We resolve the disturbance structures that destabilize steady convection rolls in favour of a time-periodic pattern in two-dimensional containers of fluid-saturated porous material. Analysis of these structures shows that instability occurs as a travelling wave propagating in a closed loop outside the nearly motionless core. The travelling wave consists of five pairs of thermal cells and four pairs of vorticity disturbances in the case of a square container. The wave speed of the thermal disturbances is determined by an average base-state velocity and their structure by a balance between convection and thermal diffusion. Interpretation of the ‘exact’ solution is aided by a one-dimensional convection-loop model which correlates (i) point of transition, (ii) disturbance wavenumber, and (iii) oscillation frequency given the base-state temperature and velocity profiles. The resulting modified Mathieu-Hill equation clarifies the role of the vertical pressure gradient, induced by the impenetrable walls, and the role of the base-state thermal layer.