Spherical wave

Under ideal conditions (e.g., point sources producing spherical waves and no multiple reflections) a rectified backscattered signal represents line integrals of the ultrasonic reflectivity over concentric arcs centered at the transducer position. To reconstruct the reflection tomo-... 

In this approximation, the wave fiinction is identical to the incident wave (first tenn) plus an outgoing spherical wave multiplied by a complex scattering factor... 

Figure 4 Interference pettern created when regularly spaced atoms scatter an incident plane wave. A spherical wave emanates from each atom diffracted beams form at the directions of constructive interference between these waves. The mirror reflection—the (00) beam—and the first- and second-order diffracted beams are shown.

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The local and partial den.sitie.s of. state have been calculated u.sing projections of the plane-wave components of the eigenstates onto spherical waves centred at the atomic sites[.53]. 

The SSW s (screened spherical waves) and their accompanying hard core spheres were defined in [3,4] and we assume the reader is familiar with their definition. 

Friedman, B., and Russek, J., 1954, Addition theorems for spherical waves. Quart. Appl. Math. 12 13. 

We can talk about the wave-front associated with this wave as being a surface of constant phase traveling at the speed of the wave. This surface is related to our previous idea of the rays in that it represents the joined perpendiculars from a series of rays emitted from a single point. A perfectly spherical wave will converge to (or diverge from) a single point in space. 

In terms of wave-fronts the conic mirror will take the spherical wave-front from one conjugate point and modify it to be another perfectly spherical wave-front heading towards the other conjugate point. 

Conic mirrors produce a perfect spherical wave-front when the light striking the mirror comes from the conic conjugates. In general, most optical systems produce an aberrated wave-front, whose departure from a perfect spherical wave-front, A x, y) can be described quantitatively. 

Monochromatic Waves (1.14) A monochromatic e.m. wave Vcj r,t) can be decomposed into the product of a time-independent, complex-valued term Ucj r) and a purely time-dependent complex factor expjojt with unity magnitude. The time-independent term is a solution of the Helmholtz equation. Sets of base functions which are solutions of the Helmholtz equation are plane waves (constant wave vector k and spherical waves whose amplitude varies with the inverse of the distance of their centers. 

The volume integral will give a higher order term in k, so for now, we focus on the surface integral. The displacement due to the phonon is conveniently expanded in terms of the spherical waves e " =... 

The probability density of an electron with amplitude (wave function) / is /2. The s-type (spherical) wave functions, / for the first few principal quantum numbers (n = 1,2,3. ..) are ... 

This ensemble of spherical waves forms a complete set. The plane-wave solution of a particle of momentum hk and energy E is therefore represented by... 

The book contains very little original material, but reviews a fair amount of forgotten results that point to new lines of enquiry. Concepts such as quaternions, Bessel functions, Lie groups, Hamilton-Jacobi theory, solitons, Rydberg atoms, spherical waves and others, not commonly emphasized in chemical discussion, acquire new importance. To prepare the ground, the... 

The second term in (6-9) expresses that nearest and next nearest neighbors dominate scattering contributions to the EXAFS signal, while contributions from distant shells are weak. The dependence of the amplitude on 1/r2 reflects that the outgoing electron is a spherical wave, the intensity of which decreases with the distance squared. The term exp(-2r/X) represents the exponential attenuation of the electron when it travels through the solid, as in the electron spectroscopies of Chapter 3. The factor 2 is there because the electron has to make a round trip between the emitting and the scattering atom in order to cause interference. 

These are shear waves. Planar waves result from stress sources of infinite dimension. In practise planar waves are considered as the sum of spherical waves. 

The extent to which a detonation will propagate from one experimental configuration into another determines the dynamic parameter called critical tube diameter. It has been found that if a planar detonation wave propagating in a circular tube emerges suddenly into an unconfined volume containing the same mixture, the planar wave will transform into a spherical wave if the tube diameter d exceeds a certain critical value dc (i.e., d > dc). II d < d.. the expansion waves will decouple the reaction zone from the shock, and a spherical deflagration wave results [6],... 

The wave function v /ex (r) of electrons at the exit face of the object can be considered as a planar source of spherical waves according to the Huygens principle. The amplitude of diffracted wave in the direction given by the reciprocal vector g is given by the Fourier transformation of the object function, i.e. 

As has already been emphasised, it is very difficult to approach a plane wave with the laboratory sources, which are properly described as spherical-wave sources. A single-crystal reflectivity profile from such a source will be dominated by the source profile rather than by the above formulae the integrated intensity expressions will be multiplied by the source profile function. 

We see that very close to the Bragg condition, where the dispersion strrface is highly cttrved, R K and the crystal acts as a powerful angrtlar amplifier. A reaches 3.5xl0 in the centre of the dispersion surface for sihcon in the 220 reflection with MoK radiation. Far from the centre, the dispersion strrface becomes asymptotic to the spheres about the reciprocal lattice points and A approaches unity. Thus when the whole of the dispersion strrface is excited by a spherical wave, owing to the amplification close to the Bragg condition, the density of wavelields will be veiy low in the centre of the Borrmann fan and... 

---