We present analytic solutions to the spatially homogeneous axion field equation, using a model potential which strongly resembles the standard anharmonic $(1-\cos N��)$ potential, but contains only a piece-wise second order term. Our exactly soluble model for $��(t)$ spans the entire range $[-��/N,��/N]$. In particular, we are able to confirm (i) Turner's numeric correction factors \cite{Turner} to the adiabatic and harmonic analytic treatments of homogeneous axion oscillations, and (ii) Lyth's estimate \cite{Lyth} valid near the metastable misalignment angle $��/N$ at the peak of the potential. We compute the enhancement of axion density fluctuations that occurs when the axion mass becomes significant at $T\sim 1$ GeV. We find that the anharmonicity amplifies density \mbox{f}luctuations, but only significantly for relatively large initial misalignment angles. The enhancement factor is $\sim$ (2,3,4,13) for $��_{\rm in}\sim (0.85,0.90,0.95,0.99)\times��$.

26 pages, 6 figures appended as a ps-file, Latex, DAMTP-94-21, VAND-TH-94-4