Beginning with Maxwell's equations and assuming only that the wall interaction can be approximated by a surface impedance, we derive formulas for the generalized longitudinal and transverse impedance in flat geometry, from which the wakefields can also be obtained. From the generalized impedances, by taking the proper limits, we obtain the normal longitudinal, dipole, and quad impedances in flat geometry. These equations can be applied to any surface impedance, such as the known dc, ac, and anomalous skin models of wall resistance, a model of wall roughness, or one for a pipe with small, periodic corrugations. We show that, for the particular case of dc wall resistance, the longitudinal impedance obtained here agrees with a known result in the literature, a result that was derived from a very general formula by Henke and Napoly. As concrete example, we apply our results to representative beam and machine parameters in the undulator region of LCLS-II and estimate the impact of the transverse wakes on the machine performance. Show Partial Abstract