## 1 Introduction  
Spatial confinement of acoustic and optical phonons in semiconductor thin films, superlattices, and nanowires changes their properties in comparison with bulk materials [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15]. Phonon confinement in nanostructures leads to emergence of the quantized energy sub-bands with corresponding modification of the phonon density of states [1; 2; 3; 4; 10; 11; 12; 13; 14; 15]. The changes in the phonon dispersion give rise to changes in the electron-phonon scattering rates [15; 16; 17; 18; 19; 20; 21], optical properties of the nanostructured materials [5; 22; 23; 24; 25; 26; 27; 28], and phonon scattering on defects, boundaries, and other phonons [10; 12; 13; 29; 30; 31]. It was predicted that the electron mobility can be increased in the phonon engineered core-shell nanowires and planar heterostructures with the barrier shell materials, which are acoustically harder than the core materials [32; 33]. The experimentally-measured reduction of the phonon thermal conductivity in thin films and nanowires usually results from the increased phonon-rough boundary scattering. Another mechanism of the thermal conductivity reduction is related to the phonon spectrum modification and decrease of the phonon group velocity in thin films [10; 29; 34] and nanowires [12; 13; 14; 30; 31; 34]. The predicted decrease of two orders of magnitude of the thermal conductivity in silicon nanowires at room temperature [31] has recently found experimental confirmation [35]. It has been determined that the thermal conductivity of the sub-20 nm diameter nanowires is suppressed by the phonon confinement effects beyond the diffusive boundary scattering limit [35]. The acoustic phonon confinement effects theoretically predicted for thin films and nanowires [6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34] have been directly observed experimentally using the Brillouin light scattering technique using suspended silicon thin films with the thickness H \(\approx\) 7 nm [36] and gallium nitride nanowires with the diameters D \(\approx\) 150 nm [37]. The initial work on the phonon confinement effects in thin films and nanowires was performed using the elastic continuum approximation [10; 11; 14; 29; 30; 31; 38]. The results obtained with the elastic continuum approach have been confirmed using other techniques such as lattice dynamics [34; 39] and molecular dynamics (MD) [40; 41; 42]. All computational approaches proved that the acoustic impedance \(\eta\) = \(\rho\)\(\times\)\(\
u\) of the barrier shells of nanostructures presents an important tuning parameter for phonon transport, which can be used together with the lateral dimensions and shape for phonon engineering of material properties (\(\rho\) is the mass density and \(\
u\) is the sound velocity of the material). In the core-shell nanostructures with the acoustically mismatched barrier shells new types of phonon modes appear. Some of them are mainly concentrated in the nanostructure core while others are localized in the shell layers [11; 14; 34; 38; 39]. Controlling the acoustic impedance mismatch and thickness of the shell layers one can tune the electron-phonon interaction and the electron mobility [32; 33]. Similarly, one can engineer the phonon group velocity and thermal conductivity in such nanostructures [11; 14; 34; 38; 39]. Further progress in phonon engineering of material properties depends on availability of nanostructures with layered structures and substantial acoustic impedance mismatch. Multilayer tubes have attracted a special attention in newly developed acoustic metamaterials and phononic crystals at the micro- and nanoscale [43; 44]. In particular, a cylindrical structure from 40 alternating layers of 0.36 mm thick natural latex rubber film and 0.38 mm thick silicone elastomers containing boron nitride particles serves as a thermal shield [45].