Nonlinear Analysis: Real World Applications 10 (2009) 798–809 www.elsevier.com/locate/nonrwa Longitudinal normals and the existence of acoustic axes in crystals Boris M. Darinskii, Dmitry A. Vorotnikov ∗ , Victor G. Zvyagin Research Institute of Mathematics, Voronezh State University, 394006, Universitetskaya Pl.,1, Voronezh, Russia Received 13 November 2006; accepted 1 November 2007 Abstract We obtain three conditions on the phase speeds of the longitudinal and transverse waves propagating along the longitudinal normals in a crystal so that each of these conditions guarantees existence of acoustic axes in this crystal. The result is based on the properties of the rational-valued topological degree and of the index of a singular point for some classes of discontinuous mappings. In addition, we give an upper estimate of the number of acoustic axes in a crystal and show some interrelation between their indices. c  2007 Elsevier Ltd. All rights reserved. Keywords: Topological degree; Acoustic axis; Longitudinal normal; Crystal; Index of an acoustic axis 0. Introduction A crystal is an anisotropic elastic medium. Three different elastic waves may simultaneously propagate in any direction within it. The displacement vectors of its atoms (polarization vectors) for these waves are mutually orthogonal. The phase speeds of these waves may coincide for some directions in a crystal. Such directions are called acoustic axes. On the other hand, for some directions in a crystal one of these waves is longitudinal, i.e. the polarization vectors are parallel to the propagation direction, and the two others are transverse, i.e. the polarization vectors are orthogonal to the propagation direction. The directions with this property are called longitudinal normals. Acoustic axes and longitudinal normals are specific propagation directions, and they play an important role in the study of wave motion in crystals (see [5, Section 17]). Acoustic axes and longitudinal normals (and the properties of the elastic waves propagating along and near the specific directions) have attracted the attention of many investigators (see, for instance, [1–5,8,12]). Various conditions under which a direction in a crystal is an acoustic axis were deduced in [8,12]. The application of topological methods turned out to be rather effective in this area of research. For acoustic axes, the topological arguments were first used by Alshits, Sarychev and Shuvalov (see [1]). They introduced the concept of index of an acoustic axis on the basis of the concept of the index of a singular point of the polarization vector field around the acoustic axis. In turn, the index of a singular point of a vector field in [1] is defined by analogy with the classical case, where the index of a singular point of a vector field is the algebraic number of revolutions of the field ∗ Corresponding author. Tel.: +7 4732208657; fax: +7 4732208755. E-mail addresses: darinskii@math.vsu.ru (B.M. Darinskii), mitvorot@math.vsu.ru (D.A. Vorotnikov), zvg@main.vsu.ru (V.G. Zvyagin). 1468-1218/$ - see front matter c  2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.11.004