8. Conclusions

The analytical theory developed above is constructed within the theory of elasticity and, within the range of its applicability, it is exact to the extent of Eqs. (5) and (8), which express the resonant reflection coefficients for an arbitrary anisotropic medium in terms of the eigenvectors of Stroh's matrix (3). Generally, the dependence of these eigenvectors on the geometrical parameters of reflection can easily be found by numerical methods. An analytical alternative is to expand the exact formulas (5) into a series in small angular parameters. Finally, an explicit analytical calculation based on these formulas for a number of geometries in highsymmetry crystals (for example, hexagonal ones) is also possible. Here, we used all three approaches: the numerical calculations based on Eqs. (5) and (8), their expansion into a series, and even an explicit representation of the results via the elastic moduli. In this case, avoiding the cumbersomeness of our calculations and the unmanageability of the analytical formulas, we retained only the first nonvanishing terms in all expansions that conveyed the key dependences and the effect being investigated on physical parameters. On the other hand, all graphical results of our analysis were obtained through computations based on the exact formulas (5) and (8).

Based on the same principles, we used the image of acoustic beams in our reasoning only for clarity. Actually, we did not go outside the plane wave approximation in our calculations by assuming it to be sufficient in the short wavelength limit of interest, <sup>λ</sup>/<sup>D</sup> <sup>10</sup><sup>3</sup> rad. Here, we also had in mind the possible manifestations of the effect in phonon physics, where the language of plane waves is more relevant.

Based on our analysis, we can probably count on the realization of resonant reflection in crystals, whereby a wide incident acoustic beam converts almost all of its energy into a narrow high-intensity reflected beam. A special choice of crystals with a definite relation between the elastic moduli is required to optimize the resonance. In addition, since the resonance region is narrow in angles of incidence,

stringent requirements for a weak divergence of the incident beam, <sup>10</sup><sup>3</sup> rad, which can be realized only at high ultrasonic frequencies 100 MHz, arise. For the same reason, the amplitude of the excitation coefficient is also limited to K<sup>2</sup> ≈ 5–10. However, in the case of retransformation of the emergent beam through its narrowing in the perpendicular dimension as well, the intensification efficiency increases many fold, to <sup>10</sup><sup>2</sup> . In the hypersonic frequency range, the amplification amplitudes can be increased significantly. In this case, however, one might expect additional restrictions due to an increase in the absorption of acoustic waves.

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The acoustic resonance considered here can also manifest itself in phonon physics as the channeling of high-density energy near the surfaces of crystals with specially chosen orientations and in accompanying nonlinear phenomena. The strategy for an experimental search of such effects can be based on existing techniques for studying the manifestations of phonon focusing [3] and the propagation of ballistic phonons [18].
