Summary: We consider smooth solutions of the Burgers-Hilbert equation that are a small perturbation \(\delta\) from a global periodic traveling wave with small amplitude \(\epsilon \). We use a modified energy method to prove the existence time of smooth solutions on a time scale of \(1/(\epsilon\delta)\), with \(0<\delta\ll\epsilon\ll1\), and on a time scale of \(\epsilon/\delta^2\), with \(0<\delta\ll\epsilon^2\ll1\). Moreover, we show that the traveling wave exists for an amplitude \(\epsilon\) in the range \((0,\epsilon^*)\), with \(\epsilon^*\sim 0.23\), and fails to exist for \(\epsilon>2/e\).