  We study the large time behaviour of the solution of a linear dispersive PDEs
posed on a finite interval, when the prescribed boundary conditions are time
periodic. We use the approach pioneered in Fokas & Lenells 2012 for nonlinear
integrable PDEs. and then applied to linear problems on the half-line in Fokas
& van der Weele 2021, to characterise necessary conditions for the solution of
such a problem to be periodic, at least in an asymptotic sense. We then fully
describe the periodicity properties of the solution in three important
illustrative examples, recovering known results for the second-order cases and
establishing new ones for the third order one.
