We study the asymptotic behaviour of the following linear growth-fragmentation equation B Bt upt, xq B Bx xupt, xq ˘ ` Bpxqupt, xq " 4Bp2xqupt, 2xq, and prove that under fairly general assumptions on the division rate Bpxq, its solution converges towards an oscillatory function, explicitely given by the projection of the initial state on the space generated by the countable set of the dominant eigenvectors of the operator. The proof relies on a general relative entropy argument in a convenient weighted L 2 space.