Ocean waves propagating beneath a sea–ice sheet encounter a variety of inhomogeneities, which normally arise because of the dynamic nature of the ice veneer over large physical scales. Zones of thinner, thicker, rougher or ridged ice, changes of material property, and abrupt transitions into and from open water, for example, each have their own distinctive scattering kernel that modifies the incoming wave energy spectrum as it progresses further into the ice cover. Here we present a theoretical analysis of wave propagation beneath sea–ice, where the ice is allowed to vary spatially. Isolated irregularities such as pressure ridges, rafted regions, ice islands that have become trapped in the sea–ice, open and refrozen leads, etc., are considered, as well as groups of such features, with their peripheries either welded to the surrounding ice sheet or separated from it by a free crack. Reflection and transmission coefficients, plotted as functions of wave period or wavelength, reveal considerable fine structure, including in some cases a comb of wave frequencies at which perfect transmission occurs. The work generalizes and extends work by Squire & Dixon, Williams & Squire and Evans & Porter, which all deal with abrupt transitions, to properly allow for inhomogeneity in the ice cover. For the multiple, randomly shaped, oriented and spaced irregularities observed in a real ice sheet, good agreement is found between the full solution, a wide–spacing approximation that neglects the evanescent parts of the wave field in subsequent interactions, and a simple serial approach where interactions between features are neglected and the effect of each irregularity is computed in sequence.