  We discuss and characterise the power spectral density properties of a model
aimed at describing pulsations in stars from the main-sequence to the
asymptotic giant branch. We show that the predicted limit of the power spectral
density for a pulsation mode in the presence of stochastic noise is always well
approximated by a Lorentzian function. While in stars predominantly
stochastically driven the width of the Lorentzian is defined by the mode
lifetime, in stars where the driving is predominately coherent the width is
defined by the amplitude of the stochastic perturbations. In stars where both
drivings are comparable, the width is defined by both these parameters and is
smaller than that expected from pure stochastic driving. We illustrate our
model through numerical simulations and propose a well defined classification
of stars into predominantly stochastic (solar-like) and predominately coherent
(classic) pulsators. We apply the model to the study of the Mira variable U
Per, and the semiregular variable L2 Pup and, following our classification,
conclude that they are both classical pulsators. Our model provides a natural
explanation for the change in behaviour of the pulsation amplitude-period
relation noted in several earlier works. Moreover, our study of L2 Pup enables
us to test the scaling relation between the mode line width and effective
temperature, confirming that an exponential scaling reproduces well the data
all the way from the main sequence to the asymptotic giant branch, down to
temperatures about 1000 K below what has been tested in previous studies.
