Let us look at some of the details of the reflection coefficient calculation.
A unit amplitude, monochromatic plane wave
incident on the side boundary generates a reflected wave
of magnitude *c*. The mathematical representation is:

(3) |

(4) |

The case of zero reflection arises
when the numerical value of *k*_{z} selected
by the interior equation at happens also to
satisfy exactly the dispersion
relation *D* of the side boundary condition.
This explains why we try to match the quarter-circle
as closely as possible.
The straight-line dispersion relation does *not * correspond
to the most general form of a side boundary condition, which is expressible
on just two end points.
A more general expression with adjustable
parameters *b _{1}*,

(5) |

The absolute stability of straight-line absorbing side boundaries for the
15 equation can be established,
including the discretization of the *x*-axis.
Unfortunately, an airtight analysis of stability seems to be outside
the framework of the Muir impedance rules.
As a consequence, I don't believe that stability has been established
for the 45 equation.

10/31/1997