Harmonic sine function

Table 11.2. Force constants for the harmonic sine function for hexacoordinate transition metal hexaamines 45. ...

Another approach for describing the geometry around metal centers is to use a harmonic sine function [89,125]. This functionhas minima at 0°, 90° and 180°, and is based on Eqs. (3.21) and (3.7) with ... 

Table 12.3 Some examples showing the improvements in structural predictions with the harmonic sine function [125, 351].

Consider a simple harmonic sine function such as that shown in figure AI.l, which might, for example, represent a plucked string. 

Figure AI.l. Representation of a simple harmonic sine function.

Figure AI.2. Representation of a simple harmonic sine function as in Figure AI.l before and after its motion along the horizontal time (t) or distance (x) axis.

Because the second term in the brackets contains 3v in the sine function, radiation at a frequency which is three times that of the incident radiation is generated. This is referred to as third harmonic generation. The first term in brackets indicates that some radiation of unchanged frequency also results. 

Visco-elastic fluids like pectin gels, behave like elastic solids and viscous liquids, and can only be clearly characterized by means of an oscillation test. In these tests the substance of interest is subjected to a harmonically oscillating shear deformation. This deformation y is given by a sine function, [ y = Yo sin ( t) ] by yo the deformation amplitude, and the angular velocity. The response of the system is an oscillating shear stress x with the same angular velocity . 

The harmonic wave may also be described by the sine function... 

The representation of tp(x, t) by the sine function is completely equivalent to the cosine-function representation the only difference is a shift by A/4 in the value of X when t = 0. Moreover, any linear combination of sine and cosine representations is also an equivalent description of the simple harmonic wave. The most general representation of the harmonic wave is the complex function... 

Mathematically, a phase-sensitive detection (PSD) is carried out by multiplying A(v, f) by a sine function of the same frequency as the stimulation or harmonics thereof, sin(/rfflt + followed by a normalized integration of the product over... 

It is shown by substitution that this equation is satisfied by the straight line, y = mx + c, with first and second derivatives Sly = m, S/2y = 0. A string which is stretched between two points is known to assume this linear shape. Such a string, when disturbed, goes into harmonic vibration with displacements described by sine functions,... 

Any function of the form 5/ (x) jx is known as a sine function. The real spectrum of this is shown in Figure 10.10b, and we can see that this has the exact shape suggested by the envelope of the harmonics in Figure 10.10. Figures 10.10c and lO.lOd show the waveform and Fourier transform for a pulse of shorter duration. The Fourier transform is now more spread-out , and this demonstrates an important of the Fourier transform known as the scaling property. 

The exact structure of the replay field distribution depends on the shape of the fundamental pixel and the number and distribution of these pixels in the hologram. The pattern we generate with this distribution of pixels is repeated in each lobe of the sine function from the fundamental pixel. The lobes can be considered as spatial harmonics of the central lobe, which contains the desired 2D pattern. For example, a line of square pixels with alternate pixels being one or zero (i.e. a square wave) would have the basic replay structure seen in Fig. 1.4. 

There is a direct analogy between the one dimensional (ID) and 2D examples. The repetition of a square wave leads to discrete sampling in the frequency domain. In the case of the square wave, there is a series of odd harmonics generated. In 2D, these harmonics appear as orders radiating out in the lobes of the sine function from the dimensions of the fundamental aperture or pixel. The more pixels we have in the hologram, the closer we get to the infinite case and spots generated become more like delta functions. 

Other modulation techniques are oscillation (tilting) of an interference filter [3] and modulation of the electron beam scan pattern in a vidicon or image-disk-sector photomultiplier spectrophotometer [34]. This was the first nonmechanical wavelength modulation. Wavelength modulation induces a synchronous modulation of the amplitude. If these intensities are expanded, for instance, in the form of a Taylor series in Aq, and the powers of the sine functions are expressed as sine and cosine functions of the corresponding multiple angles, then the derivatives can be obtained from the Fourier coefficients (see Sec. 2.1.3.3) of these series. The second derivative is obtained from the second harmonic of the induced intensity. 

For t = 0, Eq. (4.178) can be simplified because only those spherical harmonics survive whose magnetic quantum number is zero, i.e., all other spherical harmonics include a vanishing sine function, sin = 0. Nonvanishing spherical harmonics occur if wzj = l/2orifOTj = —1/2. For the first case, mj = 1/2, we deduce from Eq. (4.178) that... 

For example, the block-reactor eigenfunctions must include the sine functions as well as the cosine, and in the spherical and cylindrical cases, the angular harmonics are required. 

The ab initio calculations were carried out at the CCSD(T) level of theory with aug-cc-pVTZ basis set augmented by midbond functions (for more details, see Ref. [31]). Then, the full 3D surface of each dipole moment component was represented as a sum of spherical harmonics multiplied by corresponding expansion coefficients and a cosine/sine function. 

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