WAVES AND THEIR PROPERTIES

In a mathematical sense, a periodic wave is any function f(x) whose value varies in a repetitive and perfectly predictable manner over discrete intervals of some variable x. A physical way of describing waves is that they are some property of the medium in which they exist that changes in a regular and periodic manner as a function of the distance from some point, or as a function of time if one stands at a fixed point in space and measures the unique property. For sound, the property may be pressure for waves in the water, it may be height above or below the surface for light or X rays, the properties are electromagnetic. 

---

For more detailed description of shock waves and their properties, see Refs which follow. In these Refs are described, among others, the following items ... 

Because electrons are concentrated around atomic nuclei, knowing p(xj, yj, Zj) for all points j is essentially the same as knowing the distribution of atoms in the unit cell, which in turn means the structure of the molecules which inhabit the unit cell. This is illustrated in Figures 3.22 and 3.23. Thus another way of looking at a crystal is that it is a three-dimensional, periodic, electron density wave that repeats in a perfectly regular manner in space. This is important because several hundred years of physics and mathematics have been focused on periodic waves and their properties, and many clever mathematical tools exist that allow us to manipulate and analyze them. 

Before we delve further into the properties of the nucleus, let us momentarily shift our attention back to one of the electrons zooming around the nucleus. Just like photons, electrons exhibit both wave and particle properties. Each electron wave in an atom is characterized by four quantum numbers. The first three of these numbers can be taken as the electron s address and describe the energy, shape, and orientation of the volume the electron occupies in the atom. This volume is called an orbital. The fourth quantum number is the electron spin quantum number s, which can assume only two values, or - f. (Why J was selected rather than, say, 1 will be described a little later.) The Pauli exclusion principle tells us that no two electrons in an atom can have exactly the same set of four quantum numbers. Therefore, if two electrons occupy the same orbital (and thus possess the same first three quantum numbers), they must have different spin quantum numbers. Therefore, no orbital can possess more than two electrons, and then only if their spins are paired (opposite). 

Equation 3.80 has been verified in measurements made on a flexural-wave oscillator operated successively in air at STP (essentially zero loading) and then with various liquids on one side of the plate. Table 3.4.1 lists the liquids and their properties, and compares measured frequencies with those calculated from Equation 3.80. 

Self-excited combustion instabilities are associated with the propagation and reflection of heat-release-induced acoustic waves and their interactions. Hence, flame sound represents the main source of these acoustic waves. Therefore, sound pressure level (SPL) data for turbulent nonpremixed jet flames have been obtained for two Turbulent Nonpremixed Flame (TNF) workshop flames, DLR-A and DLR-B [1]. The exit Reynolds numbers (Re) for the two flames based on injected gas properties at room temperature were 15.200 (DLR-A) and 22,800 (DLR-B). Air was used for studying the sound emission from equivalent nonreacting jets. The flow in each case had very low exit Mach numbers (M = 0.04-... 

Cyclic voltammetry (CV) can provide information about the thermodynamics of the redox process, kinetics of heterogeneous electron transfer reactions and coupled chemical reactions [32]. The reversible electron transfer steps inform us about the compound s ability to accept electrons however, experimental conditions, such as solvent and temperature also influence the voltammogram. The structure of the lowest unoccupied molecular orbital (LUMO) levels of the compound can be determined from the number of CV waves and reduction potentials ( 1/2)- Moreover, the CV can serve as a spectroscopy as demonstrated by Heinze [32], since the characteristic shapes of the waves and their unequivocal positions on the potential scale are effectively a fingerprint of the individual electrochemical properties of the redox system. 

Similar to the present introduction, in Sections 5.2 and 5.3 mainly (but not exclusively) one-dimensional models are used to describe metal surfaces and their properties. However, in contrast to the present section, we (mostly) do not rely on an expansion of the Bloch wave function in terms of atomic orbitals, but rather use plane waves. For metals with s,p-derived valence/conduction electrons, the latter approach turns out to be particularly advantageous as often a few plane waves are suflident for a semiquantitative description. 

The concept of two-state systems occupies a central role in quantum mechanics [16,26]. As discussed extensively by Feynmann et al. [16], benzene and ammonia are examples of simple two-state systems Their properties are best described by assuming that the wave function that represents them is a combination of two base states. In the cases of ammonia and benzene, the two base states are equivalent. The two base states necessarily give rise to two independent states, which we named twin states [27,28]. One of them is the ground state, the other an excited states. The twin states are the ones observed experimentally. 

Valence bond and molecular orbital theory both incorporate the wave description of an atom s electrons into this picture of H2 but m somewhat different ways Both assume that electron waves behave like more familiar waves such as sound and light waves One important property of waves is called interference m physics Constructive interference occurs when two waves combine so as to reinforce each other (m phase) destructive interference occurs when they oppose each other (out of phase) (Figure 2 2) Recall from Section 1 1 that electron waves m atoms are characterized by their wave function which is the same as an orbital For an electron m the most stable state of a hydrogen atom for example this state is defined by the Is wave function and is often called the Is orbital The valence bond model bases the connection between two atoms on the overlap between half filled orbifals of fhe fwo afoms The molecular orbital model assembles a sef of molecular orbifals by combining fhe afomic orbifals of all of fhe atoms m fhe molecule... 

Atomic Levels and Their Decay. There are many commonalities between the properties of atomic and nuclear levels and between their respective decays. Each level has a quantum mechanical wave function which describes its properties. It is common practice to illustrate the atomic and... 

For example, the measured pressure exerted by an enclosed gas can be thought of as a time-averaged manifestation of the individual molecules random motions. When one considers an individual molecule, however, statistical thermodynamics would propose its random motion or pressure could be quite different from that measured by even the most sensitive gauge which acts to average a distribution of individual molecule pressures. The particulate nature of matter is fundamental to statistical thermodynamics as opposed to classical thermodynamics, which assumes matter is continuous. Further, these elementary particles and their complex substmctures exhibit wave properties even though intra- and interparticle energy transfers are quantized, ie, not continuous. Statistical thermodynamics holds that the impression of continuity of properties, and even the soHdity of matter is an effect of scale. 

The description of phenomena in a continuous medium such as a gas or a fluid often leads to partial differential equations. In particular, phenomena of wave propagation are described by a class of partial differential equations called hyperbolic, and these are essentially different in their properties from other classes such as those that describe equilibrium ( elhptic ) or diffusion and heat transfer ( para-bohc ). Prototypes are ... 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

---