  We theoretically study the propagation of sound waves in GaAs/AlAs
superlattices focussing on periodic modes in the vicinity of the band gaps.
Based on analytical and numerical calculations, we show that these modes are
the product of a quickly oscillating function times a slowly varying envelope
function. We carefully study the phase of the envelope function compared to the
surface of a semi-infinite superlattice. Especially, the dephasing of the
superlattice compared to its surface is a key parameter. We exhibit two kind of
modes: Surface Avoiding and Surface Loving Modes whose envelope functions have
their minima and respectively maxima in the vicinity of the surface. We finally
consider the observability of such modes. While Surface avoiding modes have
experimentally been observed (Phys. Rev. Lett. 97, 1224301 (2006)), we show
that Surface Loving Modes are likely to be observable and we discuss the
achievement of such experiments. The proposed approach could be easily
transposed to other types of wave propagation in unidimensional semi-infinite
periodic structures as photonic Bragg mirror.
