Sinusoidal variation

Assume now that the primary electromagnetic field varies with time as  

In the quasistationary approximation when both the time of observation t and the period T = 27t/w of the excitation are much greater than the relaxation time tq, we have  

Neglecting the second-order term and assuming that the field is not zero, that is, ujt is not a multiple of tt, we finally obtain the expression for the volume charge density under quasistationary harmonic conditions  

We have so far only investigated the volume charge density. Let us next consider time-varying surface charges. Combining the following equations  

Thus the equation for surface density is a differential equation of the first order similar 

---

If a sinusoidal variation of the temperature of the heat transfer medium in the jacket or coil occurs, say... 

If the material being subjected to the sinusoidal stress is elastic then there will be a sinusoidal variation of strain which is in phase with the stress, i.e. 

Fig. 2.53 Sinusoidal variation of stress and strain in viscoelastic material...

In dynamic mechanical analysis of plastics, the material is subjected to a sinusoidal variation of stress and the strain is recorded so that 1, 2 and S can be determined. The classical variation of these parameters is illustrated in Fig. 2.55. 

The importance of maintaining the flowrate in a pipeline constant may be seen by considering the effect of a sinusoidal variation in flowrate. This corresponds approximately to the discharge conditions in a piston pump during the forward movement of the piston. The flowrate Q is given as a function of time t by the relation ... 

Apply sinusoidal variations in the flow rate F3, F3AMP = 0.05, and again study the responses in h2 to changes in F3. 

Repeat Exercise 3 but with a sinusoidal variation in F3, as in Exercise 2. 

Figure 22. Response of the seawater U activity ratio to a sinusoidal variation of the U activity ratio of world rivers (adapted from Richter and Turekian 1993). Such a scenario could explain the apparent discrepancy between the theoretical mean riverine activity ratio of 1.25-1.35 and the estimated value of 1.17 (see text). The scenario could be supported by the preliminary conclusions from the study of U in Himalayan rivers (Chabaux et al. 2001), which assumed a climatic dependence of the Himalayan U flux, sufficient to induce a periodic variation of the mean U activity ratio of the world rivers on a glacial-interglacial time-scale (T = 10 y). The amplitude of variation proposed for the mean ratios of...

In order to increase the overall extraction efficiency during SFE sonication has been applied [352]. Ultrasound creates intense sinusoidal variations in density and pressure, which improve solute mass transfer. Development of an SFE method is a time-consuming process. For new methods, analysts should refer the results to a traditional sample preparation method such as Soxhlet or LLE. 

A sinusoidal input. The frequency of the sinusoidal variation is changed and the steady-state response of the effluent at different input frequencies is determined, thus generating a frequency-response diagram for the system. 

The time variations of the effluent tracer concentration in response to step and pulse inputs and the frequency-response diagram all contain essentially the same information. In principle, any one can be mathematically transformed into the other two. However, since it is easier experimentally to effect a change in input tracer concentration that approximates a step change or an impulse function, and since the measurements associated with sinusoidal variations are much more time consuming and require special equipment, the latter are used much less often in simple reactor studies. Even in the first two cases, one can obtain good experimental results only if the average residence time in the system is relatively long. 

Kramers and Alberda (6) have discussed the manner in which sinusoidal variations are analyzed, but we will discuss only the first two types of stimuli. They are sufficient for the analysis of the majority of situations that will be encountered by the chemical engineer engaged in the practice of designing chemical reactors. 

Cycled Feed. The qualitative interpretation of responses to steps and pulses is often possible, but the quantitative exploitation of the data requires the numerical integration of nonlinear differential equations incorporated into a program for the search for the best parameters. A sinusoidal variation of a feed component concentration around a steady state value can be analyzed by the well developed methods of linear analysis if the relative amplitudes of the responses are under about 0.1. The application of these ideas to a modulated molecular beam was developed by Jones et al. ( 7) in 1972. A number of simple sequences of linear steps produces frequency responses shown in Fig. 7 (7). Here e is the ratio of product to reactant amplitude, n is the sticking probability, w is the forcing frequency, and k is the desorption rate constant for the product. For the series process k- is the rate constant of the surface reaction, and for the branched process P is the fraction reacting through path 1 and desorbing with a rate constant k. This method has recently been applied to the decomposition of hydrazine on Ir(lll) by Merrill and Sawin (35). 

Conditions that may give rise to unsteadiness are changes in feed rate, composition or temperature. In problem P4.09.34, sinusoidal variations of these three properties are forced on the CSTR. The resulting outlet conditions... 

Values of Dr can be calculated from the change in shape of a pulse of tracer as it passes between two locations in the bed, and a typical procedure is described by Edwards and Richardson(27). Gunn and Pryce(28), on the other hand, imparted a sinusoidal variation to the concentration of tracer in the gas introduced into the bed. The results obtained by a number of workers are shown in Figure 4.6 as a Peclet number Pe = ucd/eDL) plotted against the particle Reynolds number (Re c = ucdp//j ). 

When a chemical or biochemical reaction takes place in the sensor area, only the light that travels through this arm will experience a change in its effective refractive index. At the sensor output, the intensity (I) of the light coming from both arms will interfere, showing a sinusoidal variation that depends on the difference of the effective refractive indexes of the sensor (Neff,s) and reference arms (Neff,R) and on the interaction length (L) ... 

Here X is the wavelength. This sinusoidal variation can be directly related to the concentration of the analyte to be measured. 

A sinusoidal variation of the 0 K energy difference between b.c.c. and close-packed structures is predicted across the transition metal series, in agreement with that obtained by TC methods (Saunders et al. 1988). For the most part magnitudes are in reasonable agreement, but for some elements FP lattice stabilities are as much as 3-10 times larger than those obtained by any TC methods (Fig. 6.6). 

FP methods inherently lead to a marked sinusoidal variation of /ffp h across the periodic table (Pettifor 1977) and for Group V and VI elements, electron energy calculations predict j/f e -c p h- of opposite sign to those obtained by TC methods. It is worth noting, however, that a sinusoidal variation is reproduced by one of the more recent TC estimates (Saunders et al. 1988) although displaced on the energy axis (Fig. 6.7). 

Figure 10 (a) Schematic representation of structural modulations showing repeated shortening and widening of Bi-Bi distances, (b) Modulations in Bi-Bi separation in adjacent layers, indicating a sinusoidal variation. 

Calculate the annual sinusoidal variation of the PCE concentration in the wells of Groundwater System S relative to the variation in River R. Compare this number with the relative variation of a nonsorbing chemical such as 2,4-dinitrophenol (see Illustrative Example 25.5). Determine the time lag of oscillation in the well relative to the variation in the river. Use all three flow regimes of Illustrative Example 25.1. 

The lower part of Fig. 6.5 shows the electric signal obtained with the aid of the described method. An exact differentation of the upper curve is obtained by the selective amplification of the fundamental frequency of the modulation. In fact, at the inflexion point of the upper curve, a sinusoidal variation of the azimuth causes a practically sinusoidal variation of intensity. At any other place, the variation of the intensity can only be described by a Fourier series with the same basic frequency. The higher frequencies are not detected electronically. The amplitude ratio AIJA

absolute value of the curve. This is the reason, why sharp edges are observed in the lower curve at the extinction positions. This forms a welcome increase of the accuracy of the determination. The advantages of this method clearly follow from Fig. 6.514. 

These geochemical tracers have been successfully applied to studies of the shells of a variety of marine organisms including bivalve and gastropod mollusks, ostracods, forams, brachiopods and solitary corals (47, 54, 57, 58). In the case of mollusk shells, for example, serial microsampling around the spiral whorls from earliest to oldest growth revealed sinusoidal variations in isotope ratios, which result from shell deposition in a seasonal environment (47, 50, 62-64). Our previous work showed this to be true of Olivella shells as well, where 8,sO levels fluctuate from warm summer temperatures to cold spring and winter temperatures (38). 

The first term describes the sinusoidal variation of the particle displacement with distance, the second term the variation with time, and the final term describes the attenuation of the wave. In most text books equation 1 is written in the following form ... 

---