  We study the asymptotics for the lengths $L_N(q)$ of the instability tongues
of Hill equations that arise as iso-energetic linearization of two coupled
oscillators around a single-mode periodic orbit. We show that for small
energies, i.e. $q\rightarrow 0$, the instability tongues have the same behavior
that occurs in the case of the Mathieu equation: $L_N(q) = O(q^N)$. The result
follows from a theorem which fully characterizes the class of Hill equations
with the same asymptotic behavior. In addition, in some significant cases we
characterize the shape of the instability tongues for small energies.
Motivation of the paper stems from recent mathematical works on the theory of
suspension bridges.
