  In this paper, we study the weakly-compressive high-frequency plasma waves
which are superposed on a large-amplitude Alfv\'en wave in a multi-fluid plasma
consisting of protons, electrons, and alpha particles. For these waves, the
plasma environment is inhomogenous due to the presence of the low-frequency
Alfv\'en wave with a large amplitude, a situation that may apply to space
plasmas such as the solar corona and solar wind. The dispersion relation of the
plasma waves is determined from a linear stability analysis using a new
eigenvalue method that is employed to solve the set of differential wave
equations which describe the propagation of plasma waves along the direction of
the constant component of the Alfv\'en wave magnetic field. This approach also
allows one to consider weak compressive effects. In the presence of the
background Alfv\'en wave, the dispersion branches obtained differ significantly
from the situation of a uniform plasma. Due to compressibility, acoustic waves
are excited and couplings between various modes occur, and even an instability
of the compressive mode. In a kinetic treatment, these plasma waves would be
natural candidates for Landau-resonant wave-particle interactions, and may thus
via their damping lead to particle heating.
