  The dynamical formulation of time-independent scattering theory that is
developed in [Ann. Phys. (NY) 341, 77-85 (2014)] offers simple formulas for the
reflection and transmission amplitudes of finite-range potentials in terms of
the solution of an initial-value differential equation. We prove a theorem that
simplifies the application of this result and use it to give a complete
characterization of the invisible configurations of the truncated
$\mathfrak{z}\,e^{-2ik_0 x}$ potential to a closed interval, $[0,L]$, with
$k_0$ being a positive integer multiple of $\pi/L$. This reveals a large class
of exact unidirectionally and bidirectionally invisible configurations of this
potential. The former arise for particular values of $\mathfrak{z}$ that are
given by certain zeros of Bessel functions. The latter occur when the
wavenumber $k$ is an integer multiple of $\pi/L$ but not of $k_0$. We discuss
the optical realizations of these configurations and explore spectral
singularities of this potential.
