River dunes and antidunes are induced by the morphological instability of stream-sediment boundary. Such bedforms raise a number of subtle theoretical questions and are crucial for many engineering and environmental problems. Despite their importance, the absolute/convective nature of the instability has never been addressed. The present work fills this gap as we demonstrate, by the cusp map method, that dune instability is convective for all values of the physical control parameters, while the antidune instability exhibits both behaviors. These theoretical predictions explain some previous experimental and numerical observations and are important to correctly plan flume experiments, numerical simulations, paleo-hydraulic reconstructions, and river works.