Velocity field Fourier-transformed

At any fixed time t, the velocity field can be recovered by an inverse Fourier transform ... 

As for the velocity field, the (discrete) Fourier-transformed scalar field AK can be defined by... 

Figure 7.1 Most modern NMR techniques are based on the fact, that the phase (p of the precessing transverse magnetisation M t) kann be measured. By use of the Fourier transformation the phase provides access to NMR spectra, images, and parameters of translational motion like velocity v and acceleration a. Spectroscopic parameters as well as components of translational velocity and acceleration can be used for generating contrast in NMR imaging. In the drawing the magnetisation M(t) has been generated from Mz by use of a 90° pulse of the B1 radio-frequency (rf) field in y direction...

The angular velocity and angular momentum acfs themselves are important to any dynamical theory of molecular liquids but are very difficult to extract directly from spectral data. The only reliable method available seems to be spin-rotation nuclear magnetic relaxation. (An approximate method is via Fourier transformation of far-infrared spectra.) The simulated torque-on acfs in this case become considerably more oscillatory, and, which is important, the envelope of its decay becomes longer-lived as the field strength increases. This is dealt with analytically in Section III. In this case, computer simulation is particularly useful because it may be used to complement the analytical theory in its search for the forest among the trees. Results such as these for autocorrelation functions therefore supplement our... 

It is usual in laminar mixing simulations to represent the flow using tracer trajectories. The computation of such flow trajectories in a coaxial mixer is more complex than in traditional stirred tank modelling due to the intrinsic unsteady nature of the problem (evolving topology, flow field known at a discrete number of time steps in a Lagrangian frame of reference). Since the flow solution is periodic, a node-by-node interpolation using a fast Fourier transform of the velocity field has been used, which allowed a time continuous representation of the flow to be obtained. In other words, the velocity at node i was approximated... 

Alternatively, the fluctuating velocity field can be characterized by the energy spectrum defined as the Fourier transform of the two-point velocity correlation integrated over a spherical shell of wavenumbers with magnitude k ... 

Here, s is the spatial coordinate along the line of sight, and and I are the mean quadratic turbulent velocity and the correlation length of the velocity field, respectively. The correlation length I defines the length scale of the stochastic velocity variation. Since the spatial power spectrum of the velocity field is the Fourier transform of the correlation function, eq. (10.13) implies a one-dimensional power spectrum... 

VmaY = the frequency at which the band has a maximum fimax = the intensity of the absorption Ho = the applied magnetic field g = the spectroscopic splitting factor r = the distance between protons P = the angle between a line joining the protons and Hq S2 = the mean-square deviation of the field firom the center of the line Hq Mn = the mass of a neutron particle A = the wave length of a neutron beam V = the partiele velocity A (= X2 - xi) = changing path distance r = is the reflection coefficient T = the transmission coefficient of the beam splitter yl(v) = the fi quency distribution 1(D) and B(n) = orthogonal fimctions F y) = the fi-equency distribution N = number of points in a Fourier Transform 9 = a set of normal coordinates... 

The Kingdon trap, an ion trapping device that consists of an outer barrel-like electrode and a coaxial inner spindle-like electrode that form an electrostatic field with quadro-logarithmic potential distribution. The frequency of harmonic oscillations of the orbitally trapped ions along the axis of the electrostatic field is independent of the ion velocity and is inversely proportional to the square root of m/z so that the trap can be operated as a mass analyser using image current detection and Fourier transformation of the time domain signal. 

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