The Gaussian wave functional method is applied to investigate the effects of the scattering of a local impurity to the one dimensional conduction electrons,whose low energy behavior can be described by the Luttinger model. Our results show that both the single particle scattering and pair particle scattering are important in this case. The competition between these two scattering mechanisms gives rise to the renormalized masses for both charge and spin density waves. The vanishing of these masses depicts the phase boundaries in the parameter space,different dynamical behaviors are thus determined.