Adding evanescent plane waves (EPWs)

Evanescent plane waves (EPWs) are given a good introduction in [26]. Vergini [194] developed a set of EPWs for the stadium which involved `families' with increasing evanescence parameter, which he arranged to become increasingly localized at the stadium curvature-discontinuity point P (see Fig. J.1). I found a single such family to be sufficient. Following Vergini's recipe, the EPWs are

$$\left. \begin{array}{r} \bar{\phi}_{2i-1} (\bar{{\mathbf r}... ..._{x,i} \bar{x} + n_{y,i} \bar{y}) \rule{0in}{0.2in} \right) ,$$ (J.5)

for $i = 1 \cdots N_{{\mbox{\tiny EPW}}}/2$. For each $i$, the propagation direction ${\mathbf n}_i$ (parallel to the wavevector ${\mathbf k}$) points along $\theta_i$, and the growth and decay directions are perpendicular to this. The evanescence parameters are given by an algebraic progression,

$$\alpha_i \ = \ \frac{3 + i}{2k^{1/3}}.$$ (J.6)

The angles (restricted to the 2nd quadrant $[\pi/2,\pi]$) are chosen so as to `balance' the EPW to roughly equal weight on the boundary either side of the point P:

$$\sin \theta_i \ = \ \frac{1}{\beta} \left( \frac{2}{k \sinh \alpha_i} \right )^{1/2},$$ (J.7)

where the balancing is controlled by the constant $\beta$. I found $\beta=1.5$ optimal (Vergini reports $\beta=2$ and uses a second family at $\beta=10$). The impact parameters $b_i$ (see Fig. J.1) are such that the maximum value of $\phi$ on the boundary is unity. The number $N_{{\mbox{\tiny EPW}}}$ is chosen so that the maximum $\alpha_i$ is about 3 (the limit on $\alpha$ is discussed in Section 6.3.3). At $k=10^3$, $N_{{\mbox{\tiny EPW}}} \approx 50$ is sufficient. So even though the EPW functions take longer to evaluate than RPWs (especially when symmetrized, see below), there are so few that their computational effort remains small.