  We study experimentally the effect of linear plasma density gradients on the
self-modulation of a 400\,GeV proton bunch. Results show that a
positive/negative gradient in/decreases the number of micro-bunches and the
relative charge per micro-bunch observed after 10\,m of plasma. The measured
modulation frequency also in/decreases. With the largest positive gradient we
observe two frequencies in the modulation power spectrum. Results are
consistent with changes in wakefields' phase velocity due to plasma density
gradient adding to the slow wakefields' phase velocity during self-modulation
growth predicted by linear theory.
