Fourier harmonics

Spherical harmonics closely resemble normal Fourier harmonics except that they are functions of both the latitude and the longitude instead of the linear abscissa on a standard axis. Bi-dimensional Fourier analysis on a plane exists but is inadequate since the most desirable property of the requested expansion is the orthogonality of its components upon integration over the surface of the Earth, assumed to be spherical for most practical purposes. 

Figure 14-6 Amplitudes of Fourier Harmonics for Various Switching Voltage Waveforms...

The direct Fourier transform of this function is two delta functions with amplitude po and pi located at = 0 and q = Inta. It is shown in the Inset to Fig. 5.18. Disregarding the zero Fourier harmonic the corresponding intensity of scattering for the infinite one-dimensional crystal is ... 

Here I is interlayer distance and is the infinite set of possible complex order parameters (amplitudes and phases of density harmonics with m = 1,2,3. ..) hi fact, usually the modulation is not deep and, in the simplest approach, we can leave only the first strongest Fourier harmonic with m = and the role of highest harmonics will be discussed later. Then,... 

For a typical situation pi > p2, it is sufficient to take only one cross-term with coefficient B. Coefficients Oi and 2 re assumed positive and, in addition, we assume T > T2 because on cooling, the first Fourier harmonic appears at higher temperature, and afterwards, at a lower temperature, the single harmonic law is violated and p2 appears. The minimization of (6.20) with respect to p2 results in... 

The Fourier harmonics and y of the director fluctuating field are represented by volume integrals (q is wavevector) ... 

Let us recall the reason for the light scattering in gas or in isotropic hquid. In that case, we deal with fluctuations of the mass density. They can be represented by a sum of normal elastic vibration modes (Fourier harmonics) with wavevector q and frequency Q. When such a particular mode interacts with hght of frequency m and wavevector k the conservation laws for energy and momentum read ... 

For instance, for T(p = 10 dyn,ii = 2 x 10 " cm,Fs = 300 statC/cm (1 mC/m ), the threshold field is 0.1 statV/cm, i.e. 3 kV/m. Due to a high value of the Frederiks type distortion in SmC can be observed at extremely low voltage across the cell Uc = dE 30 mV for 10 pm thick cell). However, independently of the field magnitude, after switching the field off, the distortion relaxes to the initial uniform structure, cp(x) = 0. The relaxation time of the distorted structure is owed to pure elastic, nematic-like torque and for small distortion only fundamental Fourier harmonic is important. 

As noted in Section II. for saturating probes, high-order spatial Fourier harmonics can be transferred, via saturated absorption (SA) processes, from the incident, trans-... 

Note that in the expression for A, the wave number k and flow velocity v always appear as a product kv. This means that all observations concerning the dispersion relations between A or Re A and fc at a fixed v can be viewed as relations between A or Re A and u at a fixed k. We can state thus, that for any perturbation which is a Fourier harmonic with a finite k, the growth rate monotonically rises from Re A(0) < 0 to an > 0 as the velocity v grows from 0 to oo. This can be interpreted in terms of disengaging activator and inhibitor as the velocity of the differential flow grows, the separation of the activator and inhibitor becomes more and more effective until the growth rate of the unstable modes reaches the rate of autocatalysis, an. Since the notion of autocatalytic growth is associated with chemistry we call this kind of instability differential flow induced chemical instability . 

The mechanism of the differential flow instability without an activator can be easily interpreted in the case when the two-variable subsystem (or one of the subsystems with more than one variable in a system with > 3 variables) is a damped oscillator (its steady state is a stable focus). Then we can think about a resonance between the frequency of the damped oscillator and the Doppler frequency u = kv of a Fourier harmonic with which the other, moving subsystem was perturbed. Although each of the separate modes tends to decay, their resonant coupling may cause them to grow. 

When the sample is large, B becomes very small then the highermode decay is comparable with that of the first one. Under these circumstances, higher modes can still be present during the time when the data are statistically valid. Harmonic analysis of the time and space decay is then necessary to obtain the decay constant of the fundamental mode alone. A Fourier harmonic analysis of a space-time neutron-flux distribution can be performed by the following method ... 

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