Outgoing Radiation Condition Research Articles

The solution of exterior interface problems using a variational method with Lagrange multipliers

A numerical method is proposed and analyzed for approximately solving a class of exterior interface problems in three dimensions. The solution satisfies the Helmholtz equation near infinity as well as an outgoing or incoming radiation condition. The interface conditions involve jumps in both the function and the normal derivative of the function across a two-dimensional surface Γ. The former condition is an essential interface condition and is treated using Lagrange multipliers. The unboundedness of the exterior domain is treated by introducing an artificial sphere Γ R of sufficiently large radius R and an approximate local radiation boundary condition on this sphere. A variational formulation is obtained for this approximate problem that takes into account the approximate boundary condition on Γ R as well as the natural interface condition (i.e., the jump in the normal derivative on Γ). The resulting variational problem is discretized using the finite element method. It is proved that the discretization error is optimal in the sense of the approximation properties of the finite element subspaces. The discrete Lagrange multiplier formulation is generalized to take into account various perturbations of the method that often occur in its implementation. The error due to the perturbation is analyzed. It is also proved that the error due to the artificial boundary and approximate boundary condition is of order O ( R −2 ) for large R .

Direct and inverse wave propagation theories and the factorized Helmholtz equation

The reduced scalar Helmholtz equation for a transversely inhomogeneous half-space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial-value plane defines a direct acoustic propagation model. This elliptic formulation admits a factorization and is subsequently equivalent to a first-order Weyl pseudo-differential equation which is recognized as an extended parabolic propagation model. Perturbation treatments of the appropriate Weyl composition equation result in a systematic development of approximate wave theories while exact inversions for several nontrivial profiles provide for an analysis of strong refractive and diffractive effects. The analysis, in a natural manner, provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic, seismic, and optical studies. Backscatter effects in both the direct and inverse problems can be included through weak-backscatter and spatial-dimensional (imbedding) perturbation methods. Moreover, the n-dimensional reduced scalar Helmholtz equation for the transversely inhomogeneous medium is naturally related to parabolic propagation models through (1) the above mentioned n-dimensional extended parabolic (Weyl pseudo-differential) equation and (2) an imbedding in an (n + 1)-dimensional parabolic (Schrödinger) equation. The first relationship provides the basis for the parabolic-based Hamiltonian phase space path integral representation of the half-space propagator. The second relationship provides the basis for the elliptic-based path integral representations associated with Feynman/Fradkin, Feynman/Garrod, and Feynman/Dewitt-Morette. The path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework in the direct and inverse formulations for dynamical approximations, resolution of the square root operator, and the concepts of an underlying stochastic process and free motion on curved spaces, the wave equation and path integral analysis foreshadows computational algorithms, extension to (1) the vector formulation appropriate for elastic media and (2) the bilinear formulation appropriate for acoustic field coherence, and application to wave propagation in random media.

Open Boundary Conditions for Dispersive Waves

Abstract Approximate outgoing radiation conditions have been widely used at open boundaries in dispersive wave computations. Exact outgoing radiation conditions are constructed here for infinitesimal surface inertia-gravity waves, barotropic Rossby waves, and non-hydrostatic internal gravity waves incident at infinite plane open boundaries. They are also constructed for infinitesimal two-dimensional surface gravity waves incident at a square open boundary, a finite straight open boundary in a channel, and a circular open boundary, The constructions use Laplace transforms, various Fourier transforms, and a wave field splitting technique. The exact outgoing radiation conditions all involve weighted space and time averages. Their numerical implementation would be most complex and would require computer storage approaching that which one is trying to avoid by the introduction of open boundaries. Alternatives are discussed.

Equations of hydrodynamics in general relativity in the slow motion approximation

By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.

Satisfaction of deDonder and Trautman conditions by radiative solutions of the Einstein field equations

We show that when the Einstein field equations for the gravitational field are modified by imposing the deDonder coordinate conditions these equations can be ‘solved’ in terms of source functions using the retarded Green's function for the d'Alembertian in flat space. The ‘solution’, which becomes an actual solution in the fast-motion approximation, is shown to satisfy the deDonder conditions if and only if the stress-energy tensor of the sources of the gravitational field is covariantly conserved. It is also shown to satisfy the Trautman outgoing radiation condition.

Gravitational waves in general relativity. VI. The outgoing radiation condition

A covariant formulation of the outgoing radiation condition for gravitational fields is proposed. The condition is based on a detailed examination of the geometry of null lines and of the algebraic and differential properties of the Riemann tensor. It relates the absence of incoming radiation, in a gravitational field with bounded sources and Euclidean topology, to the asymptotic behaviour of the Riemann tensor. Fields that are algebraically special in the Petrov classification are highly special examples of fields obeying the suggested condition.