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NASA NTRS1972

On the determination of the harmonic components of the diurnal variation of the thermosphere from satellite drag data

The accuracy of the diurnal and semidiurnal harmonics of density variation determined from satellite drag data is discussed and compared with data from the Jacchia empirical atmospheric model. The phase discrepancies between densities and temperatures, which are more pronounced in summer, and the extrema of the density data are described. The ratio of the semidiurnal to diurnal component of kinetic temperature, determined from Salah and Evans incoherent scatter radar observations, also indicates a seasonal variation. The radar data are compared with the Jacchia model, and a third harmonic is discussed.

Blum, P. W.↗

NASA NTRS1973

Phase matching in second-harmonic generation using artificial periodic structures.

The use of artificial periodic structures, consisting of spatial modulations of the linear and nonlinear susceptibilities of a nonlinear optical medium, to achieve phase matching in second-harmonic generation is analyzed. Dispersion relations and approximate formulas for the second-harmonic fields generated under various conditions are obtained and used to evaluate the experimental situation.

Tang, C. L.↗

NASA NTRS1973

The prevalence of second harmonic radiation in type III bursts observed at kilometric wavelengths.

We present the analysis of 64 type III solar bursts that drifted from 3.5 MHz down to the range 350-50 kHz between March 1968 and February 1970. Bursts arrival times were predicted by a simple model and then compared with observations. The results show that, as the bursts drift, the fundamental often disappears below a certain frequency range while the second harmonic remains. Below about 1 MHz, the second harmonic occurrence predominates. Recognizing this fact we deduce a mean velocity of 0.32c plus or minus 0.02c for the exciter particles, where the uncertainty is the standard error and c the velocity of light in vacuum; the electron density model used is comparable to a solar wind model.

Haddock, F. T.↗

NASA NTRS1974

A finite difference method for the solution of the transonic flow around harmonically oscillating wings

A finite difference method for the solution of the transonic flow about a harmonically oscillating wing is presented. The partial differential equation for the unsteady transonic flow was linearized by dividing the flow into separate steady and unsteady perturbation velocity potentials and by assuming small amplitudes of harmonic oscillation. The resulting linear differential equation is of mixed type, being elliptic or hyperbolic whereever the steady flow equation is elliptic or hyperbolic. Central differences were used for all derivatives except at supersonic points where backward differencing was used for the streamwise direction. Detailed formulas and procedures are described in sufficient detail for programming on high speed computers. To test the method, the problem of the oscillating flap on a NACA 64A006 airfoil was programmed. The numerical procedure was found to be stable and convergent even in regions of local supersonic flow with shocks.

Ehlers, E. F.↗

NASA NTRS1975

Flapping response characteristics of hingeless rotor blades by a gereralized harmonic balance method

Linearized equations of motion for the flapping response of flexible rotor blades in forward flight are derived in terms of generalized coordinates. The equations are solved using a matrix form of the method of linear harmonic balance, yielding response derivatives for each harmonic of the blade deformations and of the hub forces and moments. Numerical results and approximate closed-form expressions for rotor derivatives are used to illustrate the relationships between rotor parameters, modeling assumptions, and rotor response characteristics. Finally, basic hingeless rotor response derivatives are presented in tabular and graphical form for a wide range of configuration parameters and operating conditions.

Peters, D. A.↗

NASA NTRS1975

Harmonic generation at high field strengths - Frequency shifts and saturation phenomena

Optical harmonic generation and mixing in the gas phase has been proposed as a technique for the generation of coherent radiation in the vacuum ultraviolet and soft X-ray spectral region. At the high field strengths required by these processes the interaction between atoms and the electromagnetic field shows intensity-dependent resonances. In this paper we modify harmonic generation theory to include the effect of these frequency shifts. Closed-form expressions for generated dipole moment, absorption probability, and coherence length are presented. The most important consequences of frequency shifts on resonantly enhanced processes are that the pump laser must be tuned away from the small-field resonance frequency, that the conversion efficiency may saturate, and that the dispersion of the medium may change sign. As an example, the generation of 198-A radiation by a five-photon mixing process in Li(+) is considered.

Stappaerts, E. A.↗

NASA NTRS1976

Zonal and tesseral harmonic coefficients for the geopotential function, from zero to 18th order

Zonal and tesseral harmonic coefficients for the geopotential function are usually tabulated in normalized form to provide immediate information as to the relative significance of the coefficients in the gravity model. The normalized form of the geopotential coefficients cannot be used for computational purposes unless the gravity model has been modified to receive them. This modification is usually not done because the absolute or unnormalized form of the coefficients can be obtained from the simple mathematical relationship that relates the two forms. This computation can be quite tedious for hand calculation, especially for the higher order terms, and can be costly in terms of storage and execution time for machine computation. In this report, zonal and tesseral harmonic coefficients for the geopotential function are tabulated in absolute or unnormalized form. The report is designed to be used as a ready reference for both hand and machine calculation to save the user time and effort.

Kirkpatrick, J. C.↗

NASA NTRS1975

The motion of a satellite in resonance with the longitude-dependent harmonics

The solution to the motion of a satellite in an eccentric orbit and in resonance with one or more of the longitude-dependent harmonics of the central planet is developed. The method of solution parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order resonant perturbations due to longitude-dependent harmonics (in terms of Legendre normal elliptic integrals of the first and second kind).

Dallas, S. S.↗

NASA NTRS1977

Gravitational potential energy of the earth: A spherical harmonic approach

A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass-elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic equation agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the mantle and crust of -2.77 x 10 to the twenty-ninth power ergs, an order of magnitude. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3 x 10 to the twentieth power poises is found by assuming the total geothermal flux is due to viscous dissipation. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at one per cent efficiency, then the viscosity is ten to the twenty second power poises, a number preferred by some as the viscosity of the mantle.

Rubincam, D. P.↗

NASA NTRS1976

The motion of a satellite in resonance with the longitude-dependent harmonics

Dallas, S. S.↗

NASA NTRS1977

A harmonic analysis of lunar topography

A global lunar topographic map has been derived from existing earth-based and orbital observations supplemented in areas without data by a linear autocovariance predictor. Of 2592 bins, each 5 deg square, 1380 (64.7% by area) contain at least one measurement. A spherical harmonic analysis to degree 12 yields a mean radius of 1737.53 plus or minus 0.03 km (formal standard error) and an offset of the center of figure of 1.98 plus or minus 0.06 km toward (19 plus or minus 2) deg S, (194 plus or minus 1) deg E. A Bouguer gravity map, derived from a 12-degree free-air gravity model and the present topography data, is presented for an elevation of 100 km above the mean surface. It is confirmed that the low-degree gravity harmonics are determined primarily by surface height variations and only secondarily by lateral density variations.

Bills, B. G.↗

NASA NTRS1977

The motion of a satellite in resonance with the second-degree sectorial harmonic

The solution to the motion of a satellite in an eccentric orbit and in resonance with the second-degree sectorial harmonic of the potential field is developed. The method of solution used parallels the well known von Zeipel method of general perturbations. The solution consists of expressions for the variations of the Delaunay variables. These expressions are composed of the perturbations developed by Brouwer in 1959 for the motion of an artificial satellite plus first-order perturbations due to the second-degree sectorial harmonic (in terms of the Legendre normal elliptic integrals of the first and second kind).

Dallas, S. S.↗

NASA NTRS1979

Characteristics of a Josephson junction harmonic mixer with external pumping

Numerical calculations are carried out on harmonic mixing with an externally pumped Josephson junction. Bias conditions for optimum conversion efficiency are studied for a variety of junction parameters. The predicted noise temperature for harmonic mixing is compared with that for fundamental mixing at the same signal frequency.

Taur, Y.↗

NASA NTRS1980

The NASA/MSFC global reference atmospheric model: MOD 3 (with spherical harmonic wind model)

Improvements to the global reference atmospheric model are described. The basic model includes monthly mean values of pressure, density, temperature, and geostrophic winds, as well as quasi-biennial and small and large scale random perturbations. A spherical harmonic wind model for the 25 to 90 km height range is included. Below 25 km and above 90 km, the GRAM program uses the geostrophic wind equations and pressure data to compute the mean wind. In the altitudes where the geostrophic wind relations are used, an interpolation scheme is employed for estimating winds at low latitudes where the geostrophic wind relations being to mesh down. Several sample wind profiles are given, as computed by the spherical harmonic model. User and programmer manuals are presented.

Justus, C. G.↗

NASA NTRS1979

Gravitational potential energy of the earth - A spherical harmonic approach

A spherical harmonic equation for the gravitational potential energy of the earth is derived for an arbitrary density distribution by conceptually bringing in mass-elements from infinity and building up the earth shell upon spherical shell. The zeroth degree term in the spherical harmonic expansion agrees with the usual expression for the energy of a radial density distribution. The second degree terms give a maximum nonhydrostatic energy in the crust and mantle of -2.77 x 10 to the 29th ergs, an order of magnitude below McKenzie's (1966) estimate. McKenzie's result stems from mathematical error. Our figure is almost identical with Kaula's (1963) estimate of the minimum shear strain energy in the mantle, a not unexpected result on the basis of the virial theorem. If the earth is assumed to be a homogeneous viscous oblate spheroid relaxing to an equilibrium shape, then a lower limit to the mantle viscosity of 1.3 x 10 to the 20th P is found by assuming that the total geothermal flux is due to viscous dissipation of energy. This number is almost six orders of magnitude below MacDonald's (1966) estimate of the viscosity and removes his objection to convection. If the nonequilibrium figure is dynamically maintained by the earth acting as a heat engine at 1% efficiency, then the viscosity is 10 to the 22nd P, a number preferred by Cathles (1975) and Peltier and Andrew (1976) as the viscosity of the mantle.

Rubincam, D. P.↗

NASA NTRS1980

Orthogonality of spherical harmonic coefficients

Orthogonality relations are obtained for the spherical harmonic coefficients of functions defined on the surface of a sphere. Following a brief discussion of the orthogonality of Fourier series coefficients, consideration is given to the values averaged over all orientations of the coordinate system of the spherical harmonic coefficients of a function defined on the surface of a sphere that can be expressed in terms of Legendre polynomials for the special case where the function is the sum of two delta functions located at two different points on the sphere, and for the case of an essentially arbitrary function. It is noted that the orthogonality relations derived have found applications in statistical studies of the geomagnetic field.

Mcleod, M. G.↗

NASA NTRS1980

Corrected formula for the polarization of second harmonic plasma emission

Corrections for the theory of polarization of second harmonic plasma emission are proposed. The nontransversality of the magnetoionic waves was not taken into account correctly and is here corrected. The corrected and uncorrected results are compared for two simple cases of parallel and isotropic distributions of Langmuir waves. It is found that whereas with the uncorrected formula plausible values of the coronal magnetic fields were obtained from the observed polarization of the second harmonic, the present results imply fields which are stronger by a factor of three to four.

Melrose, D. B.↗

NASA NTRS1981

The nonlinear wave equation for higher harmonics in free-electron lasers

The nonlinear wave equation and self-consistent pendulum equation are generalized to describe free-electron laser operation in higher harmonics; this can significantly extend their tunable range to shorter wavelengths. The dynamics of the laser field's amplitude and phase are explored for a wide range of parameters using families of normalized gain curves applicable to both the fundamental and harmonics. The electron phase-space displays the fundamental physics driving the wave, and this picture is used to distinguish between the effects of high gain and Coulomb forces.

Colson, W. B.↗