On the basis of the discussion in the previous section we define normal deformation as those that are orthogonal to all special deformations, in the sense of Eq. (4.5). Obviously there are `good' normal deformations for which the WNA is an excellent approximation (P1 and W8 in Fig. 3.4, for example), and there are `bad' normal deformations for which the WNA is not a very good approximation (FR in Fig. 3.4, and the normal component in Fig. 4.8b). In this section we present numerical evidence that verifies the theoretical results of the previous section, and investigate how `bad' a normal deformation has to be for them to break down.
From what we have claimed it follows that if
normal deformations, then
We could as well write
In general we observe that the quality of the addition rule is limited by the deviation from the WNA of the better of the two deformations. In Fig. 4.3 we see that if both and are bad, then also the addition rule (4.8) becomes quite bad. Fig. 4.4 shows that the addition rule (4.8) is reasonably well satisfied also if either or is a `good' normal deformation. We have chosen as WG (good), and as SX which is almost completely dominated by the special x-translation deformation. The addition rule (4.8) is obeyed at all . This proves that our assertions Eq.(4.4) about the vanishing of is indeed correct. It holds here as a non-trivial statement ( is general and `bad').
Finally, we consider the case where is general and is special. This is illustrated in Fig. 4.5. The addition rule (4.8) becomes exact in the limit of small frequency corresponding to the vanishing of as implied by Eq.(4.3). In particular this implies that . Note that there is no condition on the orthogonality of and ). This will be the key to for improving over the WNA, which we are going to discuss in the next section.