Higher-order field

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. 

Moments and polarizabilities can also be obtained by the fixed-charge method [76]. This technique allows for the single-step incorporation of the nonuniform electric field contributions due to gradients and higher order field derivatives. One or more charges are placed around the molecule in regions where the molecular wavefunctions are negligible. It is important that the basis set used for the field-free molecule be the same as that used in the presence of the field and that the molecule basis be adequate to describe any... 

Figure 2.20. Higher order field components can be obtained by changing the geometry of a classical quadrupolar ion trap.

Higher order fields may be introduced in a completely analogous fashion bv defining... 

In summary, the strategy of these calculations is to explicitly consider products of phase-space density fields in the theory. These product fields describe the correlated motion of the solute and solvent molecules. An examination of the coupling of these higher order fields to the solute fields should then lead to a precise description of the effects on single-particle dynamics of correlated motion of many particles. 

Calculations involving higher order fields may be done in the same way. For example. 

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