  Recent observations of Jupiter and Saturn suggest that heavy elements may be
diluted in the gaseous envelope, providing a compositional gradient that could
stabilise ordinary convection and produce a stably-stratified layer near the
core of these planets. This region could consist of semi-convective layers with
a staircase-like density profile, which have multiple convective zones
separated by thin stably-stratified interfaces, as a result of double-diffusive
convection. These layers could have important effects on wave propagation and
tidal dissipation that have not been fully explored. We analyse the effects of
these layers on the propagation and transmission of internal waves within giant
planets, extending prior work in a local Cartesian model. We adopt a simplified
global Boussinesq planetary model in which we explore the internal waves in a
non-rotating spherical body. We begin by studying the free modes of a region
containing semi-convective layers. We then analyse the transmission of internal
waves through such a region. The free modes depend strongly on the staircase
properties, and consist of modes with both internal and interfacial gravity
wave-like behaviour. We determine the frequency shifts of these waves as a
function of the number of steps to explore their potential to probe planetary
internal structures. We also find that wave transmission is strongly affected
by the presence of a staircase. Very large-wavelength waves are transmitted
efficiently, but small-scale waves are only transmitted if they are resonant
with one of the free modes. The effective size of the core is therefore larger
for non-resonant modes.
