  We study low frequency waves that propagate in a region of layered
semi-convection. Layered semi-convection is predicted to be present in stellar
and planetary interiors and can significantly modify the rate of thermal and
compositional mixing. We derive a series of analytical dispersion relations for
plane-parallel layered semi-convection in the Boussinesq approximation using a
matrix transfer formalism. We find that like a continuously stratified medium,
a semi-convective staircase -- in which small convective regions are separated
by sharp density jumps -- supports internal gravity waves (g-modes). When the
wavelength is much longer than the distance between semi-convective steps,
these behave nearly like g-modes in a continuously stratified medium. However,
the g-mode period spacing in a semi-convective region is systematically {\em
smaller} than in a continuously stratified medium, and it decreases with
decreasing mode frequency. When the g-mode wavelength becomes comparable to the
distance between semi-convective steps, the g-mode frequencies deviate
significantly from those of a continuously stratified medium (the frequencies
are higher). G-modes with vertical wavelengths smaller than the distance
between semi-convective steps are evanescent and do not propagate in the
staircase. Thus, there is a lower cutoff frequency for a given horizontal
wavenumber. We generalize our results to gravito-inertial waves relevant for
rapidly rotating stars and planets. Finally, we assess the prospects for
detecting layered semi-convection using astero/planetary seismology.

Solar and Stellar Astrophysics;Earth and Planetary Astrophysics