Vector wave equations homogeneous

The electromagnetic fields (x,r) and H(x, t) associated with scattering from a microsphere satisfy Maxwell s equations. For a homogeneous, isotropic linear material the time-harmonic electrical held E and the magnetic held H satisfy vector wave equations, which in SI units are (Bohren and Huffman, 1983)... 

Let us consider in conclusion of this section the case where Cp = Cg = c and U satisfies a homogeneous vector wave equation ... 

Assuming that all external forces are located outside the homogeneous domain V c = const), we arrive at the Kirchhoff integral formula for the vector wave equation... 

We have found in Chapter 13, that the compressional and shear waves in a homogeneous domain satisfy the vector wave equations (13.44) and (13.46). Using the vector identity... 

In Chapter 11 we discussed the fundamental properties of modes on optical waveguides. The vector fields of these modes are solutions of Maxwell s source-free equations or, equivalently, the homogeneous vector wave equations. However, we found in Chapter 12 that there are few known refractive-index profiles for which Maxwell s equations lead to exact solutions for the modal fields. Of these the step-profile is probably the only one of practical interest. Even for this relatively simple profile the derivation of the vector modal fields on a fiber is cumbersome. The objective of this chapter is to lay the foundations of an approximation method [1,2], which capitalizes on the small... 

We showed above that the modes of weakly guiding waveguides are approximately TEM waves, with fields e = e, h S h, and h, related to e, by Eq. (13-1). In an exact analysis, the spatial dependence of e,(x,y) requires solution of Maxwell s equations, or, equivalently, the vector wave equation, Eq. (1 l-40a). However, when A 1, polarization effects due to the waveguide structure are small, and the cartesian components of e, are approximated by solutions of the scalar wave equation. The justification in Section 13-1 is based on the fact that the waveguide is virtually homogeneous as far as polarization effects are concerned when A 1. As we showed in Section 11-16, these effects... 

When there are no sources present, i.e. J = 0, the fields with the separable form of Eq. (30-4) satisfy the homogeneous vector wave equations... 

If we work with the cartesian field components of Eq. (30-16) and the separable fields of Eq. (30-4), then in source-free regions Eq. (30-17) reduces to the homogeneous vector wave equations with transverse and longitudinal components given by... 

The wave vector of a homogeneous wave may be written k = (k + zk")e, where k and k" are nonnegative and is a real unit vector in the direction of propagation. Equation (2.45) requires that... 

A second question concerns the existence of longitudinal components of the magnetic field. Maxwell s equations in free space are (completely ) equivalent to two homogeneous uncoupled wave equations for the vector fields E and B. The uncoupled wave equations admit longitudinal components for both fields E and B. However, longitudinal components are prohibited in the conventional interpretation of Maxwell s equations. 

In free space, an electromagnetic field of frequency w is governed by the homogeneous vector Helmholtz wave equation... 

Equations (12.4)-(12.9) describe an outgoing transverse spherical wave propagating radially with the phase velocity v = cojk and having mutually perpendicular complex electric and magnetic field vectors. The wave is homogeneous in that the real and imaginary parts of the complex wave vector kx are parallel. The surfaces of constant phase coincide with the surfaces of constant amplitude and are spherical. Obviously,... 

Combine Maxwell s equations in vacuum with Eqs. 1.39 and 1.42 to generate the homogeneous wave equation for the vector potential in the Coulomb gauge,... 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

E0exp( — k" x) and H0exp( —k" x) are the amplitudes of the electric and magnetic waves, and = k x — ut is the phase of the waves. An equation of the form K x = constant, where K is any real vector, defines a plane surface the normal to which is K. Therefore, k is perpendicular to the surfaces of constant phase, and k" is perpendicular to the surfaces of constant amplitude. If k and k" are parallel, which includes the case k" = 0, these surfaces coincide and the waves are said to be homogeneous if k and k" are not parallel, the waves are said to be inhomogeneous. For example, waves propagating in a vacuum are homogeneous. 

Up to now, we have examined how the Beltrami vector field relation surfaces in many electromagnetic contexts, featuring predominantly plane-wave solutions (PWSs) to the free-space Maxwell equations in conjunction with biisotropic media (Lakhtakia-Bohren), in homogeneous isotropic vacua (Hillion/Quinnez), or in the magnetostatic context exemplified by FFMFs associated with plasmas (Bostick, etc.). 

The reactance matrix K is ai Q. Exact solutions require the matrix to be of rank n0, implying n linearly independent null-vectors as solutions of the homogeneous equations ma = 0. Because this algebraic condition is not satisfied in general by approximate wave functions, a variational method is needed in order to specify in some sense an optimal approximate solution matrix a. 

In this equation the gasfraction in point (x,y,t) is correlated to the gasfraction in a point (x, y, t ). A condition for application of the Fourier-StieltJes transform was, that the wave field must be homogeneous. This means that all probability-densities are invariant under the addition of a constant vector to all space points. Strictly speaking this condition is not fulfilled for a wavy wall. But when the length scale on which the mean quantities change, is small in respect with the variation of Z in the separation variable r, then the wave field can be assumed to be almost homogeneous. A second condition on the transformation yields the stationarity of the wave field, so the second moment function will be only dependent on the separation variable x. This condition is satisfied when the mean quantities are independent in time. In this way the frequency, wave-number spectrum can be written as. 

This form shows explicitly that the elements of the dynamical matrix D(q) depend only on the difference L = - 5, and are therefore independent of The dimension of the dynamical matrix is 3n, where n is the number of atoms in the primitive unit cell (a, 3 = x, y, z k, k = l...n). We have therefore reduced the infinite set of equations of motion (3,16) to the -problem (3,21) which represents a set of 3n lineccr homogeneous equations in the 3n unknown amplitudes e (K q) for each wave vector q. This reduction is a direct consequence of the periodicity of the lattice as expressed by (3.19), Equation (3.21) can be rewritten as... 

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