The analysis of the transmission of pressure waves in a fluid-filled elastic tube has been extended to the case where the tube is surrounded by a fluid medium. The sound pressure inside the tube is the resultant of a number of modes, some of which are nonpropagating, while others propagate at their own characteristic phase velocities. Neglecting end effects, and for continuously generated waves, it is found that only the modes whose velocity is larger than the sound velocity of the surrounding medium radiate sound energy radially outward. These modes will be damped out by radiation losses, while modes having a phase velocity smaller than this sound velocity are propagated without attenuation (if viscous and heat-transfer losses are neglected). Consequently, if the fluid is the same in the surrounding medium and in the tube only the lowest mode, which resembles a plane wave, propagates unattenuated. In any case, the mass-loading of the surrounding fluid lowers the phase velocities of the propagating modes, particularly at intermediate frequencies. It is shown that in this application the membrane theory of shells will lead to incorrect results, even in thin-walled tubes. This is illustrated by comparison with experimental data.