Planar wave

In contrast to the planar wave solution, the penetration depth in case of a coil can be described —besides the dependence on co and a— as a function of the depth z and the coil diameter R ... 

Using Equ. (3.1), we can now compute the optimum frequency for cracks in various depths (see Fig. 3.2). For comparison, the optimum excitation frequency for a planar wave or a sheet inducer (300 x 160 mm) is also displayed. One finds that for a planar excitation source, a much lower excitation frequency is required, which causes a reducfion in the response signal of the crack of up to an order of magnitude in case of a small circular coil. 

In planar waves the amplitude A(r) = Aq is a constant. All particles situated within a plane normal to the wave vector move at the same time. Longitudinal waves are planar waves where the displacements are parallel to the wave vector (v and c are colinear). Longitudinal waves can propagate in every material media. These are compression or dilatation waves. In transverse waves, v and c are orthogonal. Transverse waves do not propagate in fluids. 

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. 

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

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],... 

Compared with the momentum of impinging atoms or ions, we may safely neglect the momentum transferred by the absorbed photons and thus we can neglect direct knock-on effects in photochemistry. The strong interaction between photons and the electronic system of the crystal leads to an excitation of the electrons by photon absorption as the primary effect. This excitation causes either the formation of a localized exciton or an (e +h ) defect pair. Non-localized electron defects can be described by planar waves which may be scattered, trapped, etc. Their behavior has been explained with the electron theory of solids [A.H. Wilson (1953)]. Electrons which are trapped by their interaction with impurities or which are self-trapped by interaction with phonons may be localized for a long time (in terms of the reciprocal Debye frequency) before they leave their potential minimum in a hopping type of process activated by thermal fluctuations. 

Figure 1 illustrates the case wherein a planar wave is pictured to be stationary with respect to the observer and the flow, assumed to be one-dimensional, is passing left to right through the wave. If subscript 1 is used to denote conditions upstream of the... 

With radio wave interferometric imaging of astronomical sources, it is typically assumed that the incoming radiation consists of planar waves. In this limit, any curvature of the incoming wavefronts is negligible. This far-field assumption is used for synthetic imaging and simplifies the imaging reconstruction process. However, most standoff THz applications do not fall in the far-field limit, and the simplified inverse Fourier transform of the electric field correlation - the far-field image reconstruction [99] - must be modified [100] to account for the curvature of the wavefronts in the near-field. 

The incoming states Rubidium atom. The collimators allow separation in space base states 10) and 20) propagate along channels 1 and 2 < i(x) and 02(x) are the scattered states at DS-1 and, as the source state is the same and we take identical slit interaction potentials, these functions differ only in space origin. After interaction with the laser beam the base states label from 1) to 10> are sufficient for discussing several possibilities... 

For purposes of further analyses of detonation structure, the shock wave may be treated as a discontinuity. Both the viscous interaction between the shock and the reaction region and the molecular transport within the reaction region are small perturbations that do not appear to exert qualitatively significant influences on the wave structure. This conclusion appears to apply not only to steady, planar waves but also to unsteady, three-dimensional structures it affords one helpful simplification in the complicated analyses of transverse wave structures. It also alters the interpretation of a detonation as a deflagration-supported shock the support provided by the chemical reactions is of a nonplanar compressible gasdynamic character with negligible molecular transport. 

The experimental prevalence of multiheaded spin has prompted categorical statements to the effect that all self-sustaining detonations exhibit significant three-dimensional structure [25]. The belief that no Chapman-Jouguet detonations are planar rests on the conclusion that all such planar waves are unstable to certain nonplanar disturbances. This conclusion is difficult to substantiate in general because of the complexity of the needed... 

Theoretical expression for xih) was derived by Sayers et al. (1971) using single scattering formalism and back scattering of planar waves from shells of atoms surrounding a given emitter and it is given by. 

These solutions are rather similar but completely different from that for a typical planar wave, since they are only dependent on the vibration mode and the external frequency perturbation. The new equilibrium position, r, can be obtained from the calculation of the potential energy and its derivation with respect to r. After performing this operation, we can obtain [9]... 

A longitudinal wave is called a planar wave if the scalar potential 4> - E dq or each parameter... 

When the propagation of the planar waves defines parallel planes that are normal to the propagation line of the wave, these planes are called rays. In any case, the general solution to the d Alembert equation is... 

The equation that corresponds to a planar wave is sinusoidal... 

The solutions in the case of 3-dimensional spaces are also of planar waves ... 

One interesting property is the group velocity of the planar waves, v = Sco/[Pg.160]

Bloch s theorem is a corollary of the lattice periodicity applied to the potential energy. The eigenfunctions of the Hamiltonian is given in Equation 6.D.1 with lJ(r I R) U(r) with R of the Bravais lattice, which can be selected as planar waves that are recurrent to the Bravais lattice. 

The expansion of the planar wave only results in planar waves with the same periodicity and wave vectors, q, of the reciprocal lattice ... 

Using Equation 6.D.21, we can obtain the rate of propagation of the planar waves of the Bloch electrons given by the band index n and the wave vector k. 

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