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The aim of this paper is to better understand the correspondence between classical plane waves propagating in each layer of an anisotropic periodically multilayered medium and Floquet waves. The last are linear combinations of the classical plane waves. Their wave number is obtained from the eigenvalues of the transfer matrix of one cell of the medium. A Floquet polarization which varies with its position in the periodically multilayered medium has been defined. This allows one to define a Floquet wave displacement by analogy with the displacement of classical plane waves, and to check the equality of the two displacements at any interface separating two layers. The periodically multilayered medium is then an equivalent material, considered as homogeneous, and one can draw dispersion curves and slowness surfaces which are dispersive. In the low-frequency range, when the relation between the Floquet wave numbers and the frequency is linear, the multilayered medium can be homogenized; the Floquet polarization at different interfaces tends to a limit which is the polarization of the classical plane wave in the homogenized medium. PACS numbers: 43.35.Cg INTRODUCTION Acoustic propagation through anisotropic multilayered media has become a subject of intensive study, because of its application to nondestructive evaluation, geophysics, etc .... Generally speaking, multilayered media are made by stacking distinct anisotropic media. These multilayered media are now studied by the use of the propagator matrix formalism which was first developed by Thomson, 1 then furthered by Haskell 2 and afterwards by Gilbert and Backus. 3 By writing boundary conditions at each interface separating two successive layers, a transfer matrix of the whole medium can be obtained. This matrix relates the stresses and displacements at the last interface to those at the first one. A very interesting case is the one of anisotropic periodically multilayered media which are P times an anisotropic multilayered medium cell, named "superlayer." As an extensive background has already been done in previous papers in Refs. 4 and 5, we will not do it again. The study of such media leads to Floquet waves which correspond to the propagation modes in the infinite periodically multilayered medium. They are linear combinations of the classical plane waves propagating in each layer of the multilayered medium. Many researchers such as Gilbert, 6 Schoenberg, 7'8 and Rousseau and Gatignol 9 have studied periodically multilayered media made up of fluid layers. Others, like Richard m and also Gatignol, Rousseau, and Moukemaha 1•'•2 have studied the case of isotropic layers. They have obtained solutions involving Floquet waves, as did Lhermitte with the propagation of an elastic shear wave, normal to the interfaces in a cross-ply fiber reinforced composite. •3 The dispersive behavior of the waves in such media has been known for a long time: Brillouin's works TM and then Haskell's works in 19532 lead to an interpretation of the behavior of periodically multilayered m...