  We report results of a combined investigation of thermal conductivity and
acoustic phonon spectra in nanoporous alumina membranes with the pore diameter
decreasing from D=180 nm to 25 nm. The samples with the hexagonally arranged
pores were selected to have the same porosity of ~13%. The Brillouin-Mandelstam
spectroscopy measurements revealed bulk-like phonon spectrum in the samples
with D=180-nm pores and spectral features, which were attributed to spatial
confinement, in the samples with 25-nm and 40-nm pores. The velocity of the
longitudinal acoustic phonons was reduced in the samples with smaller pores.
Analysis of the experimental data and calculated phonon dispersion suggests
that both phonon-boundary scattering and phonon spatial confinement affect heat
conduction in membranes with the feature sizes D<40 nm.
