MATERIAL SYMMETRIES VERSUS WAVEFRONT SYMMETRIES ANDREJ BÓNA, IOAN BUCATARU, AND MICHAEL A. SLAWINSKI ABSTRACT. We show that, in general, wavefronts are more symmetric than the medium in which they propagate. This means that we cannot determine the symmetries of the medium based solely on the symmetries of the wavefronts. However, we show that we can determine the symmetries of the medium from the symmetries of both wavefronts and polarizations. I NTRODUCTION The focus of this study is the relation between the symmetry class to which a given medium belongs and the symmetries of wavefronts and polarizations within it. Herein, we discuss the wavefronts and polarizations in a homogeneous linearly elastic continuum that are generated by a point source. Our studying the symmetries of wavefronts and polarizations is motivated by inverse problems in seismology. The problem of determining the components of the elasticity tensor from a series of experiments has been investigated by several researchers; notably, by Van Buskirk et al. [2] and Norris [10]. In this paper, we show that, in general, it is not possible to determine the symmetries of an elasticity tensor from the symmetries of wavefronts, alone; we need also information about symmetries of polarizations. We begin this paper with a brief review of the elasticity tensor and its symmetries. Then, we discuss the symmetries of the Christoffel matrix and of its eigenvalues. We show that the symmetries of the eigenvalues are equivalent to the symmetries of wavefronts. Subsequently, we show that while the symmetry of the elasticity tensor is equivalent to the symmetry of the Christoffel matrix, the eigenvalues can be more symmetric than the matrix itself. Thus, symmetry of the wavefronts is not equivalent to the symmetry of the medium. We present cases where the wavefronts are more symmetric than the medium in which they propagate. Also, we present symmetry classes for which the wavefronts cannot have higher symmetry. We conclude this paper by showing that information about the wavefront symmetries combined with information about polarization symmetries, which are associated with the eigenvectors of the Christoffel matrix, allow us to determine the symmetry of the continuum. 1. SYMMETRIES OF ELASTICITY TENSOR In this section, we review briefly the background necessary to discuss the relationship between material symmetries and symmetries of wavefronts and polarizations. A linearly elastic continuum is fully described by its elasticity tensor and mass density. All properties associated with the symmetries of such a continuum are contained in this fourth-rank tensor that linearly relates two second-rank tensors: stress and strain. Key words and phrases. anisotropy, wavefront, polarization, material symmetry, symmetry group. Department of Earth Sciences, Memorial University, St. John’s, Canada, A1B 3X5. 1