Asymptotic Wave Research Articles

Higher order asymptotic homogenization and wave propagation in periodic composite materials

We present an application of the higher order asymptotic homogenization method (AHM) to the study of wave dispersion in periodic composite materials. When the wavelength of a travelling signal becomes comparable with the size of heterogeneities, successive reflections and refractions of the waves at the component interfaces lead to the formation of a complicated sequence of the pass and stop frequency bands. Application of the AHM provides a long-wave approximation valid in the low-frequency range. Solution for the high frequencies is obtained on the basis of the Floquet–Bloch approach by expanding spatially varying properties of a composite medium in a Fourier series and representing unknown displacement fields by infinite plane-wave expansions. Steady-state elastic longitudinal waves in a composite rod (one-dimensional problem allowing the exact analytical solution) and transverse anti-plane shear waves in a fibre-reinforced composite with a square lattice of cylindrical inclusions (two-dimensional problem) are considered. The dispersion curves are obtained, the pass and stop frequency bands are identified.

Predicted Doppler shifts induced by ocean surface wave displacements using asymptotic electromagnetic wave scattering theories

Sea surface motions can produce different measured Doppler shifts with respect to instrumental configurations (incidence angle, electromagnetic wavelength, polarization). Under Gaussian statistics for the sea surface elevation and in the general framework of asymptotic theories for ocean surface electromagnetic wave scattering, Doppler shifts can be predicted. The small-slope, Kirchhoff, local curvature and resonant curvature approximations are compared in the backscatter configuration. Predicted Doppler shifts for Kirchhoff and small-slope approximations in co-polarized configuration are insensitive to the polarization state. On the other hand, the local and resonant curvature solutions, through a phase perturbation formalism, yield to significant differences between co-polarized predicted Doppler shifts. Comparisons with data are shown to confirm the polarization and wind speed sensitivities.

Strongly modulated transmission of a spin-split quantum wire with local Rashba interaction

We investigate the transport properties of ballistic quantum wires in the presence of Zeeman spin splittings and a spatially inhomogeneous Rashba interaction. The Zeeman interaction is extended along the wire and produces gaps in the energy spectrum, which allow electron propagation only for spinors lying along a certain direction. For spins in the opposite direction, the waves are evanescent far away from the Rashba region, which plays the role of the scattering center. The most interesting case occurs when the magnetic field is perpendicular to the Rashba field. Then, the spins of the asymptotic wave functions are not eigenfunctions of the Rashba Hamiltonian, and the resulting coupling between spins in the Rashba region gives rise to sudden changes of the transmission probability when the Fermi energy is swept along the gap. After briefly examining the energy spectrum and eigenfunctions of a wire with extended Rashba coupling, we analyze the transmission through a region of localized Rashba interaction, in which a double interface separates a region of constant Rashba interaction from wire leads free from spin-orbit coupling. For energies slightly above the propagation threshold, we find the ubiquitous occurrence of transmission zeros (antiresonances), which are analyzed by matching methods in the one-dimensional limit. We find that a minimal tight-binding model yields analytical transmission line shapes of Fano antiresonance type. The general angular dependence of the external magnetic field is treated within projected Schr\"odinger equations with the nondiagonal Hamiltonian matrix elements mixing different wave function components. Finally, we consider a realistic quantum wire where the energy subbands are coupled via the Rashba intersubband coupling term and discuss its effect on the transmission zeros. We find that the antiresonances are robust against intersubband mixing, magnetic field changes, and smooth variations of the wire interfaces, which paves the way for possible applications of spin-split Rashba wires as spintronic current modulators.

Asymptotic gravity wave drag expressions for non‐hydrostatic rotating flow over a ridge

AbstractAsymptotic expressions are derived for the mountain wave drag in flow with constant wind and static stability over a ridge when both rotation and non‐hydrostatic effects are important. These expressions, which are much more manageable than the corresponding exact drag expressions (when these do exist) are found to provide accurate approximations to the drag, even when non‐hydrostatic and rotation effects are strong, despite having been developed for cases where these effects are weak. The derived expressions are compared with approximations to the drag found previously, and their asymptotic behaviour in various limits is studied. Copyright © 2008 Royal Meteorological Society

Fundamental burn-up mode in a pebble-bed type reactor

This paper deals with a pebble-bed type reactor, in which the fuel is loaded from one side (top) and discharged from the other side (bottom). A boundary value problem of a single group diffusion equation coupled with simplified burn-up equations is studied, where the natural radioactive decay processes are neglected in the burn-up modelling. An asymptotic burning wave solution is found analytically in the one-dimensional case, which is called as fundamental burn-up mode. Among this solution family there are two particular cases, namely, a classic fundamental solution with a zero burn-up and a partial solitary burn-up wave solution with a highest burn-up. An example of Th–U conversion is considered and the solutions are presented in order to show the mechanism of the burning wave.

A kind of extended Korteweg-de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

This paper considers interfacial waves propagating along the interface between a two-dimensional two-fluid with a flat bottom and a rigid upper boundary. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. It just focuses on the weakly non-linear small amplitude waves by introducing two small independent parameters: the nonlinearity ratio epsilon, represented by the ratio of amplitude to depth, and the dispersion ratio mu, represented by the square of the ratio of depth to wave length, which quantify the relative importance of nonlinearity and dispersion. It derives an extended KdV equation of the interfacial waves using the method adopted by Dullin et al in the study of the surface waves when considering the order up to O(mu(2)). As expected, the equation derived from the present work includes, as special cases, those obtained by Dullin et al for surface waves when the surface tension is neglected. The equation derived using an alternative method here is the same as the equation presented by Choi and Camassa. Also it solves the equation by borrowing the method presented by Marchant used for surface waves, and obtains its asymptotic solitary wave solutions when the weakly nonlinear and weakly dispersive terms are balanced in the extended KdV equation.

Ray velocity and ray attenuation in homogeneous anisotropic viscoelastic media

Asymptotic wave quantities such as ray velocity and ray attenuation are calculated in anisotropic viscoelastic media by using a stationary slowness vector. This vector generally is complex valued and inhomogeneous, and it predicts the complex energy velocity parallel to a ray. To compute the stationary slowness vector, one must find two independent, real-valued unit vectors that specify the directions of its real and imaginary parts. The slowness-vector inhomogeneity affects asymptotic wave quantities and complicates their computation. The critical quantities are attenuation and quality factor ([Formula: see text]-factor); these can vary significantly with the slowness-vector inhomogeneity. If the inhomogeneity is neglected, the attenuation and the [Formula: see text]-factor can be distorted distinctly by errors commensurate to the strength of the velocity anisotropy — as much as tens of percent for sedimentary rocks. The distortion applies to strongly as well as to weakly attenuative media. On the contrary, the ray velocity, which defines the wavefronts and physically corresponds to the energy velocity of a high-frequency signal propagating along a ray, is almost insensitive to the slowness-vector inhomogeneity. Hence, wavefronts can be calculated in a simplified way except for media with extremely strong anisotropy and attenuation.

Interfacial capillary–gravity waves due to a fundamental singularity in a system of two semi-infinite fluids

The interfacial capillary-gravity waves due to a transient fundamental singularity immersed in a system of two semi-infinite immiscible fluids of different densities are investigated analytically for two- and three- dimensional cases. The two-fluid system, which consists of an inviscid fluid overlying a viscous fluid, is assumed to be incompressible and initially quiescent. The two fluids are each homogeneous, and separated by a sharp and stable interface. The Laplace equation is taken as the governing equation for the inviscid flow, while the linearized unsteady Navier-Stokes equations are used for the viscous flow. With surface tension taken into consideration, the kinematic and dynamic conditions on the interface are linearized for small-amplitude waves. The singularity is modeled as a simple mass source when immersed in the inviscid fluid above the interface, or as a vertical point force when immersed in the viscous fluid beneath the interface. Based on the integral solutions for the interfacial waves, the asymptotic wave profiles are derived for large times with a fixed distance-to-time ratio by means of the generalized method of stationary phase. It is found that there exists a minimum group velocity, and the wave system observed will depend on the moving speed of the observer. Two schemes of expansion of the phase function are proposed for the two cases when the moving speed of an observer is larger than, or close to the minimum group velocity. Explicit analytical solutions are presented for the long gravity-dominant and the short capillary-dominant wave systems, incorporating the effects of density ratio, surface tension, viscosity and immersion depth of the singularity.

Determination of the magnetic susceptibility of the quark condensate using radiative heavy meson decays

We use a light-cone sum rule (LCSR) analysis of the branching ratios of radiative meson decays to constrain the value of the magnetic susceptibility of the quark condensate χ(μ). For the first time, we use a complete set of three-particle distribution amplitudes that enables us to give a consistent prediction for the branching ratios. Furthermore we will make use of a very recent update of several non-perturbative parameters. Our final result for χ(μ = 1 GeV) = 2.85 ± 0.5 GeV−2 (assuming asymptotic wave functions) agrees with the currently used value of 3.15 ± 0.3 GeV−2.

Evolutionary Conditions in Dissipative MHD Systems Revisited

The evolutionary conditions of dissipative continuous magnetohydrodynamic (MHD) shocks are studied. We modify Hada’s approach to the stability analysis of the MHD shock waves. The matching conditions between perturbed shock structure and asymptotic wave modes show that all types of the MHD shocks, including intermediate shocks, are evolutionary and perturbed solutions are uniquely defined. We also adapt our formalism to MHD shocks in systems with resistivity but no viscosity, which is often used in numerical simulations, and we show that all types of shocks found in such systems satisfy the evolutionary conditions and perturbed solutions are uniquely defined. These results suggest that intermediate shocks may appear in physical systems.

On wave propagation in porous media with strain gradient effects

A porous material with large irregular holes is here studied as a hyperelastic continuum with latent microstructure, and the mechanical balance equations are derived from the general ones for a medium with ellipsoidal microstructure. This is done by imposing the kinematical constraint of microstretch bounded to the macrodeformation: in this case, the microstructure disappears, apparently, and the response of the material involves higher gradients of the displacement without incurring known constitutive inconsistencies. An application to the propagation of asymptotic waves compatible with such a model is also considered. Physical situations corresponding to an axisymmetric motion either in spherical or cylindrical symmetry are considered, and it is shown that the time evolution of the wave amplitude factor is governed by the spherical and cylindrical Korteweg–deVries equations, respectively.

Scattering of a Light Particle by a System of Harmonic Oscillators

We discuss the asymptotic wave function of a quantum system in ℝ 3 composed by heavy and light particles, in the case where the light particles are in scattering states and no interaction is assumed among particles of the same kind. We first review a recent result concerning the case of K heavy and N light particles, where the one-particle potential acting on each heavy particle decays at infinity. Then we consider the case of one light particle interacting with a system of harmonic oscillators and prove the same kind of result following, with some modification, the proof of the previous case. A possible application to the analysis of the scattering of a light particle from condensed matter is also outlined.

Frequency- and time-domain asymptotic fields near the critical cone in fluid-fluid configuration

Near the critical cone of a point source placed at the interface between two half-space fluid media, investigation is made of the asymptotic fields in the frequency domain and their synthetic wave forms in the time domain. While the leading-order (and the head-wave) components give a good description of the true fields well off the critical cone, the uniform asymptotic (UA) analysis has to be made for the approximation near the critical cone. The UA analysis splits into two cases, depending on the medium densities and wave speeds. For Case 1, the UA1 approximation is employed that takes into account the proximity of the stationary-phase point to the branch point. In Case 2, the UA2 approximation is employed in which consideration is also given to the proximity of the stationary point to the pole and to the combined effect of the stationary point, the branch point, and the pole. The validities of the asymptotic fields are checked in the time domain by comparing the asymptotic field wave forms against the wave-number-synthetic wave forms. The UA fields show good accuracy and causal behaviors, with the causality of the UA2 fields previously unreported in the literature.

Finite-frequency sensitivity of body waves to anisotropy based upon adjoint methods

We investigate the sensitivity of finite-frequency body-wave observables to mantle anisotropy based upon kernels calculated by combining adjoint methods and spectral-element modelling of seismic wave propagation. Anisotropy is described by 21 density-normalized elastic parameters naturally involved in asymptotic wave propagation in weakly anisotropic media. In a 1-D reference model, body-wave sensitivity to anisotropy is characterized by ‘banana–doughnut’ kernels which exhibit large, path-dependent variations and even sign changes. P-wave travel-times appear much more sensitive to certain azimuthally anisotropic parameters than to the usual isotropic parameters, suggesting that isotropic P-wave tomography could be significantly biased by coherent anisotropic structures, such as slabs. Because of shear-wave splitting, the common cross-correlation travel-time anomaly is not an appropriate observable for S waves propagating in anisotropic media. We propose two new observables for shear waves. The first observable is a generalized cross-correlation travel-time anomaly, and the second a generalized ‘splitting intensity’. Like P waves, S waves analysed based upon these observables are generally sensitive to a large number of the 21 anisotropic parameters and show significant path-dependent variations. The specific path-geometry of SKS waves results in favourable properties for imaging based upon the splitting intensity, because it is sensitive to a smaller number of anisotropic parameters, and the region which is sampled is mainly limited to the upper mantle beneath the receiver.

Asymptotic Stability of Traveling Wave Fronts in the Buffered Bistable System

In this paper, we study a model which describes the propagation of increased calcium concentration wave front in excitable systems with the diffusing species being buffered. Our goal is to prove the global exponential stability of the unique traveling wave front. Comparing with the unbuffered system, we conclude that multiple stationary buffers (buffers do not diffuse) cannot prevent the existence of a global asymptotic stable traveling wave front, or cannot eliminate propagated waves in the buffered bistable equation. Concerning the method of the proof, we will present a method in which only the comparison principle and suitably constructed supersolutions (subsolutions) are involved. The feature of the method is to avoid calculating the spectrum of the associated linear operator.

Pattern-generating travelling waves in a discrete multicellular system with lateral inhibition

On a one-dimensional string of cells, the juxtacrine signalling model for Delta–Notch lateral inhibition by Collier et al. [J.R. Collier, N.A.M. Monk, P.K. Maini, J.H. Lewis, Pattern formation by lateral inhibition with feedback: A mathematical model of Delta–Notch intercellular interaction, J. Theoret. Biol. 183 (1996) 429–446] exhibits a predominant alternating pattern of cells expressing either Delta or Notch, as well as many aperiodic patterns. Despite this multistationarity, in the idealised situation of no noise, travelling waves invading the unstable, homogeneous state only generate the predominant alternating pattern in their wake all over the lattice. However, this robustness is totally lost in the presence of stochastic noise because the invaded, initial state is unstable. Using linear approximations around the initial, homogeneous state and around the final, patterned state, we are able to derive analytically all essential properties of the wave: the shape of the wave front, the unique, alternating pattern generated by the wave, and the asymptotic speed of the wave front. We show that the asymptotic wave speed equals the theoretical minimum wave speed. The latter agrees extremely well with the value estimated from numerical simulations. Thus, in this system travelling waves are pulled by the leading edge of the front.

Theoretical and numerical investigations of global and regional seismic wave propagation in weakly anisotropic earth models

Smith and Dahlen demonstrated that in a weakly anisotropic earth model the relative surface wave phase‐speed perturbation δc/c may be written in the form , where An and Bn are frequency‐dependent depth integrals and ζ denotes the ray azimuth. In this approximation, surface wave anisotropy is governed by 13 elastic parameters and the azimuthal dependence of the phase speed is represented by an even Fourier series in ζ involving degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters). Jech and Ps̆enc̆ík demonstrated that in such a weakly anisotropic earth model the relative compressional‐wave phase‐speed perturbation may be expressed as δc/c= (2c2)−1B33, whereas the relative shear wave phase‐speed perturbations are given by δc/c= (4c2)−1{B11+B22±[(B11−B22)2+ 4B212]1/2}. We demonstrate that the coefficients B33, B11, B22, and B12 may be written in the generic form , where ζ denotes the local azimuth and i the local angle of incidence. For B11, B22 and B33 the coefficients an(i) and bn(i) are an even Fourier series of degrees zero, two and four in i, but for B12 they are an odd Fourier series of degrees one and three. Like surface waves, the azimuthal (ζ) dependence of body waves involves even degrees zero (five elastic parameters), two (six elastic parameters), and four (two elastic parameters), but, unlike surface waves, it also involves the odd degrees one (six elastic parameters) and three (two elastic parameters). Thus, weakly anisotropic body‐wave propagation involves all 21 independent elastic parameters. We use spectral‐element simulations of global and regional seismic wave propagation to assess the validity of these asymptotic body and surface wave results. The numerical simulations and asymptotic predictions are in good agreement for anisotropy at the 5 per cent level.

Renormalization of theNNinteraction with a chiral two-pion-exchange potential: Central phases and the deuteron

We analyze the renormalization of the $\mathit{NN}$ interaction at low energies and the deuteron bound state through the chiral two-pion-exchange potential assumed to be valid from zero to infinity. The short distance van der Waals singularity structure of the potential as well as the requirement of orthogonality conditions on the wave functions determine that after renormalization, in the ${}^{1}{S}_{0}$ singlet channel and in the ${}^{3}{S}_{1}\text{\ensuremath{-}}{}^{3}{D}_{1}$ triplet channel, one can use the deuteron binding energy, the asymptotic $D/S$ ratio, and the $S$-wave scattering lengths as well as the chiral potential parameters as independent variables. We use then the asymptotic wave function normalization ${A}_{S}$ of the deuteron and the singlet and triplet effective ranges to determine the chiral constants yielding ${c}_{1},{c}_{3}$, and ${c}_{4}$. The role of finite cutoff corrections, the loss of predictive power due to uncertainties in the input data, and the connection to one-pion-exchange distorted wave perturbative approaches is also discussed.

Transient free-surface waves due to impulsive motion of a submerged source

The problem of a viscous flow past a submerged source starting from rest and moving with a constant velocity, below and parallel to a free surface is examined. Asymptotic expressions for long-time evolution of free-surface elevation are obtained. The results show explicitly the viscous effect on the free-surface elevation and no surface tension effect on the asymptotic wave.

HIGHER TWIST EFFECTS IN PROTON-PROTON COLLISIONS

In this article, we investigate the contribution of the high twist Feynman diagrams to the large-pT pion production cross section in proton-proton collisions and we present the general formulae for the high and leading twist differential cross sections. The pion wave function where two non-trivial Gegenbauer coefficients a2 and a4 have been extracted from the CLEO data, two other pion model wave functions, P2, P3, the asymptotic and the Chernyak-Zhitnitsky wave functions are used in the calculations. The results of all the calculations reveal that the high twist cross sections, the ratios R, r, the dependence transverse momentum pT and the rapidity y of pion in the Φ CLEO (x,Q2) wave function case is very close to the Φ asy (x) asymptotic wave function case. It is shown that the high twist contribution to the cross section depends on the choice of the meson wave functions.