Upwind schemes for the wave equation in second-order form

We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d’Alembert’s exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.     

Scattering of a slow quantum particle on a central short-range potential

The linear version of the variable-phase approach in the theory of potential scattering is supplemented with a new asymptotic method. This method is intended for performing a quantum-mechanical analysis and for constructing explicit low-energy approximations for partial-wave phase shifts and radial components of the wave function for the scattering of a quantum particle on a central short-range potential. The procedure used to construct all low-energy approximations reduces to solving a recursion chain of energy-independent sets of equations, each such set consisting of two linear first-order differential equations.     

Scattering of a slow quantum particle on an axially symmetric short-range potential

A linear version of the variable-phase approach in potential-scattering theory is supplemented with a new asymptotic method. This method is intended for analyzing quantum mechanically and for constructing explicit low-energy approximations for partial-wave phase shifts, amplitudes, cross sections, and radial components of the wave function for the scattering of a quantum particle on an axially symmetric short-range potential. The procedure used to construct all low-energy approximations reduces to solving a recursion chain of energy-independent sets of equations, each such set consisting of two linear first-order differential equations.     

A multi-domain Chebyshev collocation method for predicting ultrasonic field parameters in complex material geometries

The use of ultrasound to measure elastic field parameters as well as to detect cracks in solid materials has received much attention, and new important applications have been developed recently, e.g., the use of laser generated ultrasound in non-destructive evaluation (NDE). To model such applications requires a realistic calculation of field parameters in complex geometries with discontinuous, layered materials. In this paper we present an approach for solving the elastic wave equation in complex geometries with discontinuous layered materials. The approach is based on a pseudospectral elastodynamic formulation, giving a direct solution of the time-domain elastodynamic equations. A typical calculation is performed by decomposing the global computational domain into a number of subdomains. Every subdomain is then mapped on a unit square using transfinite blending functions and spatial derivatives are calculated efficiently by a Chebyshev collocation scheme. This enables that the elastodynamic equations can be solved within spectral accuracy, and furthermore, complex interfaces can be approximated smoothly, hence avoiding staircasing. A global solution is constructed from the local solutions by means of characteristic variables. Finally, the global solution is advanced in time using a fourth order Runge-Kutta scheme. Examples of field prediction in discontinuous solids with complex geometries are given and related to ultrasonic NDE.     

The energy spectrum of fully developed turbulence generated by random stirring

For fully developed turbulence in an incompressible fluid described by the Navier-Stokes equations with Gaussian random forces the relation between the energy spectrum and the stirring mechanism is investigated within a variational approach. Therein, the effect of nonlinear mode coupling is approximated by a wave number dependent eddy viscosity determined via a nonlinear integral equation for the energy spectrum. For various stirring spectra analytic approximations are compared with the solution obtained numerically with a cutoff in the integral kernel which ensures in eddy relaxing processes that the stirring forces exert strain only on scales larger than the eddy size. The results are compared with renormalization group calculations and closure approximations. Random forces injecting energy at a ratek ^–1 into the wave number banddk aroundk lead to a Kolmogorov distribution of energy. The spectrum of small-scale velocity fluctuations is shown to be universal in the sense that it remains unchanged under variations of the long wavelength stirring spectra.     

A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.     

ON TRANSMISSION OF SOUND IN A NON-UNIFORM DUCT CARRYING A SUBSONIC COMPRESSIBLE FLOW

The differential equations governing the transmission of one-dimensional sound waves in a non-uniform duct carrying a subsonic compressible mean flow have been the subject of a recent debate [1, 2]. Of the two formulations presented, one is considered to be non-acoustical and the other as neglecting the spatial variation of the speed of sound. The present paper shows that both formulations are acoustical and represent valid approximations to correct conditions for isentropic sound propagation in a subsonic low Mach number duct. Each formulation is associated with an “error wave”, which is essentially a hydrodynamic wave when the mean flow Mach number is small. Three-port modelling is required, however, to capture this wave when the Mach number of the mean flow is relatively large and a numerical matrizant approach is described which can be used for this purpose.     

General theory for the interference of two-photon and one-photon processes in semiconductor heterostructures

We develop a general theory for the dynamics of multi-photon processes in semiconductor heterostructures. The resulting effective multi-band Bloch equations describe the dynamics of electrons in the reduced set of bands between which the optical pulses induce quasi-resonant transitions. The model is specialized to the case of interfering one- and two-photon transitions across the band gap. The withdrawn bands are included as intermediate states for an effective interaction that is quadratic in the electromagnetic fields. The benefit of this perturbative approach is to lead to equations of motion for slowly varying quantities only, in the spirit of the rotating wave approximation. Coulomb interaction and relaxation can also easily be included. Finally, a general expression for the time dependent polarization current that is consistent with the approximations involved by the effective multi-band Bloch equations is derived.     

On electromagnetic waves by an accelerated charged particle in medium

We have investigated the effects of acceleration of a charged particle on its Cerenkov emission and ionization-losses. We have considered the accelerated motion of a charged particle in an infinite medium with the acceleration parallel to the direction of its motion. We have used the method of Fourier transforms to solve the Maxwell's equations with appropriate current and charge-densities to find electromagnetic fields and hence the force experienced by the incident charge due to its interaction with the medium (dielectric or plasma). The results obtained are general and applicable to any acceleration. In the approximations of ‘small acceleration’ and ‘small interaction time’, we have solved the wave equations and determined electromagnetic potentials. It is found that the acceleration of the charged particle strongly changes both its ionization-loss and Cerenkov emission.     

Evolution of basic equations for nearshore wave field

In this paper, a systematic, overall view of theories for periodic waves of permanent form, such as Stokes and cnoidal waves, is described first with their validity ranges. To deal with random waves, a method for estimating directional spectra is given. Then, various wave equations are introduced according to the assumptions included in their derivations. The mild-slope equation is derived for combined refraction and diffraction of linear periodic waves. Various parabolic approximations and time-dependent forms are proposed to include randomness and nonlinearity of waves as well as to simplify numerical calculation. Boussinesq equations are the equations developed for calculating nonlinear wave transformations in shallow water. Nonlinear mild-slope equations are derived as a set of wave equations to predict transformation of nonlinear random waves in the nearshore region. Finally, wave equations are classified systematically for a clear theoretical understanding and appropriate selection for specific applications.     

Comparison between analytical and numerical methods as applied to calculations of excited atomic states

This article presents a comparison between two approaches for implementing a variational method when calculating excited states of atoms, namely a numerical approach in which the equations arising from the requirement of an extremum of the variational functional (the Hartree—Fock equations) are solved, and an analytical approach in which the energy functional expressed in terms of analytical test functions is minimized. Both approaches are analyzed from the point of view of the approximations used to ensure that the conditions are satisfied for the complete wave function of the excited state being sought to be orthogonal to all wave functions of lower-lying energy states having the same symmetry. The well-known ATOM package is used for numerically solving the Hartree—Fock equations and the MINMAX package is used for the analytical variational calculations. It is shown that the analytical approach based on the minimax method possesses greater possibilities for taking account of relaxation effects. A comparison is made between single-electron wave functions, the matrix elements, and the energies of dipole transitions for a number of excited states of the Ne atom, as calculated using both approaches. State Pedagogical University, Tomsk. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 120–128, July, 1998.     

Minimal atmospheric finite-mode models preserving symmetry and generalized Hamiltonian structures

A typical problem with the conventional Galerkin approach for the construction of finite-mode models is to keep structural properties unaffected in the process of discretization. We present two examples of finite-mode approximations that in some respect preserve the geometric attributes inherited from their continuous models: a three-component model of the barotropic vorticity equation known as Lorenz’ maximum simplification equations [E.N. Lorenz, Maximum simplification of the dynamic equations, Tellus 12 (3) (1960) 243-254] and a six-component model of the two-dimensional Rayleigh-Bénard convection problem. It is reviewed that the Lorenz-1960 model respects both the maximal set of admitted point symmetries and an extension of the noncanonical Hamiltonian form (Nambu form). In a similar fashion, it is proved that the famous Lorenz-1963 model violates the structural properties of the Saltzman equations and hence cannot be considered as the maximum simplification of the Rayleigh-Bénard convection problem. Using a six-component truncation, we show that it is again possible to retain both symmetries and the Nambu representation in the course of discretization. The conservative part of this six-component reduction is related to the Lagrange top equations. Dissipation is incorporated using a metric tensor.     

A General Deterministic Treatment of Derivatives in Particle Methods

A unified approach to approximating spatial derivatives in particle methods using integral operators is presented. The approach is an extension of particle strength exchange, originally developed for treating the Laplacian in advection–diffusion problems. Kernels of high order of accuracy are constructed that can be used to approximate derivatives of any degree. A new treatment for computing derivatives near the edge of particle coverage is introduced, using “one-sided” integrals that only look for information where it is available. The use of these integral approximations in wave propagation applications is considered and their error is analyzed in this context using Fourier methods. Finally, simple tests are performed to demonstrate the characteristics of the treatment, including an assessment of the effects of particle dispersion, and their results are discussed.     

The contribution of a lateral wave in simulating low-frequency sound fields in an irregular waveguide with a liquid bottom

A further development of a previously proposed approach to calculating the sound field in an arbitrarily irregular ocean is presented. The approach is based on solving the first-order causal mode equations, which are equivalent to the boundary-value problem for acoustic wave equations in terms of the cross-section method. For the mode functions depending on the horizontal coordinate, additional terms are introduced in the cross-section equations to allow for the multilayer structure of the medium. A numerical solution to the causal equations is sought using the fundamental matrix equation. For the modes of the discrete spectrum and two fixed low frequencies, calculations are performed for an irregular two-layer waveguide model with fluid sediments, which is close to the actual conditions of low-frequency sound propagation in the coastal zone of the oceanic shelf. The calculated propagation loss curves are used as an example for comparison with results that can be obtained for the given waveguide model with the use of adiabatic and one-way propagation approximations.     

Simulation of low-frequency sound propagation in an irregular shallow-water waveguide with a fluid bottom

A further development of a previously proposed approach to calculating the sound field in an arbitrarily irregular ocean is presented. The approach is based on solving the first-order causal mode equations, which are equivalent to the boundary-value problem for acoustic wave equations in terms of the cross-section method. For the mode functions depending on the horizontal coordinate, additional terms are introduced in the cross-section equations to allow for the multilayer structure of the medium. A numerical solution to the causal equations is sought using the fundamental matrix equation. For the modes of the discrete spectrum and two fixed low frequencies, calculations are performed for an irregular two-layer waveguide model with fluid sediments, which is close to the actual conditions of low-frequency sound propagation in the coastal zone of the oceanic shelf. The calculated propagation loss curves are used as an example for comparison with results that can be obtained for the given waveguide model with the use of adiabatic and one-way propagation approximations.     

On propagation in continuous random media

The analysis of wave propagation in continuous random media typically proceeds from the parabolic wave equation with back scatter neglected. A closed hierarchy of moment equations can be obtained by using the Novikov-Furutsu theorem. When the same procedure is applied in the spatial Fourier domain, one obtains a closed hierarchy of coupled moment equations for the forward- and back-scattered wavefields that is not restricted to narrow scattering angles nor to small local perturbations. The general equations are difficult to solve, but a Markov-like approximation is suggested by the form of the scattering terms. Simple algebraic solutions can be obtained if a narrow-angle-scatter approximation is then invoked. Thus, three distinct approximations are explicit in this analysis, namely closure, Markov and narrow-angle scatter.

The results show that the extinction of the coherent wavefield has a distinctly different form from the corresponding result for propagation in a sparse distribution of discrete scatteres. Furthermore, when the scatter is constrained to narrow forwardand back-scattered cones, there is no back-scatter enhancement. These results are discussed within the context of the extension of the spectral-domain formalism to discrete random media. The general continuous-media moment equations are developed but not solved. The results correct and extend an earlier analysis that used a perturbation approach to compute the scattering functions rather than the Novikov-Furutsu theorem.     

Analysis of the wave functions for accelerating Kerr-Newman metric

As is well-known, it is very difficult to solve wave equations in curved space-time. In this paper,we find that wave equations describing massless fields of the spins s≤2 in accelerating KerrNewman black holes can be written as a compact master equation. The master equation can be separated to radial and angular equations, and both can be transformed to Heun's equation,which shows that there are analytic solutions for all the wave equations of massless spin fields.The results not only demonstrate that it is possible to study the similarity between waves of gravitational and other massless spin fields, but also it can deal with other astrophysical applications, such as quasinormal modes, scattering, stability, etc. In addition, we also derive approximate solutions of the radial equation.     

A note on discretization of nonlinear differential equations

An important issue when integrating nonlinear differential equations on a digital computer is the choice of the time increment or step size. For example, it is known that if this quantity is not sufficiently short, spurious chaotic motions may be induced when integrating a system using several of the well-known methods available in the literature. In this paper, a new approach to discretize differential equations is analyzed in light of computational chaos. It will be shown that the fixed points of the continuous system are preserved under the new discretization approach and that the spurious fixed points generated by higher order approximations depend upon the increment parameter. (c) 2002 American Institute of Physics.     

Coupled scalar wave diffraction theory

A simple but accurate coupled-wave theory describing diffraction from a volume grating is developed, which applies to planar diffraction geometries with the electric field polarized normal to the plane of incidence. Modulation of the dielectric and the absorption properties of the medium are considered, and the analysis is developed for gratings nonuniform with depth and for composite (multiplexed) gratings. The theory is based axiomatically on Maxwell's equations, and no approximations or simplifying assumptions other than those requisite to the scalar wave formulation of the theory enter into the analysis (except that for computational applications, only a finite number of diffracted orders may be retained in the analysis).