Size of the reflection coefficient

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:  

$$P(x,z) \ \eq \ e^{ -\,i\, \omega t + i\, k_z z } \left(\ e^{ +\,i\, k_x x } \ +\ c \ e^{ -\,i\, k_x x } \right)$$ (3)

In equation (3) $\omega$ and k_x are arbitrary, and k_z is determined from $\omega$ and k_x using the dispersion relation of the interior region, i.e., a semicircle approximation. Assuming this interior solution is applicable at the side boundary, you insert equation (3) into the differential equation (2), which represents the side boundary. As a result, $\partial / \partial x$ is converted to $+\,i\, k_x$ on the incident wave, and $\partial / \partial x$ is converted to $-\,i\, k_x$ on the reflected wave. Also, $\partial / \partial z$ is converted to i k_z. Thus the first term in (3) produces the dispersion relation $D( \omega , k_x , k_z )$times the amplitude P. The second term produces the reflection coefficient c times $D ( \omega , -\,k_x , k_z )$times P. So (2) with (3) inserted becomes:  

$$c \ \eq \ { -\,D ( \omega , k_x , k_z ) \over D( \omega , -\,k_x , k_z ) }$$ (4)

The case of zero reflection arises when the numerical value of k_z selected by the interior equation at $( \omega , k_x )$ 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

$$D( \omega , k_x , k_z ) \ \eq \ \left( \ 1 \ -\ b_3 \ { v k... ... }\ -\ \left( \ b_1 \ -\ b_2 \ { v k_x \over \omega }\,\right)$$ (5)

The absolute stability of straight-line absorbing side boundaries for the 15$^\circ$ 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$^\circ$ equation.