  We characterize the long-time behaviour of solutions to Smoluchowski's
coagulation equation with a diagonal kernel of homogeneity $\gamma < 1$. Due to
the property of the diagonal kernel, the value of a solution depends only on a
discrete set of points. As a consequence, the long-time behaviour of solutions
is in general periodic, oscillating between different rescaled versions of a
self-similar solution. Immediate consequences of our result are a
characterization of the set of data for which the solution converges to
self-similar form and a uniqueness result for self-similar profiles.
