Wave equation dispersive

In constant pattern analysis, equations are transformed into a new coordinate system that moves with the wave. Variables are changed from ( , Xi) to ( — Xi, Xi). The new variable — Xi is equal to zero at the stoichiometric center of the wave. Equation (16-130) for a bed with no axial dispersion, when transformed to the ( — x1 xd coordinate system, becomes... 

There are many features of real waves, such as dispersion and nonlinearity, that cannot be described directly in terms of the general wave equation. A brief discussion of such effects is needed to understand solitons. 

Our main motivation to develop the specific transient technique of wavefront analysis, presented in detail in (21, 22, 5), was to make feasible the direct separation and direct measurements of individual relaxation steps. As we will show this objective is feasible, because the elements of this technique correspond to integral (therefore amplified) effects of the initial rate, the initial acceleration and the differential accumulative effect. Unfortunately the implication of the space coordinate makes the general mathematical analysis of the transient responses cumbersome, particularly if one has to take into account the axial dispersion effects. But we will show that the mathematical analysis of the fastest wavefront which only will be considered here, is straight forward, because it is limited to ordinary differential equations dispersion effects are important only for large residence times of wavefronts in the system, i.e. for slow waves. We naturally recognize that this technique requires an additional experimental and theoretical effort, but we believe that it is an effective technique for the study of catalysis under technical operating conditions, where the micro- as well as the macrorelaxations above mentioned are equally important. 

Propagation of non-stationary light beam in a nonlinear medium with material dispersion is described by the scalar wave equation for the linearly-polarized y-component of electrical field E x,z,t) ... 

Rigorous treatment of the self-action problem needs the transformation of Eq.(2.1), (2.5) into a system of integro-differential equations. However, if just some orders of group velocity dispersion and nonlinearity are taken into account, an approximate approach can be used based on differential equations solution. When dealing with the ID-i-T problem of optical pulse propagation in a dielectric waveguide, one comes to the wave equation with up to the third order GVD terms taken into account ... 

The loss of observable THG in the far field with tight focusing of the beam in homogenous normal dispersion media can be described with the paraxial wave equation [Equation (4.2)] assuming slow spatial variation of electric field amplitudes along the beam propagation direction (z direction). The solution of the paraxial wave equation for the amplitude of third harmonic (A3 J can be written as follows (Boyd 1992) ... 

The attenuation may be expressed by making the wavenumber complex (this would be k — ia in eqn (6.12)), and the velocity (= w/k) may also be written as a complex quantity. This in turn corresponds to a complex modulus, so that the relationship v - /(B/p) is preserved indeed the acoustic wave equation may be written as a complex-valued equation, without the need for the extra term in (6.11). Complex-valued elastic moduli are frequency-dependent, and the frequency-dependent attenuation and the velocity dispersion are linked by a causal Kramers-Kronig relationship (Lee et al. 1990). 

When more terms are added to the wave equation, corresponding to complex losses and dispersion characteristics, more terms of the form y(n -l, m - k) appear in (10.10). This approach to numerical simulation was used in early computer simulation of musical vibrating strings [Ruiz, 1969], and it is still in use today [Chaigne, 1992, Chaigne and Askenfelt, 1994],... 

It is true that all molecular and atomic forces ultimately find their root in the mutual behavior of the constituent parts of the atoms, viz., the nuclei and the electrons. They may theoretically all be derived from the fundamental wave equations. It is, however, convenient, as in other branches of physics and chemistry, to treat the various forms of mutual interaction of atoms as different forces, acting independently. We shall therefore follow the usual procedure and treat such forces as the nonpolar van der Waals (dispersion) forces, the forces of the electrostatic polarization of atoms or molecules by ions or by dipoles, the mutual attraction or repulsion Coulomb forces of ions and of dipoles, the exchange forces leading to covalent bonds, the repulsion forces due to interpenetration of electronic clouds, together with the Pauli principle, etc., all as different, independently acting forces. 

D. L. Hovhannisyan, Analytic solution of the wave equation describing dispersion-free propagation of a femtosecond laser pulse in a medium with cubic and fifth-order nonlinearity, Optics Commun. 196, 103 (2001)... 

The electromagnetic fields of the right- and left-propagating polaritons, respectively, follow the wave equations with the speeds and damping rates of the different frequency components dispersed according to the frequency- and wavevector-dependent complex refractive index n = v/e(k, oj). A typical example of the dispersion of these modes is shown in Fig. 1 for the case of a real permittivity e. The term Ao(r,t) represents the envelope of the wavepacket on the phonon-polariton coordinate A. Note that this phonon-polariton coordinate is a linear combination of ionic and electromagnetic displacements, which both contribute to the polarization... 

A dispersion relation such as this allows one to calculate the phase velocity of the waves, given by v = dispersion relations for Equations 2.13 and 2.14 indicate that V2 = V3 = (C44/p). ... 

When the shear waves propagate through the elastic layer, or the elastic plate and reach the steady state, the type of the wave, SH wave for example, and it s dispersion relation are determined by the boundary conditions at the plate surfaces [7]. We have assumed that the sound waves modulate the stress fields at the tip of the crack, and then solved the wave equations with the boundary conditions at the surfaces of the crack and the plate. If the analysis is extended to derive the higher order fields and the dispersion relation of the wave is then obtained, such a wave do exist in the steady state. In this case we could confirm the existence of such "new wave" associated with the crack. Much algebra is required to obtain the higher order fields, however, it is not difficult to see the structure of the fields with the boundary condition at the plate surfaces. We find the boundary conditions at the plate surfaces for the second order stress fields are satisfied by the factor, cos /5 z, in the similar manner to Eq. 

An amazing feature of shock compression is illustrated in Figs. Id-e. A driven shock front steepens up as it runs, in contrast to acoustic waves that disperse as they run [1]. Imagine a shock front that is not initially steep (Fig. Id). Think of the front as a higher pressure wave trailing a lower pressure wave. Equation (2) above shows the trailing wave moves faster. In an ideal continuous elastic medium, the shock front steepens until it becomes an abrupt discontinuity. The shock front risetime tr —> 0. 

Since in this one-dimensional case one has complete freedom to define the jc and y directions in a convenient manner, it is simplest to consider the wavevector as lying in the Jc direction. One then has two cases (i) where the E field is perpendicular to the plane formed by the two vectors q and z — Le., E is in the y direction and (ii) where the magnetic field is perpendicular to this plane. In both of these cases, denoted as -waves and -waves respectively, both the wave equation and Eq. (5.34) are satisfied and the form of the solutions is simplified. One then constructs the solutions in all the regions and finds a system of equations of the form of Eq. (5.35). This system only has a solution when the determinant of the coefficients is equal to zero this condition yields two dispersion relations, one associated with the -waves and another with the H waves ... 

Numerical solutions to the coupled heat and mass balance equations have been obtained for both isothermal and adiabatic two- and three-transition systems but for more complex systems only equilibrium theory solutions have so far been obtained. In the application of equilibrium theory a considerable simplification becomes possible if axial dispersion is neglected and the plug flow assumption has therefore been widely adopted. Under plug flow conditions the differential mass and heat balance equations assume the hyperbolic form of the kinematic wave equations and solutions may be obtained in a straightforward manner by the method of characteristics. In a numerical simulation the inclusion of axial dispersion causes no real problem. Indeed, since axial dispersion tends to smooth the concentration profiles the numerical solution may become somewhat easier when the axial dispersion terra is included. Nevertheless, the great majority of numerical solutions obtained so far have assumed plug flow. 

If the water is deeper than 200 m, the linear long wave equation should be applied. For the region shallower than 200 m, the shallow water theory with a term for bottom friction included should be used. This shallow water theory includes the first order approximation of the amplitude dependent dispersion. Under special conditions, the term for frequency dependent dispersion should be included. If the purpose of the simulation is to determine the runup height, the equations of higher order approximations are not necessary. 

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