Actual propagation constant

At any rate, the characterization of the surface waves on a finite FSS structure by their actual propagation constant appears not to have occurred until recently in papers by Munk et al. [87] and Tanning and Munk [88], as well as in the dissertation by Tanning [77] and the thesis by Pryor [76],... 

The modal propagation constant y is given in the following equation from the actual propagation constant matrix [T] explained in Chapter 1 ... 

From the earlier results, it is clear that the product Z Y or the actual propagation constant matrix is diagonal and purely imaginary at the infinite frequency or in the perfect conductor case. This results in the fact that the attenuation is zero and the propagation velocity is a light velocity in free space on any phases. Also, it is noteworthy that the modal theory is not necessary as far as only the propagation constant is concerned, because it is already diagonal. 

V) is the actual phase voltage 3. Modal propagation constant... 

In the first step, trioxan is protonated by the complex protic acid formed by interaction of boron trifluoride and water. (It has been shown that no reaction occurs in the complete absence of water [3]). The resulting oxonium ion undergoes ring-opening to give a resonance-stabilized species. This then depolymerizes to build up an equilibrium concentration of formaldehyde, which remains constant during the polymerization. The actual propagation step then involves the addition of formaldehyde rather than trioxan. This scheme accounts for the observation that the polymerization of pure trioxan involves an induction period which may be reduced or even eliminated by the addition of formaldehyde. 

In multifunctional monomer polymerizations, the mobility of radicals through segmental diffusion falls well before their mobility through reaction diffusion at very low functional group conversions (as compared to linear polymerizations). From this point in the reaction, the termination and propagation kinetic constants are found to be related, and the termination kinetic constant as a function of conversion may actually exhibit a plateau region. Figure 6 illustrates the typical behavior of kp and k, vs conversion as predicted by a kinetic based model. 

In practice, linear semilogarithmic kinetic plots and linear dependencies of molecular weight on monomer conversion require only that the rate constants of chain transfer and termination are much less than that of propagation klr kp, k,< kp). This is therefore the practical requirement for the synthesis of well-defined polymers, such that complete monomer conversion can be reached and the chain ends can be functionalized quantitatively. However, because chain-breaking reactions are actually present, we prefer to call such systems controlled polymerizations rather than living polymerizations. 

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