Sketch presenting an illustrative summary of our motivations, results and future work. North-west quarter: how density staircases are possibly created (see Sect. 1). South-west quarter: main properties of the transmission of internal waves (see Sects. 3 and 4.2). South-east quarter: additional modes that could potentially be observed thanks to the seismology of planets (see Sect. 4.2). North-east quarter: the question we address in our second paper.