This paper is concerned with the distribution of the resonances near the real axis for the transmission problem for a strictly convex bounded obstacle $𝒪$ in ${ℝ}^n$, $n\ge 2$, with a smooth boundary. We consider two distinct cases. If the speed of propagation in the interior of the body is strictly less than that in the exterior, we obtain an infinite sequence of resonances tending rapidly to the real axis. These resonances are associated with a quasimode for the transmission problem the frequency support of...