arXiv:0812.3448v1 [math-ph] 18 Dec 2008 Degenerate weakly nonlinear elastic plane waves W lodzimierz Doma´ nski ∗ and Andrew N. Norris † November 9, 2018 Dedicated to Philippe Boulanger on the occasion of his sixtieth birthday. Abstract Weakly nonlinear plane waves are considered in hyperelastic crystals. Evolu- tion equations are derived at a quadratically nonlinear level for the amplitudes of quasi-longitudinal and quasi-transverse waves propagating in arbitrary anisotropic media. The form of the equations obtained depends upon the direction of propa- gation relative to the crystal axes. A single equation is found for all propagation directions for quasi-longitudinal waves, but a pair of coupled equations occurs for quasi-transverse waves propagating along directions of degeneracy, or acoustic axes. The coupled equations involve four material parameters but they simplify if the wave propagates along an axis of material symmetry. Thus, only two parameters arise for propagation along an axis of two-fold symmetry, and one for a three-fold axis. The transverse wave equations decouple if the axis is four-fold or higher. In the absence of a symmetry axis it is possible that the evolution equations of the quasi-transverse waves decouple if the third order elastic moduli satisfy a certain identity. The theoretical results are illustrated with explicit examples. 1 Introduction We characterize and analyze degenerate weakly nonlinear elastic waves in crystals. The degeneracy considered here arises from the existence of acoustic axes in elastic materials. Acoustic axes are directions for which the phase velocities of at least two waves coincide. For classical elasticity this phenomenon typically occurs for transverse or quasi-transverse waves. In the mathematical literature the coincidence of wave speeds is called a loss of strict hyperbolicity. Conditions for the existence of acoustic axes were first derived by Khatkevich (1962), and a useful review of the topic is given by Fedorov (1968). Recent developments can be found in e.g. Boulanger and Hayes (1998); Mozhaev et al. (2001); Norris (2004). Analysis and properties of nonlinear elastic waves propagating along acous- tic axes were discussed in Shuvalov and Radowicz (2001), see also the book of Lyamov (1983). The existence of acoustic axes, i.e. the loss of strict hyperbolicity, is typically accom- panied by the local loss of genuine nonlinearity, that is, vanishing of the scalar product * Military University of Technology, Faculty of Cybernetics, Institute of Mathematics and Cryptology, Gen. S. Kaliskiego 2, 00-908 Warsaw 49, Poland, domanski.wlodek@gmail.com † Rutgers University, Department of Mechanical and Aerospace Engineering, 98 Brett Road, Piscat- away, NJ 08854-8058, norris@rutgers.edu 1