A cell's ability to move allows it to efficiently follow nutrient gradients and enables complex processes in tissues. Migration is mostly driven by the actin cytoskeleton. Spontaneous actin waves have been suggested to provide a physical mechanism for its organization during migration. We study theoretically the motion of cells driven by spontaneous acts polymerization waves. To this end, we introduce a mean-field description of actin waves. The actin field is confined to an evolving cellular domain by means of a phase-field. First, we show the emergence of spontaneous, traveling waves akin to excitable systems and quantify their dynamics. Next, the phase diagram of possible movement states is determined. In particular, we find erratic motion due to the deterministic emergence of spiral waves and compare these findings to the searching behaviour of immature dendritic cells. Lastly, we simulate cells in confinements and identify conditions of wave synchronization in neighboring cells.