Asymptotic Wave Research Articles

Theory and observations of waves on hollow-core vortices

A linear non-homogeneous analysis is presented for the standing waves produced on the hollow core of an irrotational vortex by an arbitrary obstacle on the wall of the tube containing the vortex. The group-velocity criterion based upon Kelvin's corresponding dispersion relation predicts whether a certain asymptotic wave pattern appears upstream or downstream of the obstacle. The analysis leads to amplitude singularities for the standing waves at certain critical radii of the core. The particularly interesting case of a counter-helix for which the wave energy is propagating upstream appears for a first-mode angular disturbance. For this situation it seems to be possible that the helix ends in a hydraulic jump and is continued by a counter-helix downstream, as the core size gradually diminishes due to the deceleration of the flow caused by viscous effects (not included in the analysis). The capillary-wave pattern produced by surface tension is also considered. A brief outline for the analogous wave problem is given for the case where the fluid rotates like a rigid body.Photographic observations of hollow-core vortices in water flow are presented which confirm the qualitative predictions of the analysis, both for the response to an axisymmetric area contraction and also to a 90° bend at the downstream end of the vortex tube.

Asymptotic behavior of atomic and molecular wave functions

The asymptotic form of bound-state wave functions is derived by analytic continuation of asymptotic scattering-state wave functions. The result is also regorously derived by using an approach that is independent of scattering theory. One aspect of the result is that the N electron wave function becomes the lowest accessible exact wave function for the remaining N - 1 electrons when one electron is far away from all the nuclei. This shows that the recently developed extended Koopmans' procedures are in principle exact for the first ionization energy.

The asymptotic vacuum and soliton sector wave functionals

Using a variant of the Creutz quantization method we find in the field representation the wave functionals of the "in" vacuum and soliton sector states. The quantum stability of the different vacuum sectors is explicitly proved.

Cutoff frequencies of vector wave modes in cladded inhomogeneous optical fibre

Asymptotic vector wave formulas for calculating the cutoff frequencies in optical fibres with graded-index core and uniform cladding are derived. Numerical examples are given in comparison with the scalar wave solutions, in order to show that the mode degeneracy present in the regime of scalar wave approximations is removed and the cutoff values in degenerate states split slightly.

On the use of the WKBJ density wave theory at the Inner Lindblad Resonance region of our galaxy

I have tested the reliability of certain approximations involved in the asymptotic WKBJ density wave description of the inner Lindblad resonance (=ILR) of our galaxy.

How short are long waves in the cochlea?

An asymptotic long wave technique is applied to a two‐dimensional cochlear model. The method leads to an ordinary differential equation for either the pressure difference across the cochlear partition or the partition velocity. The lowest order approximation appears to be identical to the equation describing one‐dimensional models; the complete approach is equivalent to that of Viergever and Kalker [J. Eng. Math. 9, 353–365 (1975) and 11, 11–28 (1977)] and that of Sondhi [Proc. 9th ICA, Madrid, ( 1977)], which thus are shown to be long wave methods as well. The validity of the one‐dimensional solution is confined to waves along the basilar membrane whose wavelength λ satisfies λ >> 2πh; h is the scalaheight. The presented technique is applicable when λ > 2h, which is not only a factor of π, but also an order of magnitude better. The asymptotic approach is especially useful since it makes the physical principles in the cochlea comprehensible. The limited validity however asks for a comparison of the results of the present method with exact numerical results of the two‐dimensional cochlear model, which indeed will be made in the lecture.

The Korteweg-de Vries equation and water waves. Part 3. Oscillatory waves

Water-wave experiments are presented showing the evolution of finite amplitude waves in relatively shallow water when no solitons are present. In each case, the initial wave is rectangular in shape and wholly below the still water level; the amplitude of the wave is varied. The asymptotic solution of the Korteweg-de Vries (KdV) equation in the absence of solitons (Ablowitz & Segur 1976) is compared with observed evolution. In addition, the asymptotic solution of the linearized KdV equation (a linear dispersive model) is compared with both the KdV solution and experiments. This comparison is a natural consequence of the fact that, in the absence of solitons, the asymptotic solutions of the KdV equation and its linearized version are qualitatively similar. Both the experiments and the model equations suggest that the asymptotic wave structure consists of a negative triangular wave, travelling with a speed (gh)½, followed by a train of modulated oscillatory waves which travel more slowly. Quantitative comparisons are made for the amplitude, shape and decay rate of the leading wave and the amplitude, dominant wavenumbers and velocities of the trailing wave groups. Over the parameter range of the experiments, asymptotic KdV theory predicts more closely the observed behaviour. The leading wave is observed to decay more rapidly than the trailing wave groups; hence the leading wave becomes less prominent with time. This is in agreement with the KdV solution, whereas just the opposite is predicted by linear theory. Linear predictions for the trailing wave groups are accurate only when they agree with the KdV predictions. Both models predict the evolution of short waves in the trailing wave region. When the short waves are unstable (k gt; 1·36), either group disintegration or focusing into envelope solitons is possible. Both of these phenomena are observed in the experiments; neither is predicted by long-wave models. The nonlinear Schrödinger equation is reviewed and tested as a model of these unstable wave groups. There is some evidence that the KdV equation and the nonlinear Schrödinger equation can be patched together to provide an asymptotic description of these unstable groups.

Asymptotic Wave Functions and Energy Distributions for Symmetric Hyperbolic Systems of First Order

This paper studies from the view point of L 2-theory the asymptotic behavior for £-»oo of solutions (with finite energy) of symmetric hyperbolic systems of first order with constant coefficients. For each solution of such systems the corresponding asymptotic wave function will be constructed from the initial data. The asymptotic energy distributions of the solutions will be investigated making use of the asymptotic wave functions. Wilcox [8] studied these problems for solutions of the wave equation

Transient displacement and stress in non-homogeneous elastic shells

Transient displacement and stress in thick inhomogeneous cylindrical and spherical shells is investigated for situations when the surfaces are subjected to dynamic loads consistent with the production of radial vibrations. In the first instance, transformation of the governing equations is sought to a form amenable to solution via a finite Hankel transform technique. Such reduction is available for a broad class of radial variations in the material density and elastic parameters. However, the solutions generated by this method are rather involved and two other approaches are indicated. Thus, reduction of the system to the conventional wave equation is obtained under certain restrictions on the nature of the inhomogeneities and a simple correspondence principle is presented for the imposed boundary stress boundary value problem. Finally an asymptotic wave front analysis is presented which has a wide generality of application.

Asymptotic Wave Theory (Vol. 20, North-Holland Series in Applied Mathematics and Mechanics)

Asymptotic Wave Theory (Vol. 20, North-Holland Series in Applied Mathematics and Mechanics)

Zero-mass plane waves in nonzero gravitational backgrounds

The mathematical definition of what is intuitively called a ''plane wave'' on the curved background of a black hole is clarified and discussed from the viewpoints of potentials and fields. Because of the long-range Newtonian part of the gravitational field the asymptotic wave fronts of an incident ''plane wave'' (describing a radiative perturbation for a scattering experiment) are distorted in a manner analogous to the wave fronts of an electron beam in the quantum-mechanical Coulomb scattering problem. In addition, the electromagnetic and gravitational fields can be described with either a potential formalism (i.e., the vector potential and the metric perturbation) or a field formalism (i.e., the electromagnetic field tensor and the Riemann tensor). In this paper we present a distorted ''plane wave'' prescription, necessary for the calculation of the scattering cross sections of electromagnetic and gravitational waves off of a black hole, which agrees with the accepted prescription for a massless scalar field and satisfies the intuitive notions of what constitutes a ''plane wave'' in terms of potentials and fields. (AIP)

Charge exchange betweenH(2s)and fully stripped carbon and nitrogen ions at low-keV impact energies

Approximate cross sections for charge transfer between $\mathrm{H}(2s)$ and fully stripped carbon and nitrogen ions have been obtained in the velocity range 6 \ifmmode\times\else\texttimes\fi{} ${10}^{6}$-7 \ifmmode\times\else\texttimes\fi{} ${10}^{7}$ cm/sec. In these systems, transfer takes place into high-lying orbitals of the final-state ion. The Landau-Zener theory modified for application to a multistate system is used to calculate the cross sections. The coupling-matrix elements are determined both from a study of the pseudocrossings observed in the relevant system of the adiabatic potential curves and by direct calculation using the appropriate asymptotic parabolic wave functions. The resulting cross sections are large, in the ${10}^{\ensuremath{-}14}$-${\mathrm{cm}}^{2}$ range for both ${\mathrm{C}}^{+6}$ and ${\mathrm{N}}^{+7}$, and show the relatively flat energy dependence characteristic of systems with a number of contributing curve crossings.

The 0 + states in doubly even deformed nuclei

The K π = 0 + states in deformed rare-earth nuclei are calculated within the pairing plus multipole-multipole interaction model. The monopole-monopole particle-hole interaction is taken into account with the quadrupole-quadrupole one. The Nilsson scheme with asymptotic wave functions is used. In contrast with earlier works, all the ΔN = 0,2 particle-hole transitions contributing to the giant quadrupole resonance were included. For this reason the B(E0; 0 i + → 0 g +) and B(E2; 0 i + → 2 g +) values were calculated without effective charges, i.e. with e p = 1, e n = 0 (in electron charge units).

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.

Supermultiplet expansion for nuclear reactions

Asymptotic wave functions for nuclear reactions with two fragments are expanded into supermultiplets. The expansion coefficients turn out to be essentially coefficients of fractional parentage for the spin-isospin part.

Haag-Ruelle approximation of collision states

We investigate the rate of convergence of the Haag-Ruelle approximation Ψ(t) at large timest for arbitrary collision states Ψ with finite energy. An improved estimate of the norm distance ∥Ψ−Ψ(t)∥ is given. In particular for states Ψ with smooth asymptotic wave functions it turns out that ∥Ψ−Ψ(t)∥ approaches 0 almost liket−3/4.

The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

Integral transform functions and separable potentials

AbstractGeneral forms for asymptotic wave functions are derived from properties of the relevant Green's function. The use of separable potentials constructed from the asymptotic functions is described and the relation with integral transform functions is discussed.

Angular momentum reduction for physical amplitudes in the three-body problem

We give here an approach to the angular momentum reduction that is tailed to be appropriate for the multichannel structure of the three-body problem. Wherever possible we work directly with the physical scattering amplitudes. We obtain concrete partial-wave expansions of elastic, rearrangement, and breakup amplitudes. For these amplitudes we obtain a coupled two-variable integral equation. The effects of parity, time reversal, and rotational invariance are fully discussed. Finally, we provide expressions for the multichannel partial-wave cross sections, asymptotic coordinate-space wave functions, off-shell unitarity, and the partial-wave version of the optical theorem as well as phase-shift parametrizations of the amplitudes.

Some problems of asymptotic theory of nonlinear waves

Asymptotic methods developed recently in the nonlinear wave theory and applicable to the analysis of nonsinusoidal processes are discussed. These methods are based on the assumption of sufficiently slow space-time modulation of wave parameters (shortwave approximation). The intention of this paper is neither to review all problems of asymptotic wave theory nor to give a complete bibliography. Its purpose is to show the main ideas of the methods and to present concrete procedures for the construction of the approximate solutions that correspond to the "nonlinear geometrical optics" for various types of waves. A brief classification of these processes is also given in this paper. The following questions are considered: a) the asymptotic method for quasi-stationary waves; b) a modification of this method describing slowly varying aperiodic waves (shocks and solitary waves); c) the derivation of "averaged variational principle" and its generalization for dissipative systems; d) the method of reduction of a system of equations close to the linear hyperbolic ones which is valid for nonstationary waves propagating along the characteristics; and e) the concrete problem of cylindrical finite-amplitude electromagnetic wave propagation in isotropic dielectrics.