Propagation of sound in an oceanic waveguide in the presence of a low Mach number, stratified flow is investigated. Equations for the wave-number integration (WI) and normal-mode (NM) approaches are simultaneously derived in a consistent formulation. At low Mach numbers acoustic azimuthal coupling is negligible and the effect of current can be accounted for by a simple modification of existing WI and NM, codes. Numerical results for low- and high-frequency sources in a simple waveguide show a high degree of agreement between modified versions of OASES and KRAKEN, and confirm previous theoretical results. An approximate modal closed-form solution based on the medium-at-rest mode set is then derived assuming adiabatic propagation (no current-induced mode coupling). Comparison with the modified versions of KRAKEN and OASES shows this assumption breaks down if there is bottom penetration or if the current profile exhibits somewhat sharp variations with respect to depth. An application to matched-field current tomography is eventually presented. Assuming accurate knowledge of the acoustic waveguide, a current velocity of 1.5 m/s can be measured at a range of 2 km with a resolution of 0.2 m/s using a multitone cw signal spanning 200–250 Hz.