Gibbs’ phenomena

Convergence of the actual solution to the self-similar one over time occurs in a way similar to convergence of a Fourier series to a discontinuous function as the number of terms in the sum increases the well-known Gibbs phenomenon leads to the fact that, near the discontinuity, for any number of terms, the maximum difference between the series and the function does not approach zero however, the width of the region in which the series differs noticeably from the function approaches zero as the number of terms increases. 

Due to the finite propagation time T of the wavepackets, the Fourier transformation causes artifacts known as the Gibbs phenomenon [122]. In order to reduce this effect, the autocorrelation function is first multiplied by a damping function cos jtt/IT) [81,123]. Furthermore, to simulate the experimental line broadening, the autocorrelation functions will be damped by an additional multiplication with a Gaussian function exp — t/xd)% where zj is the damping parameter. This multiplication is equivalent to a convolution of the spectrum with a Gaussian with a full width at half maximum (FWHM) of /xd- The convolution thus simulates... 

A.J. Jerri, The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations (Kluwer, New York, 1998)... 

Only the right half of one period is shown. The first partial sum only vaguely resembles the function, but 5io is a better approximation. Notice the little spike or overshoot near the discontinuity. This is a typical behavior and is known as the Gibbs phenomenon. The partial sum Sioo fits the function more closely away from the discontinuity, but it has a spike near the discontinuity that is just as high as that of 5io, although much narrower. 

A Fourier series carried through N terms is called a partial sum Sn- Fig. 7.2 shows the function f(9) and the partial sums i, S2, S3, and 5 iq. Note how the higher partial sums overshoot near the points of discontinuity 9 = 0,n, 2n. This is known as the Gibbs phenomenon. As N 00, 1.18 at these... 

To compensate for the finite time of the propagation, the autocorrelation must be multiplied by a function which ensures that it goes to zero at the end of the propagation. If this is not done, spurious structures (known as Gibbs phenomenon) will be seen in the spectrum. We choose the function... 

Gibbs phenomenon Refers to an oscillatory behavior in the convergence of the Fourier transform or series in the vicinity of a discontinuity, typically observed in the reconstruction of a discontinuous X (t). Formally stated, the Fourier transform does converge uniformly at a discontinuity, but rather, converges to the average value of the waveform in the neighborhood of the discontinuity. 

It should be noted that this discussion of the Fourier series is incomplete in that matters of convergence and the Gibbs phenomenon are not included It has been tacitly assumed that any periodic physiological signals encountered in experimental work are accurately represented by some Fourier series. For this reason the detailed mathematical arguments have been omitted for the sake of clarity and brevity (see Carslaw, 1930, for example). 

Gibbs adsorption equation is an expression of the natural phenomenon that surface forces can give rise to concentration gradients at Interfaces. Such concentration gradient at a membrane-solution Interface constitutes preferential sorption of one of the constituents of the solution at the interface. By letting the preferentially sorbed Interfacial fluid under the Influence of surface forces, flow out under pressure through suitably created pores in an appropriate membrane material, a new and versatile physicochemical separation process unfolds itself. That was how "reverse osmosis" was conceived in 1956. 

Schofield Phil. Mag. March, 1926) has recently verified this relation by direct experiment. In order to appreciate the significance of this result, it is necessary to consider in more detail the electrical potential difference V and the manner in which it arises. Instead of regarding the phenomenon from the point of view of the Gibbs equation, it has been, until recently, more usual to discuss the subject of electro-capillarity from the conceptions developed by Helmholtz and Lippmann. These views, together with the theory of electrolytic solution pressure advanced by Nemst, are not in reality incompatible with the principles of adsorption at interfaces as laid down by Gibbs. 

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