Be stars are non-supergiant B-type stars with Balmer emission lines. The emission lines arise from a cool envelope surrounding the star. Interferometric observations have confirmed that the cool envelope has a disc-like geometry with a rotational velocity distribution closer to a Keplerian distribution than to a distribution in the flow with a constant angular momentum (Mourard et al. 1989; Hirata 1994; Quirrenbach 1994; Vakili et al. 1994).
Many (but not all) Be stars exhibit long-term quasi-cyclic variations in the relative intensities of the violet and red components in double-peaked Balmer emission-line profiles (V/R variations). Typical features of V/R variations are summarized in e.g., Dachs (1987) or Hubert (1994) and references therein (see also Appendix). The periods of V/R variations range from years to a decade, which is much longer than the rotation periods of the central stars and cool disc-like envelopes. These periods do not depend on the spectral type of the central star. Moreover, these periods can vary from one cycle to the next for individual stars. In addition, the profile as a whole shifts blueward when the red component is stronger and shifts redward when the violet component is stronger. The present paper is mainly concerned with the physical mechanisms that cause the long-term V/R variations. A satisfactory model should account for (1) the range of V/R periods, (2) the dependence of the period on the properties of the Be star, and (3) the time evolution of the V/R cycle.
Okazaki (1991) constructed a model first suggested by Kato (1983) in which the long-term V/R variations are due to the phenomena caused by global one-armed (i.e., m=1) oscillations in the equatorial discs of Be stars. Here, m is the azimuthal wave number. The model is based on the theoretical result that the low-frequency, one-armed oscillations are the only possible global oscillations in geometrically thin (i.e., near Keplerian) discs (Kato 1983, 1989; Okazaki & Kato 1985; Adams et al. 1989). Studying the linear m=1 eigenmodes in isothermal equatorial discs, Okazaki (1991) found that the one-armed oscillation model well explains the observed periodicity of the long-term V/R variations.
Since then, a number of studies which support
the one-armed oscillation model
have appeared.
Based on 3D radiative transfer calculations,
Hummel & Hanuschik (1994, 1996) showed that
calculated line profiles from discs
with m=1 perturbation patterns agree with observed line-profile
variability.
Using high-resolution H
and Fe II spectroscopic data
of five selected Be stars,
Hanuschik et al. (1995) demonstrated that the one-armed oscillation
model can account for the asymmetric profile shapes as well as
the profile variations.
Hummel & Vrancken (1995) analysed the line-profile shapes
of Be stars in detail and derived several kinematical and
geometrical constraints on the structure of individual discs.
They found strong evidence that the asymmetric profiles
originate from discs with an eccentric (i.e., m=1) density
distribution.
Moreover, Telting et al. (1994) clearly demonstrated that
the long-term variability
of Balmer lines of the Be star 
Mon is caused
by a prograde one-armed density structure in the disc.
In addition, the following circumstantial evidence also seems to
support the one-armed oscillation model:
The activity of discrete absorption components in the stellar wind
of 
Cas is associated with the cyclic V/R variability
(Doazan et al. 1987; Telting & Kaper 1994).
Telting & Kaper (1994) found that
this correlation is consistent with the one-armed oscillation model.
In the one-armed oscillation model, the slow pattern speed of
the m=1 density wave results from the slight deviation
of the rotational velocity distribution in the disc
from the Keplerian distribution.
Okazaki (1991) considered the pressure force in the disc
as the only mechanism
that gives rise to this deviation.
In such a disc, the m=1 waves can propagate throughout the disc.
This means that the period of the finite-order m=1 mode
asymptotically goes to infinity as the disc size goes to infinity.
In order to make oscillation periods finite,
Okazaki (1991) needed to impose a disc outer radius and
obtained the spectrum of the retrograde eigenmodes
of which the periods depend on the size of the disc outer radius.
When the disc outer radius is 5-20 stellar radii,
the periods of the eigenmode agreed with the observed period range of
V/R variations.
At present, however,
imposing an outer radius to avoid the confinement problem
seems inadequate,
because there is no observational evidence that
the edge of the H
emitting region is also
the edge of the equatorial disc,
although the above range
covers the observed sizes of H
emitting regions quite well.
On the contrary, from IR and radio observations,
the discs seem to extend far beyond the area
where the H
emission arises (Waters et al. 1991).
Moreover, the prograde one-armed density structure
found in the disc of 
Mon (Telting et al. 1994)
cannot be explained by Okazaki's model.
Papaloizou et al. (1992, henceforth PSH) pointed out this shortcoming of Okazaki's model and suggested that prograde one-armed oscillations are confined to the inner part of Be-star discs by including the quadrupole contribution to the potential of the star which is distorted by the rapid rotation. Savonije & Heemskerk (1993, henceforth SH) constructed a model in which the effect of the rotational deformation is much stronger than the pressure effect, and obtained the results that confirm the above conclusion of PSH.
In their studies, however,
the effect of the rotational deformation of the star is
overestimated by the pressure effect being underestimated.
The range of the disc temperature they studied
is too low to apply to Be-star discs.
PSH adopted 
,
where r is the distance from the stellar center and
H the half-thickness of the disc.
This value of disc thickness corresponds to a disc temperature
for a B0-type star and
for a B5-type star,
where 
is the stellar radius.
Even in the innermost region of the disc,
the disc temperature PSH adopted
is thus about an order of magnitude lower
than typical temperatures of Be-star discs.
SH studied m=1 oscillations
in discs with 
around a star of 
and 
.
This parameter range covers
,
which is also too low to apply to Be-star discs.
Consequently, it is worth examining
the effect of the rotational deformation of the central star
to investigate whether
it is large enough to give rise to the confinement of m=1 oscillations
in the inner part, say 
, of the discs
of Be stars.
Since the proposal of the rotation hypothesis for Be stars by Struve (1931), many studies have been done to test this hypothesis [see Slettebak (1976) for a historical review]. Statistical studies of the distribution of rotation velocities of Be stars have basically confirmed the rotation hypothesis. They have also revealed, however, that the rapid rotation cannot be the sole factor responsible for the Be phenomena (Massa 1975; Slettebak 1976; Fukuda 1982; Kogure & Hirata 1982). Be stars rotate significantly below the critical velocity. The ratio of the rotation velocity to the critical velocity (the rotation parameter) is, on average, smaller for earlier-type Be stars [see Fig. 4 of Kogure & Hirata (1982), and other references]. The distribution of the rotation velocities of Be stars earlier than B3 mostly overlaps that of normal B stars, while Be stars in the spectral range B3--B9 have rotation velocities distinctly higher than normal B stars. This property, together with the observed fact that the fraction of Be stars among B stars has a maximum at B2 (Kogure & Hirata 1982), strongly suggests that another mechanism also has to work to trigger the Be phenomena, at least in the spectral range B0--B2. It is the radiative force that we consider to be the promising candidate. If the radiative force plays an important role for the Be phenomena, it should also have a strong influence on the confinement of the one-armed oscillations in Be-star discs.
The purpose of this paper is to examine the effects of the radiative force and the rotational deformation of the star on the confinement of one-armed oscillations in Be-star discs. Since the radial flow in the inner part of the Be-star disc is considered to be subsonic (Hanuschik 1994), the radiative force in the disc would arise not from the optically-thick strong lines but from an ensemble of optically-thin weak lines (and from the optically-thin continuum). Hence, for the radiative force, we adopt the parametric form proposed by Chen & Marlborough (1994). Examining the eigenvalue problem for a wide range of parameters characterizing the effects of rotation and radiation, we find that, in late-type Be stars, the effect due to rotational deformation of the star is the cause for the confinement of m=1 oscillations. Our results thus show that the mechanism proposed by PSH and confirmed by SH effectively works in discs around late-type Be stars. We also find, however, that the rotational effect is much less important in discs of early-type Be stars. For these stars, it is the weak line force that mainly contributes to the confinement of the m=1 oscillations. Finally, we compare our results with observed V/R cycles of an extensive number of stars.