Open-End Reflection Function

From the data plots given by Levine and Schwinger, it turns out the ratio $Z_L/Z_0$ may be approximated by

$$Z_L/Z_0 \approx \frac{jka}{\zeta+jka}$$

where $k=\omega/c$ is the wavenumber, and $a$ is the radius of the cylinder, and $\zeta$ is a scalar near one².

Substituting this expression for $Z_L/Z_1$ into

$$R_{op}(\omega) = \frac{Z_L(\omega)/Z_0-1}{Z_L(\omega)/Z_0 + 1},$$

yields a reflection filter approximated by

$$R_{op} = \frac{-1}{1+2jka/\zeta}.$$

The open-end reflection filter $R_{op}$ is a one-pole filter with a cut-off frequency of $\omega = \zeta c/(2a)$.