The relationship between undularity and L dependence of the nuclear optical model potential

The contribution of collective or reaction channels to a local optical model potential, OMP, can be readily calculated as a dynamical polarization potential, DPP. The resulting local DPPs commonly have undulatory (`wavy') features, often including local regions of emissivity in the imaginary component. We show here that this undularity arises from $l$-dependence of the underlying formal non-local and $l$-dependent DPP. The $l$-independent proton OMPs, that have the same $S$-matrix $S_{lj}$ as phenomenological $l$-dependent potentials, exhibit undulations that are qualitatively similar to undulations of local DPPs generated by channel coupling. The $l$-dependent phenomenological potentials studied are the potentials that give the best existing fits to the relevant elastic scattering data and the undulatory potentials presented here, being $S$-matrix equivalent (i.e.\ having the same $S$-matrix, $S_{lj}$) give exactly the same scattering. In addition, we present calculations strongly suggesting that undularity (`waviness') is a generic property of $l$-independent potentials that are $S$-matrix equivalent to $l$-dependent potentials. Implications for the validity of folding models based on a local density model are noted.