  We consider weak solutions to dispersive partial differential equations with
periodic boundary conditions and initial data with jump discontinuities. These
are already known to be continuous at irrational times and piecewise constant
at rational times; we show that as time approaches a rational value the
solution exhibits a ringing effect, with the characteristic overshoot of fixed
amplitude near the discontinuities. Furthermore this effect is the same whether
the sequence of times follows rational or irrational values.
