This paper focuses on the transient regime of wave amplitude growth and stabilization occuring in a standing wave thermoacoustic engine. Experiments are performed on a simple apparatus consisting of an open ended thermoacoustic oscillator with atmospheric air as working fluid. The results show that, even in that simple device, the transient regime leading to steady state sound exhibits complicated dynamics, like the systematic overshoot of wave amplitude before its final stabilization, and the spontaneous and periodic switch on/off of the thermoacoustic instability at constant heat power supply. A simplified model is then presented which describes wave amplitude growth from the coupled equations describing thermoacoustic amplification and unsteady heat transfer. In this model, the assumption of a one-dimensional and exponential temperature profile is retained and the equations describing heat transfer through the thermoacoustic core are substantially simplified into a set of ordinary differential equations. These equations include the description of two processes saturating wave amplitude growth, i.e. thermoacoustic heat pumping and heat convection by acoustic streaming. It is notably shown that accounting for the effect of acoustic streaming allows to reproduce qualitatively the overshoot process. (c) 2012 Elsevier Ltd. All rights reserved.