This paper begins a study of impurity pinning in general incommensurate charge-density-wave systems using the replica trick. In particular, models based on the impurity-phase interactions cos(\ensuremath{\theta}+2${k}_{f}$x) or cos(\ensuremath{\theta}+\ensuremath{\varphi}) (with \ensuremath{\varphi} a random field evenly distributed on [0,2\ensuremath{\pi}]) are investigated. With the replica method, adequate information about the low-frequency linear conductivity is contained in the diagonal component (with respect to the replica indices) of the phase propagator. Two types of fluctuations are contained in the replica-averaged action. First, for a given impurity configuration one has ordinary thermodynamic fluctuations. Second, the replica-averaged model contains the additional effects of impurity fluctuations. An exact symmetry of the averaged Lagrangian is found for certain impurity distributions. Using various approximations, the massless mode associated with this symmetry is found to be unbroken and is a manifestation of impurity fluctuations. Unfortunately, while these fluctuations dominate approximate solutions to the model, it is the thermodynamic fluctuations that are physically relevant, and it is necessary to separate the two. This is partially accomplished using a shift of variable, and the resulting model is found to resemble the random-field Ising model (which contains no zero mode). A variational method is then used to study the behaviors of the two models. For the standard model, it is found that the phason self-energy must vanish in the ${\ensuremath{\omega}}_{p}$\ensuremath{\rightarrow}0 limit under the most general quadratic trial interaction (i.e., the averaged excitation spectrum is predicted to be gapless), supporting the earlier analysis. Moreover, the extremal mass matrix is unstable in the n\ensuremath{\rightarrow}0 limit, limiting the usefulness of the result. In contrast, a nonvanishing, stable mass is found for the shifted model. The pole in the resulting propagator (in an approximation valid for \ensuremath{\omega}\ensuremath{\rightarrow}0) is in qualitative agreement with expectation, predicting a gap that initially grows linearly with the standard deviation of the impurity distribution.