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

On the instability tongues of the Hill equation coupled with a conservative nonlinear oscillator