Waves described

Fig. 1. (a) Myocardial ceU transmembrane potentials, where the numbers and letters refer respectively to the phases and waves described in the text. ECG is... 

FIGURE 4.5 Balances across a differential element in a thermal wave describing a laminar flame. 

Pi, pi) and is exactly equivalent to the shock wave described in Chapter 1. When heat q is produced in the combustion wave, the Hugoniot curve shifts in the direction indicated by the arrow in Fig. 3.2. It is evident that two different types of combustion are possible on the Hugoniot curve (1) a detonation, in which pressure and density increase, and (2) a deflagration, in which pressure and density decrease. 

The wave described by eqn 1.6 is different from that discussed above. The displacement varies sinusoidally in space and time, but the positions of maximum and minimum displacement do not move. It is known as a standing wave, as opposed to the travelling wave illustrated in Fig. 1.1. Figure 1.2 shows a standing wave at three successive times. The points of zero displacement are called nodes, and those where the displacement is maximum, antinodes. Standing waves are formed in vibrating strings which are fixed at one or more points. They form the basis for musical instruments. 

J. P. Vigier, Explicit mathematical construction of relativistic non-linear de Broglie waves described by three-dimensional (wave and electromagnetic) solitons piloted (controlled) by corresponding solutions of associated linear Klein-Gordon and Schrodinger equations, Found. Phys. 21(2) (1991). 

The Fourier series that the crystallographer seeks is p(x,y,z), the three-dimensional electron density of the molecules under study. This function is a wave equation or periodic function because it repeats itself in every unit cell. The waves described in the preceeding equations are one-dimensional they represent a numerical value/(x) that varies in one direction, along the x-axis. How do we write the equations of two-dimensional and three-dimensional waves First, what do the graphs of such waves look like ... 

When an electromagnetic wave, described by an electric field, E, impinges on a material, the absorption of the electromagnetic wave as a function of z behaves as E - exp /2, so the intensity falls off as / - exp . The immediate goal is to relate this classical absorption coefficient, a, which is the quantity of experimental interest, to the theory, wherein the quantum-mechanical mechanism responsible for absorbing a photon of a given frequency is found. To do this, a is defined as [18]... 

Figure 1.1. Notice that the wave is actually composed of two orthogonal (mutually perpendicular) waves that oscillate exactly in phase with each other. That is, they both reach peaks, nodes, and troughs at the same points. One of these waves describes the electric field vector (E) of the radiation, oscillating in one plane (e.g., the plane of the page) the other describes the magnetic field vector (B) oscillating in a plane perpendicular to the electric field. Thus, both these fields exhibit uniform periodic (e.g., sinusoidal) motion. The axis along which the wave propagates (the abscissa in Figure 1.1) can have dimensions of either time or length.

The picture of glacial waves described by stratigraphical distribution data may be seen all over Europe through the following Betfian and Templomhegyian sub-... 

Many trajectories are necessary to describe all the different events that are summed up to form a unique wave describing the global chemical reaction under observable conditions in quantum mechanics. In this respect, a set of classical trajectories which spread around a mean trajectory in classical mechanics corresponds roughly to the quantum mechanical spreading (through space or time) of the density probability function around its center. 

The phase velocity is the speed at which one must travel to keep the phase of a sinusoidal wave at a constant value. The phase of the wave described by Equation 2.IS is the quantity (m( - kx). 

What are the mechanisms of injury of a blast wave Describe each. 

A further problem in the interpretation of vibrational spectra of solid state compounds arises from the different phases of the vibrations in neighboring cells, leading to a wave described by the wave vector k. In the absence of a phase difference, k equals zero. This is the basis for the factor group analysis. If the vibrational motions are oriented parallel to the direction of the wave caused by the phase differences, a longitudinal branch results while transverse branches result from orthogonal vibrational motions. Furthermore, it is necessary to differentiate between optical and acoustic modes. In the optical mode of the NaCl lattice, Na+ and Cl ions have opposite displacements, while the acoustic modes are caused by in-phase motions of Na+ and CF. 

The wavelike behavior of particles is the fundamental observation upon which the discussion is based. If a particle has attributes of a wave, describe... 

The arbitrary longshore shape of the wave described by G x — ci) remains unchanged while propagating along the coast and can be of a localized or a sinusoidal pattern. The Kelvin wave is trapped in a coastal wave guide along the coast and is not felt offshore of this guide with the width determined by the Rossby radius. Barotropic Kelvin waves have a wide Rossby radius of the order of = 0(100 km) and baroclinic waves a Rossby radius of the order of R = 0(5 km) in the Baltic Sea. 

Tliis theorem is, of course, valid only when i// is described exactly. Because plane waves describe all of space evenly and are not centered on atomic nuclei, Eq. (20) can be used to calculate forces accurately for approximate plane-wave expansions of lA. 1 or expansions of i/f in terms of atomic centered basis functions such as Gaussians and atomic centered grids, Eq. (19) must be used. This is because space is not covered uniformly by atomic centered basis functions, and thus, errors in i r are nonuniform, and the last two terms in Eq. (19) do not add to zero. The added expense of calculating forces via Eq. (19) often prohibits the use of atomic centered basis functions in molecular dynamics simulations. 

Closer to the Flade potential, the width of the active area started to oscillate or breathe in a way similar to that of the modulated waves described in the above experiments. The difference between these two types of modulated waves is that here the wave constantly rotates around the center and possesses a localized structure, while in the above experiments it traveled once across the whole electrode and reappeared only after some time. 

It is worth considering at this stage what the overall n electron distribution will be in this conjugated system. The electron population in any molecular orbital is derived from the square of the atomic orbital functions, so that the sine waves describing the coefficients in Fig. 1.34a are squared to describe the electron distribution in Fig. 1,34b. The n electron population in the molecule as a whole is then obtained by adding up the electron populations, allowing for the number of electrons in each orbital, for all the filled n molecular orbitals. Looking only at the n system, we can see that the overall % electron distribution for the cation is... 

The second equality completes the set ik(p,R) with the plane waves describing the u-dynamics. This transformed function is approximated, in the molecular frame aoi by X k(R) Y ( p aoi) and the total wave function is naturally factored in three terms exp(-i u.Pyjh) Xik(R) Yi(p aoi). A general expansion of the exact wave function in terms of such states has to be handled with care. There is no mathematical proof that the set of all these states provides a complete basis. In practice, one always works with subspaces suggested by the physical and chemical nature of the problem. 

Dipolar waves describe the structure and topology of helices in membrane proteins. The fit of sinusoids with the 3.6 residues per turn period of ideal... 

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