Order Wave Equation Research Articles

Multi‐dimensional shock fronts for second order wave equations

On etudie l'existence au temps court et la stabilite structurelle des fronts de choc multidimensionnels pour des equations d'onde quasi lineaires scalaires d'ordre 2

Quaternionic quantum field theory

We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second order wave equation. The theory takes the form of either a functional integral with quaternion-imaginary Lagrangian, or a Schrodinger equation and transformation theory for quaternion-valued wave functions, with a quaternion-imaginary Hamiltonian. The connection between the two formulations is developed in detail, and many related issues, including the breakdown of the correspondence principle and the Hilbert space structure, are discussed.

Matrix Formulation of the Subsidiary Conditions of the Second Kind for the Pauli-Fierz Wave Equation

It is demonstrated, using the Pauli-Fierz wave equation as an example, that the subsidiary conditions o f the second kind for a first order wave equation can be expressed in matrix language.

Wave Propagation in Polyphase Transmission Lines a General Solution to Include Cases Where Ordinary Modal Theory Fails

The solution of transmission line equations is usually written as a superposition of so called natural modes of exponential type. These are obtained through the use of a suitable transformation that decouples the original sets or N simultaneous 2nd order wave equations for voltages and currents into N independent equations. For such a transformation to exist the fundamental product matrix ZY must be diagonalizable. In a previous paper it has been shown that physically realizable transmission lines are possible for which ZY is not diagonalizable and to which ordinary modal theory does not apply. In the present paper a new generalized modal theory is developed for the purpose of including non-diagonalizing cases.

Quaternionic quantum field theory.

We show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions, as well as the bosons, obey a second order wave equation. The theory takes the form of either a functional integral with quaternion-imaginary Lagrangian, or a Schrodinger equation and transformation theory for quaternion-valued wave functions, with a quaternion-imaginary Hamiltonian. The connection between the two formulations is developed in detail, and many related issues, including the breakdown of the correspondence principle and the Hilbert space structure, are discussed.

High order numerical sommerfeld boundary conditions: Theory and experiments

We develop high-order, non-reflecting boundary equations for a semidiscrete approximation of the simple (hyperbolic) advection equation U 1 + cU x = 0. These boundary equation are based on a discrete interpretation of Sommerfeld's radiation condition for a second order wave equation which is associated with the semi-discrete equation. The performance of these schemes is expressed by an exact measure of the energy reflected at the boundary. For low order cases, the discrete Sommerfeld boundary equations are identical with the standard finite difference equations, but for high orders of approximation (starting with 4 points), the discrete Sommerfeld schemes differ from standard finite differences. It is shown, and verified experimentally, that the discrete Sommerfeld schemes are optimal, in the sense that they produce the least amount of reflected energy. Moreover, it is known theoretically, and we verify experimentally, that the reflected energy remains invariat when the semi-discrete equation are time-discretized with the trapezoidal (Crank-Nicolson) method. The corresponding fully discrete boundary equations are thus also optimal in the sense that they minimize the reflected energy.

Some Aspects of the Identification of Continuous Vibrating Systems

This paper deals with the identification of systems described by the one dimensional, second order wave equation. Such equations arise commonly in various areas of mathematical physics, such as, acoustics, elastic wave propagation, and electromagnetic theory. Assuming that suitable measurements are made at only one point in the spatial domain, sufficient conditions for unique identification of the system eigenvalues and system coefficients are presented.

Epstein profiles and the Riemann P-functions

We examine the analytical implications of the direct power series solutions of the second order wave equation and the fourth order elastodynamic equations for inhomogeneous media represented by the Epstein profiles.

Non-linear wave propagation solutions by Fourier transform perturbation

A Fourier transform perturbation method is developed and used to obtain uniformly valid asymptotic approximations of the solution of a class of one-dimensional second order wave equations with small non-linearities. Multiple time scales are used and the initial-value problem on the infinite line is solved by Fourier transforming the wave equation and expanding the Fourier transform in powers of the small parameter. The non-linearity involves only the first partial derivatives of the dependent variable and the determination of the leading approximation is reduced to the solution of a pair of coupled non-linear ordinary differential equations in Fourier space. Examples are given involving a convolution non-linearity and a Van-der-Pol non-linearity.

N-Soliton Solution of the Higher Order Wave Equation for a Fluid of Finite Depth

Exact N -soliton solution of the higher order wave equation for a stratified fluid layer of finite depth is obtained by employing Hirota's bilinear transformation method. The solution reduces to the N -soliton solution of the higher order Korteweg-de Vries equation in the shallow water limit and to that of the higher order Benjamin-One equation in the deep water limit. The asymptotic form of the solution for large time is derived.

On the decoupling of P and S waves in inhomogeneous elastic media

Summary. A representation derived by Richards (1974) for P-SV wave displacements in spherically symmetric elastic media is extended to general displacements in a general inhomogeneous isotropic elastic medium, with suitably differentiable elastic constants. This representation in terms of appropriate potentials gives rise to a partial decoupling of the P and S (weighted) displacements, which satisfy simple second order wave equations with lower order coupling terms. The highest order P and S components satisfy homogeneous wave equations that depend only on the P and S velocities a and respectively and are unaffected by the density other than at the source and observer positions. In the study of small high-frequency particle displacements in homogeneous elastic media the motion may be completely separated into P and S components. Each component has a characteristic wave speed and satisfies a simpler equation than the general elastic wave equation. In inhomogeneous media, this separation of P and S waves may still be observed, for example as distinct phases on seismograms, but there is now lower order coupling between the phases. Also a modified form of the Helmholtz representation in terms of a gradient and curl is required for other than the highest order components. In this paper we examine the partial first order decoupling of the P and S components in a general isotropic inhomogeneous medium with differentiable material constants. We begin by deriving for the harmonic displacement u exp (-- iwt) a representation in terms of potentials P and S, u = V(P/p”2) t g,(P/p”Z) + v x (S/p’/Z) t g, x (S/p”2) + C(u)

A non-relativistic wave equation of second order in time II, electromagnetic interaction

In this paper it is shown that the electromagnetic interaction can be introduced according to the usual prescription of the minimal electromagnetic interaction into a field equation of second order in time and of fourth order in the space coordinates for bosons, which was proposed in a previous paper. We derive the interaction Hamiltonian for the non-relativistic boson field and the electromagnetic field, as well as the Feynman rules, which have to be used in a perturbation expansion. We have obtained in this way a non-relativistic theory of bosons with electromagnetic interaction in which pair formation or annihilation of the bosons may occur. We calculated some characteristic examples of cross sections and could show that the cross sections provide a non-relativistic limit of the results of the relativistic theory.

Atoms in jellium

We report one-electron calculations of the soft-x-ray ${L}_{2,3}$ absorption and emission of Na, Mg, and Al and the soft-x-ray $K$ absorption and emission of Li. The absorbing ion is placed at the center of a metal consisting of a valence band of electrons and of a uniform positive background which has a spherical hole removed from around the ion. Self-consistent solutions for states with and without a core hole are found using the density - functional approach. However, we eliminate self-interaction effects from the bound-state wave equation in order to get more realistic wave functions. This was originally proposed by Cowan but ignored subsequently. We think it is essential for getting good wave functions. In our solution both the core electrons bound to the central ion and the valence electrons are allowed to relax into the self-consistent solution. We find self-consistent potentials, charge densities, and threshold energies for soft-x-ray transitions. Scattering states in the relaxed potentials are used in spectrum calculations. We report cross sections found for Li, Na, Mg, and Al and compare them with experimental results. Moreover, we calculate the exponents $\ensuremath{\alpha}$, ${\ensuremath{\alpha}}_{0}$, and ${\ensuremath{\alpha}}_{1}$ essential to the threshold theory of Mahan and Nozi\`eres and de Dominicis (MND). For Li, Mg, and Al we get agreement with experimental results for $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}_{0}$. We get no agreement for Na but we attribute this not to a failure in MND theory but to an inadequacy in our model when used to find low-energy zero-angular-momentum valence-electron wave functions.

A Note on the Operator Compact Implicit Method for the Wave Equation

In a previous paper a fourth order compact implicit scheme was presented for the second order wave equation. A very efficient factorization technique was developed when only second order terms were present. In this note we implement the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present. The resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present. For this more general operator compact implicit spatial approximation the same factorization as in our previous paper is developed.

A note on the operator compact implicit method for the wave equation

In a previous paper a fourth order compact implicit scheme was presented for the second order wave equation. A very efficient factorization technique was developed when only second order terms were present. In this note we implement the operator compact implicit spatial discretization method for the second order wave equation when first order terms are present. The resulting algorithm is completely analogous to the compact implicit algorithm when lower order terms were not present. For this more general operator compact implicit spatial approximation the same factorization as in our previous paper is developed.

Continental Shelf Waves and Alongshore Variations in Bottom Topography and Coastline

The effects of alongshore variations in bottom topography and coastline on the wind-stress-forced barotropic motion over a continental shelf and slope are studied. Perturbation methods are used to obtain solutions for forced and free continental shelf waves on an idealized continental shelf and slope with small-amplitude alongshore variations in topography. The relevant alongshore scales, set by the wind stress and by the bottom and coastline topography, are assumed to be greater than the shelf-slope width. This enables the resulting motion to be treated in the long-wave nondispersive limit. As a result, the alongshore and time-dependent behavior of the perturbation flow is governed by a forced, first order wave equation, with terms from the interaction of the basic, lowest order flow with the bottom and coastline topography acting as the forcing function. To clarify the effects of topography alone, problems are considered where a uniform wind stress forces a basic unperturbed flow which is independent of the alongshore coordinate. In one example, a steady, alongshore-independent basic flow is established impulsively by a delta function application of the wind stress. The perturbation flow adjusts to the alongshore variations in topography through the propagation of disturbances as free continental shelf waves. There is an eventual establishment in the region of variable topography of a steady-state motion which follows contours of constant depth. Other problems in which single mode free shelf wave disturbances of limited alongshore extent propagate into regions of different topography are studied also. The basic disturbance is found to travel at the local wave speed with its cross shelf modal structure described by the local eigenfunctions. As a region of varying bottom topography is crossed, disturbances in other modes are generated. General features of this scattering process are examined for limiting cases where the alongshore scale of the disturbance is greater than, or less than, the scale of the topographic feature.

L p -Absch�tzungen und klassische L�sungen f�r nichtlineare Wellengleichungen. I

This paper deals with the global classical solvability of the Cauchy problem for nonlinear wave equations of higher order of the following form where Δ denotes the Laplacian, ζ a real constant, m a positive integer, and f the nonlinearity. Using results of the first part [6] concerning estimations for the kernel of the associated integral equation a priori estimates for certain LP-norms of the local regular solution are established for higher space dimensions and for a class of nonlinearities satisfying certain growth and positivity conditions. This leads to completely regular solutions by means of the Sobolev embedding theorems.

The Riemann problem for certain classes of hyperbolic systems of conservation laws

is called a Riemann problem. The classical method of solution of the Riemann problem is based on the construction of the shock and wave curves of (1.1). However, these curves can be constructed only if the shock admissibility conditions are known a priori. For this reason, the method has been applied so far only for genuinely nonlinear systems [I, 2J and the system generated by the second order wave equation [3,4]. (For results published after the completion of this work, see [9].) Our investigation here is based on a different approach, which was intro-

A Model Describing the Unsteady Transport of Substrate to Tissue from the Microcirculation

A mathematical model of the interrelationships of substrate concentrations in blood and tissue is given. This model describes (a) unsteady capillary transport by convection, (b) the unsteady substrate exchange kinetics within the capillary, (c) diffusion across a finitely permeable capillary wall and (d) unsteady diffusion and consumption of substrate within the tissue space. The mathematical model is a parabolic partial differential equation with its principal boundary condition (i.e., at the capillary wall) coupled to a system of first order wave equations. These wave equations, which describe the capillary conditions, are “stiff” along their characteristics. The model is treated indirectly by considering the solution of a related initial-boundary value problem which has the solution of the pure boundary value problem as its asymptotic (in time t) limit. An unconditionally stable implicit scheme is developed by coupling a locally one dimensional method (for the parabolic equation) with an implicit scheme for solving the transport equations. The numerical convergence is shown to be first order in the mesh size. The methods and results are illustrated with steady and unsteady flows using parameters of physiologic interest.

Magnetic moments for particles described by a class of relativistic wave equations

We compute the magnetic moments predicted by the general linear, finite-component, manifestly covariant relativistic wave equation of first order, $(\ensuremath{-}i{\ensuremath{\beta}}^{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\mu}}+1)\ensuremath{\psi}(x)=0$, with Hermitian ${\ensuremath{\beta}}^{0}$, and minimal coupling to an external electromagnetic field. Here the wave function is assumed to transform according to an arbitrarily given finite-dimensional representation of the Lorentz group. The $g$ factors depend not only on the transformation law of the wave function but also on the mass spectrum of wave equation.