The behavior of both step waves and sinusoidal waves in fluid-filled elastic vessels whose area and distensibility vary with distance is explored theoretically. It is shown that the behavior of these waves may be explained, to a large extent, by considering the effect of the continuous stream of infinitesimal reflections that is set up whenever any wave travels in a region of vessel where the local impedance, (that is, the ratio of elastic wavespeed to tube area) is not constant. It is found that in such vessels the behavior of sinusoidal waves over distances which are a fraction of a wavelength can be quite different from their average behavior over several wavelengths. Both behaviors are described analytically. The results are applied to the mammalian circulatory system, one of the most interesting results being that a longitudinal variation in the pressure and velocity amplitudes which has a wavelength roughly one-half that of standing waves is predicted. The treatment is essentially a linearized quasi-one-dimensional one, the major assumptions being that the fluid is inviscid, the mean flow is zero, and the vessel is perfectly elastic and constrained from motion in the longitudinal direction. As in the physiological situation, the ratio of fluid velocity to pulse propagation speed is assumed small. For comparison with the analytical results, the linearized equations are also solved numerically by computer.