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Results for “WKB APPROXIMATION”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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NASA NTRS1992

Theory of magnetohydrodynamic waves - The WKB approximation revisited

The present paper revisits the derivation of the WKB approximation for small-amplitude magnetohydrodynamic waves, allowing for possible spatial variation of the background entropy. The equation of variation of wave amplitude is rederived; it is a bilinear equation which, it turns out, can be recast in the action conservation form. It is shown that this action conservation equation is in fact equivalent to the action conservation law obtained from Lagrangian treatments.

Barnes, Aaron↗
NASA NTRS1994

Transport theory and the WKB approximation for interplanetary MHD fluctuations

An alternative approach, based on a multiple scale analysis, is presented in order to reconcile the traditional Wentzel-Kramer-Brillouin (WKB) approach to the modeling of interplanetary fluctuations in a mildly inhomogeneous large-scale flow with a more recently developed transport theory. This enables us to compare directly, at a formal level, the inherent structure of the two models. In the case of noninteracting, incompressible (Alven) waves, the principle difference between the two models is the presence of leading-order couplings (called 'mixing effects') in the non-WKB turbulence model which are absent in a WKB development. Within the context of linearized MHD, two cases have been identified for which the leading order non-WJB 'mixing term' does not vanish at zero wavelength. For these cases the WKB expansion is divergent, whereas the multiple-scale theory is well behaved. We have thus established that the WKB results are contained within the multiple-scale theory, but leading order mixing effects, which are likely to have important observational consequences, can never be recovered in the WKB style expansion. Properties of the higher-order terms in each expansion are also discussed, leading to the conclusion that the non-WKB hierarchy may be applicable even when the scale separation parameter is not small.

Matthaeus, William H.↗
NASA NTRS1986

Eigenfunction methods in magnetospheric radial-diffusion theory

Complete sets of orthonormal basis functions constructed according to a generalization of the quantum-mechanical WKB approximation can be used to generate a nearly-diagonal matrix representation of the radial-transport operator for ring-current ions in the presence of radial diffusion and charge exchange. The resulting eigenfunctions (constructed by weighting the basis functions in proportion to the respective components of the eigenvectors of the matrix representation) and eigenvalues provide a spatial and temporal description of the evolving phase-space density during and following a magnetospheric disturbance (e.g., a magnetic storm). A linear superposition of the basis functions can also be used to eliminate any discrepancy between the steady-state solution of the transport equation and the appropriate WKB approximation of this steady-state solution.

Schulz, Michael↗
NASA NTRS1990

Theory of biaxial graded-index optical fiber

A biaxial graded-index fiber with a homogeneous cladding is studied. Two methods, wave equation and matrix differential equation, of formulating the problem and their respective solutions are discussed. For the wave equation formulation of the problem it is shown that for the case of a diagonal permittivity tensor the longitudinal electric and magnetic fields satisfy a pair of coupled second-order differential equations. Also, a generalized dispersion relation is derived in terms of the solutions for the longitudinal electric and magnetic fields. For the case of a step-index fiber, either isotropic or uniaxial, these differential equations can be solved exactly in terms of Bessel functions. For the cases of an istropic graded-index and a uniaxial graded-index fiber, a solution using the Wentzel, Krammers and Brillouin (WKB) approximation technique is shown. Results for some particular permittivity profiles are presented. Also the WKB solutions is compared with the vector solution found by Kurtz and Streifer. For the matrix formulation it is shown that the tangential components of the electric and magnetic fields satisfy a system of four first-order differential equations which can be conveniently written in matrix form. For the special case of meridional modes, the system of equations splits into two systems of two equations. A general iterative technique, asymptotic partitioning of systems of equations, for solving systems of differential equations is presented. As a simple example, Bessel's differential equation is written in matrix form and is solved using this asymptotic technique. Low order solutions for particular examples of a biaxial and uniaxial graded-index fiber are presented. Finally numerical results obtained using the asymptotic technique are presented for particular examples of isotropic and uniaxial step-index fibers and isotropic, uniaxial and biaxial graded-index fibers.

Kawalko, Stephen F.↗
NASA NTRS1988

Tunneling near the peaks of potential barriers - Consequences of higher-order Wentzel-Kramers-Brillouin corrections

The quantum-mechanical tunneling transmission coefficient is derived for energies near the peak of a general potential barrier, to fourth order in the WKB approximation, using a method based on an eighth-order Taylor expansion of the potential near the peak. The result agrees with contour-integral formulations of WKB tunneling. For the Poeschl-Teller potential the WKB series converges rapidly to the known exact transmission coefficient. Higher-order WKB corrections may be important in attempts to determine the internuclear potential by inversion using low-energy fusion cross-section data.

Will, Clifford M.↗
NASA NTRS1973

Alfven waves in spiral interplanetary field

A theoretical study is presented of the Alfven waves in the spiral interplanetary magnetic field. The Alfven waves under consideration are arbitrary, large amplitude, non-monochromatic, microscale waves of any polarization. They superpose on a mesoscale background flow of thermally anisotropic plasma. Using WKB approximation, an analytical solution for the amplitude vectors is obtained as a function of the background flow properties: density, velocity, Alfven speed, thermal anisotropy, and the spiral angel. The necessary condition for the validity of the WKB solution is discussed. The intensity of fluctuations is calculated as a function of heliocentric distance. Relative intensity of fluctuations as compared with the magnitude of the background field has its maximum in the region near l au. Thus outside of this region, the solar wind is less turbulent.

Whang, Y. C.↗
NASA NTRS1994

Time-dependent variational approach in terms of squeezed coherent states: Implication to semi-classical approximation

A general framework for time-dependent variational approach in terms of squeezed coherent states is constructed with the aim of describing quantal systems by means of classical mechanics including higher order quantal effects with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. The Maslov phase occurring in a semi-classical quantization rule is investigated in this framework. In the limit of a semi-classical approximation in this approach, it is definitely shown that the Maslov phase has a geometric nature analogous to the Berry phase. It is also indicated that this squeezed coherent state approach is a possible way to go beyond the usual WKB approximation.

Tsue, Yasuhiko↗
NASA NTRS1987

Approximate penetration factors for nuclear reactions of astrophysical interest

The ranges of validity of approximations of P(l), the penetration factor which appears in the parameterization of nuclear-reaction cross sections at low energies and is employed in the extrapolation of laboratory data to even lower energies of astrophysical interest, are investigated analytically. Consideration is given to the WKB approximation, P(l) at the energy of the total barrier, approximations derived from the asymptotic expansion of G(l) for large eta, approximations for small values of the parameter x, applications of P(l) to nuclear reactions, and the dependence of P(l) on channel radius. Numerical results are presented in tables and graphs, and parameter ranges where the danger of serious errors is high are identified.

Humblet, J.↗
NASA NTRS1979

The adiabatic semiclassical perturbation theory for vibrationally inelastic scattering. I - Collinear calculations. II - Three-dimensional treatment

A semiclassical approximation to treat vibrationally inelastic scattering is developed. The vibrational basis set used is adiabatic with respect to a reference potential which is chosen to be as close as possible to the true potential and also gives easily obtainable solutions to the vibrational wave equation. The radial wave functions are obtained using the WKB approximation, and the coupled Schroedinger equations are solved by a first-order perturbation method to yield a phase shift matrix which is exponentiated to give the full scattering matrix. Results were obtained for all the cases computed by Secrest and Johnson and by Clark and Dickinson, and the agreement is better than 10% for half of the cross-sections and rarely off by more than a factor of 2.

Cross, R. J., Jr.↗
NASA NTRS1985

Black hole normal modes - A semianalytic approach

A new semianalytic technique for determining the complex normal mode frequencies of black holes is presented. The method is based on the WKB approximation. It yields a simple analytic formula that gives the real and imaginary parts of the frequency in terms of the parameters of the black hole and of the field whose perturbation is under study, and in terms of the quantity (n + 1/2), where n = 0, 1, 2,... and labels the fundamental mode, first overtone mode, and so on. In the case of the fundamental gravitational normal modes of the Schwarzschild black hole, the WKB estimates agree with numerical results to better than 7 percent in the real part of the frequency and 0.7 percent in the imaginary part, with the relative agreement improving with increasing angular harmonic. Carried to higher order the method may provide an accurate and systematic means to study black hole normal modes.

Schutz, B. F.↗
NASA NTRS1988

An invariant asymptotic formula for solutions of second-order linear ODE's

An invariant-matrix technique for the approximate solution of second-order ordinary differential equations (ODEs) of form y-double-prime = phi(x)y is developed analytically and demonstrated. A set of linear transformations for the companion matrix differential system is proposed; the diagonalization procedure employed in the final stage of the asymptotic decomposition is explained; and a scalar formulation of solutions for the ODEs is obtained. Several typical ODEs are analyzed, and it is shown that the Liouville-Green or WKB approximation is a special case of the present formula, which provides an approximation which is valid for the entire interval (0, infinity).

Gingold, H.↗
NASA NTRS1982

Stratospheric warmings diagnosed using the transformed Eulerian-mean equations and the effect of the mean state on wave propagation

Terms for the transformed Eulerian equations are calculated in order to characterize the phenomenom of sudden stratospheric warming. The transformed diagnostics are applied to data for warmings during Dec. and Jan. 1976-1977, as well as cross sections for the directions of the Eiliassen-Palm (EP) fluxes and residual mean meridional circulations. The convergence of the EP flux was determined to provide a strong approximation to the total effect of waves in forcing the zonal mean flow. The EP fluxes change from an upward and equatorward direction to an upward and poleward direction during the warmings, and indications are reported that the effect is due to a feedback on wave propagation of an evolving mean flow. Ray paths in the meridional plane are computed for different mean wind fields to determine the direction of wave propagation according to linear theory based on the WKB approximation.

Oneill, A.↗