Laplace's equation in a quarter space as the limit of the wave equation.
Wave equation in quarter space, showing particular solution of Laplace equation with wave velocity increase
Bobisud, L. E.↗
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Wave equation for superconductor.
Wave equation for electron pairs under self consisting field in interacting fermions system, obtaining Landau-Ginzberg type equation
Heims, S. P.↗
Noncausal effects in relativistic wave equations
Manifestly covariant wave equations describing particles with a unique mass and spin can, for certain types of external-field interactions, possess noncausal solutions. The paper reports a procedure for applying Pierce decomposition of all six couplings expressed in the Duffin-Kemmer-Petiau spin-0 formula. This treatment converts the equation to a form in which the causal properties are apparent. The conditions under which the causal properties of higher-spin equations can be made manifest are discussed.
Guertin, R. F.↗
Well-posedness of one-way wave equations and absorbing boundary conditions
A one-way wave equation is a partial differential which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. The construction of such equations can be reduced to the approximation of the square root of (1-s sup 2) on -1, 1 by a rational function r(s) = p sub m (s)/q sub n(s). Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r(s) interpolates the square root of (1-s sup 2) at sufficiently many points in (-1,1), then well-posedness is assured. It follows that absorbing boundary conditions based on Pade approximation are well-posed if and only if (m, n) lies in one of two distinct diagonals in the Pade table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived from Chebyshev or least-squares approximation.
Trefethen, L. N.↗
The existence and stability of nonlinear wave equations
Second order nonlinear wave equation solution on Hilbert space with stability and uniqueness
Pao, C. V.↗
An Operator Method for Field Moments from the Extended Parabolic Wave Equation and Analytical Solutions of the First and Second Moments for Atmospheric Electromagnetic Wave Propagation
The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.
Manning, Robert M.↗
Heating of the solar chromosphere and corona. I - Generalized inhomogeneous wave equation for magnetoacoustic motions
The generalized inhomogeneous wave equation that governs magnetoacoustic, vortical, and thermal motions in compressible fluids and that is applicable to the problem of heating of the solar chromosphere and corona is obtained. The effects of kinematic and bulk viscosity, heat conduction, Joule dissipation, and magnetic diffusivity are included. Under the usual assumptions, the generalized wave equation reduces to the well-known equations of Lighthill, Kulsrud, Phillips, and others. The major problems encountered in applying Lighthill's (1952) mechanism to sound generation in turbulent media are reviewed for both the subsonic and supersonic cases.
Anand, S. P. S.↗
Evidence for self-refraction in a convergence zone: NPE (Nonlinear progressive wave equation) model results
The nonlinear progressive wave equation (NPE) model was developed by the Naval Ocean Research and Development Activity during 1982 to 1987 to study nonlinear effects in long range oceanic propagation of finite amplitude acoustic waves, including weak shocks. The NPE model was applied to propagation of a generic shock wave (initial condition provided by Sandia Division 1533) in a few illustrative environments. The following consequences of nonlinearity are seen by comparing linear and nonlinear NPE results: (1) a decrease in shock strength versus range (a well-known result of entropy increases at the shock front); (2) an increase in the convergence zone range; and (3) a vertical meandering of the energy path about the corresponding linear ray path. Items (2) and (3) are manifestations of self-refraction.
Mcdonald, B. Edward↗
Long-Time Numerical Integration of the Three-Dimensional Wave Equation in the Vicinity of a Moving Source
We propose a family of algorithms for solving numerically a Cauchy problem for the three-dimensional wave equation. The sources that drive the equation (i.e., the right-hand side) are compactly supported in space for any given time; they, however, may actually move in space with a subsonic speed. The solution is calculated inside a finite domain (e.g., sphere) that also moves with a subsonic speed and always contains the support of the right-hand side. The algorithms employ a standard consistent and stable explicit finite-difference scheme for the wave equation. They allow one to calculate tile solution for arbitrarily long time intervals without error accumulation and with the fixed non-growing amount of tile CPU time and memory required for advancing one time step. The algorithms are inherently three-dimensional; they rely on the presence of lacunae in the solutions of the wave equation in oddly dimensional spaces. The methodology presented in the paper is, in fact, a building block for constructing the nonlocal highly accurate unsteady artificial boundary conditions to be used for the numerical simulation of waves propagating with finite speed over unbounded domains.
Ryabenkii, V. S.↗
Shock wave equation of state of muscovite
Shock wave data were obtained between 20 and 140 GPa for natural muscovite obtained from Methuen Township (Ontario), in order to provide a shock-wave equation of state for this crustal hydrous mineral. The shock equation of state data could be fit by a linear shock velocity (Us) versus particle velocity (Up) relation Us = 4.62 + 1.27 Up (km/s). Third-order Birch-Murnaghan equation of state parameters were found to be K(OS) = 52 +/-4 GPa and K-prime(OS) = 3.2 +/-0.3 GPa. These parameters are comparable to those of other hydrous minerals such as brucite, serpentine, and tremolite.
Sekine, Toshimori↗
Lyapunov functionals for a class of wave equations.
Liapunov functionals guaranteeing asymptotic stability for wave equations by constructing Liapunov functions for ordinary differential equations
Buis, G. R.↗
Stability and accuracy analysis of some fully-discrete algorithms for the one-dimensional second-order wave equation
The present investigation is concerned with some basic results for a predictor-multicorrector algorithm applied to the one-dimensional wave equation, giving particular attention to so-called 2-pass explicit schemes in which both lumped and coupled mass matrices are employed. In an assessment of the accuracy and stability properties of the algorithms, use is made of the one-dimensional, second-order wave equation. The maximum stable time step of the lumped right-hand-side mass, 2-pass explicit algorithm is twice that of the 1-pass explicit algorithm. Improved accuracy is obtained by employing higher-order, or consistent, right-hand-side, mass.
Hughes, T. J. R.↗
An approximate method for solving the wave equation in an inhomogeneous medium.
Approximate method for wave equation with dielectric variation in propagation direction
Kinney, R. A.↗
Working With the Wave Equation in Aeroacoustics: The Pleasures of Generalized Functions
The theme of this paper is the applications of generalized function (GF) theory to the wave equation in aeroacoustics. We start with a tutorial on GFs with particular emphasis on viewing functions as continuous linear functionals. We next define operations on GFs. The operation of interest to us in this paper is generalized differentiation. We give many applications of generalized differentiation, particularly for the wave equation. We discuss the use of GFs in finding Green s function and some subtleties that only GF theory can clarify without ambiguities. We show how the knowledge of the Green s function of an operator L in a given domain D can allow us to solve a whole range of problems with operator L for domains situated within D by the imbedding method. We will show how we can use the imbedding method to find the Kirchhoff formulas for stationary and moving surfaces with ease and elegance without the use of the four-dimensional Green s theorem, which is commonly done. Other subjects covered are why the derivatives in conservation laws should be viewed as generalized derivatives and what are the consequences of doing this. In particular we show how we can imbed a problem in a larger domain for the identical differential equation for which the Green s function is known. The primary purpose of this paper is to convince the readers that GF theory is absolutely essential in aeroacoustics because of its powerful operational properties. Furthermore, learning the subject and using it can be fun.
Farassat, F.↗
The wave equation and the green's dyadic for bounded magnetoplasmas.
Separation method solution of dyadic-vector Helmholtz equation representing electromagnetic wave propagation in bounded magnetoplasmas
HELMHOLTZ EQUATION↗
Well-posedness of one-way wave equations and absorbing boundary conditions
The one-way wave equations occurring in geophysics, underwater acoustics, and numerical studies involving absorbing boundary conditions are characterized analytically. The conditions under which such equations are well posed are obtained by examining the rational functions used to reduce them. Results are presented for Chebyshev, Pade, least-squares, and symmetric least-squares approximations.
Trefethen, Lloyd N.↗
Transient difference solutions of the inhomogeneous wave equation: Simulation of the Green's function
A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.
Baumeiste, K. J.↗
Transient difference solutions of the inhomogeneous wave equation - Simulation of the Green's function
A time-dependent finite difference formulation to the inhomogeneous wave equation is derived for plane wave propagation with harmonic noise sources. The difference equation and boundary conditions are developed along with the techniques to simulate the Dirac delta function associated with a concentrated noise source. Example calculations are presented for the Green's function and distributed noise sources. For the example considered, the desired Fourier transformed acoustic pressures are determined from the transient pressures by use of a ramping function and an integration technique, both of which eliminates the nonharmonic pressure associated with the initial transient.
Baumeister, K. J.↗