Fourier double series

Consider the general class of laminated rectangular plates that are simply supported along edges x = 0, x = a, y = 0, and y = b and subjected to a distributed transverse load, p(x,y). In Figure 5-8. The transverse load can be expanded in a double Fourier sine series ... 

To compute the interacting RPA density-response function of equation (32), we follow the method described in Ref. [66]. We first assume that n(z) vanishes at a distance Zq from either jellium edge [67], and expand the wave functions (<) in a Fourier sine series. We then introduce a double-cosine Fourier representation for the density-response function, and find explicit expressions for the stopping power of equation (36) in terms of the Fourier coefficients of the density-response function [57]. We take the wave functions <)),(7) to be the eigenfunctions of a one-dimensional local-density approximation (LDA) Hamiltonian with use of the Perdew-Zunger parametrization [68] of the Quantum Monte Carlo xc energy of a uniform FEG [69]. 

For the cases where the transverse load can be expanded in the double Fourier sine series, the load q (x, y) is given by ... 

The expansion of the differential cross section in a Fourier sine series allows one to obtain very accurate differential cross sections from quasiclassical trajectories. In turn, this permits precise comparison of the reaction dynamics on different potential energy surfaces and at different energies. We plan to extend this work to different energies and different systems, and we also hope to use similar techniques to obtain accurate double-differential cross sections. 

For quadnipolar nuclei, the dependence of the pulse response on Vq/v has led to the development of quadnipolar nutation, which is a two-dimensional (2D) NMR experiment. The principle of 2D experiments is that a series of FIDs are acquired as a fimction of a second time parameter (e.g. here the pulse lengdi applied). A double Fourier transfomiation can then be carried out to give a 2D data set (FI, F2). For quadnipolar nuclei while the pulse is on the experiment is effectively being carried out at low field with the spin states detemiined by the quadnipolar interaction. In the limits Vq v the pulse response lies at v and... 

Molecules that are composed of atoms having a maximum valency of 4 (as essentially all organic molecules) are with a few exceptions found to have rotational profiles showing at most three minima. The first three terms in the Fourier series eq. (2.9) are sufficient for qualitatively reproducing such profiles. Force fields which are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos (3u ) and double bonds only cos (2u))). 

Although the idea of generating 2D correlation spectra was introduced several decades ago in the field of NMR [1008], extension to other areas of spectroscopy has been slow. This is essentially on account of the time-scale. Characteristic times associated with typical molecular vibrations probed by IR are of the order of picoseconds, which is many orders of magnitude shorter than the relaxation times in NMR. Consequently, the standard approach used successfully in 2D NMR, i.e. multiple-pulse excitations of a system, followed by detection and subsequent double Fourier transformation of a series of free-induction decay signals [1009], is not readily applicable to conventional IR experiments. A very different experimental approach is therefore required. The approach for generation of 2D IR spectra defined by two independent wavenumbers is based on the detection of various relaxation processes, which are much slower than vibrational relaxations but are closely associated with molecular-scale phenomena. These slower relaxation processes can be studied with a conventional... 

Equations. For a ID two-phase structure Porod s law is easily deduced. Then the corresponding relations for 2D- and 3D-structures follow from the result. The ID structure is of practical relevance in the study of fibers [16,139], because it reflects size and correlation of domains in fiber direction . Therefore this basic relation is presented here. Let er be50 the direction of interest (e.g., the fiber direction), then the linear series expansion of the slice r7(r)]er of the corresponding correlation function is considered. After double derivation the ID Fourier transform converts the slice into a projection / Cr of the scattering intensity and Porod s law... 

The function is called a double Fourier series. 

Experiments were performed using a titanium sapphire laser oscillator capable of producing pulses with bandwidths up to 80 nm FWHM. The output of the oscillator was evaluated to make sure there were no changes in the spectrum across the beam and was compressed with a double prism pair arrangement. The pulse shaper uses prisms as the dispersive elements, two cylindrical concave mirrors, and a spatial light modulator (CRI Inc. SLM-256), composed of two 128-pixel liquid crystal masks in series. The SLM was placed at the Fourier plane [5]. After compression and pulse shaping, 200 pJ pulses were used to interrogate the samples. 

To conclude, the second dimension is introduced if the switching time ti (Fig. 2.48) is incremented in a series of single experiments so as to reach all possible double quantum frequencies vDQ within a sample molecule by the reciprocals l/t1. Again, the acquired FID signals will depend on two variable times t1 and t2, respectively. A first Fourier transformation in the t2 domain generates 13C — 13C satellite spectra. The corresponding AB or AX type doublet pairs, however, are modulated by the individual double quantum frequencies which characterize each AB or AX pair. The second Fourier transformation in the tl domain liberates the double quantum frequency as the second dimension Maximum AB or AX 13C—13C subspectra are observed at the corresponding double quantum frequencies, so that each doublet appears with unique coordinates,... 

In fact, the frequency ofthe torsional oscillation mode V4 is found to be more than double that ofthe ground state. The frequency ofthe torsional oscillation mode was reevaluated by Mukheijee et al [56], using a very accurate representation of the one-dimensional vibrational Hamiltonian of the non-rigid rotor in terms of a Fourier series [76-78], and other spectroscopic parameters calculated for the first time taking care of anharmonicity. A new assignment of the experimental spectrum was given. The results are displayed in Table 8. For reference purpose the vibrational frequencies of the ionic states are also listed... 

In a 2D experiment one or more scans are acquired with a delay tl that is incremented in subsequent acquisitions to generate a time domain tl. The time domain tl in conjunction with the acquisition time domain t2 generates a 2D data set that upon double Fourier transform gives a 2D spectrum. In a very simplified view all 2D experiments can be described as series of ID experiments but in practise the situation is rather more complicated because to achieve quadrature detection in both dimensions phase cycling or pulse field gradients must be used. Consequently the processing of 2D data sets depends upon the detection mode and the experimental setup. 

The lowest frequency occurs when n = I and is called the fundamental. Doubling the frequency corresponds to raising the pitch by an octave. Those solutions having values of n > I are known as the overtones. As mentioned previously, one important property of waves is the concept of superposition. Mathematically, it can be shown that any periodic function that is subject to the same boundary conditions can be represented by some linear combination of the fundamental and its overtone frequencies, as shown in Figure 3.8. In fact, this type of mathematical analysis is known as a Fourier series. Thus, while the note middle-A on a clarinet, violin, and piano all have the same fundamental frequency of 440 Hz, the sound (or timbre) that the different instruments produce will be distinct, as shown in Figure 3.9. 

Third term (summing over torsions) represents the energy for twisting a bond due to bond order (e.g., double bonds) and neighboring bonds or lone pairs of electrons. Note that a single bond may have more than one of these terms, such that the total torsional energy is expressed as a Fourier series. 

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