Inhomogeneous wave equations

This net symmetric regauging operation successfully separates the variables, so that two inhomogeneous wave equations result to yield the new Maxwell... 

Thus the two previously coupled Maxwell equations (1) and (2) (potential form) have been changed to the form given by Eqs. (6) and (7), to leave two much simpler inhomogeneous wave equations, one for and one for A. 

When 4> and A are so selected, then one can obtain the two inhomogeneous wave equations ... 

Therefore, the solution of the inhomogeneous wave equation is given by... 

As all polarization effects due to the vraveguide are ignored, we can decompose E, into cartesian components and solve the two inhomogeneous wave equations for these components separately. 

These are known as the inhomogeneous wave equations. The solutions of equations (2.54) and (2.55) consist of particular solutions involving integrals over the charge and... 

The parabolic equations derived in a slowly varying envelope approximation that describe the second harmonic generation (SHG) of ultrashort pulses in media with locally inhomogeneous wave-vector mismatch, have the form ... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

These new potentials are solutions of wave equations including inside the sources. To obtain the general solution, one must add a particular solution of the inhomogenous potential equations. Usually, the electromagnetic helds Eo, Eo and the potentials , C are discarded for the following reasons. Either (1), they represent transient solutions of Maxwell s equations that decay rapidly to zero or... 

When the analytic polarization P(r, t) is given, the wave equation is linear and inhomogeneous and can be solved exactly in a closed form. The general solution of Eq. Bl is [17]... 

Equation (348) is the globally invariant wave equation defining a, and Eq. (349) is its locally invariant equivalent. Using the locally invariant Lagrangian (345) in Eq. (347) gives the inhomogeneous held equation (SI units)... 

This is a system of inhomogeneous linear equations for the functions (vectors) T m ) (the mixed notation for the perturbation corrections to eigenvalues and eigenvectors is used above). The 0-th order in A yields the unperturbed problem and thus is satisfied automatically. The others can be solved one by one. For this end we multiply the equation for the first order function by the zeroth-order wave function and integrate which yields ... 

In the quasi-static case, effective frequency dependent moduli and loss factors may be calculated from Equation 8. With respect to Equation 29, a lossy matrix material implies that k is now a complex number. The new expressions for c and a differ from Equations 31 and 32, but follow straightforwardly. Equation 30 is usually cited only for elastic matrix materials, but, of course, it need not be used to interpret a. The potential problem (also with viscoelastic inclusions) is that the derivation of Equation 30 is based on homogeneous stress waves, whereas in viscoelastic materials one should, strictly speaking, consider inhomogeneous waves. The results obtained from Equation 29 are reasonable in the sense of yielding the expected superposition of scattering and dissipation effects. 

The beam propagation method is derived for a scalar field. The theory is restricted to small changes in the refractive index. The first part of the derivation assumes the propagation of a high-frequency beam through an inhomogeneous medium. It begins with the wave equation [98]... 

We have shown that the zeroth-order geometric optics approximation can be used to describe the propagation of normally incident, elliptically polarized light in an inhomogeneous, locally uniaxial medium. The approximation corresponds to finding an asymptotic solution of the wave equation in the short-wavelength limit. It is found that a set of pseudo-Stokes parameters, linearly related to the usual Stokes parameters, can be defined to characterize the propa-... 

These equations are inhomogeneous differential equations, which can sometimes be solved anal3dically for the first-, second- and higher-order corrections to the wave-function. But normally they are solved by expanding the mth-order correction to the wavefunction I o (.F)) in a complete basis of functions, which fulfill the same boundary conditions as the unknown function. The eigenfunctions of the... 

If we eliminate either the electric or magnetic fields from Maxwell s equations in Eq. (30-la), we obtain the inhomogeneous vector wave equations... 

When current sources are present within a waveguide, the total fields are related to the currents through Maxwell s equations, or, equivalently, through the inhomogeneous vector wave equations. The cartesian components E, Ey and E of the total electric field E satisfy Eq. (30-17a)... 

Consider the reflection of a normally incident time-harmonic electromagnetic wave from an inhomogeneous layered medium of unknown refractive index n(x). The complex reflection coefficient r(k,x) satisfies the Riccati nonlinear differential equation [2] ... 

A generalized oscillator-wave model is considered showing that the inhomogeneous external influence is realized naturally and does not require any specific conditions. The article considers also the presence of a small horseshoe in the dynamics of a particle under the action of two waves. Originally the problem comes from the plasma physics in despite of the existence of some other applications of the differential equation we study here. 

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