Sine 2 profile

Figure 5.34. The pulse sequence representation of (a) a single z-axis field gradient pulse and (b) two z-axis gradient pulses applied in opposite directions. Gradient pulses typically have a shaped rather than square profile, such as the sine profile illustrated here (see Section 5.5.4). RP identifies the radiofrequency pulse channel.

Legendre expansion with two terms (2) and elliptic-sine profile (3)... 

A sine-shape has side lobes which impair the excitation of a distinct slice. Other pulse envelopes are therefore more commonly used. Ideally, one would like a rectangular excitation profile which results from a sine-shaped pulse with an infinite number of side lobes. In practice, a finite pulse duration is required and therefore the pulse has to be truncated, which causes oscillations in the excitation profile. Another frequently used pulse envelope is a Gaussian frmction ... 

Combustor performanee is measured by effieieney, the pressure deerease eneountered in the eombustor, and the evenness of the outlet temperature profile. Combustion effieieney is a measure of eombustion eompleteness. Combustion eompleteness affeets fuel eonsumption direetly, sinee the heating value of any unburned fuel is not used to inerease the turbine inlet... 

The uniformity of the eombustor outlet profile affeets the useful level of turbine inlet temperature, sinee the average gas temperature is limited by the peak gas temperature. This uniformity assures adequate nozzle life, whieh depends on operating temperature. The average inlet temperature to the turbine affeets both fuel eonsumption and power output. A large eombustor outlet gradient will work to reduee average gas temperature and eonse-quently reduee power output and effieieney. Thus, the traverse number must have a lower value—between 0.05 and 0.15 in the nozzle. 

First we run the model so as to study the influenee of sereen resistanee on the overall flow patterns and on the maldistribution. The resulting profiles of inward radial veloeity at the inner sereen aeross the eatalyst bed appear in Figure 10-13 for different sereen resistanees. It ean be seen that higher sereen resistanee leads to more-uniform flow, as one would expeet. The existing sereens (with resistanee eoeffieients C2 of 2 X 10 /m) appear to be satisfaetory, sinee the deviations experieneed are less than 10%. 

Obtain the momentum equation for an element of boundary layer. If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 

Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

Obtain the boundary layer thickness and its displacement thickness as a function of the distance from the leading edge of Ihe surface, when the velocity profile is expressed as a sine function. 

Theoretical level populations. Sinee there are population variations on time seale shorter than some level lifetimes, a complete description of the excitation has been modeled solving optical Bloch equations Beacon model, Bellenger, 2002) at CEA. The model has been compared with a laboratory experiment set up at CEA/Saclay (Eig. 21). The reasonable discrepancy when both beams at 589 and 569 nm are phase modulated is very likely to spectral jitter, which is not modeled velocity classes of Na atoms excited at the intermediate level cannot be excited to the uppermost level because the spectral profile of the 569 nm beam does not match the peaks of that of the 589 nm beam. 

The m/z values of peptide ions are mathematically derived from the sine wave profile by the performance of a fast Fourier transform operation. Thus, the detection of ions by FTICR is distinct from results from other MS approaches because the peptide ions are detected by their oscillation near the detection plate rather than by collision with a detector. Consequently, masses are resolved only by cyclotron frequency and not in space (sector instruments) or time (TOF analyzers). The magnetic field strength measured in Tesla correlates with the performance properties of FTICR. The instruments are very powerful and provide exquisitely high mass accuracy, mass resolution, and sensitivity—desirable properties in the analysis of complex protein mixtures. FTICR instruments are especially compatible with ESI29 but may also be used with MALDI as an ionization source.30 FTICR requires sophisticated expertise. Nevertheless, this technique is increasingly employed successfully in proteomics studies. 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

It has been shown recently [25] that concentrations of NOj, tend to reduce with increase in the amplitude of discrete-frequency oscillations. The mechanisms remain uncertain, but may be associated with the imposition of a near-sine wave on a skewed Gaussian distribution with consequent reduction in the residence time at the adiabatic flame temperature. Profiles of NO, concentrations in the exit plane of the burner are shown in Fig. 19.6 as a function of the amplitude of oscillations with active control used to regulate the amplitude of pressure oscillations. At an overall equivalence ratio of 0.7, the reduction in the antinodal RMS pressure fluctuation by 12 dB, from around 4 kPa to 1 kPa by the oscillation of fuel in the pilot stream, led to an increase of around 5% in the spatial mean value of NO, compared with a difference of the order of 20% with control by the oscillation of the pressure field in the experiments of [25]. The smaller net increase in NO, emissions in the present flow may be attributed to an increase in NOj due to the reduction in pressure fluctuations that is partly offset by a decrease in NOj, due to the oscillation of fuel on either side of stoichiometry at the centre of the duct. 

When the small slope approximation is not fulfilled, the profile shape is expected to deviate from a sine wave and the decay kinetics are not necessarily exponential. Numerical calculations for / = 0 orientations and for not so small slopes show profiles with flattened maxima and minima as well as non-exponential decay behavior [18]. Examples of amplitude decay for several miscuts a are plotted in fig. 3. Calcnlations for f nearn/2 are also possible bnt have not been carried out as yet. 

Due to symmetry only half of the period is shown.) The curve indicated by the thick solid line is the profile after x = 131072 updates per surface site Also shown are the sine function and the predicted profile from the Lan on-Villain theory, chosen in such a way that the slopes at the steepest part of the profile agree with one another. It, is seen that the Lan on-Villain curve agrees quite well with the measured profile apart from the top one or two layers, where the latter has more rounded shape The rounding tendency at the top of the groove is more pronounced (in relative terms) for smaller systems, in agreement with earlier observations made by Selke and Duxbury (1995). 

The CC pulse train experiments in Refs [63-65] utilize shaped pulses that use a transform-limited (TL) Gaussian pulse its phase is modulated in the frequency domain with a sine function, p ( ) = a sin( -I- c), while keeping the amplitude profile intact. The parameters a, b, and c are further varied to control molecular populations. In Reference [35], the effect of different values of these parameters on the IC dynamics of pyrazine and / -carotene is investigated and the significant role of overlapping resonances is exposed. 

Laird (L3), 1954 Experimental study of pressure drop in gas stream in tubes with sine-wave oscillations of tube wall. Shows that large pressure drop is partly due to change in shape of gas velocity profiles. 

Fig. 9.1. (A) Gaussian (a) and sine (b) excitation profiles. (B) Composite (G3) Gaussian pulse. (C) Train of soft pulses modified after the DANTE sequence to achieve selective off-resonance excitation. (D) Redfield 21412 sequence. (E) Binomial 11, 121, 1331, 14641 sequences. (F) JR (a) and compensated JR (or 1111) (b) sequences. (G) Watergate sequence. (H) Weft (Superweft) sequence. (I) Modeft sequence. (J) MLEV16 sequence. (K) NOESY sequence with trim pulse. (L) MLEV17 sequence with trim pulses. (M) Clean-TOCSY sequence.

In Fourier compression, each profile (x ) is essentially decomposed into a linear combination of sine and cosine functions of different frequency. If the spectrum x is considered to be a continuous function of the variable number m, then this decomposition can be expressed as ... 

Figure 2. Time-domain excitation waveforms (left) and corresponding frequency-domain magnitude-mode spectra (right) of four excitation waveforms used in FT/ICR. A time-domain rectangular rf pulse gives a "sine" excitation spectrum in the frequency-domain. A time-domain frequency-sweep gives a complex profile described by Fresnel integrals. Single-scan time-domain noise gives noise in the frequency-domain. Finally, Stored Waveform Inverse Fourier Transform (SWIFT) excitation can provide an optimally flat excitation spectrum (see Figure 3 for details).

---