Frequency Response

• The propagation constant $\Gamma(\omega)$ gives the per unit length attenuation of waves propagating along an infinitely long tube.

• The frequency response approximating the attenuation and phase delay over a tube of length $L$ is given by
  $$e^{-\Gamma(\omega)L} = e^{-\alpha(\omega)L - j(\omega/v(\omega))L}.$$

• Removing a pure delay of duration $L/c$, we have the desired wall loss filter frequency response $\lambda (\omega )$, given by
  $$\lambda(\omega) = e^{-\alpha(\omega)L - j(\omega/v(\omega) - \omega/c) L}.$$

• The wall loss filter $\lambda (\omega )$ is seen to have a gentle low-pass characteristic, which is more pronounced with decreasing tube radius $a$ or increasing tube length $L$.

Figure 16: Wall loss filter magnitude for increasing radii.

Figure 17: Wall loss filter magnitude for increasing lengths.