  The phenomenon of double-periodic Cepheid pulsation is still poorly
understood. Recently we rediscussed the problem of modelling the
double-periodic pulsation with non-linear hydrocodes. We showed that the
published non-resonant double-mode models are incorrect, because they exclude
the negative buoyancy effects. Aims. We continue our efforts to verify whether
the Kuhfuss one-equation convection model with negative buoyancy included can
reproduce the double-periodic Cepheid pulsation. Methods. Using the direct time
integration hydrocode, which implements the Kuhfuss convection model, we search
for stable double-periodic Cepheid models. We search for models pulsating in
both fundamental and first overtone modes (F+1O), as well as in the two lowest
order overtones (1O+2O). In the latter case, we focus on reproducing
double-overtone Cepheids of the Large Magellanic Cloud (LMC). Results. We have
found full amplitude non-linear beat Cepheid models of both types, F+1O and
1O+2O. In the case of F+1O models, the beat pulsation is most likely caused by
the three-mode resonance, 2omega_1=omega_0+omega_2, while in the
double-overtone models the underlying mechanism (resonant or non-resonant)
cannot be identified beyond doubt. Double-periodic models found in our survey
exist, however, only in narrow period ranges and cannot explain the majority of
the observed double-periodic objects. Conclusions. With only little doubt left,
we conclude that current one dimensional one-equation convection models are
incapable of reproducing the majority of the observed beat Cepheids. Among the
shortcomings of current pulsation hydrocodes, the simple treatment of
convection seems to be the most severe one. Growing evidence for the presence
of non-radial modes in Cepheids suggests that the interaction between radial
and non-radial modes should also be investigated.
