Sinusoids

The instrument uses a sinusoidal driver. The spectrum is very clean as we use a 14 bits signal generator. The probe signal is modulated in amplitude and phase by a defect signal. The demodulation is intended to extract the cartesian values X and Y of this modulation. 

Maxwell s equation are the basis for the calculation of electromagnetic fields. An exact solution of these equations can be given only in special cases, so that numerical approximations are used. If the problem is two-dimensional, a considerable reduction of the computation expenditure can be obtained by the introduction of the magnetic vector potential A =VxB. With the assumption that all field variables are sinusoidal, the time dependence... 

A ring specimen is cut from the tube. Two coils are wrapped around this ring, one exciter coil and one receiver coil. The exciter coil N1 should cover the entire ring so that there are no field losses when ring is saturated. The sinusoidal exciter current which can be measured at... 

Let us imagine a solenoid traversed by an alternating sinusoidal current near a conducting piece. The tension U on the coil is the sum of the tension Rsl due to the ohmic drop of potential in the coil of resistance Rs in the absence of eddy current and of the tension e opposing to the tension e given by the LENZS law ... 

For fluorescent compounds and for times in die range of a tenth of a nanosecond to a hundred microseconds, two very successftd teclmiques have been used. One is die phase-shift teclmique. In this method the fluorescence is excited by light whose intensity is modulated sinusoidally at a frequency / chosen so its period is not too different from die expected lifetime. The fluorescent light is then also modulated at the same frequency but with a time delay. If the fluorescence decays exponentially, its phase is shifted by an angle A([) which is related to the mean life, i, of the excited state. The relationship is... 

A8, which leads to D, = 1/(2A8). The factor of two arises because a minimum of two data points per period are needed to sample a sinusoidal wavefonn. Naturally, the broadband light source will detennine the actual content of the spectrum, but it is important that the step size be small enough to acconunodate the highest frequency components of the source, otherwise they... 

Figure Bl.5.2 Nonlinear dependence of tire polarization P on the electric field E. (a) For small sinusoidal input fields, P depends linearly on hence its hannonic content is mainly tiiat of E. (b) For a stronger driving electric field E, the polarization wavefomi becomes distorted, giving rise to new hannonic components. The second-hamionic and DC components are shown.

The remarkable stability and eontrollability of NMR speetrometers penults not only the preeise aeeiimulation of FIDs over several hours, but also the aequisition of long series of speetra differing only in some stepped variable sueh as an interpulse delay. A peak at any one ehemieal shift will typieally vary in intensity as this series is traversed. All the sinusoidal eomponents of this variation with time ean then be extraeted, by Fourier transfomiation of the variations. For example, suppose that the nomial ID NMR aequisition sequenee (relaxation delay, 90° pulse, eolleet FID) is replaeed by the 2D sequenee (relaxation delay, 90° pulse, delay i -90° pulse, eolleet FID) and that x is inereased linearly from a low value to ereate the seeond dimension. The polarization transfer proeess outlined in die previous seetion will then eause the peaks of one multiplet to be modulated in intensity, at the frequeneies of any other multiplet with whieh it shares a eoupling. 

Pulsed-force mode AFM (PFM-AFM) is a method introduced for fast mapping of local stiffness and adliesion with lower required data storage than recording force-distance curves at each point on the x-y plane [115]. A sinusoidal or triangular modulation is applied between the tip and sample (either via lever or sample piezo) at a lower frequency than that of either the piezo or cantilever resonance frequency. Tip and sample then come... 

As long as AT, Ax and Ax remain small, they will be proportional to the sinusoidal pressure wave. In... 

It remains to investigate the zeros of Cg t) arising from having divided out by. The position and number of these zeros depend only weakly on G, but depends markedly on the fomi that the time-dependent Hamiltonian H(x, () has. It can be shown that (again due to the smallness of ci,C2,...) these zeros are near the real axis. If the Hamiltonian can be represented by a small number of sinusoidal terms, then the number of fundamental roots will be small. However, in the t plane these will recur with a period characteristic of the periodicity of the Hamiltonian. These are relatively long periods compared to the recurrence period of the roots of the previous kind, which is characteristically shorter by a factor of... 

The problem is heated in elementary physical chemishy books (e.g., Atkins, 1998) and leads to a set of eigenvalues (energies) and eigenfunctions (wave functions) as depicted in Fig. 6-1. It is solved by much the same methods as the hamionic oscillator in Chapter 4, and the solutions are sine, cosine, and exponential solutions just as those of the harmonic oscillator are. This gives the wave function in Fig. 6-1 its sinusoidal fonn. 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

Assuming that the xj undergo some form of sinusoidal time evolution ... 

If the applied force varies sinusoidally with time, the period of the oscillation defines the time scale. Quite different mechanical responses are expected at different frequencies. This type of experiment will be described in Secs. 3.10 and 3.11. 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Interference effects, which arise because of the extraordinary uniformity of thickness of the film over the spectrometer sample beam, superimposed on the absorption of incident light by parylene films, can be observed. Experimentally, a sinusoidal undulation of the baseline of the spectmm is seen, particularly in the spectral regions where there is Htde absorption by the sample. These so-called "interference fringe" excursions can amount to some... 

Pulsed Columns. The efficiency of sieve-plate or packed columns is increased by the appHcation of sinusoidal pulsation to the contents of the column. The weU-distributed turbulence promotes dispersion and mass transfer while tending to reduce axial dispersion in comparison with the unpulsed column. This leads to a substantial reduction in HETS or HTU values. 

Fig. 4. The effect of temperature for Mng 6 Zng 3 Fe Fe on (a) initial magnetic permeabiUty, )J., measured on a polycrystalline toroid appHed as a core for a coil driven by a low (B <0.1 mT) ampHtude, low (10 kHz) frequency sinusoidal signal and (b) magnetocrystalline anisotropy constant, measured on a monocrystalline sphere showing the anisotropy/compensation temperature Tq and the Curie temperature, T. To convert joules to calories, divide by...

The crimp imparted to the tow has a sawtooth or sinusoidal wave shape. Because the filaments are usually crimped as a group, the crimp in parallel fibers is in lateral registry, ie, with the ridges and troughs of the waves aligned, as shown in Figure 14. 

Many classes of second-order material appHcations can be envisioned by noting the sinusoidal nature of electromagnetic radiation and rewriting equation 2... 

Blood Access Devices. An investigational device called the Osteoport system allows repeated access to the vascular system via an iatraosseous iafusion directiy iato the bone marrow. The port is implanted subcutaneously and secured iato a bone, such as the iUac crest. Medications are adrninistered as ia any conventional port, but are taken up by the venous sinusoids ia the marrow cavity, and from there enter the peripheral circulation (8). 

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