to about a week. We observe a growth in I at low q as the nanoscale asphaltenes agglomerate into microscale aggregates and interpret this growth as an increase in surface scattering from the aggregates. We fit I(q,t) to an empirical model and measure the growth in the power-law exponent, α, associated with the low-q logarithmic slope of I(q). We define a time, τα, associated with the first appearance of the aggregates when α>3; τα increases as a function of the volume fraction, φm, of SACO in the mixture. The surface scattering intensity initially increases and then saturates at long times when the aggregate structures no longer evolve at the length scales we probe. Based on this saturation, we define a time scale, τI, which is larger than τα but has essentially the same dependence on φm. We interpret τα(φm) and τI(φm) in terms of a simple aggregation model based on diffusion-limited kinetics and a repulsive potential barrier that models the effective solvent quality.

A remedy for the ill-posedness of the one-dimensional two-fluid model

A one-dimensional two-fluid model with explicit momentum flux parameters is proposed as a remedy for the mathematical ill-posedness of the standard one-dimensional two-fluid model. By constructing a simplified two-phase flow using existing correlations for the distribution parameter C o and velocity profile, the momentum flux parameter is determined as a function of void fraction and density ratio for the whole range of flow regime. By performing a linear stability analysis for a two-phase flow in a channel described by the proposed model, an analytical expression for the growth factor is derived as a function of wave number, void fraction, drag coefficient, and relative velocity. From the analytical expression for the growth factor, the stability criteria are derived as a function of void faction, density ratio, and momentum flux parameters. It is shown that the two-phase flow in a channel described by the proposed model is stable in the whole range of flow regime.

Computationally determined existence and stability of transverse structures. I. Periodic optical patterns.

We present a Fourier-transform based, computer-assisted, technique to find the stationary solutions of a model describing a saturable absorber in a driven optical cavity. We illustrate the method by finding essentially exact hexagonal and roll solutions as a function of wave number and of the input pump. The method, which is widely applicable, also allows the determination of the domain of stability (Busse balloon) of the pattern, and sheds light on the mechanisms responsible for any instability. To show the usefulness of our numerical technique, we describe cracking and shrinking patches of patterns in a particular region of parameter space.

Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.

Some dynamic properties of a pre-stressed, nearly incompressible (rubber-like) elastic layer

The propagation of harmonic waves in a slightly compressible, finitely deformed, elastic plate is considered. The dispersion relation associated with harmonic waves propagating in a plate composed of such material with zero incremental surface traction is derived and asymptotic expansions, giving phase speed as a function of wave number, harmonic number and pre-stress, are obtained for high and low wave number. These expansions are shown to provide excellent approximation to the numerical solution and reveal a possible high wave number structure which differs significantly from the corresponding unstrained case. These expansions have important application to numerical inversion of transform solutions used to determine the dynamic transient response of a plate to surface, internal or edge loading. The paper concludes with a discussion of the longitudinal wavefront as the bulk modulus increases. It is shown numerically that the contribution of this wavefront to transient response decreases as the bulk modulus increases.

The effect of finite primary deformations on harmonic waves in layered elastic media

The problem of extensional wave propagation in a pre-stressed, incompressible, 4-ply symmetric layered structure is considered. The high wave number behaviour of the harmonics is shown to fall into one of four distinct cases. Each of these are examined in detail and appropriate asymptotic expansions, giving phase speed as a function of wave number, are obtained. These are shown to provide excellent agreement with the numerical solution. A surface wave front arising from the combined influence of all harmonics is observed numerically. Corresponding plots of the eigenfunctions confirm that this is indeed a surface wave with the behaviour associated with each harmonic remarkably sensitive to changes in wave number. This paper concludes with a comparison of extensional and flexural waves.

Calculation of spectra of turbulence in the energy-containing and inertial ranges

A statistical theory for the power law stage of freely decaying homogeneous and isotropic developed turbulence is proposed. Attention is focused on the velocity field statistics in the energy-containing and inertial scales. The kinetic energy spectrum $E(k,t)$ and energy transfer spectrum $T(k,t)$ are calculated as functions of wave number $k$ and decay time $t.$ The scaling properties of the spectra of the stationary model of the randomly stirred fluid have been chosen as the starting point for the approximate derivation of time-dependent spectra $E(k,t)$ and $T(k,t).$ The stationary model analyzed by means of the renormalization group and short-distance expansion methods has provided the spectra ${E(k)=C}_{K}{\ensuremath{\varepsilon}}^{2/3}{k}^{\ensuremath{-}5/3}\mathcal{F}(\mathrm{kl})$ [where ${C}_{K}$ is the Kolmogorov constant and $\mathcal{F}(\mathrm{kl})$ is a function] and $T(k)\ensuremath{\propto}\ensuremath{\varepsilon}{k}^{\ensuremath{-}1}{\ensuremath{\psi}}^{{\mathcal{F}}}(\mathrm{kl})$ [where ${\ensuremath{\psi}}^{{\mathcal{F}}}(\mathrm{kl})$ is functionally dependent on F]. The characteristic length scale of these spectra defined from the mean square root velocity $u$ and mean energy dissipation \ensuremath{\varepsilon} is the von K\'arm\'an scale ${l=u}^{3}/\ensuremath{\varepsilon}.$ We have assumed that $l,$ $u,$ and \ensuremath{\varepsilon} as well as $E(k)$ and $T(k)$ are no longer constants but unknown functions of $t.$ Scaling forms constructed in this way are consistent with the basic assumption of George's closure [W. M. George, Phys. Fluids A 4, 1492 (1992)]. Power decay laws for $\ensuremath{\varepsilon}(t),$ $l(t),$ $u(t)$ and the constituent integro-differential equation for the scaling function $F(\mathrm{kl}(t)){=E(k,t)/C}_{K}{\ensuremath{\varepsilon}}^{2/3}{k}^{\ensuremath{-}5/3}$ have been obtained using the equation of the spectral energy budget. The equation for $F(\mathrm{kl}(t))$ has been investigated numerically for the three-dimensional system with Saffman's invariant [P. G. Saffman, J. Fluid Mech. 27, 581 (1967); Phys. Fluids 10, 1349 (1967)]. The calculated longitudinal energy spectrum has been compared with the available experimental data.

On the existence of surface waves and the propagation of plate waves in pre-stressed fibre-reinforced composites

A theoretical investigation of the effect of pre-stress on wave propagation in a fibre-reinforced plate is carried out using a continuum material model. In the case of both extensional and flexural waves the high wave number phase speed limit of the fundamental mode of the dispersion relation is found to be the corresponding surface wave speed. Whenever such a wave exists it is shown to be unique. The existence of such waves is dependent on the underlying primary deformation; necessary and sufficient conditions for existence are established. For the harmonics, numerical calculations indicate the possibility of two distinct classes of material response. One of these is similar to classical linear isotropic theory. The other is quite different, with a shear wave front arising from the cumulative effect of the harmonics, the contribution of successive harmonics arising in adjacent wave number regimes, and flexural and extensional modes intersecting in both the moderate and high wave number regimes. This shear wave front is further illuminated through examination of the associated group velocity curves, in which corresponding distinct flat maxima are observed. High and low wave number representations of the dispersion relations are derived for each harmonic, giving phase speed as a function of wave number, pre-stress and the arbitrary tension and provide excellent agreement with some illustrative numerical solutions. The paper is concluded with a brief discussion of stability and bifurcation.

Some asymptotic expansions of the dispersion relation for an incompressible elastic plate

This paper is concerned with a general asymptotic analysis of the dispersion relation associated with waves propagating in a pre-stressed, incompressible elastic plate. In the high wave number limit it is well-known that, whenever a real surface wave speed exists, the fundamental modes of both symmetric and anti-symmetric motions tend to this surface wave speed, with all harmonics tending to a single shear wave speed limit. The character of the two dispersion curves in the moderate and high wave number regimes falls into one of two distinct cases, these being dependent on pre-stress. In the first case all the harmonics are monotonic decreasing functions and as such the asymptotic analysis in this case offers a modest generalisation of an earlier study, see Rogerson and Fu (Rogerson, G. A. and Fu, Y. B. (1995) An asymptotic analysis of the dispersion relation of a pre-stressed incompressible elastic plate. Acta Mechanica111, 59–77). In contrast, the second case is quite different in character with the passage to the high wave number limit accompanied by sinusoidal behaviour. This behaviour is fully elucidated by obtaining asymptotic expansions which give phase speed as a function of wave number, pre-stress and harmonic number, sinusoidal terms being found to occur at third order. Both these asymptotic expansions and ones obtained for high harmonic number are found to provide excellent agreement with numerical solutions for Varga materials in the appropriate regimes. It is envisaged that the expansions derived in this paper may well find important potential applications in the numerical inversion of the transform solution sometimes used in impact problems.

Wave‐number‐based assessment of the doubly asymptotic approximation.

The most widely implemented technique for modeling fluid–structure interaction effects associated with shock response is the doubly asymptotic approximation (DAA), which has been developed in a variety of versions. Because of a lack of analytical solutions for realistic geometries, prior validation efforts have not provided definitive guidelines regarding the accuracy of the method. The present work uses Nicholas–Vuillierme’s derivation [Numerical Techniques in Acoustic Radiation, edited by R. J. Bernhard and R. F. Keltie (American Society of Mechanical Engineers, New York, 1989), Vol. 6, pp. 7–13] of the frequency‐domain version of DAA as the basis for examining the accuracy and limitations of DAA for a slender hemi‐capped cylindrical shell. The development computes the wet surface impedance matrix relating surface pressure and velocity variables over a broad frequency band according to two DAA versions and the surface variational principle. The alternative wet surface impedances are used to predict the frequency‐domain structural response of the shell to a point force, after which alternative time‐domain responses are obtained from the FFT algorithm. Results for wave‐number amplitudes (azimuthal and meridional) are reported for moderate and large aspect ratios. An overview indicates that a higher‐order DAA version occasionally works extremely well, but that there is no consistent high/low trend as a function of wave number. [Work supported by the Office of Naval Research, Code 1222.]

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.

Effect of Pore Alignment on Surface Wave Propagation in a Liquid-Saturated Porous Layer over a Liquid-Saturated Porous Half-Space with Loosely Bonded Interface.

Dispersion of Rayleigh type surface waves have been studied in a liquid-saturated porous solid layer over a liquid-saturated porous half-space having a loosely bonded interface with the effect of pore alignment. Some special cases have been deduced by changing the depth of the layer. Frequency equation in the form of a determinant has been obtained. Dispersion curves giving phase and group velocities as a function of wave number have been plotted graphically for four different cases with a particular model.

Static structure factor and collective diffusion of globular proteins in concentrated aqueous solution

We report our measurement of the time average and the temporal autocorrelation function of the intensity of light scattered by the highly monomeric globular protein, bovine γII-crystallin, in aqueous solution as a function of wave number q, protein volume fraction φ, and temperature T. The time average intensity data is used to obtain the q→0 limit of the static structure factor S(φ,T), as a function of φ and T. We show that S(φ,T) may be well characterized by modeling the proteins as interacting through the Baxter adhesive hard sphere pair interaction potential. The temporal autocorrelation function data is used to determine the collective diffusion coefficient D̃(φ,T) of the proteins as a function of φ and T. We then obtain the experimental hydrodynamic factor H̃(φ,T)≡S(φ,T)[D̃(φ,T)/D0(T)], where D0(T) is the diffusion coefficient of the individual proteins in the φ→0 limit. We find that H̃ exhibits a different φ-dependence at low (φ≤0.016) and high (φ≳0.02) protein volume fractions. In the low φ domain our data for H̃ are consistent with the theoretical result for the collective diffusion in the q→0, t→0 limit. However, for φ≳0.02 we find a deviation from single exponential decay in the autocorrelation functions, and an unexpected, large change in the slope of the H̃ vs φ relation. This crossover at such low φ suggests the existence of a heretofore unappreciated length scale in the dynamics of colloid solutions. Clearly, further theoretical insights are required to understand the origin of this crossover behavior.

Breakdown of linear dynamics in phase-ordering kinetics.

Simulations and experiments have shown that the linear theory in phase-ordering dynamics fails first on short length scales rather than long length scales as suggested previously. By following the example of a simple coupled nonlinear system, we show that the linear theory breaks down first at the largest wave numbers due to the nonlinear slaving of the most stable Fourier modes to the larger amplitude unstable modes. The range of wave numbers in which the dynamics is nonlinear expands toward smaller wave numbers with time. We present numerical results verifying the mode slaving hypothesis and determine ${\mathit{t}}_{\mathrm{br}}$(k), the time at which the linear theory breaks down as a function of wave number k. (c) 1995 The American Physical Society

Electron–interface-phonon interaction and scattering in asymmetric semiconductor quantum-well structures

The interaction Fr\ohlich-like Hamiltonian between an electron and interface optical phonons in asymmetric single and step quantum-well (QW) structures is obtained and studied. We observe that the behaviors of electron--interface-phonon coupling functions as functions of wave number k have two anomalies. The first is that, in an asymmetric single QW and general step QW, the interaction between an electron and the lowest frequency interface mode is large in the vicinity of the Brillouin-zone center. The second anomaly is that in step QW's the interaction between an electron and the fifth interface mode has a maximum which relates to the inflection point in the dispersion (${\mathrm{\ensuremath{\omega}}}_{5\mathrm{\ensuremath{-}}}$k) plot. The physical origins of these two anomalies have been analyzed. The electron--interface-phonon-scattering rates for intrasubband and intersubband transitions in asymmetric single and step QW structures are also calculated and are given as functions of well width, step width, and step height. It is shown that the electron scattering depends strongly on the potential parameters, and the usual selection rules for these transitions break down in asymmetric heterostructures.

Time-resolved small-angle neutron scattering study of spinodal decomposition in deuterated and protonated polybutadiene blends. II. Q-dependence of Onsager kinetic coefficient

The spinodal decomposition (SD) of a critical mixture of deuterated and protonated polybutadiene of nearly equal chain lengths was investigated. This mixture has an upper critical solution temperature type phase diagram and the spinodal temperature at the critical point is 99.2 °C. Phase separation was induced by quenching a single-phase specimen at an initial temperature, Ti (=102.3, 123.9, and 171.6 °C), to a final temperature, Tf (=−7.5, 1.1, and 10.5 °C). The subsequent SD was followed by time-resolved small-angle neutron scattering. The Onsager coefficient, Λ(q;Tf), as a function of wave number q and Tf, derived from experimental growth rates, R(q;Tf), of the Fourier mode of concentration fluctuations and estimation of ST(q;Tf), was compared to the reptation model theories of Pincus and Binder. Experimental Λ(q;Tf) was found to give a q-dependence greater than that given by the theories. Here, ST(q;Tf) denotes the virtual structure factor at Tf inside the spinodal region. The reduced wave number Qm(τ) and intensity S̃m(τ) at the peak of the scattering structure factor in the early and intermediate stages of SD were found to be scalable in terms of a reduced time τ when Ti was fixed and Tf was varied, but not when Tf was fixed and Ti was varied. The failure of the scaling law in the latter instance may be attributed to the fact that the concentration fluctuation at the onset of SD has a different memory of the thermal concentration fluctuation in the single-phase region depending on Ti, which affects the subsequent SD over an extended period of time.

Interchange instability of the Earth's plasmapause

We reexamine the stability of low‐frequency electrostatic waves with k·B = 0 at the plasmapause, using an extension of the procedure of Richmond (1973), an approach that is based on computing individual particle motions and includes the line‐tying effect of the ionosphere. We derive expressions for the complex frequency as a function of wave number. The effect of the polarization (inertial) current accompanying the wave is included in an approximate way. The instability is caused by the sharp change in plasma pressure at the plasmapause, but its growth rate is limited by ionospheric conductivity and, for very short wavelengths, by the inertia of the magnetospheric particles. The growth rate generally increases with decreasing ripple wavelength, until that wavelength becomes small compared to the thickness of the plasmapause. The ring current (hot and warm plasma) particles suppress wave growth significantly only for long wavelengths. According to our linear analysis, an interchange ripple with wavelength ≲2000 km that forms on a sharp plasmapause (≲0.1 RE) should grow by ≳10 e folds as it traverses the nightside of the Earth. This result is consistent with Richmond's (1973) conclusion that the interchange instability limits the thickness of the plasmapause on the nightside of the Earth.