Waves amplitude and

An important aspect of micromechanical evolution under conditions of shock-wave compression is the influence of shock-wave amplitude and pulse duration on residual strength. These effects are usually determined by shock-recovery experiments, a subject treated elsewhere in this book. Nevertheless, there are aspects of this subject that fit naturally into concepts associated with micromechanical constitutive behavior as discussed in this chapter. A brief discussion of shock-amplitude and pulse-duration hardening is presented here. 

Champion and Rohde [42] investigate the effects of shock-wave amplitude and duration on the Rockwell C hardness [41] and microstructure of Hadfield steel over the pressure range of 0.4-48 GPa (pulse duration of 0.065 s, 0.230 ls, and 2.2 ps). The results are shown in Fig. 7.8. In addition to the very pronounced effeet of pulse duration on hardness shown in Fig. 7.8, postshoek electron microscope observations indicate that it is the final dislocation density and not the specific microstructure that is important in determining the hardness. 

The reflected pressure wave amplitude and impulse for shock waves associated with detonations are well documented, as shown in Figure A. 3 (Ref. 7, Volume II). Less information is available on reflected overpressure and impulse resulting from deflagration pressure waves. Reference 67 documents approaches for evaluating reflected overpressure from weaker blast pressure waves. Forbes (Ref. 71) suggests the following approximate relation to model the more complex relations in Reference 64 ... 

In Chapter 3 we went as far as we could in the interpretation of rocking curves of epitaxial layers directly from the features in the curves themselves. At the end of the chapter we noted the limitations of this straightforward, and largely geometrical, analysis. When interlayer interference effects dominate, as in very thin layers, closely matched layers or superlattices, the simple theory is quite inadequate. We must use a method theory based on the dynamical X-ray scattering theory, which was outlined in the previous chapter. In principle that formrrlation contains all that we need, since we now have the concepts and formtrlae for Bloch wave amplitude and propagatiorr, the matching at interfaces and the interference effects. 

These have hyperbolic form, and in a perfect crystal can be solved analytically. In a distorted crystal they must be solved by numerical integration. Anthier et al. 30 were the first to devise an algorithm for solution over a grid of points. If, in Figure 8.12, M is the point at which we require the wave amplitude and P and Q are neighbouring points at which the amplitudes are known,... 

In upward or climbing-film flow, waves or ripples are always present. At low gas rates and low liquid rates, films are thin, wave amplitude and entrainment may be relatively small, and straightforward hydrodynamic... 

A dimensional analysis of the problem of film flow (F7) has shown that in general the properties of a film flow may depend on the Reynolds, Weber, and Froude numbers of the flow, a dimensionless shear at the free surface of the film, and, for wavy flows, a Strouhal number formed from the frequency of the surface waves, and various geometrical ratios, e.g., the ratios of the wave amplitude and length to the mean film thickness. 

Early approaches to music analysis relied on a running Fourier transform to measure sine-wave amplitude and frequency trajectories. This technique evolved into a filter-bank-based processor and ultimately to signal analysis/synthesis referred to as the phase vocoder [Flanagan and Golden, 1966], This section describes the history of the phase vocoder, its principles, and limitations that motivate sinusoidal analysis/synthesis. Other formulations and refinements of the phase vocoder are given in chapter 7. 

In spite of the many successes of the phase vocoder, numerous problems have limited its use. In the applications of time-scale modification and compression, for example, it is assumed that only one sine wave enters each bandpass filter within the filter bank. When more than one sine wave enters a bandpass filter, the meaning of the input sine-wave amplitude and phase envelope is lost. A particular sine wave also may not be adequately estimated when it falls between two adjacent filters of the filter bank. In addition, sine waves with rapidly-varying frequency due to large vibrato or fast pitch change are difficult to track. A result of using a fixed filter bank is that the frequency of... 

Analysis/Synthesis With estimates of excitation and system sine-wave amplitudes and phases at the center of the new time-scaled synthesis frame, the synthesis procedure becomes identical to that of the baseline system of section 3.6. The goal then is to obtain estimates of the amplitudes, Ak (/), and phases, Ok (/), in Equation (9.42) at the center of the synthesis frame of duration Q = pQ where Q is the analysis frame interval as defined in section 3.5. Since in the time-scale modification model, the system and excitation amplitudes are simply time scaled, from Equations (9.40a) and (9.42) the composite amplitude need not be separated and therefore the required amplitude can be obtained from the sine-wave amplitudes measured on each frame m, by spectral peak-picking. 

Therefore, although the summed waveform x(n) = x a(n) + x fin) is well represented by peaks in the STFT of x(n), the sine-wave amplitudes and phases of the individual waveforms are not easily extracted from these values. To look at this problem more closely, let s (n) represent a windowed speech segment extracted from a time-shifted version of the sum of two sequences... 

In this section, the nonlinear problem of forming a least squares solution for the sine-wave amplitudes, phases, and frequencies is transformed into a linear problem. This is accomplished by assuming the sine-wave frequencies are known apriori, and by solving for the real and imaginary components of the quadrature representation of the sine waves, rather than solving for the sine-wave amplitudes and phases. The... 

Yet another unsolved problem is the separation of two voices that contain closely spaced harmonics or overlapping harmonic and aharmonic components. The time-varying nature of sine-wave parameters, as well as the synchrony of movement of these parameters within a voice [Bregman, 1990], may provide the key to solving this more complex separation problem. Section 6.3 revealed, for example, the limitation of assuming constant sine-wave amplitude and frequency in analysis and as a consequence proposed a generalization of Equation (9.75) based on linear sine-wave amplitude and frequency trajectories as a means to aid separation of sine waves with closely-spaced frequencies. 

Finally sine-wave analysis/synthesis is also suitable for extrapolation of missing data [Maher, 1994], Situations occur, for example, where a data segment is missing from a digital data stream. Sine-wave analysis/synthesis can be used to extrapolate the data across the gap. In particular, the measured sine-wave amplitude and phase are interpolated using the linear amplitude and cubic phase polynomial interpolators, respectively. In this way, the slow variation of the amplitude and phase function are exploited, in contrast with rapid waveform oscillations. 

The discussion of the previous section suggests that the linear combination of the shifted and scaled Fourier transforms of the analysis window in Equation (9.72) must be explicitly accounted for in achieving separation. The (complex) scale factor applied to each such transform corresponds to the desired sine-wave amplitude and phase, and the location of each transform is the desired sine-wave frequency. Parameter estimation is difficult, however, due to the nonlinear dependence of the sine-wave representation on phase and frequency. 

The preceding analysis views the problem of solving for the sine-wave amplitudes and phases in the frequency domain. Alternatively, the problem can be viewed in the time domain. It has been shown that [Quatieri and Danisewicz, 1990], for suitable window lengths, the vectors a andJ3 that satisfy Equation (9.75) also approximate the vectors that minimize the weighted mean square distance between the speech frame and the steady state sinusoidal model for summed vocalic speech with the sinusoidal frequency vector . Specifically, the following minimization is performed with respect to a andJ3... 

The peak charge and the half-peak width, Wy2, depend on the square wave amplitude and present the following limiting values ... 

The splitting of the SWV net current appears on increasing the square wave amplitude and the ratio co. For example, for E sw = 50 mV, the splitting appears for surface electrode reactions with co > 3. Therefore, the values of the peak... 

In order to calculate the RF PAD for a set of partial wave amplitudes and phases we use Eq. (54) to first calculate the MF PAD. The MF is defined with the z axis along the Ciy symmetry axis, the y-axis along the N—N bond, and the x axis perpendicular to the molecular plane. The RF plane is defined with the z axis along the N—N bond direction. In order to calculate the RF PAD from the MF PAD, a rotation is applied to bring the MF z axis to the RF z axis. The resulting PAD is then azimuthally averaged about the z axis (the N—N direction),... 

Hypokalemia results in decreased T-wave amplitude and ST-segment depression however, accurate QT interval measiuement is difficult. Malignant ventricular arrhythmias result when potassium concentrations become very low. Hypercalcemia shortens the QT interval while hypocalcemia produces ST-segment prolongation. 

The electron wave path of phase contrast in a TEM is illustrated by Figure 3.24. The properties of electron waves exiting the specimen are represented by an object function, /(r), which includes the distributions of wave amplitude and phase, r is the position vector from the central optical axis. The object function may be represented as a function in Equation 3.3. 

In order to complement the results of the radar backscatter measurements on the open sea, laboratory measurements of the wave amplitude and slope and of the radar backscatter at X- and Ka-band were carried out in a wind-wave tank with mechanically generated gravity waves as well as with wind-generated waves on a slick-free and a slick-covered water surface. In this paper, we concentrate on the results of the radar measurements with wind-generated waves. For a full description of the obtained results the reader is referred to Gade et al. (1998c). 

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