The validity of kinematic wave theory for linear waves with small dissipation propagating through a homogeneous medium is examined with the aid of the higher‐order theory of Chin. It is shown that both for nondissipative and weakly‐dissipative waves the kinematic theory holds with errors of order &egr;2for times of order &egr;−1times the wave period, where &egr; is a measure of the nonuniformity of the wave train. For longer times, of order &egr;−2times the period, secular terms arise, which makes the theory invalid when wave dispersion becomes vanishingly small. By selecting a complex wavenumber chosen such as to make the group velocity real, it is possible to remove the first‐order secular terms and thus produce a modified theory uniformly valid within the longer time period. This modification is also applied to wave trains propagating in a nonhomogeneous medium. General solutions are presented for wave trains or packets of arbitrary initial conditions.