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We perform spin wave analysis of classical ground states of a modelHamiltonian proposed earlier(Phys. Rev. B {\bf 90}, 104420(2014)) for$\alpha-MnO_2$ compounds. It is known that the phase diagram of the Hollanditelattice (lattice of $\alpha-MnO_2$ compounds) consists of four differentHelical 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 whichinterpolates 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 rotonminima mainly from $X$ to $M$ and $M$ to $Z$ path. Few higher modes also showroton minima. Each helical phase has its characteristic traits which can beused to determine the phases itself. The analytical expressions of eigenmodesat high symmetry points are obtained which can be utilized to extract thevalues of $J_i$'s. Density of states, specific heat and susceptibilities at lowtemperature has been studied within spin wave approximation. The specific heatshows departure from $T^{1.5(3)}$ dependence found in three dimensionalunfrustrated ferromagnetic(anti-ferromagnetic) system which seems to be thesignature of incommensurate helical phase. The parallel susceptibility ismaximum for FH phase and minimum for CH phase at low temperature. Theperpendicular susceptibility is found to be independent of temperature at verylow temperature. Our study can be used to compare experimental results onmagnon spectrum, elastic neutron scattering, and finite temperature propertiesmentioned above for clean $\alpha-MnO_2$ system

Quantum theory of spin waves for Helical ground states in Hollandite lattice