Addition of waves

The sum of a large number of waves having the same frequency, then, may be obtained by adding their corresponding vector representatives in the complex plane, which is still awkward, or by simply adding up a list of complex numbers of the form a + ib. We can always recover the resultant amplitude K and phase 0 of the wave from the corresponding 

Second, looking again at the vector representation of K, we see that 

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The simplest bonding model requires only the addition of wave functions. Later in this chapter, we show that a complete description of bonding often requires both addition and subtraction of wave functions. 

Figure 1.3 The addition of waves scattered by an angle 2 from an atom at the origin and one at a vector r from the origin. The wavevectors k q and k jj are in the directions of the incident and diffracted beams, respectively, and k o = k h =l/...

A second quantum mechanical bonding theory is molecular orbital theory. This theory is based on a wave description of electrons. The molecular orbital theory assumes that electrons are not associated with an individual atom but are associated with the entire molecule. Delocalized molecular electrons are not shared by two atoms as in the traditional covalent bond. For the hydrogen molecule, the molecular orbitals are formed by the addition of wave functions for each Is electron in each hydrogen atom. The addition leads to a bonding molecular... 

We have indicated that interference and reinforcement effects depend both on the positions of atoms in a structure and the number of electrons associated with each atom. A quantitative treatment of these effects makes use of the important structure-factor equation which represents the addition of waves (sine and cosine functions) from each atom within a unit cell. All waves are of the same lengths but amplitudes and phases may differ. The structure-factor equation, to be given here but not derived, deals with the relative intensities of the reflected rays rather than with the absolute amplitudes or intensities of reflected rays (which depend on the amplitudes and intensities of the x-rays used as a source). The relative intensity, /, of a ray of indices hkl, from a set of planes hkl, is... 

FIGURE 8.1. Four waves with the same amplitude and periodicity are combined in three different ways [(a), (b), and (c)] as a result of different relative phase angles In each case the result of the Fourier synthesis (addition of waves) is different. Shown at the top of this Figure is a crystallographer with information on amplitudes and periodicities of the electron-density waves to be summed (on cardboard strips), but no information on relative phases (how to align the cardboard strips). 

Figure 6.11 The representation of scattered waves as vectors (a) a scattered wave vector in the x, y plane (b) the wave scattered by an atom A at (x, y, z) (c) the wave scattered by atom A at the origin, (000) (d) the addition of waves scattered by five atoms, A, B, C, D, and E (e) represention of fA as a complex number on an Argand diagram...

Figure 6.13 Reflections from a body-centred unit cell (a) a body-centred unit cell (b) vector addition of waves for reflections h + k + l even, gives rise to a scattered amplitude (c) vector addition of waves for reflections h + k + l odd, gives rise to zero amplitude...

When the product of the electrode process reacts chemically with other components, the above is not true. Such examples, observed with inorganic systems, have rarely been described for organic substances. A more complicated example of this t)rpe of system (i.e. the non-additivity of waves) is formaldehyde and acetaldehyde (Fig. 5). Here, probably, the radicals formed as intermediates in the formaldehyde reduction, react with the acetaldehyde diffusing towards the electrode. The amount of acetaldehyde reaching the surface of the electrode is thus diminished and the acetaldehyde wave is decreased (Fig. 5, curve 2), compared with the wave obtained with solutions containing the same concentration of acetaldehyde, but no formaldehyde (Fig. 5, curve 3). 

The constructive interaction—that is, the addition of wave functions—gives a sigma (ct) molecular orbital. The electron density between two nuclei is located in this cyhndrically symmetrical region. 

The use of the surface ultrasonic waves seems to be convenient for these purposes. However, this method has not found wide practical application. Peculiarities of excitation, propagation and registration of surface waves created before these time great difficulties for their application in automatic systems of duality testing. It is connected with the fact that the surface waves are weakened by soil on the surface itself In addition, the methods of testing by the surface waves do not yield to automation due to the difficulties of creation of the acoustic contact. In particular, a flow of contact liquid out of the zone of an acoustic line, presence of immersion liquid, availability of chink interval leads to the adsorption and reflection of waves on tlie front meniscus of a contact layer. The liquid for the acoustic contact must be located only in the places of contact, otherwise the influence on the amplitude will be uncontrolled. This phenomenon distorts the results of testing procedure. 

In (a), two photon waves combine to give a new waveform, which has the same appearance and frequency as the initial separate waves. The photons are said to be coherent, and the amplitude of the waves (light intensity) is simply doubled. In (b), the two photon waves are shown out of step in time (incoherent). Addition of the two waveforms does not lead to a doubling of amplitude, and the new waveform is more complex, composed of a doubled overlapping frequency. In (c), the two waveforms are completely out of step (out of phase) and completely cancel each other, producing darkness rather than light (an interference phenomenon). 

A new chemical sensor based on surface transverse device has been developed (99) (see Sensors). It resembles a surface acoustic wave sensor with the addition of a metal grating between the tranducer and a different crystal orientation. This sensor operates at 250 mH2 and is ideally suited to measurements of surface-attached mass under fluid immersion. By immohi1i2ing atra2ine to the surface of the sensor device, the detection of atra2ine in the range of 0.06 ppb to 10 ppm was demonstrated. 

Other Applications. The refractive index of siUcate or borosiUcate glass can be modified by the addition of cesium oxide, introduced as cesium nitrate or carbonate. Glass surfaces can be made resistant to corrosion or breakage by surface ion exchange with cesium compound melts or solutions. This process can also be used for the production of optical wave guides (61). 

T vo main streams of computational techniques branch out fiom this point. These are referred to as ab initio and semiempirical calculations. In both ab initio and semiempirical treatments, mathematical formulations of the wave functions which describe hydrogen-like orbitals are used. Examples of wave functions that are commonly used are Slater-type orbitals (abbreviated STO) and Gaussian-type orbitals (GTO). There are additional variations which are designated by additions to the abbreviations. Both ab initio and semiempirical calculations treat the linear combination of orbitals by iterative computations that establish a self-consistent electrical field (SCF) and minimize the energy of the system. The minimum-energy combination is taken to describe the molecule. 

In the MO-CI language, the correct dissociation of a single bond requires addition of a second doubly excited determinant to the wave function. The VB-CF wave function, on the other hand, dissociates smoothly to the correct limit, the VB orbitals simply reverting to their pure atomic shapes, and the overlap disappearing. 

A completely different type of property is for example spin-spin coupling constants, which contain interactions of electronic and nuclear spins. One of the operators is a delta function (Fermi-Contact, eq. (10.78)), which measures the quality of the wave function at a single point, the nuclear position. Since Gaussian functions have an incorrect behaviour at the nucleus (zero derivative compared with the cusp displayed by an exponential function), this requires addition of a number of very tight functions (large exponents) in order to predict coupling constants accurately. ... 

For coherent sound waves addition of values is possible. It will be apparent that as the scale is logarithmic, values cannot merely be added to one another. Intensities can, however, be added and thus the equation becomes ... 

The accuracy of the method depends upon the precision with which the two volumes of solution and the corresponding diffusion currents are measured. The material added should be contained in a medium of the same composition as the supporting electrolyte, so that the latter is not altered by the addition. The assumption is made that the wave height is a linear function of the concentration in the range of concentration employed. The best results would appear to be obtained when the wave height is about doubled by the addition of the known amount of standard solution. This procedure is sometimes referred to as spiking. 

To determine die diffusion current, it is necessary to subtract the residual current. This can be achieved by extrapolating the residual current prior to the wave or by recording die response of the deaerated supporting electrolyte (blank) solution. Addition of a standard or a calibration curve are often used for quantitation. Polarograms to be compared for this purpose must be recorded in the same way. 

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