  Using stationary phase methods, we provide an explicit formula for the
Melnikov function of the one and a half degrees of freedom system given by a
Hamiltonian system subject to a rapidly oscillating perturbation. Remarkably,
the Melnikov function turns out to be computable without an explicit knowledge
of the separatrix and in the case of non-analytic systems. This is related to a
priori stable systems coupled with low regularity perturbations. It also
applies to perturbations controlled by wave-type equations, so in particular we
also illustrate this result with the motion of charged particles in a rapidly
oscillating electromagnetic field. Quasiperiodic perturbations are discussed
too.
