The demagnetization of a hard ferromagnetic layer via the fringing fields of domain walls created by reversing the moment of a neighboring soft ferromagnetic layer is explored experimentally. An unusual oscillatory decay of the magnetic moment of the hard layer is observed using structures in which the demagnetization occurs after a few hundred cycles. This surprising observation is confirmed on a microscopic scale by detailed imaging of the magnetization of the hard layer using high resolution photoemission electron microscopy and by micromagnetic simulations.