1 Introduction

The progress of stellar seismology is strongly dependent on our understanding of the basic properties of the stationary waves in the interior of the star. In particular, one of the cornerstones of theoretical stellar seismology is the determination of a correct classification scheme of the eigenmodes that can be present for a given equilibrium structure. Usually there are two ways of investigating the classification scheme of stellar nonradial oscillations, where one complements the other. The numerical method is used to calculate the lower overtones and the asymptotic methods (Shibahashi 1979; Tassoul 1980; Gough 1996; Provost & Berthomieu 1986; Smeyers & Tassoul 1988; Tassoul 1990; Vorontsov 1991; Gough 1993) are applicable to the higher overtones. This formal mathematical classification is crucial to progress in the development of the analytical dispersion relation of stationary waves (Gough 1993), and in the improvement of the inversion methods.

Cowling (1941) was the first to introduce a classification scheme for waves, based on the two restoring forces present inside a star in hydrostatic equilibrium, without magnetics fields or differential rotation: the pressure and the gravity. The Cowling classification scheme divides stationary waves into three types: the gravity modes (g-modes), the acoustic modes (p-modes) and the f-mode (Deubner & Gough 1984; Gough 1993; Gough 1996; Unno et al. 1989). This classification is based on the local properties of the waves which attribute to each mode a order number determined by the number of nodes of the radial eigenfunction. In particular, the f-mode is a mode with a radial eigenfunction with no nodes. It is easy to classify nonradial oscillations for simple stellar equilibrium structures, as the less condensed polytropes or even the zero-age main-sequence stars. This is because the propagation zones for each restoring force are clearly separated and they can be seen as a simple generalization of the Sturm-Liouville type problem. For more evolved stars, the propagation zones of acoustic and gravity waves overlap with each other. In some cases, there are even more complicated propagation diagrams with three or more propagation zones for a given wave, as for evolved stars of $10\; M_\odot$ (Unno et al. 1989). In such cases, the classification of nonradial modes is not trivial. Different generalizations have been made by Scuflaire (1974) and Osaki (1975), where the Cowling scheme has been generalized. However, some problems have still remained in these new schemes and the determination of an unambiguous classification method has not yet been achieved. More recently, in a tentative to obtain a scheme that better represents the properties of modes, Shibahashi & Osaki (1976a, 1976b) introduced a classification scheme based on the behaviour of the modes in the trapping zone. These classification schemes present two main problems that difficult their use as a general classification procedure. First, the fourth-order system of nonradial adiabatic stellar oscillations has been reduced to a second-order system by approximative methods, which makes impossible to obtain a general scheme classification for the complete full problem. Second, the classification scheme is dependent of the eigenfunction that has been chosen to characterize the oscillatory motion because the order of the mode is determined by counting the number of nodes of this radial eigenfunction (Cox 1980). The classification scheme presented in this article will try to overcome this second problem.

Our goal in this article is to present a classification scheme, for which the number associated to the order of the modes is independent of the eigenfunction chosen to characterize the oscillatory motion. We will analyse with particular interest the low overtones which present some peculiar behaviour for some complicated stellar structure models. The method presented here is a particular phase representation that is made for the second-order differential equation, written in an appropriate self-adjoint form. This new method is a generalization of the method developed by Prüfer (1926), to study the classical Sturm-Liouville problem. The technique presented can be successfully applied for all the overtones from the smaller to the higher ones. An analogous procedure has already been used by Gabriel & Scuflaire (1979). This type classification scheme as also being study by Lee (1985) in the specific case of dipole modes. Lopes & Gough (2001) and Lopes et al. (1997) as also used this technique to determine the phase shift produced in the outer layers of the Sun and solar-like stars by the partial ionization of hydrogen and helium. A new variant of this method is used to calculate the eigenfrequency equation of low degree modes of acoustic oscillations (Lopes 2001).

The next section contains a brief presentation of the motion equation of stellar oscillations for a spherically symmetrical background state, where some approximative results are mentioned. This is followed by the presentation of the new phase method to study the second-order equation of oscillatory motion, where the mathematical and physical properties of the acoustic-gravity waves are discussed. In Sect. 5, we present the new classification scheme, where we discuss the relation between this new scheme and the ones currently used. In the last section, a summary and conclusion of the main results here established are presented and we also point out the usefulness of the results obtained for the general study of stellar oscillations.