THE EUROPEAN PHYSICAL JOURNAL B Number of longitudinal normals and degenerate directions for triclinic and monoclinic media

We solved the problem of finding longitudinal acoustic directions of monoclinic media using the eliminant method. By extending Khatkevich’s approach and using the Bezout theorem, we proved that the number of longitudinal normals for mechanically stable monoclinic media can not be larger than 13. Both longitudinal normals (n1, n2, n3) lying in and out of plane perpendicular to the two-fold axis (n3 = 0) of monoclinic media are considered. Closed-form equations for ratios x = n1/n3 y = n2/n3 are derived and exactly solved by the eliminant method. With the help of this method, we show that in the case of the CDP (CsH2PO4) crystal, the number of longitudinal normals equals three. Their components are given. For media of higher symmetries (rhombic, trigonal, tetragonal, hexagonal and cubic), our approach yields well-known results obtained mainly by Borgnis and Khatkevich. For triclinic elastic media, we proved that the number of degenerate directions can not be greater than 132. PACS. 43.40.+s Structural acoustics and vibration – 62.30.+d Mechanical and elastic waves; vibrations