Stability Fourier modes

A linear stability analysis of (A3.3.57) can provide some insight into the structure of solutions to model B. The linear approximation to (A3.3.57) can be easily solved by taking a spatial Fourier transfomi. The result for the Ml Fourier mode is... 

The equations (12-20)-( 12-24) are the so-called linear stability equations for this problem in the inviscid fluid limit. We wish to use these equations to investigate whether an arbitrary, infinitesimal perturbation will grow or decay in time. Although the perturbation has an arbitrary form, we expect that it must satisfy the linear stability equations. Thus, once we specify an initial form for one of the variables like the pressure p, we assume that the other variables take a form that is consistent with p by means of Eqs. (12-20)-(12-24). Now the obvious question is this How do we represent a disturbance function of arbitrary form For this, we take advantage of the fact that the governing equations and boundary conditions are now linear, so that we can represent any smooth disturbance function by means of a Fourier series representation. Instead of literally studying a disturbance function of arbitrary form, we study the dynamics of all of the possible Fourier modes. If any mode is found to grow with time, the system is unstable because, with a disturbance of infinitesimal amplitude, every possible mode will always be present. 

An arbitrary disturbance form in the x and y directions could be expressed as a sum of the Fourier modes of wave number a x and a y, but because the governing equations are linear with coefficients that are independent of x, y, it is enough to consider the stability of these disturbance quantities one mode at a time, for arbitrary values of a x and a y. The fimctions of z must be chosen to satisfy boundary conditions on the fluid interface. The stability is determined by the sign of the real part of a. The reader is reminded that the primes on all of the symbols mean that they are dimensional. 

We assume that the flat interface between the two fluids, now designated as z = 0, is perturbed with an arbitrary infinitesimal perturbation of shape. As usual, for a linear stability analysis, we consider only a single Fourier mode in each of the x and y directions, with the wave number (or wavelength) as a parameter in the stability analysis. Hence we consider a perturbation of the form... 

Moreover, through the von Neumann method along with the use of discrete Fourier modes [1], the stability condition for any member of the (2, M) family is extracted. For instance, the (2, 4) case has... 

We next consider multi-headed spins. We first refer to the branch labeled TW2 in Figure 1. These solutions are replicates of analogous TWl modes. Thus, for any value of R the TW2 solution consists of two replicates of the TWl mode at R/2 and separated by vr radians. We have tested the stability of these solutions by imposing perturbations in the initial conditions which do not correspond to replicated perturbations. (We note that replicated conditions correspond in Fourier space to even order modes. Thus, we impose perturbations which have odd Fourier modes.) We have found stable replicated TW2 modes for 3.5 < E < 11, corresponding to TWl modes for 1.75 < R < 5.5. We note that for R = 3.36, corresponding to the TWl at E = 1.68, the smallest value of E for which we found stable TW1 modes, we were unable to compute a TW2 solution. Rather, the perturbed replicated initial conditions evolved to a TWl solution. Thus, our computations indicate that the replicated TW2 branch is stable, but not for all values of E for which a stable TW 1 solution exists for E/2. The replicated TW2 solutions near E = 3.5 have the same large mean speed as the TWl mode near E = 1.75, suggesting that these modes would be less prone to extinction and thus they may be more readily observed in experiments. 

The Stability of the homogeneous reference state with respect to spatially distributed perturbations is tested by expanding the perturbations in Fourier modes x(r, t) = / Xfc(t) dk and obtaining the following system of equations for the mode amplitudes ... 

The von Neumann stability analysis can be used to examine the long term time stability of the different finite difference approximations. This technique only applies to linear partial differential equations with constant coefficients, but much can be learned from such simple cases. This analysis begins by assuming that the solution of the finite difference system can be expressed as a superposition of Fourier modes having the form... 

The design of EPR spectrometers resembles that of a field-sweep NMR instrument (Section 3.3.2), though pulsed-mode (Fourier transform Section 3.4) EPR spectrometers are now available. Many of the considerations (such as field stability, lineshape, saturation, relaxation, etc.) that were discussed in Chapters 2 and 3 for NMR are also important in EPR,1 but there are some significant differences. 

In order to characterize the MgO sites where the Pd atoms are stabilized after deposition by soft-landing techniques, we used CO as a probe molecule [61]. The adsorption energy, Eb, of CO has been computed and compared with results form thermal desorption spectroscopy (TDS). The vibrational modes, (o, of the adsorbed CO molecules have been determined and compared with Fourier transform infrared (FTTR) spectra. From this comparison one can propose a more realistic hypothesis on the MgO defect sites where the Pd atoms are adsorbed. 

Although we are interested in the response of the cylinder to a perturbation of arbitrary shape, such a perturbation can be constructed as a Fourier sine series, and it is sufficient to look at the stability of a single mode for all possible values of k. If the shape-perturbation mode grows for any k, the system is unstable. Now, if / has the form (12-25), we see from the normal-stress condition (12-24) that p must be equal to... 

The aging of crude oils or the interfacially active fractions from crude oils are able to enhance the stability of emulsions (23, 24). Sjoblom et al. (23) found that the Fourier-transform infrared spectra reveal that the car-bonyl peak grows markedly on accoimt of the C = C mode. At the same... 

To describe mathematically the process of thin liquid film instability the shape of the corrugated film surfaces is presented as a superposition of Fourier-Bessel modes, proportional to Jo(kr/R), for all possible values of the dimensionless wave number k (Jo is the zeroth order Bessel function). The mode, which has the greatest amplitude at the moment of film breakage, and which causes the breakage itself, is called the critical mode, and its wave number is denoted by cr- The stability-instability transition for this critical mode happens at an earlier stage of the film evolution, when the film thickness is equal to Atr - the so-called transitional thickness, h r > her (Ivanov 1980). The theory provides a... 

The general method for solving Eqs. (11) consists of transforming the partial differential equations with the help of Fourier-Laplace transformations into a set of linear algebraic equations that can be solved by the standard techniques of matrix algebra. The roots of the secular equation are the normal modes. They yield the laws for the decays in time of all perturbations and fluctuations which conserve the stability of the system. The power-series expansion in the reciprocal space variables of the normal modes permits identification of relaxation, migration, and diffusion contributions. The basic information provided by the normal modes is that the system escapes the perturbation by any means at its disposal, regardless of the particular physical or chemical reason for the decay. 

In Section 5.7.2 we discussed a general problem of stability of one, two- and three-dimensional phases. Here, we shall analyze stability of the smectic A liquid crystal, which is three-dimensional structure with one-dimensional periodicity. The question of stability is tightly related to the elastic properties of the smectic A phase. Consider a stack of smectic layers (each of thickness Z) with their normal along the z-direction. The size of the sample along z is L, along x and y it is L, the volume is V = Lj L. Fluctuations of layer displacement u(r) = u(z, r i) along z and in bofli directions perpendicular to z can be expanded in the Fourier series with wavevec-tors q and q (normal modes) ... 

To identify EVA formulation, degradation and even stabilizer difiusion and consumption, the Fourier Transform Infrared (FTIR) spectroscopy in ATR mode is a simple and satisfying method. The FTIR measurements were carried out with a... 

For some tasks in ultrahigh-resolution spectroscopy, the residual finite linewidth AyL, which may be small but nonzero, still plays an important role and must therefore be known. Furthermore, the question why there is an ultimate lower limit for the linewidth of a laser is of fundamental interest, since this leads to basic problems of the nature of electromagnetic waves. Any fluctuation of amplitude, phase, or frequency of our monochromatic wave results in a finite linewidth, as can be seen from a Fourier analysis of such a wave (see the analogous discussion in Sects. 3.1,3.2). Besides the technical noise caused by fluctuations of the product nd, there are essentially three noise sources of a fundamental nature, which cannot be eliminated, even by an ideal stabilization system. These noise sources are, to a different degree, responsible for the residual linewidth of a single-mode laser. 

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