Numercial tests of the improved formula

In Fig. 4.8 we use the IFIF to estimate $\nu$ for two examples. The first is a deformation (CO + W16) whose normal component is `good', due its oscillatory nature. The deviation from a flat white power spectrum is $\sim 20$% for the normal component. The IFIF result Eq.(4.11) is accurate to a few percent. It is a much better estimate of the actual $\nu$ compared with the naive WNA Eq.(3.12) which overestimates the correct value by a factor of 2.2. In the second example the deformation is SX. The resulting normal component is `bad'. Its power spectrum fluctuates by a factor of about 10 in the $\omega$ range shown. Consequently the IFIF is limited in its accuracy, and the correct value for $\nu$ is underestimated by a factor of 2.5. However, it is still a great improvement over the naive result Eq.(3.12). In this second example we can extract another prediction about $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$. The special component is a factor $\sim 10$ larger than the normal component. Therefore the $\omega^2$ behavior at small $\omega$ is almost entirely due to the `rotation' component. The prefactor of the $\omega^2$ behavior need only be found once for each billiard shape. This saves computation and gives an extra information about the dissipation rate at finite driving frequency.

Figure 4.7: a) A deformation of the stadium which moves the `center of mass' (centroid) of the cavity to the right (from the dot to the crosshairs symbol). This deformation is orthogonal (in the sense of (4.5)) to all special deformations, in particular, all translations. b) An example volume-preserving deformation of an elongated approximately-rectangular cavity ($\beta \ll 1$) which nevertheless has a large overlap with dilation. It can be shown that this results in an IFIF estimate of $\approx 4\beta$ times that of the naive WNA. In both diagrams the undeformed shape is shown as a heavy line, the deformed one as a thin line.

Figure 4.8: Decomposition of general deformations $D(s)$ into orthogonal `normal' and `special' components. The general deformation is CO + W16 in subfigure (a), and SX in subfigure (b). The naive WNA Eq.(3.12) is indicated by short solid line. The improved (IFIF) result Eq.(4.11) is indicated by long dashed arrow. In (a) the normal component is quite `good', giving an accurate IFIF result, but in (b) the normal component of SX is `bad', limiting the accuracy of the IFIF to 40% of the actual $\nu$.