Aside from the equations of motion, wave propagation depends on boundary and initial conditions. When considering mechanical vibrations a broad class of problems spanning the range from macro- to nanometer scales, both of a practical and of a fundamental interest, can be described within linear elasticity theory, e.g., [1, 2]. In nanostructures, where surface-to-volume ratio is large, boundaries play an important role and multiple wave scattering results in properties largely different from bulk materials. One of the general methods in solving such problems is to express the wave amplitudes as a decomposition into partial waves, each satisfying the respective differential equation of motion. Then boundary conditions are used to determine the spectrum and eigenmodes of vibration from the respective secular equation and normalization conditions. The complexity of the equations naturally follows that of the material structure which is being studied as more and more internal boundaries of the composite system are taken into account. In the proposed new approach the complexity may be reduced by choosing the form of the partial waves which would automatically (i.e., identically) satisfy the boundary conditions on the outer surfaces, so that one actually has to solve only for the internal boundaries.  Thus, for a free standing plate composed of two rigidly bonded layers the standard approach using, e.g., plane partial waves leads to a (8 x 8) matrix equation, while the new approach produces a (4 x 4) system corresponding to the 4 interface continuity equations for the displacement and stress tensor components. Due to this simplification it becomes possible to obtain an explicit analytic description of the long wavelength excitations, both gapless and gapped (also called cutoff modes). The obtained solutions generalize the known results for a single uniform plate where Lamb-Rayleigh modes are classified by their symmetry with respect to the middle plane. Here the symmetry is absent, however it is shown that solutions of the secular equation obey a transformation with respect to interchange of material parameters. The latter becomes a symmetry transformation of the Lamb modes in the uniform limit and thus allows to univocally identify the flexural and dilatational waves even when the antisymmetric or symmetric wave-patterns are not observed.  On the other side, existence of an interface creates a possibility for the existence of a different kind of modes, the so called Stoneley or interface guided modes. These have been known for macroscopic systems when the considered wavelengths are much smaller compared to the scale of the system. The approach allows to answer the long-standing question of the relation of these guided modes to the behavior at long wavelengths by following the dispersion of the Stoneley waves towards small wavenumbers. A surprising phenomenon is found in relation to the evolution of their wave-pattern: there exist several material parameter‘s dependent ―critical‖ points at intermediate wavelengths where the wave-pattern is transferred from one dispersion branch to the other. So that the typical Stoneley mode pattern (exponential decrease of the amplitude with distance from the interface) undergoes several such critical jumps until it fades away in one of the dilatational or flexural patterns of the lowest cutoff modes at long wavelengths.