Wave equation periodic solutions

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. 

The free electron resides in a quantized energy well, defined by k (in wave-numbers). This result Ccm be derived from the Schroedinger wave-equation. However, in the presence of a periodic array of electromagnetic potentials arising from the atoms confined in a crystalline lattice, the energies of the electrons from all of the atoms are severely limited in orbit and are restricted to specific allowed energy bands. This potential originates from attraction and repulsion of the electron clouds from the periodic array of atoms in the structure. Solutions to this problem were... 

Since the one-dimensional wave equation is linear, the general solution periodic in x with period 2n is the linear superposition... 

Figure 12.3b shows the widths of the Bragg layers as a function of the layer number of the CBNL depicted in Fig. 12.3a. There are two notable properties of the Bragg layers (1) the width of the high-index layers is smaller than the width of the low-index layers, and (2) the width of the layers decreases exponentially as a function of the radius, converging asymptotically to a constant value. The first property exists in conventional DBRs as well and stems from the dependence of the spatial oscillation period, or the wavelength, on the index of refraction. The second property is unique to the cylindrical geometry and arises from the nonperiodic nature of the solutions of the wave equation (Bessel or Hankel functions) in this geometry.

In cylindrical resonant cavities there exist Electric (E) and Magnetic (B) fields orthogonal to each other. Eigenvalue solutions of the wave equation subjected to proper boundary conditions are called the modes of resonance and are labeled as either transverse electric (TEfom) or transverse magnetic (TM/mn). The subscripts l,m,n define the patterns of the fields along the circumference and the axis of the cylinder. Formally, these l,m,n values are the number of full-period variations of A... 

The direction of the displacement comes from the periodic boundary condition of the solution of the wave equation. It is normal to the flat of the crystal... 

For positive values of the control parameter , stationary, spatially periodic solutions y/s(x) = y/x(x I 2n/q) of (53) may be found with and without forcing. However, in the case of a vanishing forcing amplitude (a = 0) in (53), this equation has a i//-symmetry and one has a pitchfork bifurcation from the trivial solution l/r = 0 to finite amplitude periodic solutions as indicated in Fig. 19. In the unforced case, however, periodic solutions of (53) are unstable for any wave number q against infinitesimal perturbations that induce coarsening processes [114, 121],... 

In a real metallic crystal, however, the potential field is not constant but periodic and increases, as we have seen Figure 56), to a maximum at each metal ion and falls to a minimum between the ions. The solution of the appropriate wave equation shows that the electrons cannot assume any value between zero and the maximum energy value, but that there are certain permitted zones of energy values between which there are discontinuities or bands of forbidden energies. Within the limits of any one zone the arrangement of the electronic energy levels does not differ significantly from that of the free electron in a constant potential field. 

This paper deals with thermal wave behavior during frmisient heat conduction in a film (solid plate) subjected to a laser heat source with various time characteristics from botii side surfaces. Emphasis is placed on the effect of the time characteristics of the laser heat source (constant, pulsed and periodic) on tiiermal wave propagation. Analytical solutions are obtained by memis of a numerical technique based on MacCormack s predictor-corrector scheme to solve the non-Fourier, hyperbolic heat conduction equation. 

We have thus found the formalism, according to which any mechanical problem can be treated. What we have to do is to find the one-valued and finite solutions of the wave equation for the problem. If in particular we wish to find the stationary solutions, i.e. those in which the wave function consists of an amplitude function independent of the time and a factor periodic in the time (standing vibrations), we make the assumption that ijj involves the time only in the form of the factor Schrodinger s equation, we find... 

In principle it can be stated that kinetic equations containing one state variable may represent stationary states, autocatalytic processes and the phenomena related to multistability. Equations in two variables can describe periodical oscillations with time and periodical spatial structures (accounting for diffusion). Equations in three variables enable a description of chaotic processes. State variables are the reagent concentrations and, when diffusion is taken into account, additional state variables associated with the wavelengths of waves propagating in solution appear. 

Bloch wave function - A solution of the Schrodinger equation for an electron moving in a spatially periodic potential used in the band theory of solids. 

There are no analytical forms for the radial functions, / ni(r), as solutions of the radial wave equation. Hartree, in 1928, developed the standard solution procedure, the self-consistent field method for the helium atom by using the simple product forms of equation 1.10 to represent the two-electron wave function. Herman and Skillman (4) programmed a very useful approximate form of the Hartree method in the early 1960s for atomic structure calculations on all the atoms in the Periodic Table. An executable version of this program, based on their FORTRAN code, modified to output data for use on a spreadsheet is included with the material on the CDROM as hs.exe. 

The distribution curve of Fig. 38 is highl idealised. In the actual lattice the positive nuclei cause a periodic variation in the potential encountered by an electron moving through the lattice. The solution of the wave equation for such a potential distribution leads to the result that the electrons cannot assume any energies from zero to a maximum, but that there are bands or zones of perm.itted energies alternating with... 

Let us return to harbor oscillations and consider some important resonant properties of semi-closed basins. First, it is worthy to note that expressions (9.7)-(9.9) and Table 9.3 for open-mouth basins give only approximate values of the eigen periods and other parameters of harbor modes. Solutions of the wave equation for basins of simple geometric forms are based on the boundary condition that a nodal line (zero sea level) is always exactly at the entrance of a semi-closed basin that opens onto a much larger water body. In this case, the free harbor modes are equivalent to odd (antisymmetric) modes in a closed basin, formed by the open-mouth basin and its... 

Here the dielectric permittivity is spatially periodic, e(x + ti) = e(x), which means that the Kronig-Penney model is applicable. If we introduce the Bloch vector, i.e., the crystal momentum k, the periodicity of the electric field will be described by E x + d) = e E x)—Bloch or Floquet condition. The solution of the wave equation for an infinite ID lattice with periodically changing dielectric permittivity should have the form of a sum of a direct and a reflected wave... 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

With the help of asymptotic analysis and numerical methods, Hagan computed the wavenumber Qs of the spiral wave solution of Equation (1). Hagan could obtain a graph of the function Qs as a function of parameter /3, for the case where a = 0. On the other hand, this author indicated how to compute the spiral wavenumber in the case where o / 0. These indications allow us to express the wavenumber q s(a, / ) in function of the values (O, / ). The principle of the computation relies on a scaling property of the time-periodic solutions of Equation (1), which says that any solution W — cT g r a, /)) of Equation (1) can generate a family of solutions corresponding to other parameters (a, P) of an equation of the same form. A member of this family Has the general form ... 

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