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
and
are orthogonal
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.