Double Fourier Transformation Research Articles

Double fourier transformation in high-resolution NMR

Techniques are described for displaying NMR responses as a function of two independent frequency parameters, with particular emphasis on the presentation of carbon-13 spectra with the chemical shift information in one dimension and the proton-carbon splittings in the second dimension. The resolving power for these spin multiplets can be significantly better than in conventional NMR, since the linewidths are determined principally by spin-spin relaxation effects rather than by field inhomogeneity. Modulation is imposed on carbon-13 spin echoes by gating or pulsing the proton-irradiation field, and is analyzed by two successive stages of Fourier transformation. A theoretical analysis, confirmed by experiment, indicates that the profiles of the resulting resonances are an admixture of absorption and dispersion modes in a proportion that changes rapidly as a function of the second frequency parameter. Echoes acquired in the presence of wideband proton decoupling lead to a superposition of resonances which (in the case of weak proton-proton coupling) presents an absorption mode profile in both frequency dimensions.

Comment on "Coherent and incoherent scattering laws of liquid argon"

Results on the width of the self-correlation function $\ensuremath{\rho}(t)$ for liquid argon as determined recently by the method of isotopic mixtures and earlier by the technique of double Fourier transformation are compared and commented on. The excellent agreement found between $\ensuremath{\rho}(t)$ obtained by the two different methods is used to demonstrate the reliability of the latter technique. The fact that this technique can be applied for practically all monatomic liquids in contrast to the method of isotopic mixtures, is emphasized.

Two-dimensional transport in a quarter space: Uniform-source problem

Abstract Double Fourier transformation, the Boucher decomposition, and extensive application of contour integration in two complex variables are used to determine the unique solution to the quarter-space problem with a uniform source. Numerical calculations for a moderately absorbing quarter space are presented. Diffusion-theory results are also given for comparison.

Milne's problem for two-dimensional transport in a quarter space

Double Fourier transformation of the one-speed transport equation is used to determine a family of solutions to the homogeneous (Milne's) problem for the quarter space. All contributions to the density asymptotically away from the corner are determined explicitly by considering two appropriate half-space problems and by extensive use of contour integration in two complex variables. An iterative scheme previously derived may be used to determine the correction near the corner.

A derivation of the Langevin equation for shear viscosity in monatomic liquids

The Langevin equation is derived for a system of interacting molecules by a double Fourier transformation of the Hamiltonian. The transformations are with respect to the molecular equilibrium positions and the deviations, respectively. The potential energy term is then the generator of mode interactions to all orders. For the regime of large anharmonicities part of the “force” is shown to be proportional to the time derivative of the momentum. This leads to the Langevin equation.

Body waves as normal and leaking modes. Part I: Introduction

abstract Realistic artificial seismograms may be computed by considering body waves as sums of normal or leaking modes of surface waves: the S wave and those arriving after S may be considered as sums of higher normal modes of Rayleigh waves (RiN) and Love waves (LiN); in this case the apparent velocity c < βn. Earlier arrivals are generally due to the first kind of leaking modes of Rayleigh waves (RiL1) for which βn < c < αn. Deep reflections in seismic prospecting are RiL2 for which c > αn. Synthetic seismograms can be computed by double Fourier transformation in those two last cases. Alternately the method of residues followed by a single Fourier (or Laplace) transformation may be used in all cases. Earth-stretching approximations should give excellent results for SH waves and may give satisfactory results for P and SV waves.

Rectangular Plate with Elastic Hinges and Splits

This investigation considers the problem of deflection and strength of the rectangular plate with elastic hinges and splits under general distribution of loading, and under the boundary conditions in which one pair of opposite edges normal to the direction of elastic hinges and splits are simply supported, the other pair of edges are elastically supported or elastically built-in.We express the influence of elastic hinges on the rectangular plate in impulse function of 3rd order, and that of splits on the plate in impulse function of 4th. order. And we apply double finite Fourier Transformation to solve the equation of equilibrium of the rectangurar plate under the above boundary conditions.

Transient Response of a Dipole Antenna

The transient current in a long dipole antenna excited by a step-function voltage across an infinitesimal center gap has recently been calculated by Wu. A simpler derivation of Wu's result is given using double Fourier transformation.

The Scattering of Thermal Neutrons by Moderators

A comprehensive scheme for the analysis and use of thermal neutron scattering data is described. Experimental work was carrie out with a 4-rotor high speed chopper system operated in conjunction wlth a multicounter multlchannel neutron time-of-flight system. The results are reduced to a scattering law, which is a function S of two variables ( alpha and beta ) representing the momentum and energy transferred in the scattering process. The scattering law can be divided into interference and self-terms; for the application to neutron spectrum calculations it is sufficient to consider the lnterference term as a small correction to - the self-term. The self-term is the double Fourier transformation of the self-correlation function of Van Hove, which represents the motlon of an lndividual atom in the system. This function can be represented as a gaussian distribution of the atomic position plus correction terms. The width of this gaussian distribution is a function of time and its double differential with respect to time is the velocity correlation function for an atom in the system. Thus the velocity correlation function is the function which makes the major contribution to the scatterlng cross sections. In estimating the accuracy to which measurements, analysis, and calculationsmore » of the scattering law need be made, it is necessary to evaluate a sensitivity function for each neutron spectrum problem, and thus for each scattering law a variety of sensitivity functions need to be evaluated. Two slmple examples are discussed which show, separately, the effect of absorption and temperature gradients. (auth)« less

Rectangular Plate with Reinforcements

This investigation considers the problem of deflections and strength of the rectangular plate with reinforcements under general distribution of loading, and under the boundary conditions in which one pair of opposite edges normal to reinforcements are simply supported, the other pair of edges are elastically supported, by reinforcements. And all reinforcements have rigidity against twist as well as against deflection, and those are built in with brackets which have elastic constants for angular displacements.We express the influences of reinforcements on the rectangular plate in Dirac's impulse func tion δ(x, xi)and δ'(x, xi), and we apply the double or single Fourier Transformation to solve the equation of equilibrium of the rectangular plate under the above boundary conditions.

X-ray diffraction by random layers: ideal line profiles and determination of sructure amplitudes from observed line profiles

Warren's calculation of the line profile for diffraction from random layers depends on the use of an approximation function. This can be. avoided, and expressions found for (a) the ideal line profile for slow variation of the structure amplitude, and (b) the variation of the intensity of diffraction as a function of position along the `rod' of high intensity in reciprocal space, in terms of the layer shape and the observed line profile. The ideal line profile is, within a trigonometrical factor, {I(\sigma) = FF{^*}S{_0^{1 \over 2}} \int_{-\infty}^{\infty} A(t)\,\, |t|{^{1 \over 2}} (\cos 2\pi\sigma t + \sin 2\pi\sigma\,\, |t|)\,\, dt,} where S0(= 2 sin θ0/λ) is the perpendicular distance from the origin of the reciprocal lattice to the centre line of the `rod', σ = (sin2 θ − sin2 θ0)/λ sin θ0, F is the structure amplitude, and A(t) is the area common to a layer and its `ghost' shifted a distance t parallel to S0. This expression, evaluated for various layer shapes, gives intensities of diffraction slightly lower than those found by Warren and depending somewhat on the layer shape. In particular, the intensity for large negative volumes of σ is proportional to |σ|−3/2 multiplied by the maximum breadth of the layer. If w is measured along the `rod' from the foot of S0, F(w) F*(w) + F(−w) F*(−w) can be obtained by `unfolding' the observed I(σ) by means of a double Fourier transformation. Diffraction by random layers can thus give more information than diffraction by a perfect crystal, as the latter gives FF* only for integral values of the indices.