  In this paper we suggest a macroscopic toy system in which a potential-like
energy is generated by a non-uniform pulsation of the medium (i.e. pulsation of
transverse standing oscillations that the elastic medium of the system tends to
support at each point). This system is inspired by walking droplets experiments
with submerged barriers. We first show that a Poincar\'e-Lorentz covariant
formalization of the system causes inconsistency and contradiction. The
contradiction is solved by using a general covariant formulation and by
assuming a relation between the metric associated with the elastic medium and
the pulsation of the medium. (Calculations are performed in a Newtonian-like
metric, constant in time). We find ($i$) an effective Schr\"odinger equation
with external potential, ($ii$) an effective de Broglie-Bohm guidance formula
and ($iii$) an energy of the `particle' which has a direct counterpart in
general relativity as well as in quantum mechanics. We analyze the wave and the
`particle' in an effective free fall and with a harmonic potential. This
potential-like energy is an effective gravitational potential, rooted in the
pulsation of the medium at each point. The latter, also conceivable as a
natural clock, makes easy to understand why proper time varies from place to
place.
