Nonuniformities slowly varying

J. W. Dold, Flame propagation in a nonuniform mixture Analysis of a slowly varying triple flame. Combust. Flame 76 71-88,1989. 

The ultimate application of the Doppler reactivity coefficients is a calculation of the reactivity change when the temperature of the reactor changes in either a controlled or accidental manner but in either case usually in a nonuniform manner. There are at least three important causes of non-uniform temperature changes. Two are the nonuniform neutron flux and coolant temperature distribution, slowly varying nonuniformities that can be accounted for by the procedures developed by Foussoul (72A). 

Here we examine a second case of practical importance for which the transit time can be evaluated approximately. We assume that the change in the nonuniformity along the fiber is so slow that the refractive-index profile is virtually uniform, i.e. independent of z, over the distance Zp required for a ray to undergo a half-period, i.e. Zp fln(r,z)/flz 1. An example of a slowly varying fiber is shown in Fig. 5-2. The nonuniformities may be arbitrarily large, e.g. a tapered fiber whose core radius at one end is many multiples of the radius at the other end, provided the taper angle is everywhere small. 

In Chapter 3 we showed that on a uniform fiber, pulse spreading is proportional to distance z along the fiber. Aceordingly, the only influenee slowly varying nonuniformities can have on this result is to modify the coefficient multiplying z, in addition to changing the pulse shape and redueing pulse power by radiation. The latter is discussed in Section 5-13. However, as we show below, these effects tend to be small when the fiber variations are of small amplitude. 

The local-mode concept also applies to slowly varying composite waveguides, such as the two identical fibers in Fig. 19-3(a) and the pairs of nonidentical fibers in Fig. 19-4, and is therefore a powerful method for studying the properties of nonuniform couplers. 

At each position z along a nonuniform, multimode fiber, a high-order local mode is equivalent to a single family of rays, as is clear from Section 36-2. Each ray follows a path which changes slowly over the local half-period Zp(z) of Eq. (5-12). This is the ray analogue of the multimode-fiber discussion in Section 19-2. Furthermore, the equivalence of mode and ray transit times, which is demonstrated in Section 36-9, is readily extended to slowly varying fibers, for which the transit time is given by Eq. (5-11). 

In Chapter 19 we introduced local modes to describe the fields of waveguides with large nonuniformities that vary slowly along their length. As an individual local mode is only an approximation to the exact fields, it couples power with other local modes as it propagates. Our purpose here is to derive the set of coupled equations which determines the ampUtude of each mode [10]. First, however, we require the relationships satisfied by the fields of such waveguides. 

The asymmetry potential is a small potential across the membrane that is present even when the solutions on both sides of the membrane are identical. It is associated with factors such as nonuniform composition of the membrane, strains within the membrane, mechanical and chemical attack of the external surface, and the degree of hydration of the membrane. It slowly changes with time, especially if the membrane is allowed to dry out, and is unknown. For this reason, a glass pH electrode should be calibrated from day to day. The asymmetry potential will vary from one electrode to another, owing to differences in construction of the membrane. 

In nonuniform fibers many problems of practical interest can be easily solved by using local modes, as we demonstrate in the examples below. However, the locahmode fields will be an accurate approximation to the exact fields only if the nonuniformities vary sufficiently slowly along the fiber. Since the localmode fields are constructed from the modal fields of the locally equivalent, cylindrically symmetric fiber, the appropriate slowness condition is determined by the largest distance over which the total field of the cylindrically symmetric fiber changes significantly due to phase differences between the various modes. 

So far in this chapter we have discussed power redistribution due to slight perturbations of fibers that are translationally invariant. Here we show how the induced-current representation can be used to determine power redistribution and radiation losses from fibers with arbitrary nonuniformities, provided only that the nonuniformities vary slowly along the fiber. 

In Chapter 19 we introduced the concept of local modes to describe propagation on fibers with arbitrary nonuniformities. It is clear from the method of construction in Section 19-1 that the local-mode fields are an accurate approximation to the exact fields of the fiber provided the nonuniformities vary sufficiently slowly with z, as discussed in Section 19-2. Nevertheless, the local-mode fields are not an exact solution of Maxwell s equations, and the slight error can be described by induced currents. 

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