Wave equations. Phase

Molecular structure is best represented in terms of quantum mechanics. Quantum mechanical calculations are quite difficult. Therefore, approximation methods have been evolved which are result of mathematical simplifications. Molecular orbitals are centered around all the nuclei present in the molecule. Relative stabilities of molecules depend upon how electrons are distributed in them. In order to understand molecular symmetry it is essential to understand wave equations, phases of waves originated by the movement of electrons if we consider them as waves and also what are bonding and antibonding molecular orbitals. 

In this chapter we define what is meant by a shock-wave equation of state, and how it is related to other types of equations of state. We also discuss the properties of shock-compressed matter on a microscopic scale, as well as discuss how shock-wave properties are measured. Shock data for standard materials are presented. The effects of phase changes are discussed, the measurements of shock temperatures, and sound velocities of shock materials are also described. We also describe the application of shock-compression data for porous media. 

For wave functions like = exp[if x,t)], the squared operator would mask the phase information, since = <3> 2, and to avoid this, a linear Schrodinger operator would be preferred. This has the immediate advantage of a wave equation which is linear in both space and time derivatives. The most general equation with the required form is... 

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

resulting wave can be written as ... 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

Many phenomena such as dislocations, electronic structures of polyacetylenes and other solids, Josephson junctions, spin dynamics and charge density waves in low-dimensional solids, fast ion conduction and phase transitions are being explained by invoking the concept of solitons. Solitons are exact analytical solutions of non-linear wave equations corresponding to bell-shaped or step-like changes in the variable (Ogurtani, 1983). They can move through a material with constant amplitude and velocity or remain stationary when two of them collide they are unmodified. The soliton concept has been employed in solid state chemistry to explain diverse phenomena. 

The Maxwell-Heaviside theory of electrodynamics has no explanation for the Sagnac effect [4] because its phase is invariant under 7 as argued already, and because the equations are invariant to rotation in the vacuum. The d Alembert wave equation of U(l) electrodynamics is also 7 -invariant. One of the most telling pieces of evidence against the validity of the U(l) electrodynamics was given experimentally by Pegram [54] who discovered a little known [4] cross-relation between magnetic and electric fields in the vacuum that is denied by Lorentz transformation. 

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]

Equation (5.18) tells us how to calculate p(jc,y,z) simply construct a Fourier series using the structure factors Fhkl. For each term in the series, h, k, and 1 are the indices of reflection hkl, and Fhkl is the structure factor that describes the reflection. Each structure factor Fhkl is a complete description of a diffracted ray recorded as reflection hkl. Being a wave equation, Fhkl must specify frequency, amplitude, and phase. Its frequency is that of the X-ray source. Its amplitude is proportional to (- j /)1/2, the square root of the measured intensity Ihkl of reflectionhkl. Its phase is unknown and is the only additional information the crystallographer needs in order to compute p(x,y,z) and thus... 

Therefore, the equation of the interface is a wave equation, reflecting that the flow is in an unsteady wave motion. Furthermore, because the corresponding coefficient of each term in these four equations should be equal, several relations among phase quantities result ... 

Wallis (1969) defined the dynamic wave in one-phase flow as being that which occurs whenever there is a net force on the flowing medium produced by a concentration gradient. For a two-phase flow, i.e., gas-solid flow, the flow medium refers to the gas phase and the concentration refers to the solids holdup. Thus, to analyze dynamic waves, one can examine the wave equation obtained from the perturbation of the momentum and mass balance equations for the gas and solid phases. The analyses given later for both 6.5.2.2 and 6.5.2.3 follow those of Rietema (1991). 

An ordinary plane wave of definite wavelength A spreads over all space and can obviously not be used to describe the motion of a particle-like pulse, which is localized over a comparatively narrow region. One way to construct a pulse or wave packet that resembles an extended particle is to combine a number of waves with slightly different wavelengths and with amplitudes and phases chosen so that the waves interfere constructively over a limited region of space. The principle is readily demonstrated in one dimension, using the real part of the general wave equation... 

For a uniform charge distribution within a spherical atom the Fourier transform of the density has been shown (equation 5.6) to be of the form sin a/a, for a wave of phase a in momentum space. From the Bragg equation (Figure 2.9), A = 2dsin0, it follows that electrons at a distance d = A/2sin0 apart, scatter in phase, i.e. with phase difference 27T. At a separation r the relative phase shift a, is given by ... 

In an anisotropic dielectric the phase velocity of an electromagnetic wave generally depends on both its polarization and its direction of propagation. The solutions to Maxwell s electromagnetic wave equations for a plane wave show that it is the vectors D and H which are perpendicular to the wave propagation direction and that, in general, the direction of energy flow does not coincide with this. 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

A dispersion relation such as this allows one to calculate the phase velocity of the waves, given by v = dispersion relations for Equations 2.13 and 2.14 indicate that V2 = V3 = (C44/p). ... 

The paraxial approximation is essentially a Taylor series expansion of an exact solution of the wave equation in powers of p/w, terminated at ( p/tv), that allows us to exploit the rapid decay of a Gaussian beam away from the optical axis. We will develop a more precise criterion in the sequel. We will also show that the phase and amplitude modulation of the underlying plane wave structure of the electromagnetic field is a slowly varying function of distance from the point where the beam is launched. 

Gazdag, J., 1978, Wave equation migration with the phase-shift method Geophysics, 43, No. 7, 1342-1351. 

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