This paper discusses nonlinear thermally induced oscillations in quasi- one dimensional flows in ducts. In practice such oscillations are frequently observed in furnaces and combustion chambers. The problem considered involves entropy disturbances which are convected through a nozzle at the end of a tube, thereby producing an acoustic wave which propagates upstream and leads to a modulation of the mass flow at the inlet of the tube. Alternatively, if the rate of heat addition (i.e. the rate of fuel addition in the case of a combustion chamber) responds only weakly or not at all to the oscillating pressure at the inlet, the modulated air flow produces an entropy oscillation (due to the oscillating equivalence ratio in the case of a combustor) downstream of the zone of heat addition (reaction zone). To obtain general stability limits for this kind of self-induced oscillation, a second-order analysis is developed which leads to a nonlinear wave equation. The convection of entropy disturbances introduces nonlinear memory effects which are responsible for a non-local character of the wave equation. The wave equation is solved with the help of a numerical evolution scheme, making use of a suitable scaling transformation which does not change the form of the equation.