The wake function of an accelerator device appears to have a constant term if the geometry of the device is asymmetric or when the beam passes off axis in a symmetric geometry. Its contribution can be significant and has to be taken into account. In this paper a generalized definition of the impedance/wake is presented to take into account also this constant term. An example of a device where the constant term appears is analyzed. Moreover, the impact of a constant wake on the beam dynamics is discussed and illustrated by a HEADTAIL simulation.