Some properties of waves

This section reviews some basic ideas on the properties of waves, and is an introduction to the wave properties of radiation discussed later. 

A wave is a disturbance which travels and spreads out through some medium. Examples include ripples on the surface of water, vibrations in a string, and vibrating electric and magnetic fields (light waves). The wave disturbance can take many mathematical forms, but the simplest is the sinusoidal wave shown in Fig. 1,1. This illustrates how the displacement of the medium (y) varies with position (x) at three successive times. 

At any particular time, the displacement varies with position according to 

As A gives the distance between successive wave maxima, and v the number passing each point per unit time, the velocity with which the wave travels is 

Waves transmit energy, which travels and spreads out with the wave motion. The energy at any point is proportional to the intensity of the wave motion there, which is equal to the square of the amplitude  

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Besides the modules of maximum values of ig-function radial part were compared with Po-parameter values, and the line dependence between these values was found. Using some properties of wave function for P-parameter, the wave equation of P-parameter was obtained. 

But light is also a particle. Some properties of light cannot be explained by the wave-like nature of light, such as the photoelectric effect and blackbody radiation (see Section 9.4), so we also need to think of light comprising particles, i.e. photons. Each photon has a direction as it travels. A photon moves in a straight line, just like a tennis ball would in the absence of gravity, until it interacts in some way (either it reflects or is absorbed). 

Williams, G. C., and C. W. Shipman. 1953. Some properties of rod-stabilized flames of homogeneous gas mixtures. 4th Symposium (International) on Combustion, Combustion and Detonation Waves Proceedings. Baltimore The Williams and Wilkins Co. 733-42. 

PhysRev 69, 514-22(1946) (Some properties of very intense shock waves) 9) S.B. Ratner, ZhFizKhimii 20, 1377-80(1946) ... 

In the general case the proposed form of the wave function corresponds to the MP form but with matrices of infinite size. However, for special values of parameters of the model it can be reduced to the standard MP form. In particular, we consider a spin-1 ladder with nondegenerate antiferromagnetic ground state for which the ground state wave function is the MP one with 2x2 matrices. This model has some properties of ID AKLT model and reduces to it in definite limiting case. 

You should recall from your general chemistry course that electrons have some of the properties of waves. Chemists use the equations of wave mechanics to describe these electron waves. Solving these wave equations for an electron moving around the nucleus of an atom gives solutions that lead to a series of atomic... 

The wave function is, therefore, a function of the coordinates of the parts of the system that completely describes the system. A useful characteristic of the quantum mechanical way of treating problems is that once the wave function is known, it provides a way for calculating some properties of the system. 

Although the detailed solution of the Schrodinger equation for the hydrogen atom is not appropriate in this text, we will illustrate some of the properties of wave mechanics and wave functions by using the wave equation to describe a very simple, hypothetical system commonly called the particle in a box, a situation in which a particle is trapped in a one-dimensional box that has infinitely high sides. It is important to recognize that this situation... 

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. 

Modulation is defined as the changing of some property of a carrier wave by the desired signal in such a way that the carrier wave can be used to convey information about the signal. Properties that are typically altered are frequency, amplitude, and wavelength. In AAS. the source radiation is amplitude modulated, but the background and analyte emission are not and are observed as dc signals. 

The generalized oscillator strength formulation of the plane wave Born approximation to the calculation of stopping power is modified by introducing radial Green s functions in place of the infinite sums over bound excited states and integrations over the continuum. Some properties of the resulting expressions are examined. 

Sinusoidal Forcing. From the standpoint of control analysis the most useful forcing function is the steady state since wave. If a steady-state, low amplitude sinusoidal variation is imposed on some property of an inlet process stream, the same property of the corresponding outlet stream will also vary sinusoidally and at the same frequency. For most process components the output wave will lag behind the input wave, and the output amplitude will be less. 

The existence of this relation should be no surprise since, as we have demonstrated, the tensor ejj(u ,k) determines the frequencies of all normal electromagnetic waves in a condensed medium. But this relation, as we show below, permits a simplified consideration of some properties of Coulomb excitons including the dependence of Coulomb exciton energies on s for k —> 0, which by using microscopic theories is quite tedious. 

A microparticle is defined as a physical object whose wave properties can be registered. This class includes elementary particles, atomic nuclei, atoms (atomic ions), molecules (molecular ions) and more complex assemblies (like clusters and macromolecules). Some properties of microparticles belong to the universal physical constants (energy, mass, linear momentum, angular momentum, electric charge, magnetic moment) some, on the contrary, are exclusively specific for microparticles (spin, parity, life-time). Macroscopic state properties (such as temperature, pressure, volume, entropy, etc.) are irrelevant for a single microparticle. 

We have seen that the wave function can be complex, so we now review some properties of complex numbers. 

Now we would like to determine some properties of the periodic solution, whose existence we have just discussed (Tyson, 1975). To develop expressions for the amplitude, period and wave form of the oscillations we will exploit the fact that and q are small quantities, whereas p is large (see p. 42). 

A discrete kinetic system modelling some properties of retrograde fluids is proposed. Plane shock waves corresponding to the model Euler, Navier-Stokes and kinetic approximations are studied. It turns out that in some cases the number density must decrease in order to obtain a stable shock wave The shock structure auid its thickness in the kinetic approximation are determined and are consistent those of Cramer and Kluwick [l4- ... 

The lowest frequency occurs when n = I and is called the fundamental. Doubling the frequency corresponds to raising the pitch by an octave. Those solutions having values of n > I are known as the overtones. As mentioned previously, one important property of waves is the concept of superposition. Mathematically, it can be shown that any periodic function that is subject to the same boundary conditions can be represented by some linear combination of the fundamental and its overtone frequencies, as shown in Figure 3.8. In fact, this type of mathematical analysis is known as a Fourier series. Thus, while the note middle-A on a clarinet, violin, and piano all have the same fundamental frequency of 440 Hz, the sound (or timbre) that the different instruments produce will be distinct, as shown in Figure 3.9. 

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