This work examines the bell reflection/transmission by using a combination of a piecewise waveguide model and an open-end reflection (along with the assumed amplitude-complementary transmission) function, modeled as a first-order shelf filter. The wind instrument bell is first modeled by deriving the chain scattering matrix for a piecewise cylindrical model, constructed using reflection coefficient vectors that have a useful structure of being interleaved with zeros (due to the round-trip delay of two samples in each section). The matrix is then used to yield the instrument (or instrument bell) transfer function which is shown to have both poles and zeros if the open-end reflection/transmission is made frequency dependent. If the open-end is modeled as a first-order filter and incorporated into the instrument model using matrix convolution, filter coefficients can be estimated from values in the reflection coefficient vectors that would otherwise be zero.