  Using our non-local time-dependent theory of convection, the linear
non-adiabatic oscillations of 10 evolutionary model series with masses of
1--3M$_\odot$ are calculated. The results show that there is a red giant
instability strip in the lower temperature side of the Hertzsprung-Russel (HR)
diagram which goes along the sequences of the red giant branch (RGB) and the
asymptotic giant branch (AGB). For red giants of lower luminosities, pulsation
instability are found at high order overtones, the lower order modes from the
fundamental to the second overtone are stable. Towards higher luminosity and
lower effective temperature, instability moves to lower order modes, and the
amplitude growth rate of oscillations also grows. At the high luminosity end of
the strip, the fundamental and the first overtone become unstable, while all
the modes above the 4th order become stable. The excitation mechanism have been
sdudied in detail. It is found that turbulent pressure plays a key role for
excitating of red variables. The frozen convecttin approximation is
unavailaable for the low temperature stars with extended convective envelopes.
In any case, this approximation can explain neither the red edge of the Cepheid
instability strip, nor the blue edge of the pulsating red giant instability
strip. An analytic expression of a pulsation constant as a function of stellar
mass, luminosity and effective temperature is presented from this work.
