Plane waves propagation

It should be possible to achieve greater penetration into a load material using the fringing field of a slow-wave than can be achieved by plane-wave propagation (68). There are, however, no reports of practical appHcation of these principles. 

Relativistic corrections. Before the possible effects of TBF are examined, one should introduce relativistic corrections in the preceding nonrelativistic BHF predictions. This is done in the Dirac-Brueckner approach [4], where the nucleons, instead of propagating as plane waves, propagate as spinors in... 

Consider a plane wave propagating in a nonabsorbing medium with refractive index N2 = n2, which is incident on a medium with refractive index A, = w, + iky (Fig. 2.4). The amplitude of the incident electric field is E(, and we assume that there are transmitted and reflected waves with amplitudes E, and Er, respectively. Therefore, plane-wave solutions to the Maxwell equations at... 

Consider an arbitrary particle illuminated by a plane wave propagating in the z direction (Fig. 6.1). The contribution of a volume element At located at a point O to the field scattered by the particle in a direction specified by the unit vector er is... 

The electric vector for a plane wave propagating in the positive z-direction can be represented as ... 

Exercise. Show that a monochromatic electromagnetic plane wave, propagating in the x-direction through a medium whose dielectric constant varies with x, obeys an equation of the form (1.3). 

The amplitude of the shock wave decreases with time. The solution of the problem should provide an answer to the interesting question of the maximum possible rate of amplitude decay of a plane wave propagating in an Fig. 1... 

First, we consider a plane-wave propagating in the Z direction, so that... 

In Maxwell s theory, this dispersion of energy is considered to be negligible, and no damping occurs during the propagation of an electromagnetic wave. Let us consider the plane waves propagating in the z direction ... 

For a harmonic plane wave propagating in the -direction one obtains... 

The plane waves propagating inside medium 2 will have the following spatial dependence ... 

Conclusions. Our theoretical and experimental work has shown conclusively that acoustic agglomeration does result in shifting the particle size distribution from submicrometer sizes into the 10 micrometers and above size range. In order to achieve the desired 150 to 160 dB, specific acoustic powers of from 0.1 to 1 watt/cm2 are required for plane wave propagation and less than that for standing wave chambers. 

A plane wave propagating in the z-direction would therefore have an electric field given by ... 

Further, the first term in the above approximation, which is an exact propagation constant for a plane wave propagating along the z-axis, is reexpressed as a sum of its two lowest-order Taylor expansion terms plus the rest ... 

To examine the interactions, the origin for the slits is given on the y-axis. The plane wave propagates along the x-axis so that Spy is zero by construction and Sy must be infinite in such a way that an equivalent to inequality Eq. (9) holds (replace z-axis by y-axis only). 

For longitudinal plane waves propagating in the x direction through a homogeneous medium of mass density p, the wave equation is... 

Figure 2.2 Plane waves propagating in a solid, showing (a) compressional and (b) shear waves.

Example 2.3 Derive the set of wave equations corresponding to plane wave propagation along the x direction of a cubic crystal. 

Example 2.9 If a plane wave propagates in a medium in which the mass density changes, how is wave velocity effected ... 

We consider a plane wave propagating along the z-axis in a material characterized by its speed of sound Cm and density p. Letting p... 

This happens when total internal reflection of the plane waves propagating with various propagation constants p in the fiber occurs. is the angle between... 

Having obtained expressions for the dielectric susceptibility and the dielectric response functions in terms of microscopic variables, we may proceed to express other observables in microscopic terms. Consider an electromagnetic mode whose electric component is described by a plane wave propagating in the x direction in an isotropic medium, and assume that the field is weak enough to make linear response theory valid. The field is given by... 

Figure 33 Geometry of a plane wave propagating along k (upper diagram) and scattering along fcj by two centres positioned at points O and R of an incident wave defined by (lower diagram).

Classical and quantum treatments of electromagnetic fields [7], near a plane dielectric interface, show that the probability of photo-ionization or photo-excitation of an atom exhibits a pronounced peak at the critical angle of incidence of a plane wave propagating in the dense medium. The probability of absorption or emission of a photon is proportional to the photometric intensity of the electromagnetic field. In particular, consider a plane wave propagating in the dense medium with refractive index /2i, at angle of incidence Ox. The intensity of the transmitted field at a distance dy from the interface is given by... 

Let us choose for P a solution of Eqs. IV. 17 and 1V.18 representing a transverse plane wave propagated in an arbitrary direction. We then have... 

Hence, for an f e sin(/ v), cos(kx), e jkx), correction function (3.3) achieves dxejkx = dnst[ yfc] an j procluces a zero-error finite-difference technique with regard to plane waves propagating in the prefixed direction ofwavevector k. Unfortunately, this directional anisotropy restrains the expediency of the ensuing scheme, thus calling for a more effective exploitation of... 

For the evaluation of coefficients a, 02, and b, in (3.52), the Fourier analysis is selected. Assume that f represents a plane wave propagating toward the -axis, namely f has the exponential form of ejdx-a>t) Application of (3.52) to this function gives the dispersion relation of... 

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