In this paper, we study the weakly-compressive high-frequency plasma waveswhich are superposed on a large-amplitude Alfv\'en wave in a multi-fluid plasmaconsisting of protons, electrons, and alpha particles. For these waves, theplasma environment is inhomogenous due to the presence of the low-frequencyAlfv\'en wave with a large amplitude, a situation that may apply to spaceplasmas such as the solar corona and solar wind. The dispersion relation of theplasma waves is determined from a linear stability analysis using a neweigenvalue method that is employed to solve the set of differential waveequations which describe the propagation of plasma waves along the direction ofthe constant component of the Alfv\'en wave magnetic field. This approach alsoallows one to consider weak compressive effects. In the presence of thebackground Alfv\'en wave, the dispersion branches obtained differ significantlyfrom the situation of a uniform plasma. Due to compressibility, acoustic wavesare excited and couplings between various modes occur, and even an instabilityof the compressive mode. In a kinetic treatment, these plasma waves would benatural candidates for Landau-resonant wave-particle interactions, and may thusvia their damping lead to particle heating.
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