  We perform spin wave analysis of classical ground states of a model
Hamiltonian proposed earlier(Phys. Rev. B {\bf 90}, 104420(2014)) for
$\alpha-MnO_2$ compounds. It is known that the phase diagram of the Hollandite
lattice (lattice of $\alpha-MnO_2$ compounds) consists of four different
Helical phases(FH, A2H, C2H, CH phase) in the space of model parameters
$J_1,~J_2,~J_3$. The spin wave dispersion shows presence of gapless mode which
interpolates between quadratic to linear depending on phases and values of
$J_i$'s. In most cases, the 2nd lowest mode shows the existence of a roton
minima mainly from $X$ to $M$ and $M$ to $Z$ path. Few higher modes also show
roton minima. Each helical phase has its characteristic traits which can be
used to determine the phases itself. The analytical expressions of eigenmodes
at high symmetry points are obtained which can be utilized to extract the
values of $J_i$'s. Density of states, specific heat and susceptibilities at low
temperature has been studied within spin wave approximation. The specific heat
shows departure from $T^{1.5(3)}$ dependence found in three dimensional
unfrustrated ferromagnetic(anti-ferromagnetic) system which seems to be the
signature of incommensurate helical phase. The parallel susceptibility is
maximum for FH phase and minimum for CH phase at low temperature. The
perpendicular susceptibility is found to be independent of temperature at very
low temperature. Our study can be used to compare experimental results on
magnon spectrum, elastic neutron scattering, and finite temperature properties
mentioned above for clean $\alpha-MnO_2$ system
