Solution to Paradox

• It turns out the reflection transfer function looking into the cone from the cylinder has two poles and two zeros at dc

• The dc poles and zeros cancel and leave a dc cone reflectance equal to $+1$ (the physically obvious answer)

• We can't just set the reflection filter to its dc equivalent to figure out the dc behavior of the overall model

• Instead, a more careful limit must be taken

In the $s$ plane, the conical cap pressure reflectance, seen from the cylinder, can be derived to be

$$H(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}\frac{1 + R(s) (1+2st_x)}{2st_x-1-R(s)}$$

where $t_x$ is the time (in seconds) to propagate across the cone, and

$$R(s) = - e^{-2s t_x}$$

is the reflectance of the cone at its entrance. We have

$$\begin{eqnarray*} \lim_{s\to 0} R(s) &=& -1 \\ \lim_{s\to 0} H(s) &=& +1 \end{eqnarray*}$$