Cavity quantum electrodynamics (QED) provides an important platform to study the interaction between light and matter for quantum information processing. In the solid-state cavity QED system, a semiconductor quantum dot serves as an artificial atom. Electrons in the quantum dot interacts with the phonon reservoir in the solid environment besides interacting with the cavity mode. Three effects can be induced by the phonon bath: the coupling renormalization, the off-resonance assisted feeding and the pure dephasing. Phonon spectrum functions are essential in the calculation of this three effects which are the focus of this study. Spherical-Gaussian wave function is used to model the confinement of electrons and holes in the quantum dot. Analytical expressions of phonon spectrum has been developed including both deformation potential coupling and piezoelectric coupling. The confinement volume of quatnum dot plays a dominating role in the phonon spectrum and it’s dependent dynamic parameters.