  We consider Hamilton-Jacobi equations in one space dimension with
Hamiltonians of the form $H(p,x,\omega) = G(p) + \beta V(x,\omega)$, where
$V(\cdot,\omega)$ is a stationary and ergodic potential of unit amplitude. The
homogenization of such equations is established in a 2016 paper of Armstrong,
Tran and Yu for all continuous and coercive $G$. Under the extra condition that
$G$ is a double-well function (i.e., it has precisely two local minima), we
give a new and fully constructive proof of homogenization which yields a
formula for the effective Hamiltonian $\overline H$. We use this formula to
provide a complete list of the heights at which the graph of $\overline H$ has
a flat piece. We illustrate our results by analyzing basic classes of examples,
highlight some corollaries that clarify the dependence of $\overline H$ on $G$,
$\beta$ and the law of $V(\cdot,\omega)$, and discuss a generalization to
even-symmetric triple-well Hamiltonians.
