A hasty analysis based on the reflection and transmission filters in Equations (C.154) and (C.155) might conclude that the reflectance of the conical cap converges to at dc, since and . However, this would be incorrect. Instead, it is necessary to take the limit as of the complete conical cap reflectance :
We already discovered a root at in the denominator in the context of the preceding stability proof. However, note that the numerator also goes to zero at . This indicates a pole-zero cancellation at dc. To find the reflectance at dc, we may use L'Hospital's rule to obtain
and once again the limit is an indeterminate form. We therefore apply L'Hospital's rule again to obtain
Thus, two poles and zeros cancel at dc, and the dc reflectance is , not as an analysis based only on the scattering filters would indicate. From a physical point of view, it makes more sense that the cone should ``look like'' a simple rigid termination of the cylinder at dc, since its length becomes small compared with the wavelength in the limit.
Another method of showing this result is to form a Taylor series expansion
of the numerator and denominator:
Both series begin with the term which means both the numerator and denominator have two roots at . Hence, again the conclusion is two pole-zero cancellations at dc. The series for the conical cap reflectance can be shown to be
which approaches as .
An alternative analysis of this issue is given by Benade in .
A Class of Well Posed Damped PDEs