The present interest in aerodynamic propulsive devices, such as the pulse-jet and the ram-jet, has stimulated recent research in the field of combustion. One important aspect of the combustion problem considered herein is the unsteady gas dynamic effects associated with the burning process.Nonlinear differential equations of the Riemann type are derived for the solution of problems involving the propagation of one-dimensional waves in flows in tubes of slowly varying cross section w r ith heat addition or entropy variation. These equations can be solved graphically or numerically with the method of characteristics; the method is described in the Appendix. The transient flows arising when heat is added to a section (as in a combustion chamber) of an initially isentropic flow in a tube have been calculated. The results of this calculation afford an insight into the gas dynamic aspects of intermittent heat addition in a flowing gas, as well as to the apparently anomalous behavior at sonic velocity of steady gas flows with heat addition. The equations derived are well suited to the calculation of the nonlinear pulse-jet cycle with heat addition, once a suitable model of combustion has been, selected.