Motivated by several works on the stabilization of the oscillator by on-off feedbacks, we study the related problem for the one-dimensional wave equation, dampedby an on-off feedback $a(t)u_t$ .We obtain results that are radically different from those known in the caseof the oscillator. We consider periodic functions a: typicallya is equal to 1 on (0,T),equal to 0 on (T, qT) and is qT-periodic.We study the boundary case and next the locally distributed case,and we give optimal results of stability. In both cases,we prove that there are explicit exceptional values of Tfor which the energy of some solutions remains constant with time. IfT is different from those exceptional values, the energy of all solutionsdecays exponentially to zero. This number of exceptional values is countable in the boundary case andfinite in the distributed case.When the feedback is acting on the boundary,we also study the caseof postive-negative feedbacks: $a(t) = a_0 >0$ on (0,T),and $a(t) = -b_0 <0 $ on (T,qT), and we give the necessary and sufficient conditionunder which the energy (that is no more nonincreasing with time) goes to zero or goes to infinity.The proofs of these resultsare based on congruence properties and on a theorem of Weyl in the boundary case, and onnew observability inequalities for the undamped wave equation,weakening the usual “optimal time condition” in the locally distributed case.These new inequalities provide also new exact controllability results.