## 1 Introduction
In a two-layered tube with the weak interfaces between the layers, the dispersion characteristics of longitudinal guided acoustic wave provide a tool for detectingand exploring defects [46]. The microstructures are shown to play a decisive role in the dispersion of both flexural and longitudinal waves in single- and multiwall carbon nanotubes [47]. Acoustic metamaterials and phononic crystals are formed by periodic variation of the acoustic properties of the materials (elasticity and/or density), what leads to the occurrence of the phononic band gaps and provides powerful tools to control the phonon velocity spectra. The phonon crystals were proposed theoretically for elastic waves propagating in a composite material consisting of identical spheres [48] and infinite cylinders [49, 50] with parallel axes embedded in a periodic way within a host. First, sonic materials with effective negative elastic constants were fabricated as lead-coated spheres arranged in a simple cubic crystal [51]. They acted as total wave reflectors within certain adjustable acoustic frequency ranges. Split-ring resonator periodic arrays [52] and double-negative (with the negative effective bulk modulus and the negative effective density) acoustic metamaterials [53] were suggested in a close analogy with electromagnetic metamaterials. For diverse experimental realizations of phononic crystals and their applications, see [44]. A novel method of self-assembly of micro- and nanoarchitectures was designed on the base of the strain-driven roll-up procedure [54, 55]. It paved the way for novel classes of metamaterials: single semiconductor micro- and nanotubes (or radial crystals) [56] and multilayer spiral micro- and nanotubes (or radial superlattices) [57]. A comprehensive structural study was provided for semiconductor/oxide, semiconductor/organic, as well as semiconductor/metal hybrid radial superlattices [58]. A combined \"roll-up press-back\" technology has been recently presented to fabricate novel acoustic metamaterials/mechanically-joined nanomembrane superlattices [59], which reveal a significant reduction of the measured cross-sectional phonon transport compared to a single nanomembrane layer. The optical phonon spectra in multilayer cylindrical quantum wires manifest a geometric structural effect [4], which is of immanent importance for understanding of the pairing of charge carries in quantum wires [60], as well as the electron-phonon phenomena in multilayer coaxial cylindrical \(\mathrm{Al}_{\mathrm{X}}\mathrm{G}_{1\mathrm{-}x}\mathrm{As}\mathrm{GaAs}\) quantum cables [61] and double-coupled nanoshell systems [62]. The aim of the present work is to investigate the feasibility of controlling the acoustic phonon energy spectra and corresponding phonon velocity dispersion in rolled-up micro- and nanoarchitectures. Of fundamental importance in this context is the experimental evidence [63], that due to oxide formation during fabrication, a single period of a radial superlattice is represented by a semiconductor/amorphous oxide/polycrystalline metal/amorphous oxide layer rather than a semiconductor/metal layer. This implies a necessity to investigate multilayer tubes. The elastodynamic boundary conditions on spiral interfaces of a rolled-up microtube with multiple windings or on cylindrical interfaces of a multilayer tube, which consists of coaxial cylindrical shells, (multishell) immediately affect the acoustic phonon energy spectrum and, hence, phonon group velocities for propagation along the tube. Since the effect of these boundary conditions depends on the number of shells along with the geometric parameters, multishells are qualified into acoustic metamaterials. This introduces, in particular, extra capability for tuning the phonon spectrum, engineering the phonon transport and advancement of thermoelectric materials [9].