We know from the above that the denominator of the cone reflectance has at least one root at . In this subsection we investigate this ``dc behavior'' of the cone more thoroughly.

A hasty analysis based on the reflection and transmission filters in Equations (C.151) and (C.152) 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 :

(C.162) |

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

(C.163) |

and once again the limit is an indeterminate form. We therefore apply L'Hospital's rule again to obtain

(C.164) |

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:

(C.165) | |||

(C.166) |

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

(C.167) |

which approaches as .

An alternative analysis of this issue is given by Benade in [37].

[How to cite this work] [Order a printed hardcopy] [Comment on this page via email]

Copyright ©

Center for Computer Research in Music and Acoustics (CCRMA), Stanford University