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:
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(3) |
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(4) |
The case of zero reflection arises
when the numerical value of kz 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 b1, b2, and b3,
which fits even better, is
![]() |
(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.