Sine waves phase shift

We mentioned that the exponential function has some interesting properties. But a sine wave too has rather similar properties. For example, the rate of change of a sine wave is a cosine wave — which is just a sine wave phase shifted by 90°. It therefore comes as no surprise that we have the following relationships ... 

Figure 8.4 Sum and product of two equal amplitudes (=1) equal frequency sine waves, phase shifted by various amounts 30° (top), 60° (middle), and 90° (bottom). Notice the DC component of the product (right hand side). Example is with amplitude = 1 (e.g., volt), f= 1 Hz so that...

Elastic response This occurs when the maximum of the stress amplitude is at the same position as the maximum of the strain amplitude (no energy dissipation). In this case, there is no time shift between the stress and strain sine waves. Viscous response This occurs when the maximum of the stress is at the point of maximum shear rate (i.e., the inflection point), where there is maximum energy dissipation. In this case, the strain and stress sine waves are shifted by (referred to as the phase angle shift, 5, which in this case is 90°). 

In words, a cosine wave, phase shifted by n 2 radians (90°) is the same thing as a sine wave. Thus, in the EXSY experiment the effect of changing the phase of the first pulse by 90° can be described as a phase shift of the signal by 90°. 

The system transmits 5 bits/s with a phase-shift-keying system. Four sine waves are necessary to define the phase with a negligible chance of error. Assume a perfect pipe, drill collar ID same as the drill pipe and no wave reflections at the drill pipe ends. 

It was shown in the preceding section that PECD can be anticipated to have an enhanced sensitivity (compared to the cross-section or p anisotropy parameter) to any small variations in the photoelectron scattering phase shifts. This is because the chiral parameter is structured from electric dipole operator interference terms between adjacent -waves, each of which depends on the sine of the associated channels relative phase shifts. In contrast, the cross-section has no phase dependence, and the p parameter has only a partial dependence on the cosine of the relative phase. The distinction between the sine... 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

In-plane components (a22 and a33). The feature of conformation dependence of the g22 and g33 components is better understood by taking p-IONONE as an example. Here, the conformation-dependent shifts of C5 and C8 are examined. As can be seen from Figure 6, the isotropic shieldings of both carbons exhibit clear periodic dependence on the twist angle about the C6-C7 bond,cp6.7. For example, the a-cp6.7 curve of C5 appears to be approximated by a sine wave whose period is 180°. The curve for C8 shifts in phase by a half period relative to that for C5. These curves are expressed by using the Fourier series of expansion as follows ... 

How Many Fractional Phase Register Bits are Needed. The choice of how many bits to make the phase register is an important issue in computer music design. While other authors have covered this issue in relation to traditional sine wave oscillators, there are some subtle differences in the design of sample playback oscillators. Here, the fractional part of the phase register essentially determines how much pitch resolution is available, while the integer part determines how many octaves up the waveform can be transposed (pitch shifted). 

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... 

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 angle ip is the phase shift between the applied sine-wave voltage and the resulting sine wave current. 

It is not necessary for the physicist to know how to compute the Coulomb functions. They are found in subroutine libraries, for example Barnett et al. (1974). A sufficient idea of their form is obtained by putting j = L = 0 in (4.62), when they are seen to be sinp and cosp respectively. The potential terms dilate or compress the sine and cosine waves, resulting in an overall phase shift at long range. 

The most popular dynamic test procedure for viscoelastic behavior is the application of an oscillatory stress of small amplitude. This shear stress applied produces a corresponding strain in the material. If the material were an ideal Hookean body, the shear stress and shear strain rate waves would be in phase (Fig. 14A), whereas for an ideal Newtonian sample, there would be a phase shift of 90° (Fig. 14B), because for Newtonian bodies the shear strain is at a maximum, when a maximum of stress is present. The shear strain, when assuming an oscillating sine fimction, is at a maximum in the middle of the slope, because there is the steepest increase in shear strain due to the change in direction. For a typical viscoelastic material, the phase shift will have a value between >0° and <90° (Fig. 14C). 

Figure 1.1 Graphical illustration of the phase shift between two sine waves of equal amplitude.

The detected intensity, /, is the square of the amplitude, A, of the sine wave. With two waves present, the resulting amplitude is not just the sum of the individual amplitudes but depends on the phase shift Sep. The two extremes occur when S(p = 0 (constructive interference), where I= Ai + A2f, and Sep = 71 (destructive interference), where I= Ai—A. In general, I=[Ai A2Qxp(iS(p)f . When more than two waves are present, this equation becomes ... 

Dynamic (oscillatory) measurements A sinusoidal stress or strain with amphtudes (Tjj and is appHed at a frequency a> (rads ), and the stress and strain are measured simultaneously. For a viscoelastic system, as is the case with most formulations, the stress and strain amplitudes oscillate with the same frequency, but out of phase. The phase angle shift S is measured from the time shift of the strain and stress sine waves. From a, y and S, it is possible to obtain the complex modulus j G, the storage modulus G (the elastic component), and the loss modulus G" (the viscous component). The results are obtained as a function of strain ampHtude and frequency. 

Consider the case of a viscoelastic system, for which the sine waves of strain and stress are shown in Figure 20.11. The frequency co is in rads , and the time shift between the strain and stress sine waves is At. The phase angle shift S is given by (in dimensionless units of radians). 

In (c) the phase shift is 90°. Now it is Sy which takes the form of a damped cosine wave, whereas Sx is a sine wave. The Fourier transform gives a spectrum in which the absorption mode signal now appears in the imaginary part. Finally in (d) the phase shift is 180° and this gives a negative absorption mode signal in the real part of the spectrum. 

Fig. 5.3 Illustration of how a 90° phase shift takes us from cosine to sine modulation. In (a) the vector starts out along x as it rotates in a positive sense the x component, Sx, varies as a cosine wave, as is shown in the graph on the right. In (b) the vector starts on -y the x component now takes the form of a sine wave, as is shown in the graph. Situation (b) is described as a phase shift,

So, if the B field is modulated by a cosine wave the result is a pulse about jc, whereas if it is modulated by a sine wave the result is a pulse y. We see that by phase shifting the RF we can apply pulses about any axis there is no need to install more than one coil in the probe. 

The change from cosine to sine modulation in the EXSY experiment can be though of as a phase shift of the signal in tl. Mathematically, such a phase shifted cosine wave is written as cos(f2jtj +(/)), where (f) is the phase shift in radians. This expression can be expanded using the well known formula cos( + B) = cos A cos B - sin A sin B to give... 

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