Conceptually, tidal rivers are seen as narrow channels along which the cross-section geometry remains constant and the bed is horizontal. As tidal waves propagate along such a channel, they decrease exponentially in height. The more rapid the decrease, the stronger the river flow. Near the coast, the tidally averaged width and depth change little throughout the year, even if the river discharge varies strongly between the seasons. However, further upstream, the water depth varies considerably with the river discharge. Recent observations from the Kapuas River, Indonesia, show that the water surface forms a backwater profile when the river flow is low. In this case, the depth converges, i.e. it gradually decreases between the river mouth and the point where the bed reaches sea level. This effect distinctly influences how tidal waves propagate up river so that their wave height does not decrease exponentially any more. We present a theoretical analysis of this phenomenon, which reveals several so far overlooked aspects of river tides. These aspects are particularly relevant to low river flow. Along the downstream part of the tidal river, depth convergence counteracts frictional damping so that the tidal range is higher than expected. Along the upstream parts of the tidal river, the low depth increases the damping so that the tide more rapidly attenuates. The point where the bed reaches sea level effectively limits the tidal intrusion, which carries over to the overtide and the subtidal water level set-up.