Sines. Curve

Figure 18.14 The diffraction pattern of helices in fiber crystallites can be simulated by the diffraction pattern of a single slit with the shape of a sine curve (representing the projection of a helix). Two such simulations are given in (a) and (b), with the helix shown to the left of its diffraction pattern. The spacing between the layer lines is inversely related to the helix pitch, P and the angle of the cross arms in the diffraction pattern is related to the angle of climb of the helix, 6. The helix in (b) has a smaller pitch and angle of climb than the helix in (a). (Courtesy of W. Fuller.)...

Sinus-kurve, -linie, /. sine curve, sinuskurvenartlg, a. sinusoidal, siphonieren, v.t. siphon. 

The basic assumption of constant atmospheric X4C activity in radiocarbon dating is not strictly valid. We now have a record of the fluctuation of atmospheric 14C variations for the last 8,400 years B.P. obtained by measurement of the isotopes of carbon in dendrochron-ologically dated wood. Prior to contamination of atmospheric 14C activity by fossil fuel combustion and nuclear technology in the 20th century, the first-order secular variation can be closely approximated by a sine curve with a period of 10,600 years and an amplitude of ... 

All of the sonic disturbances in question take the form of progressive (forward moving) waves in air, in the manner of sine curves whose characteristics are governed by the laws of physics. The wave motion commences at the source of the energy release, i.e. the exploding firework, and is caused by the sequential disturbance (i.e. vibration) of the individual particles in the air. 

After the initial 90° pulse, the M vector is in the xy plane giving the detectable signal. Due to the precession and the relaxation processes, the signal (FID) is a sine curve with exponentially decreasing amplitude.11... 

The vector model cannot be interpreted in such a simple way in the case of a spin system with more than one nucleus. For weakly coupled spin systems, the single spin vector model may be applied for each nucleus, one after the other. Thus the coupling with the other nuclei can be incorporated into its precession frequency, since the definition of the weak coupling (J -C vM v,/1) means that the transitions of a nucleus only depend on the spin states of the other nuclei in the first order. The detected signal is the sum of the sine curves provided by the individual environment of the nuclei. 

This separation of the cr framework and the re bond is the essence of Hiickel theory. Because the re bond in ethylene in this treatment is self-contained, we may treat the electrons in it in the same way as we do for the fundamental quantum mechanical picture of an electron in a box. We look at each molecular wave function as one of a series of sine waves, with the limits of the box one bond length out from the atoms at the end of the conjugated system, and then inscribe sine waves so that a node always comes at the edge of the box. With two orbitals to consider for the re bond of ethylene, we only need the 180° sine curve for re and the 360° sine curve for re. These curves can be inscribed over the orbitals as they are on the left of Fig. 1.23, and we can see on the right how the vertical lines above and below the atoms duplicate the pattern of the coefficients, with both c and c2 positive in the re orbital, and c positive and c2 negative in re. ... 

We can easily give numerical values to these coefficients for the linear conjugated system. The coefficients are proportional to the sine of the angle, as defined by the position of the atom within the sine curve. Equation 1.12 is the algebraic expression for this idea, illustrated in Fig. 1.31 with the atomic orbitals inscribed within the sine curves ... 

The energies of the molecular orbitals can be deduced by inscribing the conjugated system inside a circle of radius 2(3. There is no need for dummy atoms, since the sine curves go right round the ring, and the picture is therefore... 

Figure 2.15 Simulation of elemental periodicity by a sine curve. The periodic part occurs between atomic numbers 3 and 99...

Following this excitation, the magnetisation will revert to the value corresponding to the Boizmann distribution with, for a single and unique type of nucleus (for example, the H of benzene) present in the sample, a behaviour M - f t) in the form of a damped sine curve from which it is immediately possible to find the resonance frequency. On the other hand, if several hydrogens with different n are present, the curve soon becomes uninterpretable (cf. Fig. I3.4B). All that is required is to apply the Fourier transfonn mathematical operation to M = fit) in order to obtain M =/(v) which is the usual NMR spectrum. 

Sine curves are relevant to pharmaceutical solid dosage manufacturing unit operations such as modeling the di.splacemeni behavior of a roll on a tablet press or roller compactor. Since sine curves have natural (more gradual) acceleration and deceleration regions within their waveforms compared to linear waveforms, it is easier for a compaction simulator to respond accurately to the command profile. 

A simple combination of sine curves with periods of 23, 41 and 100 ky (Fig. 7.8) illustrates how difficult it is to deconvolute the cycles by eye. This problem is compounded in natural samples because of their imperfect nature. Climate scientists use spectral analysis filtering as a tool to deconvolute the 8 0 records into their predominant cyclic components. When this is done the location of the main 23,41 and 100 ky cycles can be identified (Fig. 7.9) and assigned ages. Because the 100 ky forcing is weak, climate scientists primarily rely on the 23 and 41 ky cycles to date 8 0 records. With the 41 l[Pg.233]

Demonstration of the complex curves that result from combination of sine waves with amplitudes of 100,41 and 23 ky, the three predominant Milankovitch cycles. Combinations of the three sine curves illustrate the difficulty in backing out cycles from the data. The compilation here does not look like 5 0 vs. depth curves in ocean sediments because the amplitudes are the same in the illustration. Redrawn from Ruddiman (2001). 

Fig. 2-15 Coefficients in the molecular orbitals of the allyl system (a) as sine curves, (b) in elevation and (c) in plan...

In Figure 8.3, (a) shows a fluid in a deformable host (stippled) initiaUy the host is of uniform thickness, with a uniform content of fluid, and flow can only occur to left or right in the plane of the diagram (no flow perpendicular to the page). The host is instantaneously loaded with a set of heavy rods whose lengths form a sine curve then the pressure in the fluid-filled layer also must vary sinusoidally along x ... 

The sine ducts were chosen for this analysis because they appear to be a very good representation of the wrapped or corrugated structure (1). The derivations of the geometric surface area and open frontal area of this structure are based on the equation for a sine curve ... 

The annual variation of the total ozone (see Fig. 10) can be approximated at the mid-latitudes by a sine curve. It should be noted that the tropospheric 03 level shows similar annual changes. The only difference is that maximum concentrations are observed in the troposphere 1-2 months later (see Fig. 13).-On the basis of this time lag Junge (1962) estimated that the residence time of 03 in the troposphere is 2 months. In contrast the atmospheric residence time of 03 is calculated to be 1-2 years (Junge, 1963) that is, the lifetime of this species is very different in the stratospheric and tropospheric reservoirs. 

Examples.—(1) Show that the graph of y = cos a has the same form as the sine curve and would be identical with it if the y-exis of the sine curve were shifted a distance of far to the right. [Proof sin ( + x) = cos x, etc.] The physical meaning of this is that a point moving round the perimeter of the circle according to the equation y = cos a is just ir, or 90° in advance of one moving according to y — sin a. 

The resulting curve is the harmonic or sine curve shown in Fig. 52, page 186. 

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