  We demonstrate using micromagnetic simulations and a theoretical model that a
gradient in the saturation magnetization ($M_s$) of a perpendicularly
magnetized ferromagnetic film induces a non-reciprocal spin wave propagation
and, consequently an asymmetric dispersion relation. The $M_s$ gradient adds a
linear potential to the spin wave equation of motion consistent with the
presence of a force. We consider a transformation from an inertial reference
frame in which the $M_s$ is constant to an accelerated reference frame where
the resulting inertial force corresponds to the force from the $M_s$ gradient.
As in the Doppler effect, the frequency shift leads to an asymmetric dispersion
relation. Additionally, we show that under certain circumstances,
unidirectional propagation of spin waves can be achieved which is essential for
the design of magnonic circuits. Our results become more relevant in light of
recent experimental works in which a suitable thermal landscape is used to
dynamically modulate the saturation magnetization.
