Cosine wave

For our purposes we can neglect the imaginary part of Equation (3.16), i sin27ivt, and then it is apparent that f t) is a sum of cosine waves, as we had originally supposed. Fourier transformation allows us to go from f t) to F(v) by the relationship... 

In a more usual emission experiment the source contains many wavelengths, the detector sees intensity due to many cosine waves of dilferent wavelengths and the detected intensity is of the form... 

If a single sharp absorption occurs at a wavenumber v, as shown in the wavenumber domain spectmm in Figure 3.15, the cosine wave corresponding to is not cancelled out and remains in the I 5) versus 5 plot, or interferogram, as it is often called. For a more complex set of absorptions the pattern of uncancelled cosine waves becomes more intense and irregular. 

In FT-Raman spectroscopy the radiation emerging from the sample contains not only the Raman scattering but also the extremely intense laser radiation used to produce it. If this were allowed to contribute to the interferogram, before Fourier transformation, the corresponding cosine wave would overwhelm those due to the Raman scattering. To avoid this, a sharp cut-off (interference) filter is inserted after the sample cell to remove 1064 nm (and lower wavelength) radiation. 

After the pulse, we wait for a short whilst (typically a few microseconds), to let that powerful pulse ebb away, and then start to acquire the radio frequency signals emitted from the sample. This exhibits itself as a number of decaying cosine waves. We term this pattern the free induction decay or FID (Figure 1.3). 

Free Induction Decay (FID) Interference pattern of decaying cosine waves collected by Fourier Transform spectrometers, stored digitally prior to Fourier Transformation. 

Thus, (he Intensity of the standing wave at the surface increases smoothly from 0 to 4 , as 0 is increased from 0 to as shown in Figure 279. At distances >O,/(0.z) modulates as a cosine wave between 0 and 4 , the number of modulations being directly proportional to Thus, at a distance S0C above the mirror surface there will be (iV + 1/2) modulations between 0 0 and Oq. The case for r =- 2i>v is also shown in Figure 2.79. 

Let us describe the displacement of the nth atom from its equilibrium rest position by a cosine-wave with amplitude w0, angular frequency and wave vector q = 2n/X 1... 

James Clerk Maxwell predicted the existence of electromagnetic waves in 1864 and developed the classical sine (or cosine) wave description of the perpendicular electric and magnetic components of these waves. The existence of these waves was demonstrated by Heinrich Hertz 3 years later. 

Fig. 5.21. (A) shows a conventional cosine wave of 500 Hz and (B) a cosine wave of 333 Hz, (C) the algebraic sum of (A) and (B), (D) represents the cosine wave in (A) in the frequency domain, (D) the representation of (C) in the frequency domain.

Now the first term on the right is in phase with the applied strain, i.e. it has the form of a sine wave. This can be equated with the storage modulus. Conversely the phase difference between the second term on the right and the applied signal is the difference between sine and cosine waves which can be equated with the loss modulus ... 

In this demonstration of a Fourier series we will use only cosine waves to reproduce the shadow image of the black squares. The procedure itself is rather straightforward, we just need to know the proper values for the amplitude A and the index h for each wave. The index h determines the frequency, i.e. the number of full waves trains per unit cell along the a-axis, and the amplitude determines the intensity of the areas with high (black) potential. As outlined in Figure 4, the Fourier synthesis for the present case is the sum of the following terms ... 

Figure 5. Changing the sign of the amplitude from minus to plus causes a phase shift of 180° in the Fourier map. Every cosine wave with a positive amplitude starts at the origin of the unit cell with a maximum (high potential) cosine waves with negative amplitudes on the other hand produce low (zero) potential at the origin.

Since cosine waves naturally start at the origin (X = 0) with a maximum, we can generally say that changing the sign of the amplitude from plus to minus causes a 180 degree (AX = a / 2h) shift of the wave maxima. If we want to... 

The direction and the periodicity of each cosine wave are given by its index u = hkl), the amplitude of the cosine wave is 2 F(u), proportional to the structure factor amplitude F(u). More importantly, the positions of the maxima and minima of the cosine wave (in relation to the unit cell origin) are determined by the structure factor phase ( )(u). If both the amplitudes F(u) and the phases ( )(u) of the structure factors for all reflections u are known, the potential cp(r) can be obtained by adding a series of such cosine waves. 

If the Fourier synthesis is carried out by adding in the strong reflections first, we will see how fast the Fourier series converges to the projected potential. The positive potential contribution from the reflection is shown in white, whereas the negative potential contribution is shown in black. Most of the atoms are located in the white regions of each cosine wave, but the exact atomic positions will not become evident until a sufficiently large number of structure factors have been added up. 

The cosine waves generated by the second strongest reflection (3 5 0) and its symmetry-related reflection (3 -5 0) are shown in Figs. Id and e. Both cosine waves cut the a axis 3 times per unit cell and the b axis 5 times, however, they are oriented differently. The summation of the two symmetry-related cosine waves is shown in Fig. If). 

Equation 20-10 says that the value of y for any value of x can be expressed by an infinite sum of sine and cosine waves. Successive terms correspond to waves with increasing frequency. 

Figure 20-25 shows how sequences of three, five, or nine sine and cosine waves give better and better approximations to the curve in Figure 20-24. The coefficients an and bn required to construct the curves in Figure 20-25 are given in Table 20-3.

A graph of output light intensity versus retardation. 3, is called an interferogram. If the light from the source is monochromatic, the interferogram is a simple cosine wave ... 

As discussed in Section VI.A, any simple sine or cosine wave can therefore be described by three constants — the amplitude F, the frequency h, and the... 

I will write Fourier series in this form throughout the remainder of the book. This kind of equation is compact and handy, but quite opaque at first encounter. Take the time now to look at this equation carefully and think about what it represents. Whenever you see an equation like this, just remember that it is a Fourier series, a sum of sine and cosine wave equations, with the full sum representing some complicated wave. The hth term in the series, Fh 1 ni hx, can be expanded to Fj/cos 2ir(hx) + i sin 2-tt(/ )], making plain that the hth term is a simple wave of amplitude Fh, frequency h, and implicit phase [Pg.88]

Now let s look briefly at just enough of the mathematics of fiber diffraction to explain the origin of the X patterns. Whereas each reflection in the diffraction pattern of a crystal is described by a Fourier series of sine and cosine waves, each layer line in the diffraction pattern of a noncrystalline fiber is described by one or more Bessel functions, graphs that look like sine or cosine waves that damp out as they travel away from the origin (Fig. 9.3). Bessel functions appear when you apply the Fourier transform to helical objects. A Bessel function is of the form... 

Interestingly, the requirement for an non-clicking backwards-forwards loop is even symmetry around the loop points. A signal x[n] is referred to as even if x[n] = x n. A signal is odd if x[-n = —x[ri. A cosine wave is even around the origin and a sine wave has odd symmetry around the origin. 

Don t let the definition based on tracking around a circle fool you—sine and cosine waves appear in many problems in chemistry and physics. The motion of a mass... 

We performed adiabatic connection calculations for cosine-wave jellium using six values of A 0,0.2,0.4,0.6,0.8,1. The many-body wavefunctions for A > 0 were optimized by fixed-density variance minimization using 10000 independent N—electron configurations at each A. These configurations were regenerated several times. The weight factor in expression (27) was set equal... 

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