Summation of waves

FIGURE 1.1. The importance of relative phases on the result of the summation of waves. Sets of waves of equal amplitude and wavelength are shown only the relative phases of the waves differ, (a) Waves exactly in phase (the wave crests coincide so that the waves reinforce each other), (b) exactly out of phase (troughs and crests coincide giving no resultant wave), and (c) partially out of phase. On the left is shown the set of waves and on the right the result of interference [constructive in (a) or destructive in (b)], with surfers to accentuate the result,... 

The numerical summation of waves, illustrated in Figure 6.4, is quite straightforward. At each point along the horizontal axis, the displacement of each wave can be measured as the distance from this axis in a... 

A Fourier synthesis, illustrated in Figure 6.7, is the reverse of a Fourier analysis. A Fourier synthesis involves the summation of waves of known frequency, amplitude and phase in order to obtain a more complicated, but still periodic function. The relationship between a Fourier synthesis and a Fourier analysis is evident from the use of the same set of waves in both Figures 6.6 and 6.7. In a Fourier synthesis, if everything except the relative phases of the component waves are known, there will still be an almost infinite number of ways in which the waves can be combined. 

Electron-density map A contour representation of electron density in a crystal structure. Peaks appear at atomic positions. The map is computed by a Fourier synthesis, that is, the summation of waves of known amplitude, periodicity, and relative phase. The electron density is expressed in electrons per cubic A. 

Electron-density wave A term in the Fourier summation of waves of different amplitudes and frequencies to give the electron density in the crystal. Each electron-density wave represents the contribution of a single structure factor F(hkl) to the total electron density map. 

A theoretical interpretation of this exponential dependence is that depends on a summation of wave functions, which in free space are calculated to decline exponentially but although this consideration might apply to an isolated D-A complex, it may be doubted whether it can be applied when a bridge and a solvent are included in the system. 

It is easy to see that the summation of waves described above with fourfold enhancement of intensity corresponds to the displacement of the two parts of the splintered wave from each other by the difference in lengths equal to the integer wavelengths (or, accordingly, to the phase difference Acp = 2nm). The complete extinction of the wave s intensity is observed at the displacement of the two wave parts on the wavelength half (on an odd number of the wavelength half, i.e., Acp = (2m + )n). 

Assuming kinematical diffraction theory to be applicable to the weakly scattering CNTs, the diffraction space of SWCNT can be obtained in closed analytical form by the direct stepwise summation of the complex amplitudes of the scattered waves extended to all seattering centres, taking the phase differenees due to position into aeeount. 

In the VB method, a wave equation is written for each of various possible electronic structures that a molecule may have (each of these is called a canonical form), and the total )/ is obtained by summation of as many of these as seem plausible, each with its weighting factor ... 

It appears that the voltage waves recorded in the EEG represent the summation of synaptic potentials in the apical dendrites of pyramidal cells in the cortex. These cells generate sufficient extracellular current for it to reach, and be recorded from, the cranium and scalp. Although these waves originate from the cortex rather than the SCN, the distinctive REM and non-REM phases of sleep still remain after destruction of the SCN but they then occur randomly over the 24-h cycle. This is a further indication that the SCN is at least partly responsible for setting the overall circadian rhythm of the sleep cycle. 

The more synchronised the activity of the cortical neurons, the greater the summation of currents and the larger and slower the EEG wave, as in the sleep pattern (Fig. 22.4). While there are some dissociations between EEG pattern and behavioural states, the EEG offers one way of determining experimentally the pathways (and neurotransmitters) that control arousal and sleep, and can be regarded as an important objective measurement of the cortical correlates of sleep and waking. 

Fig. 3.47 is comparable to Fig. 3.41 for sinusoidal ac polarography if the tilted shape provides a net compensation of the charging current one obtains a symmetric bell-shaped curve of I in the square-wave polarogram, similar to that depicted in Fig. 3.42. In fact, virtually all of the statements made before on the sinusoidal technique are valid for the square-wave mode except for the rigid shape of its wave this conclusion is according to expectation, especially as Fourier analysis reveals the square wave to be a summation of a series of only...

What is the role of intracellular Ca2+ waves due to Ca2+ release from the SR In many cell types stimulation results in Ca2+ waves rather than a maintained increase of [Ca2+] . Summation of such waves in many cells can result in a maintained contraction. Indeed, recent work suggests that these waves are implicated in the genesis of vascular tone (Peng et al 2001). It is important for us to consider how widespread in smooth muscle excitation-contraction coupling are such waves. 

Henceforth all summations of indices i and j span the range [1, /. ) The wave function (2.2) is normalized, i.e. (VP VIl) = Zic2i =1, and its configuration interaction (Cl) functional form describes best the idea of a solute... 

The cosine waves generated by the second strongest reflection (3 5 0) and its symmetry-related reflection (3 -5 0) are shown in Figs. Id and e. Both cosine waves cut the a axis 3 times per unit cell and the b axis 5 times, however, they are oriented differently. The summation of the two symmetry-related cosine waves is shown in Fig. If). 

For a continuous electron distribution p(r), the summation over waves of different phase must be replaced by an integration leading to a diffraction amplitude ... 

Phase control provides an additional avenue for fine tuning of the interference patterns [34-36]. Interference of several coherent beams can be simulated numerically by summation of plane waves according to Eqs. 1 and 2. This is a helpful method for designing structures with desired architecture. 

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