During stellar core collapse, which eventually leads to a supernova explosion, the stalled shock is unstable due to the standing accretion shock instability (SASI). This instability induces large-scale non-spherical oscillations of the shock, which have crucial consequences on the dynamics and the geometry of the explosion. While the existence of this instability has been firmly established, its physical origin remains somewhat uncertain. Two mechanisms have indeed been proposed to explain its linear growth. The first is an advective–acoustic cycle, where the instability results from the interplay between advected perturbations (entropy and vorticity) and an acoustic wave. The second mechanism is purely acoustic and assumes that the shock is able to amplify trapped acoustic waves. Several arguments favouring the advective–acoustic cycle have already been proposed; however, none was entirely conclusive for realistic flow parameters. In this paper we give two new arguments which unambiguously show that the instability is not purely acoustic and should be attributed to the advective–acoustic cycle. First, we extract a radial propagation time-scale by comparing the frequencies of several unstable harmonics that differ only by their radial structure. The extracted time matches the advective–acoustic time but strongly disagrees with a purely acoustic interpretation. Secondly, we present a method to compute purely acoustic modes by artificially removing advected perturbations below the shock. All these purely acoustic modes are found to be stable, showing that the advected wave is essential to the instability mechanism.