Resultant waves

FIGURE 2 2 Interference between waves (a) Constructive interference occurs when two waves combine in phase with each other The amplitude of the resulting wave at each point is the sum of the amplitudes of the original waves (b) Destructive interference decreases the amplitude when two waves are out of phase with each other... 

Interference of Waves. The coherent scattering property of x-rays is used in x-ray diffraction appHcations. Two waves traveling in the same direction with identical wavelengths, X, and equal ampHtudes (the intensity of a wave is equal to the square of its ampHtude) can interfere with each other so that the resultant wave can have anywhere from zero ampHtude to two times the ampHtude of one of the initial waves. This principle is illustrated in Figure 1. The resultant ampHtude is a function of the phase difference between the two initial waves. 

In each of the three cases the resultant wave has the same wavelength, X, as the initial waves. 

Figure 4.10. Type of Hugoniot necessary to produee a two-wave shoek strueture and resulting wave profile. This type of Hugoniot will in general give a loeus as shown, with a flat region of eonstant shock velocity. Point 2 will not be observed with techniques that measure only the first arrival of the shock wave. (After McQueen et al. (1970).)...

Fig. 2.1. The traditional approach to the study of mechanical responses of shock-compressed solids is to apply a rapid impulsive loading to one surface of a diskshaped sample and measure the resulting wave propagating in the sample. As suggested in the figure, the wave shapes encountered in shock-loaded solids can be complex and may require measurements with time resolutions of a few nanoseconds.

Fig. 2.11. Strength behavior of solids at pressure can be probed with reshock or release measurements. The resulting wave profiles of such measurements on a 6061-T6 aluminum alloy with VISAR instrumentation are shown. Strength behavior indicated on many solids reveals behavior not accurately described by simple materials models (after Lipkin and Asay [77L02]).

Since in those forms of the UHF wave functions, one drops a constraint (either the need of a pure spin state in the first case or the Pauli antisymmetry rule in the second case), it is expected that the resulting wave function will give a lower energy than in the RHF case and thus introduce a part of the correlation energy. As shown in the table above, there is... 

The wave function for the particle is obtained by joining the three parts ipi, tpii, and fill such that the resulting wave function f(x) and its first derivative f x) are continuous. Thus, the following boundary conditions apply... 

Reinforce a crest meets a crest (waves of the same phase sign meet each other) => add together => resulting wave is larger than either individual wave. 

The first step beyond the statistical model was due to Hartree who derived a wave function for each electron in the average field of the nucleus and all other electrons. This field is continually updated by replacing the initial one-electron wave functions by improved functions as they become available. At each pass the wave functions are optimized by the variation method, until self-consistency is achieved. The angle-dependence of the resulting wave functions are assumed to be the same as for hydrogenic functions and only the radial function (u) needs to be calculated. 

The micrograph or the image obtained on an EM screen, photographic film, or (more commonly today) a CCD is the result of two processes the interaction of the incident electron wave function with the crystal potential and the interaction of this resulting wave function with the EM parameters which incorporate lens aberrations. In the wave theory of electrons, during the propagation of electrons through the sample, the incident wave function is modulated by its interaction with the sample, and the structural information is transferred to the wave function, which is then further modified by the transfer function of the EM. 

The boundary conditions on ipni(r) are determined by the boundary conditions of R i(r). Because R,/(r) is finite in the origin, then i/rn/(0) = 0. Furthermore, as we have a potential wall of infinite height, similar to that found in the PIAB, the resulting wave function on the surface of this wall must vanish. Thus, we have the Dirichlet boundary conditions for this problem... 

It is important to note that the velocity of the wave in the direction of propagation is not the same as the speed of movement of the medium through which the wave is traveling, as is shown by the motion of a cork on water. Whilst the wave travels across the surface of the water, the cork merely moves up and down in the same place the movement of the medium is in the vertical plane, but the wave itself travels in the horizontal plane. Another important property of wave motion is that when two or more waves traverse the same space, the resulting wave motion can be completely described by the sum of the two wave equations - the principle of superposition. Thus, if we have two waves of the same frequency v, but with amplitudes A and A2 and phase angles

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The nature of the resulting wave depends on the phase difference (2) is 0 degrees, or 360 degrees, then the two waves are said to be in phase, and the maximum amplitude of the resultant wave is A1 + A2. This situation is termed constructive interference. If the phase difference is 180 degrees, then the two waves are out of phase, and destructive interference occurs. In this case, if the amplitudes of the two waves are equal (i.e., if A = A2), then the two waves cancel each other out, and no wave is observed (Fig. 12.1). Standing waves, such as those seen when the string on a musical instrument vibrates, are caused when the reflected waves (from the bridge of the instrument) are in phase and thus interfere constructively. 

Now consider the effect of anomalous scattering on the relative intensities of the diffracted rays in Scheme 2a and b when atom Y scatters anomalously with an intrinsic phase lead A< >(Y), and atom W scatters normally. Under such circumstances, the wave scattered by atom Y in Scheme 2a would lead that of atom W by a phase difference of + A< >(Y), and the wave scattered by atom Y in Scheme 2b would lag behind that of atom W by - + A(Y). These two phase differences are unequal in magnitude, so the corresponding amplitudes of their resultant waves, and the subsequent intensities, will be different, leading to a breakdown of Friedel s law. 

The values for the dipoles, polarizabilities, and hyperpolarizabilities of the H2 series were obtained using (a) a 16-term basis with a fourfold symmetry projection for the homonuclear species and (b) a 32-term basis with a twofold symmetry projection for the heteronuclear species. These different expansion lengths were used so that when combined with the symmetry projections the resulting wave functions were of about the same quality, and the properties calculated would be comparable. A crude analysis shows that basis set size for an n particle system must scale as k", where k is a constant. In our previous work [64, 65] we used a 244-term wave function for the five-internal-particle system LiH to obtain experimental quality results. This gives a value of... 

Equation (1.10) represents the Hartree Hamiltonian and Eq. (1.8) has to be solved by iteration, in the sense that a guessed trial wave function 1) is introduced in Eq. (1.10) and the Schrodinger equation Eq. (1.8) solved. The resulting wave function is again introduced in Eq. (1.10) and Eq. (1.8) is again solved until self-consistency is achieved. 

We further proceed, as previously, by building a one-electron wave function from the associated atomic orbitals, Eq. (1.14), but imposing the Bloch condition expressed in Eq. (1.25). This is the well-known TB approximahon (Ashcroft Mermin, 1976). The resulting wave funchon is often called the crystal orbital and has the form ... 

In 1888, the American, Professor C.E. Munroe (Ref la), while working at the Naval Torpedo Station at Newport, Rl, observed that if a block of cast high explosive with letters indented on the surface was placed with the letter-side against a metal plate and exploded, the letters would be reproduced indented on the metal plate. This phenomenon was explained by the fact that two or more explosive waves will resolve into a resultant wave which is of much greater force than any of the original waves. 

If the CISD wave functions for two identical molecules are multiplied, to give the wave function for the pair of molecules, there are terms in the resulting wave function in which both molecules are doubly excited. Since these terms represent quadruple excitations from the HF configuration, they are not included in the CISD wave function for two identical molecules at infinity. Consequently, the CISD energy for a pair of identical molecules is higher than twice the CISD energy for an individual molecule. 

It would seem to be an unresolvable problem—to calculate the structure factors we need the atomic positions and to find the atomic positions we need both the amplitude and the phase of the resultant waves, and we only have the amplitude. Fortunately, many scientists over the years have worked at finding ways around this problem, and have been extremely successliil, to the extent that for many systems the solving of the structure has become a routine and fast procedure. 

Fig. 6.14. (a) If the path difference of two waves is 0, X, 2X,. .. the resultant wave is the addition of both waves, (b) However, two waves with the same amplitude but with a path difference equal to XJ2 cancel each other. 

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