Harmonic equation

The harmonic equation provides the closest answer (within approximately 5%) therefore, that equation is most representative. By use of the rate/time relationship for the harmonic equation, the recovery curve may be projected to predict recovery rates at future dates after peak production. 

We proceed to estahUsh the addition theorem for four-dimensional spherical harmonics. Equation (9.19) is an identity with respect to r. Expanding the integrand in powers of r... 

Diatom-diatom pairs. We are often concerned with dipoles induced by diatomic molecules, or on such molecules. In such cases, the Wigner rotation matrices reduce again to spherical harmonics. Equations 4.7, 4.8, and 4.13 can thus be written [316, 317, 189]... 

Without the second term on the right-hand side this is the equation of a harmonic oscillator, w is the (mass-loaded) vector of the relative elongation of the two sublattices. For dimensional reasons bn must be the square of a frequency. From the solution of the harmonic equation one obtains... 

Now comes our crucial point. It is tempting to think of the strains as the physical variables in a strain representation for free energy minima then since the minima of the harmonic Equations (4) are e, =es=0, one could further minimize /(0, ,0) However, this is incorrect as strains are all components of a single, symmetric strain tensor that moreover has too many components d(d + 1)/2> d, the number of displacement degrees of... 

For a (qq) transition process, we use the method and formidas of spherical harmonics equations given above. For quadrupolar coupling, spin-orbit coupling is necessary to break the spin selection rule (see... 

The distinction between the case v = I and those where v I is due to the fact that the angular dependence of the first term in the r.h.s. of eqn. 5.2.12 is that of the first spherical harmonic. Equation 5.2.15 is analogous to that of Fuoss and Onsager (eqn. 5.2.13) with the T terms set equal to zero. The terms grad-grad, Uiygrad/,5 which give contribu-... 

With the restriction of harmonicity (Equation 9.118) and the equivalence (Equation 9.119) between pulsations, the phase angle definition in Equation 9.85 can now be written as... 

Another popular and useful approach for many practical engineering problems that can be reduced to two dimensional plane strain or plane stress approximations involves an auxiliary stress potential. In this approach, a bi-harmonic equation is developed based on the stresses (in terms of the potential) satisfying both the equilibrium equation and the compatibility equations. The result is that stresses derived from potentials satisfying the biharmonic equation automatically satisfy the necessary field equations and only the boundary conditions must be verified for any given problem. A rich set of problems may be solved in this manner and examples can be found in many classical texts on elasticity. In conjunction with the use of the stress potential, the principle of superposition is also often invoked to combine the solutions of several relatively simple problems to solve quite complex problems. 

Bessel function A type of function denoted by the letter J to represent the solution to a type of differential equation that has independent solutions which can be expressed as infinite series. An example is the harmonic equation ... 

Many other problems are governed by an equation of the same form as Eq. 25.34, which is known as the quasi-harmonic equation. Table 25.1 lists different classes of problems governed by the quasi-harmonic equation. 

Classes of problem governed by the quasi-harmonic equation... 

We ignore the derivatives of Pi, P2 and C compared with the remaining terms, since P2 and C are slowly varying, and consequently gi satisfies the harmonic equation... 

Because the radial and angular parts are separable and the molecule rotates freely in space, the angular part of Equation 6-6 is identical to the Particle-on-a-Sphere model problem developed in Section 3.2. Hence, the angular functions Yim are the spherical harmonics (Equation 3-19). The solution of the A operator applied to Y is known and given in Equation 3-20. 

Making use of the coupled one-electron wavefunctions given by equation (3.64) and the recursion relations for the spherical harmonics, equations (5.1) and (5.12), show, by detailed calculation, that the matrix element of the z-component of the electric dipole operator between the states ib 7 4 7.., /-i and... 

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