Reflectance of the Conical Cap

• Let ${t_{\xi}}\mathrel{\stackrel{\mathrm{\Delta}}{=}}\xi/c$ denote the time to propagate across the length of the cone in one direction
• Reflectance of complete (lossless) cone is $-1$ for pressure waves
  (reflects like an open-ended cylinder)
• Round-trip transfer function from cone entrance to tip and back is
  $$R_{t_{\xi}}(s) \mathrel{\stackrel{\mathrm{\Delta}}{=}}- e^{-2s{t_{\xi}}} = e^{-s{t_{\xi}}} (-1) e^{-s{t_{\xi}}}$$

(reflectance seen inside the cone)